Berry Phase Physics in Free and Interacting Fermionic Systems
TTHE UNIVERSITY OF CHICAGOBERRY PHASE PHYSICS IN FREE AND INTERACTING FERMIONIC SYSTEMSA DISSERTATION SUBMITTED TOTHE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCESIN CANDIDACY FOR THE DEGREE OFDOCTOR OF PHILOSOPHYDEPARTMENT OF PHYSICSBYJINGYUAN CHENCHICAGO, ILLINOISJUNE 2016 a r X i v : . [ c ond - m a t . m e s - h a ll ] J un ABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivLIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background, Problems and Results . . . . . . . . . . . . . . . . . . . . . . . 11.2 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Conventions in the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 BERRY FERMI GAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Review of Symplectic Formulation of Classical Mechanics . . . . . . . . . . . 132.1.1 Worldline Parametrized by Time . . . . . . . . . . . . . . . . . . . . 132.1.2 Worldline Parametrized by Intrinsic Parameter . . . . . . . . . . . . . 182.2 Semi-Classical Particle with Berry Phase . . . . . . . . . . . . . . . . . . . . 202.3 Ensemble of Berry Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.1 Boltzmann Equation and Current . . . . . . . . . . . . . . . . . . . . 232.3.2 Anomalous Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.3 Equilibrium Current . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4 Derivation from Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5 Example: Chiral Kinetic Theory of Weyl Fermion . . . . . . . . . . . . . . . 372.5.1 Single Weyl Fermion . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5.2 Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.5.3 Chiral Magnetic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 442.6 Chiral Anomaly in Chiral Kinetic Theory . . . . . . . . . . . . . . . . . . . . 472.7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 LORENTZ SYMMETRY IN CHIRAL KINETIC THEORY . . . . . . . . . . . . 533.1 Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Non-Trivial Frame Dependence . . . . . . . . . . . . . . . . . . . . . . . . . 553.2.1 Infinitesimal Transformation . . . . . . . . . . . . . . . . . . . . . . . 573.2.2 Finite Transformation On-Shell . . . . . . . . . . . . . . . . . . . . . 593.2.3 Current, Stress-Energy Tensor and Physical Interpretation . . . . . . 593.3 Relation to Wigner Translation . . . . . . . . . . . . . . . . . . . . . . . . . 623.4 Chiral Kinetic Theory with Collisions . . . . . . . . . . . . . . . . . . . . . . 663.4.1 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.4.2 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.4.3 Entropy Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.4.4 Equilibrium and Chiral Vortical Effect . . . . . . . . . . . . . . . . . 73ii.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 BERRY FERMI LIQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.1 Invitation: (2 + 1) D Dirac Fermion with Weak Contact Interaction . . . . . 814.2 Review of Landau Fermi Liquid Theory . . . . . . . . . . . . . . . . . . . . . 864.3 Berry Fermi Liquid Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.3.1 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.3.2 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.3.3 Chemical Potential dependence of Kinetic Energy . . . . . . . . . . . 954.3.4 Chemical Potential dependence of Hall Conductivity Tensor . . . . . 964.3.5 Correspondence between Kinetic Theory and Field Theory . . . . . . 1024.4 Derivation from Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . 1034.4.1 Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.4.2 Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.4.3 Electromagnetic Coupling . . . . . . . . . . . . . . . . . . . . . . . . 1204.4.4 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.4.5 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.4.6 Cutkosky Cut and Quasiparticle Collision . . . . . . . . . . . . . . . 1414.4.7 Some Cancellation of Diagrams . . . . . . . . . . . . . . . . . . . . . 1534.5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166iii
IST OF FIGURES (cid:126)E · (cid:126)B > (cid:126)L and no total spin angular momentum (cid:126)S , and hence no total angular momentum (cid:126) J ; so are the outgoing particles. If we boost along the direction of the incomingparticles, then the incoming particles still have no (cid:126)L and no (cid:126)S and hence (cid:126) J = 0;however, the out-going particles have (cid:126)S (cid:54) = 0, and in order for (cid:126) J to be conserved,they must have non-zero (cid:126)L = − (cid:126)S . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 A schematic illustration of the non-local collision of massless spinning particlesin a non-center-of-mass frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1 Around some discrete values of chemical potential, the Fermi surface may developnew disconnected components, which may lead to a quantum phase transition.The behavior of the interacting system around such values of chemical potentialremains unknown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96iv IST OF TABLES
CKNOWLEDGMENTS
Slightly more than five and a half years ago, when I was applying to the University ofChicago for graduate study in physics, I was far more ignorant than I should about physicsresearch: at the time I had not heard of condensed matter physics, I barely had any ideathat there is an important subject called quantum field theory to be learned. . . Despite suchblindness, I had the great luck to be admitted by the Physics Department at the University ofChicago. During my five years here, the Department provided an incredible environment forlearning physics knowledge, for exploring different research areas in physics, and for fosteringstimulating discussions. I have to thank all of those who have worked towards establishing,sustaining and developing such a wonderful academic environment.I would like to express my deepest gratitude towards Dam Thanh Son, my Ph.D. advisorand thesis committee chair. His broad interest over different fields of physics, his insight tocatch interesting problems, his clarity in presenting ideas. . . have all deeply influenced meduring these years, and will continue to exert larger impact on me in the future. He is alsovery nice and interesting to get along with. I feel privileged to have him as my advisor, andI really enjoy these years working with him.I am honored to have William Irvine, Michael Levin and Lian-Tao Wang as my thesiscommittee members. I am interested in their research works and I am looking forward tomore discussions with them in the future.For a period of time, Misha Stephanov was almost like my secondary advisor. I hadmany long and fruitful discussions with him, and received many helpful advices from him.Some of my works presented in this thesis were done in collaboration with him, as well aswith Ho-Ung Yee and Yi Yin.I am also grateful to Lian-Tao Wang and Rocky Kolb, who were my Ph.D. advisors atan early stage and gave me valuable first guidance into physics research, although my workson dark matter done with them, as well as with Michael Fedderke, are not presented in thisvihesis.I benefited a lot from the lectures and discussions with the faculty members, most thank-fully Cheng Chin, Jeffery Harvey, Leo Kadanoff, Kathryn Levin, Michael Levin, Emil Mar-tinec, Savdeep Sethi, Dam Thanh Son, Robert Wald, Lian-Tao Wang and Paul Weigmann. Ihave had valuable discussions with many postdoc fellows and students, most thankfully Jo˜aoCaldeira, Chien-Hung Lin, Eun-Gook Moon, Matthew Roberts, Chengjie Wang, Yizhi Youand Hao Zhang; I especially appreciate Caner Nazaroglu, Michael Geracie and Dung XuanNguyen for many stimulating discussions over the years. The Kadanoff Center Journal Club,initiated by Siavash Golkar, has played an extremely important role in broadening my per-spective in physics. I should also thank some friends I knew from on-line through discussingphysics: Zhen Gao, Wei Gu, Jiajun Li, Jin-Bo Yang and Yi-Zhuang You. Tracing backfurther, I would like to thank Andreas Blass for being my undergraduate research advisoron set theory, and Dragan Huterer on cosmology.Finally, I want to thank my father Shao-Gang Chen for his initiation, and my motherYue-Lan Mo for her long-term support, of my interest in science, and more generally for theirshaping of me as an individual person. And I owe everything to my beloved Ying Zhao, forher wit, affection and understanding. Without her my five years here could not have beenso colorful. vii
BSTRACT
Berry phase plays an important role in many non-trivial phenomena over a broad rangeof many-body systems. In this thesis we focus on the Berry phase due to the change ofthe particles’ momenta, and study its effects in free and interacting fermionic systems. Westart with reviewing the semi-classical kinetic theory with Berry phase for a non-interactingensemble of fermions – a Berry Fermi gas – which might be far-from-equilibrium. We partic-ularly review the famous Berry phase contribution to the anomalous Hall current. We thenprovide a concrete and general path integral derivation for the semi-classical theory. Thenwe turn to the specific example of Weyl fermion, which exhibits the profound quantum phe-nomenon of chiral anomaly; we review how this quantum effect, and its closely related chiralmagnetic effect and chiral vortical effect, arise from Berry phase in the semi-classical kinetictheory. We also discuss how Lorentz symmetry in the kinetic theory of Weyl fermion, seemlyviolated by the Berry phase term, is realized non-trivially; we provide a physical interpre-tation for this non-trivial realization, and discuss its mathematical foundation in Wignertranslation. Next, we turn towards interacting fermionic systems. We consider Fermi liq-uid near equilibrium, and propose the Berry Fermi liquid theory – the extension to LandauFermi liquid theory incorporating Berry phase (and other) effects. In our proposed BerryFermi liquid theory, we can show the Berry phase is a Fermi surface property, qualitativelyunmodified by interactions. But there also arise new effects from interactions, most notablythe emergent electric dipole moment which contributes to the anomalous Hall current inaddition to the usual Berry phase contribution. We prove our proposed Berry Fermi liquidtheory from quantum field theory to all orders in Feynman diagram expansion under verygeneral assumptions.The discussion of Berry Fermi gas is based on the previous literature and the author’sworks [14] and [13]. The Berry Fermi liquid theory follows from the author’s work [15].viii
HAPTER 1INTRODUCTION1.1 Background, Problems and Results
As we learn to do quantum mechanics, we learn to diagonalize the Hamiltonian: H = U D U † . (1.1)The diagonal matrix D is the spectrum of energy eigenvalues, while the unitary matrix U has its columns as energy eigenstates. We have learned to pay much attention to D whenconsidering the dynamics, or time evolution, of the system. But it would be hard to believeif the U part has no effect on dynamics.Indeed, the U part has effects that people have noticed in different problems. But it wasnot until 1984 that a unified, general picture appeared. This is Berry’s picture of geometricphase [8], or Berry phase as we call it. Suppose the Hamiltonian depends on a few adjustableparameters Q I . For simplicity, we let | u ( Q ) (cid:105) be an energy eigenstate (a column of U ( Q ))whose corresponding energy eigenvalue E ( Q ) (an entry of D ( Q )) is well gapped with othereigenvalues. Now we adjust the parameters Q I in time, Q I = Q I ( t ); the variation in timeis so slow compared to the energy gap over (cid:126) . Then, according to the quantum adiabatictheorem, if the initial state is prepared in u ( Q ( t = 0)), the state of the system will stay inthe instantaneous eigenstate | u ( Q ( t )) (cid:105) of the Hamiltonian, up to a complex phase. That is,the state at time t is | ψ u ( t ) (cid:105) = e iθ ( t ) | u ( Q ( t )) (cid:105) with θ ( t = 0) = 0. Now let’s compute thephase θ ( t ). Using the Schr¨odinger’s equation i (cid:126) ∂ t | ψ u ( t ) (cid:105) = H ( Q ( t )) | ψ u ( t ) (cid:105) , we have i (cid:126) ∂ t | u ( Q ( t )) (cid:105) − (cid:126) ∂ t θ ( t ) | u ( Q ( t )) (cid:105) = E ( Q ( t )) | u ( Q ( t )) (cid:105) (1.2)1ontracting with (cid:104) u ( Q ( t )) | on the left, and integrating over time, we find the phase is θ ( t ) = − (cid:90) t dt (cid:48) E ( Q ( t (cid:48) )) (cid:126) − (cid:90) Q ( t ) Q (0) dQ I a I ( Q ) , a I ( Q ) ≡ − i (cid:104) u ( Q ) | ∂ Q I | u ( Q ) (cid:105) (1.3)The first term is the usual dynamical phase from the diagonal D part of the Hamiltonian. Thesecond term is the geometric phase due to the U part, and a I is called the Berry connection in the space of Q I . But we are not done yet. The definition of | u ( Q ) (cid:105) is ambiguous. Wecan redefine it by | u ( Q ) (cid:105) → e iφ ( Q ) | u ( Q ) (cid:105) (corresponding to redefining U by multiplying aunitary diagonal matrix to its right), and the Berry connection will shift by a I → a I + ∂ Q I φ ,so its contribution to θ ( t ) will be shifted by the boundary term φ ( Q ( t )) − φ ( Q (0)). Thus,it seems the Berry connection contribution is just arbitrary. It indeed is, except if the weadjust the system back to its initial setup Q ( t ) = Q (0). In that case, the phase θ Berry = (cid:90) Q ( t )= Q (0) Q (0) dQ I a I ( Q ) (1.4)is independent of the choice of φ , but only depends on the loop Q I ( t ) traces out (due toour physical adjustment) in the Q space. This unambiguous phase is the Berry phase . Notethat the Berry phase does not depend on how Q I ( t ) changes with time, but only on the looptraced out. A closely related concept is the Berry curvature b IJ ( Q ) ≡ ∂ Q I a J ( Q ) − ∂ Q J a I ( Q ) (1.5)which represents “the density of Berry phase per unit area in the Q space”, and is unam-biguously independent of the choice of phase φ ( Q ). If we compare the Berry phase θ Berry tothe magnetic flux Φ, then the Berry connection a I ( Q ) is analogous to the vector potential A i ( (cid:126)x ), while the Berry curvature b IJ ( Q ) is analogous to the magnetic field F ij ( (cid:126)x ).The phenomenon of Berry phase is rooted in the mismatching between the inner product2tructure of the Hilbert space (“dagger-ing”) and the detailed way the Hamiltonian dependson Q ; this mismatching is captured in the definition of a I . This is analogous to anotherphenomenon of geometric phase – the Foucault pendulum, where the precession angle isdue to the mismatching between the pendulum’s tendency to move under inertia and thetransport of the pendulum as the Earth rotates. In both cases adiabaticity plays a crucialrole. In the Berry phase case, the time variation of Q must be slow enough for the state | ψ (cid:105) to stay proportional to the instantaneous eigenvector | u ( Q ( t )) (cid:105) . In the Foucault pendulumcase, the transport of the pendulum must be slow enough that it does not exert force on thependulum’s motion.In the above we have considered the parameters Q being externally adjustable parameters.But they can also be quantum numbers, and that is the scenario we focus on in this thesis.Consider the Weyl Hamiltonian as an example: ˆ H αβ = ˆ p i ( σ i ) αβ . The Hilbert space hasan infinite dimensional subspace, on which the momentum operator acts, as well as a twodimensional subspace, the internal space, on which the Pauli matrices act. If we diagonalizethe momentum operators first, we can use the quantum numbers p i to label the left-over partof the Hamiltonian, which now acts only on the two dimensional internal space (spinor space) H αβ ( p ) = p i ( σ i ) αβ . Another example is the Bloch wave in a lattice. We can diagonalize thelattice momentum part first, and then the Hamiltonian becomes a matrix, labelled by thelattice momentum, acting on the discrete band index. In this thesis, the Q I we consider arethe spatial momentum p i , so the Berry curvature is a curvature in the momentum space.Of course, if p i are good quantum numbers, the momentum will be constant in time andthere will be no Berry phase effect. Therefore, to see the effects of the momentum spaceBerry curvature, we must perturb the system so that the particle’s momentum changes slowlyin time.As we can see from the above, Berry curvature effects are very common; they generallyexist if there is an internal Hilbert space of more than one dimension (i.e. more than one3ermionic systems Neglect Berry Phase Include Berry PhaseFree simple Fermi gas Berry Fermi gasInteracting Landau’s Fermi liquid Berry Fermi liquidnon-Fermi liquid. . .Table 1.1: Free and interacting fermionic systems, with and without Berry phase.band). However, their effects are order (cid:126) suppressed compared to the usual phase due tothe energy eigenvalue. Therefore, Berry curvature effects are only visible if we consider thenext-to-leading order effects in (cid:126) expansion, or equivalently low frequency / long wavelengthexpansion. This is why they are usually overlooked in more traditional studies of bands.In this thesis, we consider multi-band fermionic systems, both free and interacting ones,in which the particles’ momenta change in time due to external electromagnetic field, mani-festing the effects of the momentum space Berry curvature. Let’s organize the different casesas Table 1.1. In this thesis we study the two cell on the right, the non-interacting BerryFermi gas, and the interacting Berry Fermi liquid. Below we introduce the contents in thistable in some details.The upper left cell of simple Fermi gas is the most elementary case. The key concept isPauli’s exclusion principle, leading to phenomena such as degeneracy pressure, the existenceof Fermi surface. . . which are fundamental to our present day understanding of the physicalworld.In simple Fermi gas, Pauli exclusion is crucial to the macroscopic ensemble behavior;however, the particles’ dynamics is completely classical. If we go one order higher in (cid:126) expansion, we will see the quantum effects on the dynamics due to the momentum spaceBerry curvature. This brings us into the regime of Berry Fermi gas theory. Berry phase effectin fermionic system made its first appearance as early as 1950s, when Karplus, Luttingerand Kohn studied the anomalous Hall effect in ferromagnets and attributed the effect toan “anomalous velocity” (explained below) of the electrons in the lattice [36, 39, 51, 50].4f course, this was before the formulation of the general concept of Berry phase [8]. Theformal connection between the anomalous velocity and the Berry phase was realized byChang, Niu and Sundaram in the 1990s [12, 11, 72]. The anomalous velocity is a quantumcorrection to the particle’s velocity perpendicular to the external force (electric field), muchlike the Lorentz force is a force perpendicular to the particle’s velocity. Chang, Niu andSundaram made this analogy mathematically concrete; in particular, the role played by themagnetic field (a curvature of magnetic flux in position space) in Lorentz force is played bythe momentum space Berry curvature in the anomalous velocity.Berry curvature in Fermi gas leads to other interesting phenomena besides anomalousHall effect. Most amazingly, the profound phenomenon of chiral anomaly can be reproducedfrom Berry curvature. The chiral anomaly is the phenomenon that, in a system of Weylfermions, the right-handed fermion number and the left-handed fermion number are sepa-rately conserved classically, but quantum mechanically they are not conserved, only theirsum, the total fermion number (total electric charge) is conserved. This phenomenon hasimportant experimental consequences (e.g. the pion decay rate) and deep mathematicalstructure. In recently years it was shown this phenomenon can be reproduced from theBerry curvature effect in the kinetic theory of Weyl fermions [67, 68, 69, 29, 16]. Two closelyassociated phenomena, chiral magnetic effect (CME) [77, 28] and the chiral vortical effect(CVE) [76, 66], can also be reproduced from the Berry curvature in kinetic theory.The above aspects of Berry Fermi gas will be reviewed in Chapter 2 of this thesis.At the same time as Berry Fermi gas theory achieves such success in describing Weylfermions, a sharp problem arises. Weyl fermion is a relativistic particle respecting Lorentzinvariance. On the other hand, the entire story of Berry curvature explicitly breaks Lorentzinvariance, since it is a curvature in the space of spatial momentum (cid:126)p , i.e. it depends onthe choice of a reference frame. In Chapter 3 of this thesis we provide the resolution to thisproblem [14]. For the semi-classical theory with Berry curvature to be Lorentz invariant,5r equivalently frame independent, it turns out the “spacetime position” x µ and the “fourmomentum” p µ of the Weyl fermion must be frame dependent. This sounds odd, but it is aconsequence of the fact that the semi-classical Weyl fermion has non-point-like feature dueto its spin, and therefore the physical meaning of x µ and p µ must be interpreted carefully. Itwas also known that the frame dependence of x and p is related to Wigner translation [25, 71],the non-compact part of the little group of a massless spinning particle; in this thesis we willpresent the relation in a way somewhat different from the presentation in the literature.We will take one step further, and consider how collisions of Weyl fermions can be includedin the semi-classical kinetic theory in compatible with Lorentz invariance [13]. Collisionswill relax a far-from-equilibrium ensemble, described by our chiral kinetic theory, to localequilibrium, described by chiral hydrodynamics. This is an important step, establishing thebridge from microscopic quantum mechanics to locally-equilibrium hydrodynamics, via thefar-from-equilibrium kinetic theory.The above explains the first row in Table 1.1. But in real systems interactions aregenerally present, and usually not small. Therefore we need to turn to the second row.Landau’s theory of Fermi liquid [41, 40], established in the 1950s, is one of the cor-nerstones of condensed matter physics. A Fermi liquid is an interacting fermionic systemsatisfying two conditions: its ground state is described by a Fermi surface much like that of anon-interacting Fermi gas, and moreover its low energy spectrum is described by “quasiparti-cle” excitations which are qualitatively similar to non-interacting particles. With his superbinsight, Landau realized there is only one effect of interaction at low energy – a local interac-tion potential energy between quasiparticles. Quasiparticle decay and collision arising frominteractions are suppressed by the low energy, due to the limited availability of decay / colli-sion channels. Thus, the low energy behavior of a Fermi liquid can be described by a kinetictheory very similar to that of a Fermi gas, with the addition of a local interaction potentialenergy. This simple picture, the Landau Fermi liquid theory, has been extremely successful6n describing a large class of interacting fermionic systems, most notably Helium-3 [6] andelectrons in normal metal.Landau’s intuitive theory was soon proven by matching with Feynman diagram expansionto all orders in perturbations theory, by Landau himself and others [43, 59, 49, 1]. In the1990s, Landau’s original insight was finally concretely casted in the language of low energy ef-fective field theory [60, 61]. Except for a possible instability in the Bardeen-Cooper-Schriefferchannel, Landau’s Fermi liquid theory provides a truly universal low-energy effective descrip-tion of Fermi liquid systems with short-ranged interactions.We shall now turn to the motivation of our Berry Fermi liquid theory [15]. As mentionedabove, people have derived many interesting results from Berry Fermi gas theory, but howmuch of those survives once interaction is included (as in real systems)? Also, are thereany new effects arising from interactions that are not present in Berry Fermi gas theory?To answer these two questions, we have to extend Landau’s Fermi liquid theory into next-to-leading order in low energy / long wavelength expansion. This extension, we call
BerryFermi liquid theory , is presented in Chapter 4. We will also provide a proof to the BerryFermi liquid theory by resumming Feynman diagram expansion to all orders. Our mainresults are the followings. First, it is known that for a Fermi gas, the Berry curvature effectscan be written as Fermi surface integral [32]. It would be important if the same can be donefor a Fermi liquid, because for a Fermi liquid the system’s properties far from the Fermisurface are generally complicated and non-universal. We show this indeed can be done inFermi liquid, and the form of the Fermi surface integral is almost the same as in Fermi gas.Second, in addition to the usual Berry curvature contribution, there is a new contributionto the anomalous Hall effect from the quasiparticles’ emergent electric dipole moment dueto interactions. Third, at next-to-leading order in low energy / long wavelength expansion,we have to consider quasiparticle decay and collisions; however, they have no contributionto the “interesting physics” such as anomalous Hall effect, chiral magnetic effect, etc.7n the previous literature, the work that has the most overlap with our Berry Fermi liquidtheory is Ref. [62] where the interplay between Berry curvature and interaction has beenstudied in a very general context. The authors of Ref. [62] showed, via the Keldysh formalism,that the quasiparticles’ motion has an anomalous velocity due to the Berry curvature, as inthe non-interacting case, but the content of the Berry curvature is modified by interactions.There are four main differences between Ref. [62] and our Chapter 4. First, in contrastto Ref. [62], we only study linear response, which does not see the effect of the anomalousvelocity. Second, in our theory the Berry curvature effects show up in (the non-quasiparticlecontribution to) the current, which is not computed in Ref. [62]. Third, we are able to takeinto account the effect of the quasiparticle collisions and the finite quasiparticle lifetime.Last, we are able to answer the question whether interesting transport phenomena suchas the anomalous Hall effect involve Fermi surface contribution only, or involve Fermi seacontribution as well.One can notice at the bottom of Table 1.1, there is also “non-Fermi liquid”. Fermi liquiddescribes a large class of interacting fermionic systems, but there are also many non-Fermiliquid systems, not satisfying one or both assumptions about Fermi liquid. The study ofthese systems is an important subject, but far beyond the scope of this thesis.Finally, we emphasize that impurities / disorders are completely neglected in this thesis.This is a purely theoretical idealization. There are extra non-trivial effects if disorders areincluded. For instance, there will be extra contribution to anomalous Hall effect from skew-scattering and side-jump; see Ref.[53] for review.The chapters of this thesis are related to the author’s research works as the following.Chapter 2 is mostly a summarization of previous literature, except for Section 2.4, which is ageneralized and refined version of the path integral derivation in the author and collaborators’work Ref.[14]. The first two sections of Chapter 3 are also based on [14]; the third section8s from some unpublished work of the author; Section 3.4 is an elaboration of the authorand collaborators’ work [13]. Chapter 4 is mainly a reproduction of the author and advisor’srecent work [15], with the example in Section 4.1 added for demonstration purpose.Needless to say, the references included in this thesis are far from a complete list onthis rich subject. Here the author only included those which were more familiar to theauthor during the course of study. There must be a lot of important works missing from thereferences due to the author’s limitation.
Chapters 2 and 3 are devoted to Berry Fermi gas system without interaction. In Chapter 2,we start with a review of the symplectic formulation of classical mechanics. Then we presentthe semi-classical action of a single particle with Berry curvature in external electromagneticfield, and extract the implications from the action using the symplectic formulation. We thenconsider a non-interacting ensemble of such particles – we focus on fermions in particular –and discuss the general effects of Berry curvature on the ensemble’s macroscopic behavior;we will pay particular attention to the anomalous Hall current due to Berry curvature. Tosupport the validity of the semi-classical action, in Section 2.4 we derive it from path integralunder very general setting, and get to understand the microscopic origin of the particle’sBerry curvature as well as its magnetic dipole moment. We then turn from the generalcase to the specific example of Weyl fermion as a concrete demonstration of the generalformalism. The example of Weyl fermion is important in its own. It has band touching, andexhibits chiral anomaly as well as the associated chiral magnetic effect and chiral vorticaleffect. We will present the computation of chiral magnetic effect and chiral anomaly fromthe semi-classical theory of Berry Fermi gas, and discuss the physical picture.The semi-classical theory of Weyl fermion seems to violate Lorentz invariance, but it mustnot. In Chapter 3 we resolve this puzzle. We will first provide a physical argument arguing9hat the notion of “position” of a semi-classical massless spinning particle must be framedependent. Then we present in detail how Lorentz invariance is realized non-trivially in thesemi-classical theory – both the notion of “position” and “momentum” are frame dependent.We will show that, despite the frame dependence of position and momentum, the currentand stress-energy tensor are frame independent (Lorentz covariant), and therefore they arethe legitimate basic physical observables. We will give a physical interpretation to the non-trivial realization of Lorentz invariance based on the helical (spinning) feature of the current.In Section 3.3 we will explore the mathematical foundation of the non-trivial realization ofLorentz invariance – it is due to Wigner translation, the non-compact part of the little groupof a massless spinning particle. Finally, in Section 3.4 we will consider one step beyond Fermigas theory. We assume the Weyl fermions collide, and study the restrictions on the formof the collisionful kinetic theory due to Lorentz invariance. We show that such collisionfulkinetic theory of Weyl fermions will relax to the familiar chiral hydrodynamical limit, inwhich we will compute the chiral vortical effect.Chapter 4 is devoted to interacting fermionic systems with Berry curvature. We startwith a simple example, Dirac fermion with weak contact interaction in 2 spatial dimensions,to demonstrate the appearance of new contribution to anomalous Hall current not presentin non-interacting systems. This example shows the inclusion of interaction must be anon-trivial story. To proceed, we first provide a quick review on the computation of linearresponse in Landau’s Fermi liquid theory. Then we present our proposed Berry Fermi liquidtheory, which incorporates Berry curvature effects (as well as other effects) in extension toLandau’s theory. In Section 4.4 we derive the Berry Fermi liquid theory from perturbativequantum field theory under very general assumptions. Our derivation is valid to all orders inFeynman diagram expansion, as long as we stay in the long wavelength limit. The quantumfield theoretic derivation itself is quite technical and length, involving heavy use of Cutkoskycut, Ward-Takahashi identity and combinatorial diagrammatic techniques; the organization10f the derivation is provided at the beginning of Section 4.4.At the end of each of these three chapters there is a “Summary and Outlook” section,summarizing the main results and ideas in that chapter, and discussing unresolved problemsand future directions of study. These discussions will not be repeated in the final Conclusionchapter. The final Conclusion chapter will contain brief discussions in broader perspective.
To make the computations readable, it is important to clarify the conventions.We generally consider systems in d spatial dimensions, i.e. ( d + 1) spacetime dimensions,with d ≥ x µ , with x being time t (insteadof ct in usual relativistic context) and x i ( i = 1 , . . . , d ) being spatial position. Spatialmomentum is denoted as p i = p i , while kinetic energy is denoted as p = − p . For derivativeswith respect to momentum, we denote ∂ p ≡ ∂/∂p = − ∂/∂p = ∂ p = − ∂ p while ∂ ip ≡ ∂/∂p i = ∂/∂p i = ∂ ip = ∂ p i . This rule of raising and lowering indices on momentum iscompatible with, but does not rely on the Minkowski metric; it follows from the canonicalstructure of classical / quantum mechanics.For relativistic systems, our convention of Minkowski metric is η µν = diag( − , , . . . , (cid:15) ...d = − (cid:15) ...d = 1.We set the Boltzmann constant to k B = 1. For relativistic systems the speed of light c = 1 is understood. However, we will generally leave (cid:126) explicit to keep track of the quantumeffects in the semi-classical Berry curvature framework. We will set (cid:126) = 1 in the quantumfield theory computations in Sections 4.1 and 4.4.The validity of the semi-classical picture relies on low frequency / long wavelength ex-pansion, i.e. the expansion of q µ ∼ (cid:126) ∂ x µ over the typical energy / momentum scale of thesystem. It is equivalent to keep track of (cid:126) or q ; in Chapters 2 and 3 we keep track of the11ormer, and in Chapter 4 the latter.The notation X [ i ...i n ] means X i ...i n with all the indices antisymmetrized; there are n !terms in total, and in our convention there is an overall factor of 1 /n !. The notation X { i ...i n } means X i ...i n with all the indices symmetrized; again there are n ! terms in total and in ourconvention there is an overall factor of 1 /n !. 12 HAPTER 2BERRY FERMI GAS
In this chapter, we discuss the semi-classical theory of a single particle with Berry curvature,and extract its implications on an ensemble of non-interacting fermions – a Berry Fermi gas.The particular strength of the Berry Fermi gas theory, as a Boltzmann type kinetic theory,is that it describes macroscopic systems far-from-equilibrium, and therefore has broaderapplication (when interactions are small) than the commonly used hydrodynamics theorywhich assumes local equilibrium. We will see how the Berry phase-related interesting physicsof anomalous Hall effect, chiral magnetic effect and chiral anomaly are computed in the BerryFermi gas theory.This field has been very well-developed, therefore this chapter is mostly a summarizationof previous works. Section 2.4 contains a path integral derivation of the semi-classical theorywith full generality, which is not written down explicitly in the previous literature.
To most efficiently convey the Berry phase physics, we use the symplectic form formulationof classical mechanics. Below is a brief review of the formulation.
Consider the action of a single particle moving in d -dimensional space under Hamiltonian H ( (cid:126)p, (cid:126)x, t ): S [ (cid:126)p ( t ) , (cid:126)x ( t )] = (cid:90) dS, dS = p i dx i − H ( (cid:126)p, (cid:126)x, t ) dt (2.1)where the integration is along an arbitrary worldline of the particle in the 2 d -dimensionalphase space. We regard the first term of dS as the symplectic part , and the second term as13he Hamiltonian part . Now we denote ( x i , p i ) collectively as ξ I , so the action can be writtenas dS = ω I dξ I − H ( ξ, t ) dt (2.2)where ω I is called the symplectic 1-form , ω x i = p i , ω p i = 0. Adding to ω I a total ξ I derivativechanges the action by a boundary term and affects no dynamics.Consider arbitrary infinitesimal variations δξ I : δdS = δξ I ω IJ dξ J − δξ I (cid:16) ∂ ξ I H + ∂ t ω I (cid:17) dt, (2.3)where dropping total derivative terms is always implicitly understood. Here ω IJ ≡ ∂ [ ξ I ω J ] is the symplectic 2-form . Explicitly, ω IJ = ω x i x j = 0 ω x i p j = − δ ji ω p i x j = δ ij ω p i p j = 0 . (2.4)On the other hand, ∂ t ω I = 0 in the present case; we will see below why we formally keep it.The equation of motion (EoM) is derived by requiring dξ to be such that δdS is totalderivative for arbitrary δξ . This requires ω IJ dξ J = (cid:16) ∂ ξ I H + ∂ t ω I (cid:17) dt, i.e. dξ I /dt (cid:12)(cid:12)(cid:12) EoM = ω IJ (cid:16) ∂ ξ J H + ∂ t ω J (cid:17) , (2.5)where ω IJ ≡ ( ω − ) IJ is the inverse symplectic 2-form whose components are those of the14 oisson brackets : ω IJ ≡ ( ω − ) IJ = { x i , x j } = 0 { x i , p j } = δ ij { p i , x j } = − δ ji { p i , p j } = 0 . (2.6)Similar, the symmetries of an action can be derived by requiring δξ to be such that δdS is total derivative for arbitrary dξ . This requires the existence of some function Q ( (cid:126)x, (cid:126)p, t ),called the Noether charge of the corresponding symmetry, such that δξ I ω IJ = − ∂ ξ J Q, δξ I (cid:16) ∂ ξ I H + ∂ t ω J (cid:17) = ∂ t Q. (2.7)It is easy to see when the EoM (2.5) is satisfied, dQ/dt | EoM = − δdS/dt = 0, i.e. the Noethercharge is a conserved quantity at EoM. Finally, if the symmetry transformation involves thetransformation of some non-dynamical parameter α on which H depends, then the secondequation in (2.7) should have an extra term δα∂ α H on the left-hand-side.All these seem quite trivial – all we have done is to state the textbook Hamiltonianmechanics [30] in some fancy language. But the key point is the following: Now it is no longernecessary to use ( x i , p i ) to parametrize the phase space; ξ I can be arbitrary parametrizationof the phase space, and (2.3), (2.5) and (2.7) still hold, as long as we require ω IJ to transformas a (0, 2)-tensor on the phase space. That is, under reparametrization ξ I → ξ (cid:48) I (cid:48) , ω I (cid:48) J (cid:48) = ω IJ ∂ ξ (cid:48) I (cid:48) ξ I ∂ ξ (cid:48) J (cid:48) ξ J . (2.8)If ω IJ in a parametrization ξ I still takes the “standard form” (2.4), then the parametrization ξ I are called canonical variables , otherwise ξ I are non-canonical; for the study of Berrycurvature physics it is convenient to use non-canonical variables, as we will see later. Clearly,in any parametrization, ω is always a non-degenerate, closed 2-form – in fact, these two15roperties are the defining properties of a symplectic 2-form. Non-degeneracy means thephase space is physical and involves no redundant gauge degree of freedom; the dynamicsis completely determined by the EoM (2.5). The meaning of closed-ness will be discussedlater.The reparametrization of ξ I → ξ (cid:48) I (cid:48) can even be time dependent, as long as we requirethe Hamiltonian to transform accordingly: H (cid:48) = H + ω J ∂ t ξ J (2.9)so that the action (2.2), or equivalently its variation (2.3), remains invariant up to totalderivative.Let’s see a simple example of non-canonical variables. A particle moving in externalelectromagnetic field has action dS = p i dx i + A i ( (cid:126)x, t ) dx i − H ( (cid:126)p, (cid:126)x, t ) dt (2.10)(we have absorbed the electric potential − A into H ). Here p i is the physical momentumand is not canonical to x i . Explicitly, ω IJ = ω x i x j = F ij ω x i p j = − δ ji ω p i x j = δ ij ω p i p j = 0 (2.11)where F ij ≡ ∂ [ x i A j ] is the magnetic field, and its corresponding block ω x i x j gives rise tothe Lorentz force. One can define the canonical momentum P i ( t ) ≡ p i ( t ) + A i ( (cid:126)x ( t )) , (2.12)16o that ( x i , P i ) are canonical variables and ω is brought back into the standard form (2.4),at the price that H would now depend on A through p i = P i − A i .The transformation law (2.8) implies the volume element d d ξ (cid:112) det ω IJ = d d ξ (cid:48) (cid:112) det ω I (cid:48) J (cid:48) (2.13)is the natural volume element in the phase space – this is called the Liouville volume element.For canonical variables clearly it is just d d ξ . An important property of the volume elementis the Liouville’s theorem: ∂ t √ det ω + ∂ ξ I (cid:32) √ det ω dξ I dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) EoM (cid:33) = − √ det ω ∂ [ ξ I ω JK ] ω IJ ω KL (cid:0) ∂ ξ L H + ∂ t ω L (cid:1) = 0 (2.14)where the second equality follows from the closed-ness ∂ [ ξ I ω JK ] = 0. One can understandthis theorem as: a fluid (ensemble of independent particles) is flowing in the phase space intime according to the EoM, but the Liouville volume of this fluid remain the same over time.A question arises. In general relativity, we know not all metrics are locally reparametrization-equivalent to the Minkowski metric, and the local inequivalence is measured by the Riemanncurvature. Are all symplectic 2-forms locally reparametrization-equivalent to the standardone (2.4)? The answer is Yes: All non-degenerate closed 2-form is locally reparametrization-equivalent to the form (2.4). This is the Darboux theorem and can be easily proven byconstructing a reparametrization using interpolation [23]. Thus, in classical mechanics, lo-cally there is no non-trivial background in the phase space. The most familiar example is(2.12).But globally there is non-trivial information in the symplectic 2-form. While closed 2-forms are locally exact, globally they may not be if the phase space is not simply-connected;if so, ω I cannot be a globally continuous 1-form over the phase space. The failure of ω IJ In the above, we have reviewed the symplectic formulation of single particle classical mechan-ics. While the formalism is mostly beautiful, it has one ugliness: Time t , which we use toparametrize the worldline, is a physical quantity, so the system may have time dependence,and the phase space reparametrization ξ → ξ (cid:48) might also have time dependence. Thesetime dependence makes the symplectic 1-form appear explicitly in (2.5) and (2.9). But thesymplectic 1-form is contains non-physical information – the phase space gauge freedom ofadding total ξ derivative. Is there a formulation that can get rid all these troubles due totime dependences?There is another motivation for an alternative formulation. If we consider relativisticsystems (which we will), Lorentz invariance puts time and space at equal footing. But usingtime to parametrize the worldline makes time special.With these motivations, it is now clear how we shall reformulate the physics. Instead ofusing the physical time t to parametrize the particle’s worldline, we use some intrinsic timeparameter τ . Since τ is intrinsic to the worldline only, the physical system shall not dependon τ , and the any physically meaningful transformation should not depend on τ . The pricewe pay is we promote the physical time and the associated energy to dynamical quantities.For example, the action (2.10) is reformulated as S = (cid:90) dS, dS = p µ dx µ + A µ ( x ) dx µ − H ( p, x ) λdτ (2.15)18here x ≡ t is time and − p = p is energy. λ is the einbein on the worldline suchthat λdτ = λ (cid:48) dτ (cid:48) under worldline reparametrization. λ is also a Lagrange multiplier – thevariation with respect to it demands H ( p, x ) = 0, from which the energy p can be solved inthe form p = H ( (cid:126)p, (cid:126)x, t ) (2.16)where H is the Hamiltonian in our old formulation. The EoM for x relates the intrinsic λdτ to the physical dx , while the EoM for p will be compatible with dp /λdτ = dH/λdτ . Fornon-relativistic particle, we can choose H = p − | (cid:126)p | / m . For relativistic particle, a nicechoice is H = p µ p µ + m so that the entire action is manifestly Lorentz invariant (demanding p > d + 1)-dimensional, and instead of the ( x µ , p µ ) parametrization,we can consider arbitrary parametrization ξ I ; any parametrization of the phase space shouldbe independent of the worldline parameter τ . The formal results in the old formulation arestraightforwardly carried over, except dt are replaced by λdτ , and now nothing has the ∂ τ dependence. The only unobvious modification is the Liouville phase space volume nowbecomes (cid:90) d d ξ (cid:112) det ω IJ = (cid:90) d d +2 ξ (cid:112) det ω IJ δ ( H ) dλdτ . (2.17)(This can be proven by invoking the EoM for dx /λdτ .) In particular, the right-hand-side is invariant under reparametrization ξ → ξ (cid:48) , because the reparametrization must be τ independent and hence H (cid:48) = H . For relativistic systems, this expression of Liouville phasespace volume is very useful because it is manifestly Lorentz invariant as long as we havechosen a Lorentz invariant expression for H (e.g. H = p µ p µ + m ).19 .2 Semi-Classical Particle with Berry Phase Having had the formalism of symplectic 2-form introduced, we are ready to present thesemi-classical mechanics of a single particle with Berry curvature. The quantum mechanicaljustification of this formalism is left to Section 2.4. To be consistent with the literature, wewill use the physical time as the worldline parameterization in this chapter. The intrinsicparametrization will be used in the next chapter when we discuss the Lorentz invariance ofsemi-classical Weyl fermions.The action is [10, 83] S = (cid:90) dS, dS = p i dx i + A i ( (cid:126)x, t ) dx i − (cid:126) a i ( (cid:126)p ) dp i − H ( (cid:126)p, (cid:126)x, t ) dt + A ( (cid:126)x, t ) dt (2.18)where we have separated A from H , so that H = H − A . a i ( (cid:126)p ) is the Berry connectionin the momentum space. Microscopically it is given by (cid:126) a i ( (cid:126)p ) = − i (cid:126) u † α ( (cid:126)p ) ∂ ip u α ( (cid:126)p ) = i (cid:126) ∂ ip u † α ( (cid:126)p ) u α ( (cid:126)p ) (2.19)where u α ( (cid:126)p ) is the normalized Bloch state or spinor of the quantum mechanical particlewhen A µ = 0. The Berry curvature is given by (cid:126) b ij ( (cid:126)p ) ≡ (cid:126) ∂ [ ip a j ] ( (cid:126)p ) = − i (cid:126) ∂ [ ip u † α ( (cid:126)p ) ∂ j ] p u α ( (cid:126)p ) . (2.20)We leave (cid:126) explicit to keep track of semi-classical effects. Quantum mechanically, we canmultiply u α by a (cid:126)p -dependent complex phase, and no physics should change; indeed, sucha transformation adds to a i a total p i -derivative, under which the action and the Berrycurvature are left invariant.What is the range of applicability of this action? As we will see in the derivation inSection 2.4, the semi-classical dynamics is valid to first order in (cid:126) (compared to the typical20ction of a worldline). This means (cid:126) ∂ x / | (cid:126)p | is kept to first order, so the expansion over (cid:126) isequivalent to the low frequency / long wavelength expansion over ∂ x µ .The symplectic 2-form is given by ω IJ = ω x i x j = F ij ω x i p j = − δ ji ω p i x j = δ ij ω p i p j = − (cid:126) b ij . (2.21)Since the semi-classical theory is valid to order (cid:126) , we only need to find the determinant andinverse of ω IJ to order (cid:126) . The determinant, and hence Liouville volume, is modified by theBerry curvature [84, 26] √ det ω = 1 + (cid:126) F ij b ij / . (2.22)The inverse, or Poisson bracket, is given by ω IJ = { x i , x j } = − (cid:126) b ij { x i , p j } = δ ij + (cid:126) b ik F kj { p i , x j } = − δ ji − (cid:126) F ik b kj { p i , p j } = F ij + (cid:126) F ik b kl F lj . (2.23)The EoM (2.5) written explicitly to order (cid:126) will read dx i dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) EoM = ∂ ip H + (cid:126) b ij (cid:16) F j + F jk ∂ kp H − ∂ x j H (cid:17) ,dp i dt (cid:12)(cid:12)(cid:12)(cid:12) EoM = F i + F ij dx j dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) EoM − ∂ x i H = (cid:16) δ ji + (cid:126) F il b lj (cid:17) (cid:16) F j + F jk ∂ kp H − ∂ x j H (cid:17) (2.24)(recall that F ij is the magnetic field and F i = − F i is the electric field). The (cid:126) term in21 x i /dt , given by the Berry curvature contracting with the force, is the famous anomalousvelocity [36, 72].Here we work with Berry curvature in the momentum space only. Of course, if there issome external adjustable parameter, on which the Hamiltonian depends, that varies slowly inspace and time, there would be extra Berry curvature components b p i x j , b x i p j and b x i x j [72,10, 83]. In this thesis we would not consider such generality.Although the Darboux theorem mentioned in Section 2.1 guarantees the existence ofcanonical variables, they are rarely used in the semi-classical Berry phase physics literature.Here we mention them for completeness. If Berry curvature is absent, x i and P i ≡ p i + A i are canonical variables. In the presence of Berry curvature, a convenient choice of canonicalvariables ( X i , P i ) valid to order (cid:126) is x i = X i − (cid:126) a i ( (cid:126) P − (cid:126)A ( (cid:126) X , t )) , p i = P i − A i ( (cid:126) X , t ) − (cid:126) a j ( (cid:126) P − (cid:126)A ( (cid:126) X , t )) F ij ( (cid:126) X , t ); X i = x i + (cid:126) a i ( (cid:126)p ) , P i = p i + A i ( (cid:126)x, t ) + (cid:126) a j ( (cid:126)p ) ∂ x i A j ( (cid:126)x, t ) . (2.25)(Here we expand in (cid:126) which is the natural expansion parameter in semi-classical physics; in[83] the authors expanded in A .) According to (2.9), the Hamiltonian should shift by H cano. ( P , (cid:126) X , t ) = H ( (cid:126)p, (cid:126)x, t ) − (cid:126) a i ( (cid:126)p ) ∂ t A i ( (cid:126)x, t )= H ( (cid:126) P − (cid:126)A, (cid:126) X , t ) − A ( (cid:126) X , t ) + (cid:126) a i ( (cid:126) P − (cid:126)A ) × (cid:16) F i + F ij ∂ P j H ( (cid:126) P − (cid:126)A, (cid:126) X , t ) − ∂ (cid:126) X i H ( (cid:126) P − (cid:126)A, (cid:126) X , t ) (cid:17) (2.26)where H = H − A , and the argument of A µ is ( (cid:126) X , t ). The Liouville volume in canonicalvariables is the usual d d P d d X . 22 .3 Ensemble of Berry Fermi Gas In the previous section we considered the Berry curvature effects for a semi-classical singleparticle. Now we study the effects in an ensemble of non-interacting particles – in thisthesis we focus on fermions. One of the important consequence is the relation between theanomalous Hall current and the anomalous velocity due to Berry curvature [36, 72].
We let ρ denote the particle density per unit d d ξ = d d x d d p : ρ ( (cid:126)p, (cid:126)x, t ) ≡ dNd d x d d p . (2.27)If particles are moving independently and not being created or annihilated, the continuityequation ∂ t ρ + ∂ ξ I (cid:16) ρ dξ I /dt (cid:17) = 0 (2.28)must hold whether or not EoM is satisfied. Now we define the distribution function f ( (cid:126)p, (cid:126)x, t )to be the particle density per Liouville volume: f ≡ ρ √ det ω = dNd d ξ √ det ω . (2.29)With the aid of the Liouville’s theorem (2.14), the continuity equation implies the collisionlessBoltzmann equation ∂ t f + (cid:16) dξ I /dt (cid:17)(cid:12)(cid:12)(cid:12) EoM ∂ ξ I f = ∂ t f + ∂ ξ I f ω IJ (cid:16) ∂ ξ J H + ∂ t ω J (cid:17) = 0 . (2.30)For fermions, Pauli exclusion requires 0 ≤ f ≤ ∂ f = 0, if ω IJ , H are time independent and f = f ( H ).However, in reality there is always some small collision (whose relaxation time is perhapsmuch longer compared to our time scale of interest), therefore really the only equilibriumdistribution is the Fermi Dirac distribution f = f F D ( H ).The usual physical observables for an ensemble are the current and the stress-energytensor. First consider the current of a single particle with worldline ( z i ( t ) , p i ( t )) in thephase space satisfying the EoM: J µsp ( (cid:126)x, t )[ (cid:126)p, (cid:126)z ] ≡ δS [ (cid:126)p, (cid:126)z ] δA µ ( (cid:126)x, t ) (cid:12)(cid:12)(cid:12)(cid:12) EoM = (cid:90) dt (cid:48) (cid:18) dz µ dt (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) EoM − ∂H ( (cid:126)p, (cid:126)z, t (cid:48) ) ∂F νµ ( (cid:126)z, t (cid:48) ) ∂ z ν (cid:19) δ d ( (cid:126)z − (cid:126)x ) δ ( t (cid:48) − t ) (2.31)where the integral is along the physical worldline (satisfying EoM) of the particle in thephase space, and z ≡ t (cid:48) , x ≡ t . For an ensemble of non-interacting particles, the totalcurrent is just the integral of the single particle currents over the particle density in thephase space: J µ ( (cid:126)x, t ) ≡ (cid:90) d d p d d z (cid:112) det ω ( (cid:126)p, (cid:126)z, t )(2 π (cid:126) ) d f ( (cid:126)p, (cid:126)z, t ) J µsp ( (cid:126)x, t )[ (cid:126)p, (cid:126)z ]= (cid:90) d d p (cid:112) det ω ( (cid:126)p, (cid:126)x, t )(2 π (cid:126) ) d (cid:20) dx µ dt (cid:12)(cid:12)(cid:12)(cid:12) EoM f ( (cid:126)p, (cid:126)x, t ) + ∂ x ν (cid:18) − ∂H ( (cid:126)p, (cid:126)x, t ) ∂F µν ( (cid:126)x, t ) f ( (cid:126)p, (cid:126)x, t ) (cid:19)(cid:21) . (2.32)The J component is interpreted as the particle density at ( (cid:126)x, t ), while the J i components arethe current density. The conservation ∂ x µ J µ = 0 just follows from the U (1) gauge invarianceas usual.Clearly the first term in (2.32) is the transport current. What is the nature of thesecond term? Note that − ∂H/∂F ij can be understood as the magnetic dipole moment and24 ∂H/∂F i the electric dipole moment, so the second term really corresponds to the electricpolarization density (when µ = 0) and magnetization / electric polarization current (when µ = i ). This term plays an important role in, for instance, defining the chiral vortical currentfor Weyl fermions, as we will see in Section 3.4.4.We emphasize that although in the definition of the current, we made use of the electro-magnetic field, the result also holds for neutral particles, as we can take the charge of theparticle (which has been absorbed in A ) to the zero limit.If we couple the action to a spacetime structure (general relativity geometry for relativisticparticle or Newton-Cartan geometry for non-relativisitic particle), we can also derive thestress-energy tensor. This is beyond the scope of this thesis. We just comment the followings.The energy density T should take the form (cid:90) d d p (cid:112) det ω ( (cid:126)p, (cid:126)x, t )(2 π (cid:126) ) d [ H ( (cid:126)p, (cid:126)x, t ) f ( (cid:126)p, (cid:126)x, t ) + (terms with ∂ x f )] , (2.33)while the momentum density T i should take the form (cid:90) d d p (cid:112) det ω ( (cid:126)p, (cid:126)x, t )(2 π (cid:126) ) d [ p i f ( (cid:126)p, (cid:126)x, t ) + (terms with ∂ x f )] , (2.34)as one would expect. Moreover, the conservation ∂ x ν T µν = F µλ J λ must hold. When wediscuss the specific example of Weyl fermion in Section 2.5, we will present the full form ofits stress-energy tensor, omitting the derivation which requires either coupling to spacetimemetric or performing the Belinfante procedure to the Noether stress-energy tensor.All the discussion so far has been completely general. Now we focus on Berry fermiongas in particular. We impose some additional restrictions: • The electric dipole moment vanishes. This is a reasonable assumption for non-interactingfermions; in particular this is true for non-interacting electrons in a lattice. However,as we will consider in Chapter 4, interactions in general lead to emergent electric dipole25oment. • The magnetic dipole moment of a single fermion is order (cid:126) , and depends on (cid:126)p only butnot on (cid:126)x . This is the case because the only characteristic scale of a particle that hasdimension [ x ] (needed to cancel the [ x ] − from ∂ x in electromagnetic field) should be (cid:126) / | (cid:126)p | . • The only x dependence in H is the electric potential and the magnetic dipole term.Physically, this corresponds to no force other than electromagnetic force is acting onthe particle, or alternatively, we have effectively absorbed all other forces acting on theparticle into the electric potential A .With these assumptions, we can write H ( (cid:126)p, (cid:126)x, t ) = H (cid:48) ( (cid:126)p, (cid:126)x, t ) − A ( (cid:126)x, t ) = E ( (cid:126)p ) − (cid:126) µ ij ( (cid:126)p ) F ij ( (cid:126)x, t ) / − A ( (cid:126)x, t ) (2.35)where (cid:126) µ ij ( (cid:126)p ) is the magnetic dipole moment; its microscopic relation to the Bloch state orspinor u α will be given in the next section when we perform the path integral derivation.Now, the current (2.32) satisfying the EoM (2.24) reads J ( (cid:126)x, t ) = (cid:90) d d p (2 π (cid:126) ) d (cid:32) (cid:126) F ij b ij (cid:33) f, (2.36) J i ( (cid:126)x, t ) = (cid:90) d d p (2 π (cid:126) ) d (cid:34) ∂ ip H f + (cid:126) b ij F j f + (cid:126) F jk b [ ij ∂ k ] p H f + (cid:126) µ ij ∂ x j f (cid:35) . (2.37)Below we look at some interesting implications from these expressions. It is these implicationsthat drove people into the study of Berry Fermi gas systems.26 .3.2 Anomalous Hall Effect First we consider the anomalous Hall current in a spatially uniform electric field oscillating intime (so that the particles do not keep accelerating), without magnetic field. The anomalousHall current, i.e. that perpendicular to the electric field, is given by [12, 11, 72] δJ iH ( (cid:126)x, t ) = F j ( (cid:126)x, t ) (cid:90) d d p (2 π (cid:126) ) d (cid:126) b ij ( (cid:126)p ) f ( (cid:126)p, (cid:126)x, t ) . (2.38)This is correct in the semi-classical framework presented so far. However, recall that when wedefined the Berry curvature, the spinor / Bloch state u α is multi-component, which meansthere exists bands besides the u -band, and each of these bands has its own distribution f and Berry curvature b ij . To get the full anomalous Hall current, we must sum up (2.39) forall bands.More specifically, suppose u is the conducting band in a lattice system, there shouldalso be some valence band(s) and some empty bands. Let’s further assume the system isoscillating around in thermal equilibrium, i.e. f ( (cid:126)p, (cid:126)x, t ) equals f F D ( E ( (cid:126)p )) plus oscillation dueto the electric field. If we consider linear response, in (2.39) we can approximate f = f F D .Finally, we assume the temperature is very small compared to the gap between the u -bandand the other bands, so we can approximately take f = 1 for all valence bands and f = 0for all empty bands. The total physical anomalous Hall current in linear response δJ iH tot. = F j (cid:32)(cid:90) d d p (2 π (cid:126) ) d (cid:126) b ij f F D ( E ) + (cid:88) w (cid:90) d d p (2 π (cid:126) ) d (cid:126) b ijw (cid:33) (2.39)where w runs over the valences bands, and the integration is over the Brillouin zone (BZ)for a lattice system.For d = 2 spatial dimensions, each valence band w contributes to the Hall conductivity (cid:15) ij C w / (2 π (cid:126) ), where C w is an integer called the Chern number of the w -band. The proof isthe following [74, 37]. When b ijw ( (cid:126)p ) is an exact 2-form over the BZ, by Stoke’s Theorem the27alence band contribution vanishes. b ijw is non-exact when the complex phase of the spinor/ Bloch state w α ( (cid:126)p ) cannot be chosen smoothly over the BZ. In such case, we can dividethe BZ into multiple patches. The complex phase of u α ( (cid:126)p ) is smooth within each patch, butdiscontinuous over the boundaries (which are topologically S circles) between the patches.By Stoke’s Theorem, (cid:90) d d p (2 π (cid:126) ) d (cid:126) b ijw = (cid:15) ij (cid:126) (2 π (cid:126) ) (cid:88) B (cid:90) π dθ B d ∆ φ w B ( θ B ) dθ B (2.40)where B labels the boundaries between patches, and θ B parametrizes the boundary B (topo-logically an S circle), and ∆ φ w B ( θ B ) is the discontinuity of the complex phase of w α over B . Since w α ( (cid:126)p ) is single valued in each patch, ∆ φ w B must change by a multiple of 2 π as θ B goes from 0 to 2 π . Therefore each B contributes an integer to C w , and hence C w is aninteger. Thus, in d = 2 each valence band contributes a quantized Hall conductivity, whilstthe conducting band contributes a non-quantized part. Now we look at the (cid:126) corrections to the equilibrium current. We have f = f F D ( H ), and wecan choose the gauge so that A µ is time independent. Using the expression (2.35) for theHamiltonian, we find J ( (cid:126)x, t ) = (cid:90) d d p (2 π (cid:126) ) d (cid:20) f F D ( E − A ) + (cid:126) F ij (cid:16) b ij − µ ij ∂ E (cid:17) f F D ( E − A ) (cid:21) , (2.41) J i ( (cid:126)x, t ) = (cid:90) d d p (2 π (cid:126) ) d (cid:104) v i f F D ( E − A ) + (cid:126) F j (cid:16) b ij − µ ij ∂ E (cid:17) f F D ( E − A )+ (cid:126) F jk b [ ij v k ] f F D ( (cid:15) − A ) (cid:35) (2.42)28here v i ( (cid:126)p ) ≡ ∂ ip E ( (cid:126)p ) is the band velocity. In J , we see both the Berry curvature cor-rection [84, 26] and the magnetic dipole moment correction to the fermion density. In J i (whose first term is zero is usual systems), the anomalous Hall current receives an additionalmagnetic dipole moment contribution compared to to the spatially uniform case (2.39) (notethat the equilibrium current depends on µ ij only through this anomalous Hall current term).The last term of J i is a magnetic field induced equilibrium current; as we will see in Section2.5, in the case of Weyl fermion, this term gives rise to the chiral magnetic effect.Again, since u α is multi-component, there exist other bands. We have to add up thecontributions to J µ from all bands. The total equilibrium Hall current is (not assumingsmall temperature) δJ iH tot. = F j (cid:88) u (cid:90) d d p (2 π (cid:126) ) d (cid:126) (cid:16) b iju − µ iju ∂ E (cid:17) f F D ( E u − A ) . (2.43)Again, in d = 2, if for any band w we can approximately take f F D ( E w − A ) = 1 (valenceband), then its contribution is quantized. The total magnetic field induced equilibriumcurrent is δJ iM tot. = F jk (cid:88) u (cid:90) d d p (2 π (cid:126) ) d (cid:126) b [ iju v k ] u f F D ( E u − A ) . (2.44)We may write v ku f F D = ∂ p k (cid:82) E u − A const dε f F D ( ε ) and integrate (cid:126)p by parts, so that the abovewould be proportional to ∂ [ kp b ij ] w which vanishes. But there is an important caveat, as wewill discuss through the example of Weyl fermion in Section 2.5. By now we have presented the semi-classical formalism of particle with Berry curvature; theformalism is valid to first order in (cid:126) ∂ x / | (cid:126)p | . We also derived some interesting implications29n non-interacting fermionic systems. In this section we provide the quantum mechanicalderivation of the semi-classical formalism. The original derivation is to evolution a wavepacket in time, see [10] for review. This method has an intuitive physical picture; however,its drawback is the steps in the derivation are not so straightforward, and the role of theassumed shape of the original wave packet is not entirely clear. Alternatively, there arederivations from field theory, which have the strength of being generalizable to interactingsystems ([62] and Section 4.4 of this thesis). But such derivations have a significant drawback:they are limited to the near-equilibrium situation, i.e. f = f F D + perturbations, while fornon-interacting systems the semi-classical formalism should hold far-from-equilibrium. Thereis also a non-commutative coordinate method [31] which is quite general, but may appearunfamiliar at first sight. The method we want to present here is a path integral derivation,based on [69, 14] and refined to be more systematic and generalized. This method is verygeneral, straightforward and familiar compared to other methods.The idea of the method is the following [69]. The single particle time-ordered propagator(i.e. the unitary time evolution) from t = 0 to t = T is given byˆ G αβ ( T,
0) = T exp (cid:32) − i (cid:126) (cid:90) T dt ˆ H ( ˆ P , ˆ x, t ) (cid:33) α β (2.45)where ˆ x i is the position operator and ˆ P i is the canonical momentum operator, and ˆ H αβ ( ˆ P , ˆ x, t )is the multi-component Hamiltonian operator and we assume its (ˆ x, t ) dependence is entirelydue to A µ . Let u α be an eigenvector (band) of ˆ H αβ when A µ = 0, well gapped with otherbands. We want to evaluate ˆ G αβ in between initial and final states in the u -band, and expressthe amplitude as (cid:68) u † α ( (cid:126)p T , T ) (cid:12)(cid:12)(cid:12) ˆ G αβ ( T, (cid:12)(cid:12)(cid:12) u β ( (cid:126)p , (cid:69) = (cid:90) D ( (cid:126)p, (cid:126)x ) exp (cid:20) i (cid:126) (cid:16) S [ (cid:126)p, (cid:126)x ] + O ( (cid:126) ) (cid:17)(cid:21) + (band hopping contributions) . (2.46)30e want to show: • The action S [ (cid:126)p, (cid:126)x ], to order (cid:126) , is given by (2.18), with Hamiltonian (2.35). • The path integral measure D ( (cid:126)p, (cid:126)x ) is the product of Liouville volume measure overinfinitely many time slices. • The band hopping contributions are order O ( (cid:126) F / ∆ ) suppressed, where ∆ is theenergy gap between the u -band and other bands. This is analogous to the quantumadiabatic theorem.Below we present the derivation in full details.Microscopically the Hamiltonian should take the form of the “quantum version” of (2.35).ˆ H αβ ( ˆ P , ˆ x, t ) = ˆ E αβ ( ˆ P − A (ˆ x, t )) − δ αβ A (ˆ x, t ) − (cid:126) (ˆ µ αβ ) ij ( ˆ P − A (ˆ x, t )) F ij (ˆ x, t )2 + O ( (cid:126) ) (2.47)Here (ˆ µ αβ ) ij is the intrinsic (bare) magnetic dipole matrix that the fermion might have. Nowwe use the commutation between ˆ x and ˆ P to rewrite the operators in “anti-commutatorordering”: every term is ordered as the anti-commutator of a purely ˆ P -dependent operatorand a purely ˆ x -dependent operator. That isˆ H αβ ( ˆ P , ˆ x, t ) = (cid:18) ˆ E αβ ( ˆ P ) − (cid:110) ∂ ˆ P i ˆ E αβ ( ˆ P ) , A i (ˆ x, t ) (cid:111) + O ( (cid:126)A ) (cid:19) − δ αβ A (ˆ x, t ) − (cid:18) (cid:26) (cid:126) (ˆ µ αβ ) ij ( ˆ P ) , F ij (ˆ x, t )2 (cid:27) + O ( (cid:126) (cid:126)A ) (cid:19) + O ( (cid:126) ) (2.48)(all the higher order terms are also in anti-commutator ordering). We use the anti-commutatorordering because it is the simplest ordering rule that preserves EM U (1) gauge invariance toorder (cid:126) in all intermediate steps. 31or the purpose of deriving semi-classical physics, the O ( (cid:126) ) terms can be dropped. Fornow, let’s also neglect the order A terms. They will be restored easily at the end using U (1)gauge invariance.As usual, we discretize the period from t = 0 to t = T into many time slices of duration δt , so small that ˆ H δt / (cid:126) can be neglected. The time slice labeled by t lasts from t − δt/ t + δt/
2. At the beginning time t − δt/ = (cid:90) d d x t − δt/ (2 π (cid:126) ) d/ | x t − δt/ (cid:105) (cid:104) x t − δt/ | . (2.49)Then we are left with ˆ P operators residing in the time slice, and we decompose them intoˆ P i = (cid:90) d d P t (2 π (cid:126) ) d/ | P t (cid:105) ( P t ) i (cid:104) P t | . (2.50)This procedure keeps the EM U (1) gauge invariance A i ( (cid:126)z, t ) → A i ( (cid:126)z, t ) + ∂ z i f ( (cid:126)z ) , ( P t ) i → ( P t ) i + ∂ X i f ( x it + δt/ ) + ∂ X i f ( x it − δt/ )2 (2.51)manifest. The propagator now becomesˆ G ( T,
0) = T (cid:89) t (cid:20) ˆ1 − i δt (cid:126) ˆ H ( ˆ P , ˆ x, t ) (cid:21) = T (cid:89) t d d P t d d x t − δt/ (2 π (cid:126) ) d exp (cid:18) i (cid:126) ( P t ) i (cid:16) x t + δt/ − x t − δt/ (cid:17) i (cid:19) × (cid:34) ˆ1 − i δt (cid:126) ˆ E ( P t ) + i δt (cid:126) ∂ iP ˆ E ( P t ) A i ( x t + δt/ , t ) + A i ( x t − δt/ , t )2+ i δt (cid:126) ˆ1 A ( x t + δt/ , t ) + A ( x t − δt/ , t )2+ i δt (cid:126) (cid:126) ˆ µ ij ( P t )2 F ij ( x t + δt/ , t ) + F ij ( x t − δt/ , t )2 (cid:35) (2.52)32here the α, β indices are hidden in the matrix multiplication, with ˆ1 being δ αβ .Here we remind that the canonical commutation relation is realized in path integral asthe following [27]. Denote δP t + δt/ ≡ P t + δt − P t . It is important that δP t + δt/ should notbe regarded as of order δt . Rather, we have (cid:90) d d x t + δt/ exp (cid:18) − i (cid:126) x it + δt/ ( δP t + δt/ ) i (cid:19) f ( x t + δt/ ) ( δP t + δt/ ) k = (cid:90) d d x t + δt/ exp (cid:18) − i (cid:126) x it + δt/ ( δP t + δt/ ) i (cid:19) ( − i (cid:126) ) ∂ x kt + δt/ f ( x t + δt/ ) (2.53)via integration of x t + δt/ by parts. Therefore, effectively − δP t + δt/ → i (cid:126) ∂ x t + δt/ . Similarly,denote δx t ≡ x t + δt/ − x t − δt/ , effectively we have − δx t → − i (cid:126) ∂ P t .We need to reduce the multi-band problem to a single band problem. Recall we areregarding A as a perturbation such that each A µ ( x t + δt/ , t ) can be kept to linear order(we will easily restore O ( A ) terms by gauge invariance at the end). We decompose the“unperturbed part” ˆ1 − ( iδt/ (cid:126) ) ˆ E ( P t ) into eigenvectors of ˆ E ( P t ): δ αβ − iδt (cid:126) ˆ E αβ ( P t ) = (cid:18) − iδt (cid:126) E ( P t ) (cid:19) u α ( P t ) u † β ( P t ) + (other bands) (2.54)where u ( P ) is an eigenvector with energy eigenvalue E ( P ), and in the u band lies the initialand final states. The crucial step towards the semi-classical picture is, in the decompositionabove we can keep the uu † term and drop all other bands. The reason is the following. It isnot hard to see, after the decomposition, any matrix element of hopping between the u -bandand another band is of order A δP/ ∆ ∼ (cid:126) F/ ∆ (2.55)where ∆ is the energy gap between the u -band and the other hand; this is completelyanalogous to the quantum adiabatic theorem mentioned in the Introduction, except now33here is no externally adjustable parameter, but a quantum number P which is not strictlyconserved due to the EM field. Our initial and final states are both in the u -band, so ifhopping occurs, it must occur at least twice – hop out and eventually hop back, yielding anamplitude of order (cid:126) F / ∆ , which we can neglect. Therefore, it is legitimate to order (cid:126) tomove towards the semi-classical picture by keeping uu † in the decomposition above. Thisalso explains why the semi-classical formalism is usually not applicable to higher order in (cid:126) – because then we will have to consider hopping which is a non-classical behavior.After the projection to the u -band, the projected propagator G uu reads G uu (0 , T )= (cid:89) t d d P t d d x t − δt/ (2 π (cid:126) ) d exp (cid:18) i (cid:126) ( P t ) i (cid:16) x t + δt/ − x t − δt/ (cid:17) i (cid:19) × (cid:34) u † α ( P t ) u α ( P t − δt ) (cid:18) − i δt (cid:126) E ( P t ) (cid:19) + i δt (cid:126) u † α ( P t + δt ) ∂ iP ˆ E αβ ( P t ) u β ( P t − δt ) A i ( x t + δt/ , t ) + A i ( x t − δt/ , t )2+ i δt (cid:126) u † α ( P t + δt ) u α ( P t − δt ) A ( x t + δt/ , t ) + A ( x t − δt/ , t )2 − i δt (cid:126) u † α ( P t + δt ) (cid:126) (ˆ µ αβ ) ij ( P t ) u β ( P t − δt )2 F ij ( x t + δt/ , t ) + F ij ( x t − δt/ , t )2 (2.56)Now, we expand those EM perturbation terms in the square bracket in powers of δP t + δt/ and δP t − δt/ , and then, as mentioned before, replace them by − i (cid:126) ∂ x t + δt/ and − i (cid:126) ∂ x t − δt/ .34e define the “band magnetic dipole moment” µ ijband ≡ − i ∂ [ iP u † α (cid:16) δ αβ E − ˆ E αβ (cid:17) ∂ j ] P u β = i (cid:16) ∂ jP u † α ∂ iP ˆ E αβ u β − u † α ∂ iP ˆ E αβ ∂ jP u β ” (cid:17) − a j ∂ iP E (2.57)and the “bare magnetic dipole moment” µ ijbare ≡ u † α (cid:16) ˆ µ αβ (cid:17) ij u β . (2.58)The total magnetic dipole moment is defined as µ ij ≡ µ ijband + µ ijbare . We find, to linear orderin A and in (cid:126) , G uu (0 , T )= (cid:89) t d d P t d d x t − δt/ (2 π (cid:126) ) d exp (cid:18) i (cid:126) ( P t ) i (cid:16) x t + δt/ − x t − δt/ (cid:17) i (cid:19) × (cid:34) u † α ( P t ) u α ( P t − δt ) (cid:18) − i δt (cid:126) E ( P t ) (cid:19) + i δt (cid:126) (cid:126) µ ij ( P t )2 F ij ( x t + δt/ , t ) + F ij ( x t − δt/ , t )2+ i δt (cid:126) ∂ iP E ( P t ) (cid:16) − (cid:126) a j ( P t ) ∂ x j (cid:17) A i ( x t + δt/ , t ) + A i ( x t − δt/ , t )2+ i δt (cid:126) (cid:16) − (cid:126) a j ( P t ) ∂ x j (cid:17) A ( x t + δt/ , t ) + A ( x t − δt/ , t )2 (cid:35) = (cid:89) t d d P t d d x t − δt/ (2 π (cid:126) ) d exp (cid:18) i (cid:126) ( P t ) i (cid:16) x t + δt/ − x t − δt/ (cid:17) i (cid:19) × u † α ( P t ) u α ( P t − δt ) (cid:34) − i δt (cid:126) (cid:32) E ( P t ) − (cid:126) µ ij ( P t )2 F ij ( x t + δt/ , t ) + F ij ( x t − δt/ , t )2 − ∂ iP E ( P t ) A i ( x t + δt/ , t ) + A i ( x t − δt/ , t )2 − A ( x t + δt/ , t ) + A ( x t − δt/ , t )2 (cid:33)(cid:35) (2.59)35ote that µ ijband has no direct counter-part in the original Hamiltonian operator (2.47); itsemergence is a purely quantum effect. For example, in particle physics, the famous g = 2 ofa Dirac fermion (say electron) is its µ ijband , and there is no µ ijbare as the usual Dirac equationhas no Pauli term. On the other hand, as we move on to solid state physics, the electron’s g = 2 serves as its µ ijbare in Bloch’s band theory, and the observed magnetic dipole momentusually involves an additional µ ijband due to the lattice.Finally we recognize the consecutive product of uu † as the momentum space Wilson linewith connection a i , and exponentiate the iδt/ (cid:126) terms. Passing over to the continuum limit,we find G uu (0 , T ) = (cid:89) t d d P t d d x t − δt/ (2 π (cid:126) ) d × exp (cid:32) i (cid:126) (cid:90) T (cid:16) P i dx i − (cid:126) a i ( (cid:126)P ) dP i − H ( (cid:126)P − (cid:126)A, (cid:126)x, t ) dt (cid:17)(cid:33) (2.60)where the Hamiltonian takes the form (2.35), with (cid:126)P − (cid:126)A ( (cid:126)x, t ) in place of (cid:126)p . To connecttowards (2.18), we define the physical momentum( p t ) i = ( P t ) i − A i ( X t + δt/ ) + A i ( X t − δt/ )2 (2.61)which is invariant under (2.51). Note that d d P d d x = d d p d d x , howeverexp (cid:18) − i (cid:126) (cid:126) a i ( (cid:126)P ) dP i (cid:19) = exp (cid:18) − i (cid:126) (cid:126) a i ( (cid:126)p ) dp i − i (cid:126) (cid:126) b ij ( (cid:126)p ) A i ( (cid:126)x, t ) dp j (cid:19) = (cid:18) − i (cid:126) (cid:126) b ij ( (cid:126)p ) A i ( (cid:126)x, t ) dp j (cid:19) exp (cid:18) − i (cid:126) (cid:126) a i ( (cid:126)p ) dp i (cid:19) (2.62)(total derivative dropped). The second term is of order (cid:126) A , so we can view it as a pertur-bation, and replace A i dp j → − i (cid:126) ∂ x i A j without affecting other terms (to order (cid:126) and order A ). This gives the Berry curvature correction to the Liouville volume element (2.22).36n summary, we have proven the u -band propagator G uu (0 , T ) = (cid:90) D ( (cid:126)p, (cid:126)x ) exp (cid:18) i (cid:126) S [ (cid:126)p, (cid:126)x ] (cid:19) (2.63)where S is given by (2.18) with Hamiltonian (2.35), and the path integral measure is D ( (cid:126)p, (cid:126)x ) ≡ (cid:89) t d d p t d d x t − δt/ (2 π (cid:126) ) d (cid:32) (cid:126) b ij ( p t )2 F ij ( x t + δt/ , t ) + F ij ( x t − δt/ , t )2 (cid:33) , (2.64)where the second term is the Berry curvature correction to the Liouville volume element.Band hopping contributions are order (cid:126) F / ∆ suppressed compared to G uu , therefore wedrop them; on the other hand, this shows one cannot improve the semi-classical action toorder (cid:126) , because then one has to include hopping, which is a non-classical behavior.Our derivation above assumed linear response and kept each A µ ( x t − δt/ , t ) to leadingorder. However, it is now obvious that our conclusion remains unchanged when we includehigher orders in A . This is because the EM U (1) gauge invariance must be respected. Anyorder A n ( n ≥
2) term will just enter the higher order expansion of E ( (cid:126)P − (cid:126)A ), any order A n δP or order (cid:126) F A n − term will enter the higher order expansion of µ ij ( (cid:126)P − (cid:126)A ), and anynon-trivial new effects (similar to the emergence of µ ijband ) are at least order A δP ∼ (cid:126) F which are negligible at order (cid:126) anyways. So far we have introduced the semi-classical formalism of Berry Fermi gas, and establishedits microscopic foundation from path integral. Here we demonstrate the formalism in aconcrete example. The example we choose is that of Weyl fermion, which have receivedmuch attention in recent years. This example is a two band system that is easy to solve, butit has very rich physics due to the band touching point (Weyl node) at (cid:126)p = 0 – most notably37he chiral anomaly and its associated effects. In high energy physics, in particular in thestudy of hot, dense quark gluon plasma, light quarks can be approximately viewed as Weylfermions, and the semi-classical theory can be useful in describing its far-from-equilibriumstate [67, 69, 16].Moreover, thanks to its simplicity of Weyl Hamiltonian, its appearance is very general –if a multi-band system has two bands touching, then almost always it can be approximated(unless there is some prohibition due to symmetries) by the Weyl fermion band structurefor energy and momenta near the Weyl node. If such band touching occurs in a solid statesystem, such material is called a
Weyl semimetal [78, 86, 75], which is one of the mostfocused area of experimental study in recent years [33, 85, 52]. A closedly related solidstate system, in which two Weyl nodes of opposite chiralities reside at the same point inmomentum space (usually due to symmetry; band degeneracy near the Weyl node required),called
Dirac semimetal [87], is also under extensive experimental study [46, 45, 55, 44].
We consider a right-handed Weyl fermion. The quantum mechanical Hamiltonian isˆ H αβ = ( P i − A i ) ( σ i ) αβ − A δ αβ (2.65)for i = 1 , , − σ i in place of σ i ). When A µ = 0, theeigenvalues are ±| (cid:126)p | , whose associated eigenvectors (spinors) are respectively u α ( (cid:126)p ) = cos( θ/ e iφ sin( θ/ , w α ( (cid:126)p ) = − e − iφ sin( θ/ θ/ (2.66)38here ( | (cid:126)p | , θ, φ ) are the spherical coordinates in the momentum space. We view the negativeenergy w -band as the Dirac sea; a hole in the Dirac sea is recognized an anti-particle. Wecan compute the Berry curvature in the u -band according to (2.20) (cid:15) ijk (cid:126) b k ( (cid:126)p ) ≡ (cid:126) b ij ( (cid:126)p ) = (cid:126) (cid:15) ijk p k | (cid:126)p | , (2.67)and the magnetic dipole moment in the u -band according to (2.57) (cid:15) ijk (cid:126) µ k ( (cid:126)p ) ≡ (cid:126) µ ij ( (cid:126)p ) = (cid:126) (cid:15) ijk p k | (cid:126)p | . (2.68)In both expressions, the (cid:126) / w -band), it will be − (cid:126) /
2. (For left-handed, the ± (cid:126) / µ k being pointed along p k is related to the thewell known fact that the direction of spin of a relativistic massless spinning particle is lockedwith the direction of momentum [80].The semi-classical action and Hamiltonian are dS = p i dx i + A µ dx µ − a i dp i − H dt, H = H + A = | (cid:126)p | − (cid:126) B k p k / | (cid:126)p | (2.69)where (cid:15) ijk B k ≡ F ij is the magnetic field and E k = F k is the electric field. (Note thataccidentally one can write H = | (cid:126)p | / √ det ω .) A puzzle arises immediately. The microscopictheory of Weyl fermion is Lorentz invariant, but the semi-classical action is not; more broadly,the entire Berry curvature formalism is not Lorentz invariant because Berry curvature bydefinition is purely spatial. Where did the Lorentz invariance go? We put this aside for now.The entire next chapter will be devoted on this problem.Needless to say, the expressions of b k and µ k become problematic as (cid:126)p →
0. But theyshould, because (cid:126)p → u -band and the w -band meet, and according to Section39.4, the semi-classical single particle picture is legitimate only when ( (cid:126) F/ ∆) negligible,where in the present case ∆ = 2 | (cid:126)p | . Therefore, physically we should place some infrared (IR)cutoff | (cid:126)p | IR such that (cid:126) F/ | (cid:126)p | IR (cid:46)
1, and the semi-classical theory is only valid if the particlemomentum | (cid:126)p | (cid:29) | (cid:126)p | IR . The most important fact associated with this band touching isthe Berry curvature obeys the “inverse square law” as if there is Berry curvature monopoleresiding at the band touching point (Weyl node) (cid:126)p = 0. Mathematically (cid:126) ∂ [ kp b ij ] = (cid:126) (cid:15) ijk ∂ lp b l = 4 π(cid:15) ijk (cid:126) δ ( (cid:126)p ) . (2.70)(This is related to the fact there is no way to choose u and w so that their phases arecontinuously well-defined all over the momentum space. For example, in the phases we havechosen above, u and w are undefined along θ = π ( | (cid:126)p | = − p ).) Consequently the symplectic2-form has Liouville flow monopole ∂ [ ip ω p i p j ] = − π(cid:15) ijk (cid:126) δ ( (cid:126)p ) . (2.71)This does not mean the symplectic 2-form is not closed, because (cid:126)p = 0 hidden under the IRcutoff is not part of the classical phase space anyways. Rather, this means the symplectic2-form is not exact in the classical phase space with the | (cid:126)p | < | (cid:126)p | IR region removed (andhence becomes topologically non-trivial); there can be symplectic flow (Liouville flow) flowingin and out through the | (cid:126)p | = | (cid:126)p | IR classical-quantum interface. This is closely related toNielsen and Ninomiya’s famous spectral flow interpretation of chiral anomaly [58], as we willelaborate on in the next section. 40he EoM (2.24) for Weyl fermions are explicitly dx i dt = p i | (cid:126)p | (cid:16) (cid:126) B k b k (cid:17) + (cid:15) ijk (cid:126) b k E j , (2.72) dp i dt = (cid:16) (cid:126) B l b l (cid:17) (cid:15) ijk B k p j | (cid:126)p | + (cid:16) − (cid:126) B k b k (cid:17) E i + (cid:126) µ k ∂B k ∂x i + (cid:126) b i E j B j . (2.73)Notably, since b i = p i / | (cid:126)p | , the last term in dp i /dt is radial in the momentum space when E j B j (cid:54) = 0. If we have an ensemble of particles, collectively there will be a total number ofparticles flowing through the | (cid:126)p | = | (cid:126)p | IR the interface, so that they appear up in / disappearfrom the classical phase space. Of course this is again related to the chiral anomaly to beexplored in the next section.Before we move on, we comment about the generality of the Weyl Hamiltonian (2.65).In a multi-band system, if we are interested in two bands which are touching, we can alwayswrite the Hamiltonian about this two bands asˆ H αβ = h i ( (cid:126)p )( σ i ) αβ + h ( (cid:126)p ) δ αβ . (2.74)Suppose the two bands are touching at some momentum (cid:126)p . Clearly h has no businessto do with the touching. Moreover, since there are three h i ’s and at the same time three p j components, generically the Jacobian det( ∂ jp h i ) is non-zero near the Weyl node (unlessdictated by some symmetry), that is, the mapping of the 3-dimensional momentum spaceinto the 3-dimensional space of h i is generically non-degenerate. Therefore, up to redefinitionof variables, the Weyl Hamiltonian is very generic in band touching systems. This is whythis example is very useful. In a lattice system, there is the famous Weyl fermion doublingtheorem [56, 57] stating that the number of Weyl nodes with det( ∂ jp h i ) > ∂ jp h i ) < The above discussion focuses on a single particle. Now we consider the behavior of anensemble of non-interacting Weyl fermion gas. The kinetic theory is called chiral kinetictheory [69]. The chiral kinetic theory enables us to study macroscopic ensemble of Weylfermions far-from-equilibrium, and hence (in systems where interactions effects are small)has broader application then the chiral hydrodynamics. In particular, the chiral anomalycomputation in the next section is legitimate even when the system is far-from-equilibrium.The density and current (2.37) are explicitly J = (cid:90) d p (2 π (cid:126) ) (cid:32) (cid:126) B k p k | p | (cid:33) f, (2.75) J i = (cid:90) d p (2 π (cid:126) ) (cid:34)(cid:32) (cid:126) B k p k | (cid:126)p | (cid:33) p i | (cid:126)p | f + (cid:126) (cid:15) ijk E j p k | (cid:126)p | f + (cid:126) (cid:15) ijk p k | (cid:126)p | ∂ x j f (cid:35) . (2.76)The anomalous Hall current term will be non-vanishing only if the distribution function isin a rotationally non-symmetric configuration.With this expression for J i , we can picture the single particle current. Let f ( (cid:126)p, (cid:126)x, t )be a narrow distribution localized in position space and momentum space (can be thoughtof as representing a Gaussian wave packet). If the particle is spinless, the current J i can42e viewed as a narrow bunch of arrows pointing along its direction of velocity, and overtime these arrows will trace out a narrow bunch of parallel lines. However, with spin,and in particular for massless spinning particle whose direction of spin is locked to thedirection of momentum [80], the magnetization current term will make the bunch arrowswind around the direction of velocity, and over time the arrows trace out a narrow bunchof helices, as illustrated by Figure 2.1. Therefore, in the sense of the “shape” of the singleparticle current, we really can view the quantum mechanical spin as the particle’s current“physically” spinning around its direction of velocity. This helical feature of the current isclosely related to the Lorentz invariance of the semi-classical Weyl fermion, as we will see inthe next chapter.We will write down the stress-energy tensor without derivation, as the derivation requireseither coupling to spacetime metric or performing the Belinfante procedure to the Noetherstress-energy tensor, both of which are beyond the scope of this thesis. The energy densityis T = (cid:90) d p (2 π (cid:126) ) | (cid:126)p | f (2.77)which is the usual integration of the Hamiltonian (without the A part), with the magneticdipole correction to the energy and the Berry curvature correction to the Liouville measurecancelled out. The momentum density and energy flux is T i = T i = (cid:90) d p (2 π (cid:126) ) (cid:34)(cid:32) (cid:126) B k p k | p | (cid:33) p i f + (cid:126) (cid:15) ijk p k | (cid:126)p | ∂ x j f (cid:35) . (2.78)43he spatial stress tensor is T ij = (cid:90) d p (2 π (cid:126) ) (cid:34)(cid:32) (cid:126) B k p k | p | (cid:33) p i p j | (cid:126)p | f + (cid:126) (cid:32) p { i (cid:15) j } kl E k p l | (cid:126)p | + B { i p j } | (cid:126)p | − δ ij B k p k | (cid:126)p | (cid:33) f + (cid:126) p { i (cid:15) j } kl p l | (cid:126)p | ∂ x k f (cid:21) . (2.79)The terms in T i and T ij with ∂ x f arise from the spin 1 / − T + T ij δ ij = 0,as the Weyl fermion is massless.With some work one can verify the energy and momentum conservation ∂ x µ T µν = F νλ J λ (2.80)is satisfied. On the other hand, the charge conservation ∂ x µ J µ is violated due to the chiralanomaly proportional to (cid:126) E k B k , as we will see in the next section. Although the chiral kinetic theory is originally purposed for describing far-from-equilibriumsystems, from the theory we can learn interesting effects in an equilibrium state too. Mostnotable are the chiral magnetic effect (CME) [77, 28] and the chiral vortical effect (CVE) [76,66]. Both of them are tightly related to the chiral anomaly [66, 54, 47], and equivalently,the Berry curvature monopole [67, 69, 16, 14]. Here we will discuss the CME, which hasbeen observed in a recent experiment [44] on Dirac semimetal. On the other hand, the CVEwill be discussed later in Section 3.4.4 when we find the most general form of equilibriumdistribution in the absence of external field.Consider a Weyl fermion ensemble in a static (but might be spatially non-uniform) mag-44etic field. Assume the system is in Fermi-Dirac distribution. By (2.42), we find the equi-librium current from the u -band is J i = (cid:126) F jk (cid:90) d d p (2 π (cid:126) ) d b [ ij ∂ k ] p E f F D ( E )= (cid:126) F jk (cid:90) d d p (2 π (cid:126) ) d b [ ij ∂ k ] p (cid:90) Eε min dε f F D ( ε ) (2.81)where the lower limit ε min of the ε integral is arbitrary. Now we integrate p k by parts, whichyields [67] J i = (cid:126) F jk (cid:90) d d p (2 π (cid:126) ) d ∂ [ ip b jk ] (cid:90) ε surf E dε f F D ( ε ) . (2.82)In particle physics, ε surf ≡ E ( (cid:126)p → ∞ ) = ∞ arises as the boundary term from the integrationby part. In solid state physics, the momentum space a is boundary-less BZ, so we can let ε surf be our arbitrarily chosen ε min . The current J i is physical, so it must be independentof our arbitrarily chosen ε min . Indeed, this is due to (cid:90) d d p (2 π (cid:126) ) d ∂ [ ip b jk ] = 0 (2.83)being a total derivative in a boundary-less BZ. For d = 3 this is just the fermion doubletheorem, since right-handed and left-handed Weyl nodes have opposite Berry curvaturecharge [56, 57].A question about the general computation above is, we know the semi-classical picturefails near the band touching region where ∂ [ ip b jk ] (cid:54) = 0, so how can we trust the integral of f F D near such region? Recall the system is in equilibrium, so there shall be no net bandhopping, and we can extrapolate the validity of semi-classical theory arbitrarily close to theWeyl nodes. Even in non-equilibrium state, as long as a sufficiently large neighborhood of theband touching region is approximately in equilibrium, we should still be able to extrapolate.45he discussion above is completely general. Specifically for our system of Weyl fermion, ε surf = ∞ , and the monopole is given by (2.70) occurring at E = 0. To get the total current,we also need to first add the w -band contribution, and then subtract the contribution whenthe w -band is full. The subtraction is due to the regularization of the definition of J i againstthe infinite Dirac sea. This is equivalent to subtracting the contribution of the anti-particles,which are under the same temperature as the particles but at opposite chemical potential, andhave opposite Berry curvature and opposite electric charge (couples oppositely to magneticfield). Therefore J itot = 4 π (2 π (cid:126) ) (cid:126) B i (cid:90) ∞ dε ( f F D ( ε ; µ, T ) − f F D ( ε ; − µ, T ))= 4 π (2 π (cid:126) ) (cid:126) B i µ (2.84)regardless of temperature. One can also compute the momentum density / energy flux andfind T i tot = 4 π (2 π (cid:126) ) (cid:126) B i (cid:18) µ π T (cid:19) . (2.85)This is the well-known chiral magnetic effect in current and in momentum density (for sure,the two are related [66, 54, 47]).On the other hand, let’s consider a two band Weyl semimetal instead. In Weyl semimetal,there is no regularization against any Dirac sea, so we just add up the contributions fromboth bands – the fermions in both bands have the same electric charge and are subjected tothe same chemical potential and energy, but see opposite Berry curvature monopoles chargeat each Weyl node. Therefore J itot = (cid:126) F jk (cid:90) d d p (2 π (cid:126) ) d ∂ [ ip b jk ] (cid:90) ε usurf ε wsurf dε f F D ( ε ) = 0 (2.86)46ince the ε integral is independent of (cid:126)p and the (cid:126)p integral is a total derivative integralover a boundary-less BZ. This proves that there is no CME for any Weyl semimetal inequilibrium [5]. In order to observe CME in a Weyl semimetal, the regions near the right-handed Weyl node(s) and the left-handed Weyl node(s) must be prepared to fill up todifferent energies; the simplest way to do so is by the chiral anomaly [65] introduced in thenext section. This is how the recent experiment on the observation of CME [44] (in Diracsemimetal) is performed. Anomaly is one of the most fascinating subjects of quantum mechanics. It refers to the factthat a symmetry in classical mechanics may not be consistent with quantum mechanics, andhence must be broken by order (cid:126) . Anomalies have significant experimental consequences,and also profound mathematical structure. Here we will not review the subject; see e.g. [81]for good pedagogical introduction.The earliest and simplest example of anomaly is the chiral anomaly [2, 7], originallystudied in the context of quarks and successfully explained the experimentally observed pionlifetime. This anomaly can be stated as the following. Consider a left-handed Weyl fermionspecies and a right-handed Weyl fermion species, uncoupled with each other, and have thesame coupling to the electromagnetic (EM) field. One would expect both of them have theirown fermion number conservation U (1) L and U (1) R ; the sum of their fermion numbers is thetotal EM U (1) charge, and the difference is called the axial U (1) A charge. The statementof chiral anomaly is, quantum mechanically it is inconsistent to have U (1) L , U (1) R and theleft- right-handed parity all in one theory. In reality the spacetime parity (this is built-inif we view the left- and right-handed Weyl fermions as a single Dirac fermion) and the EM U (1) gauge invariance should be respected, but then the axial U (1) A must be broken bythe anomaly, i.e. the left- and right-handed fermion numbers are not separately conserved,47lthough their sum, the total physical electric charge, is conserved. More precisely, ∂ x µ J µR = − ∂ x µ J µL = 4 π (2 π (cid:126) ) (cid:126) (cid:15) µνρσ F µν F ρσ = 4 π (2 π (cid:126) ) (cid:126) E k B k (2.87)(remarkably, this is the exact result [3]) so that their sum vanishes but their difference doesnot. Can we avoid this if we have only right-handed Weyl fermion to start with? As longas we want to it to behave as “right-handed” at the quantum mechanical level, its fermionnumber must be non-conserved as the above.But where does the fermion number go? In solid state system, Nielsen and Ninomiyaprovided a very intuitive spectral flow interpretation with concrete computation [58]. Sup-pose we have an electron in the valence band. If we turn on E k B k >
0, the electron rises intothe conducting band through the right-handed Weyl node so that ∂ x µ J µR >
0. Eventually, inthe conducting band, the electron must move towards in the left-handed Weyl node (whoseexistence is guaranteed the doubling theorem mentioned before), through which it sinks backinto the valence band, and hence ∂ x µ J µL <
0, but the total electron number is of course con-served. In high energy physics, there is no doubling theorem and no path connecting theright-handed Weyl mode to the left-handed Weyl mode. However, there are infinite Diracseas. With E k B k >
0, right-handed Weyl fermions in their Dirac sea will increase in energyso that the top ones will pop up as right-handed fermion excitations, while left-handed Weylfermions in their Dirac sea will decrease in energy, leaving some holes in the Dirac sea, whichare left-handed anti-fermion excitations (there is some language ambiguity here – this is ananti-fermion of the Weyl mode of left-handed chirality, but the anti-fermion itself has right-handed helicity). Of course, this statement requires ultra-violet (UV) regularization, and forthe regularization to be consistent with parity and EM U (1) gauge invariance, the rate ofcreating right-handed fermion excitations and left-handed anti-fermion excitations must beequal.The remarkable success of chiral kinetic theory is that such spectral flow can even to48igure 2.2: The spectral flow under chiral anomaly with (cid:126)E · (cid:126)B > ∂ x µ on the current (2.32).We can ignore the magnetization current term because ∂H/∂F is order (cid:126) and ∂ x √ det ω isalso order (cid:126) . We are left with the velocity term: ∂ x µ J µ ( x ) = (cid:90) d d p (2 π (cid:126) ) d ∂ x µ (cid:18) √ det ω dx µ dt (cid:12)(cid:12)(cid:12)(cid:12) EoM f (cid:19) = (cid:90) d d p (2 π (cid:126) ) d (cid:34) ∂ t (cid:16) √ det ω f (cid:17) + ∂ ξ I (cid:32) √ det ω dξ I dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) EoM f (cid:33)(cid:35) . (2.88)In the second line we added a total (cid:126)p derivative which equals zero, because there is noboundary term as we assumed we can include the vicinity of the Weyl node as part of theclassical phase space. The integrand of the second line is just the left-hand-side of thecontinuity equation (2.14), which usually vanishes. However, in the spirit of spectral flow,we allow it to be violated at the Weyl node, but we assume the Boltzmann equation (2.30)still holds. The difference between the continuity equation and the Boltzmann equation isgiven by (2.14) which is non-zero at the Weyl node where the symplectic 2-form fails to be49losed. Therefore ∂ x µ J µ ( x ) = − (cid:90) d d p (2 π (cid:126) ) d f √ det ω ∂ [ ξ I ω JK ] ω IJ ω KL (cid:0) ∂ ξ L H + ∂ t ω L (cid:1) . (2.89)This is the general formula relating the chiral anomaly to the non-closedness of the symplectic2-form at band touching. For a right-handed Weyl fermion, we apply (2.71). Since the non-closedness is order (cid:126) , we can ignore any (cid:126) term in (2.23) for ω IJ , and also take ∂ x i H + ∂ t ω L (cid:39)− E i . Thus, for a right-handed Weyl fermion, ∂ x µ J µ = 4 π (2 π (cid:126) ) (cid:126) E k B k f ( (cid:126)p = 0 , (cid:126)x, t ) . (2.90)But can we make clear sense of f at the Weyl node? We do not need to worry about this,because to get ∂ x µ J µR , we also need to subtract the anti-fermion contribution, which hasopposite Berry curvature. Therefore ∂ x µ J µR = ∂ x µ J µ − ∂ x µ (cid:101) J µ = 4 π (2 π (cid:126) ) (cid:126) E k B k (cid:16) f ( (cid:126)p = 0 , (cid:126)x, t ) + (cid:101) f ( (cid:126)p = 0 , (cid:126)x, t ) (cid:17) . (2.91)The factor in the parenthesis is 1 because anti-fermion density is just hole density in the w -band, and at (cid:126)p = 0 the w -band meets the u -band. Thus, we have computed the chiralanomaly (2.87) in semi-classical (order (cid:126) ) chiral kinetic theory. No assumption of nearequilibrium is needed.Finally we justify the left-over subtleties. When we add the total (cid:126)p derivative term in(2.88), there really should be a boundary term due to the | (cid:126)p | IR cutoff. By the Stoke’sTheorem, clearly (2.90) is just computing this boundary term, but instead of f ( (cid:126)p = 0 , (cid:126)x, t ),we should integrate about f ( (cid:126)p, (cid:126)x, t ) over the | (cid:126)p | IR interface; similarly for the anti-fermions.However, f and (cid:101) f at the | (cid:126)p | IR interface not necessarily add up to 1 because they are indifferent bands. How shall we interpret our computation then? With the spectral flow50icture, it is easy to see that, in the presence of the chiral anomaly flow, f + (cid:101) f aroundthe | (cid:126)p | IR interfaces (one in the u band and the other in the w band) cannot stay awayfrom 1 because the momentum space below these interfaces has limited volume of order √ (cid:126) F . Instead, f + (cid:101) f around the | (cid:126)p | IR interfaces must average to 1 over the spacetimescale (cid:38) (cid:112) (cid:126) /F . Therefore, our computation must be interpreted as the effect averaged overa spacetime scale (cid:38) (cid:112) (cid:126) /F . But this indeed is the regime of validity of our semi-classical (cid:126) ∂ x / | (cid:126)p | expansion. Therefore the computation is justified within our framework. In this chapter, we have reviewed formal aspects of the Berry Fermi gas theory, the semi-classical theory of non-interacting Fermi gas with Berry curvature, and presented a generaland conceptually simple derivation of the semi-classical theory from single particle path in-tegral. We demonstrated the idea by the concrete example of Weyl fermion, which, despiteits computational simplicity, has rich and deep physical implications and broad applications.Most notably, the chiral anomaly and the associated chiral magnetic effect can be compu-tation from the Berry curvature monopole residing at the band touching point. We keptour computations in the most formal expressions, so that it is clear that these effects arerobust and do not depend on the details of the system. We also presented the relation be-tween anomalous Hall effect and Berry curvature, which was the earliest relation that drewpeople’s attention toward Berry curvature in fermionic systems.For sure, the formalism we presented here does not encompass the full story of Berryphase physics in non-interacting fermionic system. We considered momentum space Berryphase only, but clearly there can also be other components of Berry phase [72, 10, 83] dueto, e.g. extra externally adjustable parameter in the system. More importantly, even formomentum space Berry phase only, we restricted to particles that have no internal degreesof freedom. That is, we assumed our particle is microscopically described by a single band51 . We made this assumption for simplicity. In general physical systems there might beband degeneracy so that the particle must be described by multiple degenerate bands, andtherefore has internal degrees of freedom. In such case, the Berry curvature would haveinternal indices and becomes a non-abelian curvature [21, 63, 83], and leads to some newinteresting implications. The path integral and the semi-classical formalism we presentedhere can be straightforwardly generalized to the case with internal degrees of freedom.There are not much formal, general, outstanding puzzles lingering around the BerryFermi gas theory, because the whole formalism can be derived by the quantum mechanics ofsingle particle (e.g. our path integral derivation) due to the absence of interactions. But forapplications to specific systems, there certainly are remaining problems.The next problem we will discuss is, in the particular system of Weyl fermions, how is theLorentz symmetry realized in the Berry curvature formalism, which intrinsically separatestime and space. This is clearly a sharp problem in chiral kinetic theory. The resolution tothis problem, however, also deepens our understanding of the Berry curvature formalism ingeneral. 52
HAPTER 3LORENTZ SYMMETRY IN CHIRAL KINETIC THEORY
In this chapter we study how Lorentz symmetry is realized in the semi-classical theory ofWeyl fermion (2.69). Our conclusion is that, there is nothing wrong with the semi-classicaltheory, but the realization of Lorentz symmetry is non-trivial. More particularly, we shouldnot regard the spacetime position ( t, (cid:126)x ) and energy-momentum ( H , (cid:126)p ) as Lorentz covariantphysical observables; their definitions are frame dependent. Rather, the frame independentobservables are (as one might expect), the current and stress-energy tensor in Section 2.5.2.We start with stating a puzzle motivating the idea that the notion of position mustbe frame dependent. Then we use the intrinsic-time parameter formulation of the actionintroduced in Section 2.1 and see how the frame independence of the action is realized.We provide a physical explanation based on the single particle current Figure 2.1, and amathematical explanation relating the non-trivial frame independence with the little groupof a massless spinning particle. Finally, we discuss if the Weyl fermions have a collision termin the Boltzmann equation, how the collision should be realized to be compatible with thenon-trivial frame independence. We will see this colliding theory relaxes to the well-knownchiral hydrodynamics, and we compute the famous chiral vortical effect (CVE) [66].This chapter resolves the puzzle of Lorentz symmetry in the semi-classical Berry curvatureformalism. However, resolving this problem also sheds light on the interpretation of theformalism in general, as we will see along the text. The most direct puzzle we see is that the action (2.69) for a semi-classical Weyl fermionmanifestly breaks Lorentz invariance. In particular, the Berry curvature depends on thespatial momentum only, how can the action possibly be Lorentz invariant at all?53igure 3.1: Consider a head-on collision between two massless spinning particles. In thecenter-of-mass frame, the incoming particles have no orbital angular momentum (cid:126)L and nototal spin angular momentum (cid:126)S , and hence no total angular momentum (cid:126) J ; so are the out-going particles. If we boost along the direction of the incoming particles, then the incomingparticles still have no (cid:126)L and no (cid:126)S and hence (cid:126) J = 0; however, the out-going particles have (cid:126)S (cid:54) = 0, and in order for (cid:126) J to be conserved, they must have non-zero (cid:126)L = − (cid:126)S .A resolution would be that, in addition to the usual Lorentz boost, (cid:126)x and (cid:126)p must transformwith additional order (cid:126) corrections; this is indeed how the problem is resolved. However,such modification to the Lorentz boost would mean the four vectors x µ and p µ are framedependent quantities, which sounds strange. Now we present a physical argument why thishas to be the case [14].Recall that the a spinning massless particle’s spin angular momentum is always lockedwith its momentum, (cid:126)S = s (cid:126) ˆ p , where s = ± / (cid:126)L , zero total spinangular momentum (cid:126)S and hence zero total angular momentum (cid:126) J ; so are the outgoing ones.Now we boost along the direction of the incoming particles. The (cid:126)L , (cid:126)S and (cid:126) J of the incomingparticles are still zero as before. However, the outgoing particles now have non-zero (cid:126)S sincetheir momenta are not in opposite directions. In order for (cid:126) J to still be conserved, we mustconclude that the outgoing particles have (cid:126)L = − (cid:126)S (cid:54) = 0, which means they do not fly outdirectly from the “collision point” of the incoming particles. Thus, a local collision viewed in54igure 3.2: A schematic illustration of the non-local collision of massless spinning particlesin a non-center-of-mass frame.the center-of-mass frame becomes a non-local collision in the boosted frame, schematicallyshown in Figure 3.2. This enforces the idea that the notion of “position” must be framedependent.Clearly the collision here can be a fictitious one just to facilitate the argument that theposition of a massless spinning particle must be frame dependent. The same must holdfor non-colliding particles, for example photons (which we will mention later). A massiveparticle does not have this ambiguity because we can always go to its own center-of-massframe to define its position; on contrary, there is no center-of-mass frame for a masslessparticle. To understand the Lorentz invariance in the semi-classical theory of Weyl fermion (2.69), wefirst use the intrinsic-time parameter formulation of the action introduced in Section 2.1 toexpress (2.69) “as Lorentz invariant as possible”: S = (cid:90) dS, dS [ x, p, λ ] = p µ dx µ + A µ ( x ) dx µ − a µ ( (cid:126)p ) d(cid:126)p µ − H ( p, x ) λdτ (3.1)55here (cid:126)p µ ≡ p µ + p ν n ν n µ , | (cid:126)p | = (cid:113) (cid:126)p µ p µ (3.2)and n µ is a normalized future time-like frame vector, which is taken to be (1 , , ,
0) in thenon-relativistic formalism. But in the relativistic formalism we do not fix its components;in fact, the formalism we are presenting here works even if n µ is spacetime dependent, andthis fact will be useful when we consider the chiral vortical effect later. The Hamiltonian isgiven by H ( p, x ) = p µ p µ − (cid:126) F µν ( x )Σ µν ( (cid:126)p ) , (cid:126) Σ µν ( (cid:126)p ) ≡ − (cid:126) (cid:15) µνρσ p ρ n σ | (cid:126)p | (3.3)where (cid:126) Σ µν is the spin angular momentum of Weyl fermion (the Pauli-Lubanski pseudo-vector is proportional to (cid:15) µνρλ (cid:126) Σ µν p ρ ); we can see it is orthogonal to the direction of mo-mentum, which is a feature of massless spinning particle. Clearly the Weyl fermion’s spingenerates its magnetic dipole moment. The Berry curvature (2.67) now reads (cid:126) b µν ( (cid:126)p ) = − (cid:126) (cid:15) µνρσ p ρ n σ | (cid:126)p | . (3.4) λ serves a Lagrange multiplier that demands H = 0, which leads to the “on-shell condition” − p · n ≡ − p µ n µ = | (cid:126)p | − (cid:126) F µν Σ µν / | (cid:126)p | (3.5)(taking the − p · n > H in (2.69). We canderive the EoM from the action. The EoM relating the physical time to intrinsic time is − n µ dx µ / (2 λdτ ) = − n µ p µ = | (cid:126)p | − (cid:126) F µν Σ µν / | (cid:126)p | . (3.6)56he other EoMs just reduce to (2.73) when n µ = (1 , , ,
0) and are not reproduced here(the EoM for energy is to relate the d/ (2 λdτ ) of (3.5) to d(cid:126)p/ (2 λdτ ) and dx/ (2 λdτ )).In the form (3.1) and (3.3), Lorentz invariant is manifest as everything is written ascontraction of Lorentz indices. The problem now becomes (the order (cid:126) terms of) the actiondepends on a reference frame n µ . In fact, under an infinitesimal change of reference frame, if we let x and p transform by order (cid:126) terms, the action can remain invariant to order (cid:126) . The symplectic part p · dx + A · dx − a · dp is invariant under infinitesimal change of frame n µ → n µ + β µ , β · n = 0 (3.7)accompanied with [14] x µ → x µ + (cid:126) b µν ( (cid:126)p ) β ν ( p · n ) , p µ → p µ + F µν (cid:126) b νλ ( (cid:126)p ) β λ ( p · n ) . (3.8)This frame dependence of x was originally found in [64] while that of p is new. Substitutingthese into the Hamiltonian (3.3), we find it transforms by an overall factor H → H (cid:16) (cid:126) n λ F λµ b µν β ν (cid:17) . (3.9)We can make the action invariant by letting λ to transform with the opposite factor so that H λ remains invariant. (This is valid even when n µ is x dependent, because the x dependencewill only add to the transformation of n an order (cid:126) term, but n itself appears in the actionin order (cid:126) terms only, so the difference will the O ( (cid:126) ).)The invariance of the Hamiltonian part seems quite trivial, for we can always use λ to57ancel the transformation of H . This is not the case; in fact, this is the most non-trivial partof the story. Note that H transforms by an overall factor, so the on-shell condition H = 0is frame independent. Were it not transforming by an overall factor, the transformation of λ must involve 1 / H which would be problematic. This condition is highly restrictive.What is the most general Lorentz invariant semi-classical action with non-zero Berrycurvature? Rotational invariance and point-like monopole dictates the Berry curvature to bethat of Weyl fermion (the overall factor is fixed quantum mechanically). Then one can showframe independence dictates the most general dispersion relation to be given by (3.3) [68, 14]up to a mass term m so small that m (cid:126) F/ | (cid:126)p | can be neglected. The allowance of suchsmall mass term is compatible with the fact that a high speed massive Dirac or Majoranafermion can be approximately seen as a Weyl fermion. The exclusion of other possibilitiesreflects the fact that any non-interacting Lorentz invariant term that can be added to theWeyl Lagrangian must be at least order (cid:126) F .One can compute the commutation of two frame transformations by β and β (cid:48) . It isnon-vanishing: x µ → x µ + (cid:126) (cid:15) λκρσ n λ p κ β ρ β (cid:48) σ | (cid:126)p | p µ | (cid:126)p | , p µ → p µ + (cid:126) (cid:15) λκρσ n λ p κ β ρ β (cid:48) σ | (cid:126)p | F µν p ν | (cid:126)p | . (3.10)One can see the transformation is proportional to the EoM evolution of the variables [14].A transformation proportional to the EoM evolution always leaves the action invariant, as isobvious from (2.3) letting δξ be proportional to (2.5). Therefore, the frame transformationtogether with the EoM evolution transformation form a group. The mathematical origin ofthis algebra is discussed in the next section. 58 .2.2 Finite Transformation On-Shell For the frame transformation to be useful, we must not only have its infinitesimal form (3.8)but also the finite form. We would only need it when the “on-shell condition” H = 0 issatisfied. We can integrate the infinitesimal transformation imposing H = 0, and find, under n → n (cid:48) , [25, 71, 13] x (cid:48) µ = x µ + ∆ µnn (cid:48) ( p ) , p (cid:48) µ = p µ + F µν ( x )∆ νnn (cid:48) ( p ) , ∆ µnn (cid:48) ≡ (cid:126) (cid:15) µνρσ p ν n ρ n (cid:48) σ ( p · n )( p · n (cid:48) ) . (3.11)When F µν = 0, the total angular momentum J µν ≡ x µ p ν − x ν p µ + (cid:126) Σ µν (3.12)is frame independent, but we are not aware of an analogy that is frame independent when F µν (cid:54) = 0.One can easily see Figures 3.1 and 3.2, obtained based on physical arguments, agree withthe mathematical form of the frame dependence. If the position and momentum are frame dependent quantities, what are the physical observ-ables of the Weyl fermion that are frame independent? They are the current and stress-energytensor, for a single particle or for an ensemble. Understanding the frame independence ofthem gives us a clear physical interpretation for the frame dependence of position and mo-mentum.The single particle current, defined as J µsp ( x )[ z, p, λ ] ≡ δS [ z, p, λ ] δA µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) EoM , (3.13)59s certainly frame independent, because both the action and the field A µ ( x ) are (in thisexpression the particle’s frame dependent position is z ; x is an arbitrary point in spacetimethat has nothing to do with the particle). For an ensemble of particles, we need to integrateover the phase space with measure (2.17). The measure is also frame independent – theinvariance of the symplectic part of the action leads to the invariance of d z d p √ det ω (thesymplectic 2-form with time and energy is to expand (2.21) so that the i, j indices become µ, ν indices), while the invariance of H λ leads to the invariance of δ ( H ) /λ . Finally, we needthe distribution function to depend on frame so that f (cid:48) ( (cid:126)p (cid:48) , z (cid:48) ) = f ( (cid:126)p, z ) , i.e. f (cid:48) ( (cid:126)p (cid:48) , x ) = f ( (cid:126)p, x ) − ∆ nn (cid:48) · ∂ x f ( (cid:126)p, x ) (3.14)(because here z is the particle’s position, while x is just a generic point in the spacetimewhich does not depend on the frame). Then, the ensemble current J µ ( x ) = (cid:90) d z d p (2 π (cid:126) ) δ ( H ) θ ( − p · n )2 λdτ √ det ω f ( (cid:126)p, x ) J µsp ( x )[ z, p, λ ] (3.15)must be frame independent J (cid:48) µ ( x ) = J µ ( x ).The current (2.76) can be written as J µ ( x ) = (cid:90) d (cid:126)p (2 π (cid:126) ) ( − p · n ) (cid:20)(cid:18) (cid:126) F ρσ b ρσ (cid:19) p µ + (cid:126) b µν F νλ n λ ( − p · n )+ (cid:126) Σ µν ∂ x ν (cid:21) f ( (cid:126)p, x ) . (3.16)One can manifestly check its frame independence under (3.8) with the aid of the collisionlessBoltzmann equation. 60he stress-energy tensor can be written as T µν = (cid:90) d (cid:126)p (2 π (cid:126) ) ( − p · n ) (cid:20)(cid:18) (cid:126) F ρσ b ρσ (cid:19) p µ p ν + (cid:126) (cid:16) F ρσ p { µ − F { µρ p σ (cid:17) b ν } ρ n σ ( − p · n )+ (cid:126) p { µ Σ ν } λ ∂ x λ (cid:21) f ( (cid:126)p, x ) , (3.17)It can be obtained by coupling the action to spacetime metric and then integrating over thephase space, therefore by the same argument it must also be frame independent T (cid:48) µν ( x ) = T µν ( x ).Now we can give a physical interpretation to the frame dependence of the particle’sposition and momentum. As we did in Section 2.5.2, consider a single particle current where f is localized in space and momentum space, i.e. a narrow wave packet; for concretenessone can take f ( (cid:126)p, x ) = (2 π (cid:126) ) δ ( (cid:126)x − (cid:126)z ( t )) δ ( (cid:126)p − (cid:126)q ( t )) where the particle’s position (cid:126)z ( t ) andmomentum (cid:126)q ( t ) satisfy the EoM. The current is helical as illustrated in Figure 2.1. It is easyto see the particle’s “position” (cid:126)z must be interpreted as “the spatial center of the helix”.While the helical current itself is frame independent, it is an elementary relativity fact thatthe notion of of “spatial center” of a rotating body is frame dependent, in a way qualitativelyagree with (3.11). Similarly, (cid:126)p must be interpreted as the “momentum carried at the spatialcenter of the helix”, and therefore is also frame dependent when there is EM force curvingthe helical trajectory. A more pictorial analysis of this interpretation is found in [71]. Whatwe have seen is the massless, quantum version of the phenomenon that was first learned inthe context of massive rotating bodies [24, 20].Clearly what we described here did not rely on the particle being a Weyl fermion. Similarframe dependence of the “position” or “center” occurs in photons too, except the helicity ± (cid:126) / ± (cid:126) . The associated experimental phenomenon for photon is that,when a circular polarized light beam (or photon) undergoes a refraction, the center of thebeam (or photon wave packet) undergoes an order (cid:126) slight shift, much like the scenario in61igure 3.2. See [9] for a review on this subject. In the previous section we have seen the frame dependence of the definition of a masslessspinning particle’s position and momentum. This ambiguity only arises for massless particlewhich does not have a natural frame, the rest (center-of-mass) frame, to define its positionand momentum. In this section we see the mathematical origin of the frame dependence – inparticular, its relation to the Wigner translation [82] for massless particle. This relation canbe seen via various methods [25, 71]. Here we demonstrate a method based on an alternative,group theoretic expression of the action.First we briefly review the concept of little group and Wigner translation. Suppose we ahave time-like four vector, say ( m, , , SO (3)rotation in the time-like vector’s rest frame. Such Lorentz subgroup that leaves a four vectorinvariant is called the vector’s little group , or stablizer . What if the four vector is null, say k a = ( κ, , , κ ) ( κ has the dimension of energy)? The little group is the Euclidean group ISO (2) generated by [80] J , A ≡ J − K , B ≡ − J − K (3.18)where J i are rotation generators and K i are boost generators, whose non-zero componentsare ( J i ) jk = ( J i ) jk = (cid:15) ijk , ( K i ) j = ( K i ) j = − ( K i ) j = ( K i ) j = − δ ij . (3.19)One can easily check J , A , B annihilate k a . Clearly J is the rotation along the spatialdirection of the null vector. A , B are certain combinations of rotation and boost; the62ransformation generated by them are called Wigner translation .Now we assume A µ = 0 and consider the following action S [ x, L ] = (cid:90) (cid:32) L aµ k a dx µ dτ − (cid:126) J ) ab L µa dL bµ dτ (cid:33) dτ (3.20)where [ x µ ( τ ) , L µa ( τ )] should be viewed as a worldline lying in the Poincar´e group R , (cid:111) SO (3 , A µ = 0. Wewill interpret L aµ k a as the momentum p µ and ( (cid:126) / L µa L νb ( J ) ab as the spin matrix (cid:126) Σ µν .Note that the physical conditions p µ p µ = 0 and p µ Σ µν = 0 are already built-in.The action (3.20) has two obvious properties. It is independent of the choice of k a aslong as it stays null and ( J ) ab stays the rotational generator annihilating k a . This can beseen by performing a Lorentz transformation on the ab indices (involving a redefinition of L ).Moreover, it has the physical Poincar´e invariance of adding a constant to x and performinga Lorentz transformation on the µν indices.If we assert the action above describes the Weyl fermion, we must resolve the following:A Weyl fermion has six physical degrees of freedom ( (cid:126)x, (cid:126)p ), while the action (3.20) has tendegrees of freedom ( x µ , L µa ) of the Poincar´e group. If they can match, the action (3.20)must have four gauge (unphysical) degrees of freedom along the worldline. This is indeedthe case. If we consider an generic infinitesimal Poincar´e transformation x µ → x µ + L µa χ a , L µa → L µb ( δ ab + λ ab ) , λ ab = − λ ba (3.21)63e can easily find the following gauge transformations leave the action invariant: χ a = d ( τ ) k a + ( (cid:126) /
2) ( a ( τ ) ( e ) a + b ( τ ) ( e ) a ) ,λ ab = b ( τ ) ( A ) ab − a ( τ ) ( B ) ab + c ( τ ) ( J ) ab (3.22)where ( e ) a = δ a and ( e ) a = δ a are the unit spatial vectors orthogonal to k a , and a, b, c, d are four arbitrary infinitesimal gauge parameters along the worldline. The d transformationis a version of the transformation proportional to the EoM evolution that we mentionedat the end of Section 3.2.1, and has nothing special to do with massless spinning particle.The c transformation is an obvious one. Most interesting is the gauge Wigner translationparametrized by a, b – it involves a transformation of x , which means the “position x ” isnot a gauge invariant physical quantity. This hints the gauge Wigner transformation isrelated to the frame dependence (3.8). Another evidence is that the momentum p µ and thetotal angular momentum J µν are gauge invariant under Wigner translation, in agreementwith their frame independence when A µ = 0. Yet another evidence is, one can check thecommutation of two Wigner translations is a d transformation, also in agreement with thealgebra mentioned at the end of Section 3.2.1.How to explicitly realize the relation? Note that the benefit of (3.20) over (3.1) is thatthe former is manifestly Lorentz invariant and frame independent (no frame vector n µ isinvolved), but the price payed is it involves gauge degrees of freedom as mentioned above.To reduce (3.20) to (3.1), we need to use a frame vector n µ to fix the Wigner translationgauge. More particularly, note that in (3.3), the spin matrix satisfies both p µ Σ µν = 0 as wellas n µ Σ µν = 0, while in the present formalism of (3.20), we only have the former constraintbut not the latter, because there is no frame n µ . To fix the Wigner gauge, we use an arbitraryframe vector n µ (can be spacetime dependent), and impose the gauge fixing condition n µ Σ µν = 0 . (3.23)64quivalently, we are gauge fixing L to take the form of “first perform a boost (with respectto our frame n ) along the direction of (cid:126)k so that κ is boosted to | (cid:126)p | , then rotate the vector intothe direction of (cid:126)p ”. Under such gauge fixing, the action (3.20) reduces to the Weyl action(3.1) with H = 0. If we pick another reference frame n (cid:48) µ , the gauge fixing condition willchange, so we must perform a Wigner translation to satisfy the new gauge fixing condition,and the transformation is precisely (3.11) at A µ = 0 (the easiest way to see this is to use theWigner gauge invariance of J µν [13]).Everything said above did not rely on the smallness of (cid:126) , because there has been noexternal field A µ . How to extend the connection between Wigner translation and framedependence to the case with A µ (cid:54) = 0? If we just add to (3.20) the minimal coupling A µ ( x ) dx µ ,the gauge invariance of Wigner translation is clearly broken at order (cid:126) , which is undesired.To remedy this to order (cid:126) , it can be shown that in addition to the minimal coupling, onemust also replace k a → k a − (cid:126) F ρσ Σ ρσ ¯ k a κ (3.24)in the first term of (3.20), where ¯ k a = ( κ, , , − κ ); moreover, the gauge Wigner transfor-mation of L must also be accompanied with a correction term to λ ab by − (cid:126) κ (cid:34) ( K ) ab F ρσ L ρc L σd ( a ¯ A + b ¯ B ) cd (cid:0) b ¯ A − a ¯ B (cid:1) ab F ρσ Σ ρσ (cid:35) (3.25)where ¯ A = J + K and ¯ B = − J + K are the Wigner translation generators for ¯ k . Thereplacement (3.24) leads to the “on-shell condition” H = 0 with the magnetic dipole term in(3.3). Performing the gauge fixing as before, one can show under a change of frame choice n µ → n (cid:48) µ , the Wigner translation needed to accommodate the new gauge fixing condition isprecisely (3.11).This explains the mathematical original of the frame dependence of x and p , valid to65rder (cid:126) . Furthermore, in the presence of A µ , one can show that no further correction canmake the action (3.20) Wigner translation invariant, or equivalently make the action (3.1)frame independent, at order (cid:126) . This is the symmetry based argument towards our commentmade in Section 2.4, that the semi-classical theory generally cannot be extended to order (cid:126) ,because at that order we have to consider the presence of other bands. We have already completed the discussion of Lorentz invariance in non-interacting chiralkinetic theory, with both the physical interpretation and mathematical structure explored.In this section we take one step further [13]. We would like to consider chiral kinetic theorywith collisions, and study what constraints Lorentz invariance places on it. We will constructan entropy current, and determine what the most general form of hydrodynamical limit (localequilibrium) is. As expected, it is the well known chiral hydrodynamics [66, 54, 47], fromwhich we will compute the famous chiral vortical effect (CVE).So far we have only been able to study chiral kinetic theory with collisions in the absenceof EM field. Moreover, we have to point out the formalism presented here is solely basedon symmetry principles, and lacks of a microscopic derivation. Microscopically, collisionsare due to interactions between particles, so this section is beyond the non-interacting BerryFermi gas theory. It is well-known that microscopic interactions lead to many effects inthe kinetic theory other than collisions, for example an interaction potential energy; in thissection we assume collisions is the dominating effect, while other effects can be ignored.
Collisions in chiral kinetic theory must be non-trivial. As we have seen from Figures 3.1 and3.2, the collision must be non-local in general. However to construct a non-local collisionkernel respecting Lorentz invariance is not obvious. Before we resolve this problem, though,66e shall resolve another, perhaps more straightforward problem.In the absence of EM field, the momentum p µ is a frame independent variable. Thecurrent (3.16) reduces to J µ ( x ) = (cid:90) ( p ) j µ ( p, x ) ≡ (cid:90) d p (2 π (cid:126) ) δ ( p · p ) θ ( − n · p ) j µ ( p, x ) (3.26)where the “phase space current” (the single particle current integrated over z ) is j µ ( p, x ) ≡ p µ f ( p, x ) + (cid:126) Σ µν ( p ) ∂ x ν f ( p, x ) . (3.27)Clearly the integral (cid:82) ( p ) is frame independent. We expect j µ to be frame independent too.But it is not. Recall that the distribution transforms as (3.14), i.e. f (cid:48) ( p, x ) − f ( p, x ) = − (cid:126) ∆ nn (cid:48) · ∂ x f ( p, x ) . (3.28)(recall that here x is not “a particle’s position” but a generic point in the spacetime, which isframe independent). Because (cid:126) Σ µν transforms as − x [ µ p ν ] does (invariance of total angularmomentum), we find j (cid:48) µ ( p, x ) − j µ ( p, x ) = (cid:126) , ∆ µnn (cid:48) ( p ) ( − p · ∂ x f ( p, x )) . (3.29)When the ensemble is collisionless, the collisionless Boltzmann equation (2.30) reads p · ∂ x f =0, and therefore j µ and hence J µ are indeed frame independent.But once we include collisions, the Boltzmann equation becomes p · ∂ x f = (cid:90) BCD C ABCD [ f ] + O ( (cid:126) ) . (3.30)Here we are considering all possible collisions AB ↔ CD with p = p A , and integrating over67he momenta of the other particles, (cid:82) B ≡ (cid:82) ( p B ) etc. The collision kernel is C ABCD = W CD → AB − W AB → CD (3.31)where W is the collision rate whose detailed form will be given later; it should be frameindependent at zeroth order in (cid:126) . The Boltzmann equation at order (cid:126) will also be given later.The issue now is, the collision term makes j µ and hence J µ ( x ) become frame dependent atorder (cid:126) .The physical observable J µ being frame dependent is certainly unacceptable. But howcan it be, given that we obtained J µ from the action, which is itself frame independent?The problem is, the action is frame independent only up to boundary terms. For collision-less ensemble, the worldlines have no end points and therefore there would be no boundaryterms. With collisions, however, the Weyl fermion action varies by a boundary term of A µ (cid:126) ∆ µnn (cid:48) (cid:12)(cid:12)(cid:12) f inalinitial , resulting in the frame dependence of j µ . This means, although our descrip-tion of the particle’s helical current is frame independent along the worldline, the descriptionof the ends of the worldline depends on a frame, which causes the problem.The problem is now reduced to, which frame should we choose to describe the end points?We do not need one frame for all the worldlines. The choice of frame can be different foreach collision event. We assume, for each 2-to-2 collision event, there is a “special frame” ¯ n µ that, if we view the collision event in this special frame, then each particle’s current is justthe usual single particle current, ending (for incoming particles) or starting (for outgoingparticles) at a common collision point, just as in the spinless case. If we view the event inanother frame n µ , then we transform each single particle current according to the boundaryterm of the action, i.e. the single particle current transforms by (cid:126) ∆ µn ¯ n with a delta functionlocalized at the collision point. In the context of 2-to-2 collision, there is a natural choice for68he “special frame” ¯ n µ – the center-of-mass (CoM) frame of the particular 2-to-2 collision:¯ n µ = ( p + p B ) µ (cid:112) − ( p + p B ) = ( p + p B ) µ √− p · p B = ( p C + p D ) µ (cid:112) − ( p C + p D ) = ( p C + p D ) µ √− p C · p D . (3.32)In the following we will make this assumption.This is just the idea behind Figure 3.1 made concrete. Based on this reasoning, thepresence of collision modifies the collisionless phase space current (3.27) into j µ ≡ p µ f + (cid:126) Σ µν ∂ x ν f + (cid:90) BCD C ABCD (cid:126) ∆ µ ¯ nn (3.33)where the last term captures the frame dependent “end point current” localized at eachcollision point. From our reasoning, j µ must be frame independent by construction. Then,in retrospect, it is in fact natural to “define” the phase space distribution by the timecomponent of the phase space current: f = − n · j − n · p . (3.34)Now that j and p are frame independent, we can find the frame dependence of f is, inaddition to (3.28), f (cid:48) − f = − (cid:126) ∆ nn (cid:48) · ∂ x f + (cid:90) BCD C ABCD (cid:126) ∆ nn (cid:48) · ¯ np · ¯ n , (3.35)where we have used the nice identity∆ µnn (cid:48) + ∆ µn (cid:48) n (cid:48)(cid:48) + ∆ µn (cid:48)(cid:48) n = p µ ∆ nn (cid:48) · n (cid:48)(cid:48) p · n (cid:48)(cid:48) (3.36)that follows from the definition (3.11) of ∆. Now, one can compute j (cid:48) µ − j µ using (3.35)and find j (cid:48) µ − j µ = 0 as it should. This checks the consistency of our formalism.69ow that we have a frame independent phase space current j µ , we assert the physicalcurrent and the stress-energy tensor are give by J µ ( x ) = (cid:90) ( p ) j µ ( p, x ) , T µν ( x ) = (cid:90) ( p ) p { µ j ν } ( p, x ) . (3.37)In the collisionless limit, they reduce to (3.16) and (3.17) at A µ = 0. We can check ∂ x µ J µ = 0and ∂ x µ T µν = 0 after we developed the Boltzmann equation. Now we have found how to consistently express J µ and T µν in terms of f in the presenceof collisions. But we still need to figure out how the distribution f evolves in time, that is,what is the collisionful Boltzmann equation (3.30) to order (cid:126) . The collision kernel we findmust reflect the non-local collision in Figure 3.2.In usual kinetic theory, i.e. the theory at zeroth order in (cid:126) , the collision rate for identicalfermions is given by W AB → CD [ f ] = 12! |M (s , t) | (2 π ) δ ( p A + p B − p C − p D ) × f ( p A , x ) f ( p B , x ) (1 − f ( p C , x )) (1 − f ( p D , x )) (3.38)where M is the scattering amplitude computed from quantum field theory, with Mandelstamvariables s ≡ ( p A + p B ) , t ≡ ( p A − p C ) . But at order (cid:126) , the f ’s are frame dependent, sothe collision rate becomes ambiguous. Fortunately, again, for each collision event we have anatural choice of frame, the CoM frame ¯ n µ . Therefore we are led to consider the distributionin the CoM frame for each collision event¯ f = − ¯ n · j − ¯ n · p , (3.39)70nd we propose the collision rate for each collision event should be W AB → CD [ ¯ f ] = 12! |M (s , t) | (2 π ) δ ( p A + p B − p C − p D ) × ¯ f ( p A , x ) ¯ f ( p B , x ) (cid:0) − ¯ f ( p C , x ) (cid:1) (cid:0) − ¯ f ( p D , x ) (cid:1) . (3.40)With this proposition, we have a fix the right-hand-side of (3.30) to order (cid:126) . Since the right-hand-side does not depend on an arbitrarily chosen reference frame now, the left-hand-sidemust not be. The most natural choice of the left-hand-side would be ∂ x · j , which reducesback to p · ∂ x f at zero order in (cid:126) . Therefore, we propose the Boltzmann equation withcollisions at A µ = 0 should be ∂ x · j ( p, x ) = (cid:90) BCD C ABCD [ ¯ f ] = (cid:90) BCD (cid:0) W AB → CD [ ¯ f ] − W CD → AB [ ¯ f ] (cid:1) (3.41)where p = p A . Both side are now manifestly frame independent.In practice, it is more convenient to use f in some lab frame n µ , instead of having toconsider ¯ f in the CoM frame of each collision. For this purpose, we may rewrite (3.41) as p · ∂ x f = (cid:90) BCD C ABCD [ f ] (cid:18) − (cid:90) B (cid:48) C (cid:48) D (cid:48) dC AB (cid:48) C (cid:48) D (cid:48) [ f ] df (cid:126) ∆ ¯ nn · ¯ n (cid:48) p · ¯ n (cid:48) (cid:19) (3.42)where ¯ n (cid:48) is the COM frame of the collision AB (cid:48) ↔ C (cid:48) D (cid:48) .As usual, integrating the Boltzmann equation (3.41) with either (cid:82) ( p ) or (cid:82) ( p ) p µ yields zero.This reflects the conservation of charge and momentum under collisions. More precisely, thisleads to ∂ x µ J µ = 0 and ∂ x µ T µν = 0 (with the aid of (3.42)). Once collision is included, we are led to the most important concept of Boltzmann’s kinetictheory – the existence of an entropy current S µ which is non-decreasing ∂ x · S ≥
0. This is the71amous Boltzmann’s H-Theorem, revealing the fact the effects of collisions are irreversibleand drive the systems towards disorder. When entropy is maximized, i.e. ∂ x · S → j µφ = p µ φ ( f ) + (cid:126) Σ µν ∂ x ν φ ( f ) + (cid:90) BCD C ABCD (cid:126) ∆ µ ¯ nn dφ ( f ) df (3.43)where φ ( f ) is an arbitrary smooth function of f ; clearly the charge current j is the j φ when φ ( f ) = f . Its frame independence j (cid:48) µφ = j µφ can be verified by (3.35) and (3.36). Moreover,its divergence is ∂ x · j φ = (cid:90) BCD C ABCD [ ¯ f ] dφ ( ¯ f ) d ¯ f (3.44)using (3.42).With the above, if we know in usual kinetic theory (zeroth order in (cid:126) ) what the φ corresponding to entropy current is, we immediately have the entropy current at first orderin (cid:126) . It is elementary that in usual kinetic theory, the φ corresponding to entropy current isthe phase space local entropy s = − f ln f − (1 − f ) ln(1 − f ) . (3.45)Therefore, to first order in (cid:126) , the entropy current is S µ = (cid:90) ( p ) j µ s . (3.46)72o see the entropy current is non-decreasing, we abbreviate r ≡ W CD → AB W AB → CD = f C f D (1 − f A )(1 − f B ) f A f B (1 − f C )(1 − f D ) . (3.47)Note that C ABCD = W AB → CD ( r − C ABCD is even under A ↔ B and C ↔ D but odd under AB ↔ CD , we find ∂ x · S = 14 (cid:90) ABCD W AB → CD ( r −
1) ln r ≥ , (3.48)The above is vanishing only when r = 1, i.e. when there is no net collision C ABCD = 0(“detailed balance”). This is the local equilibrium state.
Let’s work out the most general local equilibrium state. We are going to show that the mostgeneral equilibrium distribution function, viewed in any frame n µ , is f ( p, x ) = 1exp (cid:0) U µ ( x ) p µ − Y + (cid:126) Σ µν ( p ) ∂ x µ U ν ( x ) / (cid:1) + 1 (3.49)with Y constant and U µ ( x ) a future time-like vector satisfying ∂ x µ U ν ( x ) = ∂ [ x µ U ν ] ( x ) + η µν ∂ x λ U λ ( x ) . (3.50)We will also show (3.49) is compatible with the frame transformation (3.35). In usual nota-tions one would write U µ ( x ) = u µ ( x ) /T ( x ) and Y = µ ( x ) /T ( x ) where u is the normalizedlocal fluid velocity, T > µ is the local chemical potential.As usual, it is convenient to define g such that f = 1 / ( e g + 1) so that r defined abovecan be written as r = exp(¯ g A + ¯ g B − ¯ g C − ¯ g D ). To achieve local equilibrium, we need r = 1, so ¯ g must be a linear combination of conserved quantities during a collision. The73onserved quantities are particle number (charge), momentum p µ and angular momentum J µν . Since we are viewing g in the CoM frame, the orbital angular momentum vanishes, sothe conservation of J µν becomes the conservation of spin viewed in the CoM frame (cid:126) ¯Σ µν .Therefore, in local equilibrium ¯ g must take the form¯ g eq ( p, x ) = U µ ( x ) p µ − Y ( x ) + (cid:126) ¯Σ µν ( p ) Ω µν ( x ) / . (3.51)where U µ , Y and Ω µν are some coefficients. This is the constraint from C ABCD = 0 on ¯ g .But the coefficients U, Y,
Ω are certainly not arbitrary. They need to satisfy two condi-tions: • We have expressed g eq in the CoM frame ¯ n of a particular collision AB ↔ CD .However, physically f = 1 / ( e g + 1) describes the distribution of the particle A , andis unrelated to the other particles B, C, D . This means, if we transform (3.51) intoan arbitrary lab frame n using (3.35), in the lab frame the expression of g eq must beindependent of ¯ n , and hence independent of p B , p C , p D . • The distribution f = 1 / ( e g + 1) must satisfy the Boltzmann equation (3.41).To satisfy the second condition, now that (3.51) guarantees C ABCD = 0, the Boltzmann’sequation (3.42) implies p · ∂ x g = 0. This means ∂ x µ U ν p µ p ν = 0 , p µ ∂ x µ Y = 0 , (cid:126) ¯Σ µν p λ ∂ x λ Ω µν = 0 . (3.52)The equations for U and Y reduce to (3.50) (the antisymmetric part is generic, while thetrace part relies on the Weyl fermion being massless) and Y = const. respectively; we willleave the Ω equation for later. For the first condition, using (3.35) we find, in a lab frame n , g eq = (cid:0) U µ − (cid:126) ∆ ν ¯ nn ∂ x ν U µ + (cid:126) ∆ ν ¯ nn Ω νµ (cid:1) p µ − ( Y − (cid:126) ∆ ν ¯ nn ∂ x ν Y ) + (cid:126) Σ µν Ω µν / Y is a constant and U satisfies (3.50), the condition g eq beingindependent of ¯ n reduces to Σ µν Ω µν = Σ µν ∂ x µ U ν ; this in turn solves the equation for Ωin (3.52). Finally, in order for f to approach 0 at large energy, U must be future time-like.Thus, we have shown (3.49) with constant Y and future time-like U satisfying (3.50) is themost general equilibrium distribution, and it is compatible with the change of frame (3.35).The phase space current (3.33) in equilibrium is j µeq = p µ f eq + df eq dg eq (cid:126)
32 Σ [ µν p λ ] ∂ x ν U λ = p µ f eq + df eq dg eq (cid:126) (cid:15) µνρσ p ν ∂ x ρ U σ (3.54)where f eq is f eq dropping the (cid:126) term, i.e. the Fermi-Dirac distribution for spinless particle. j µeq is explicitly frame independent. The phase space current j µ s is similar, but with s ( f eq )in place of f eq .We would like to compute J µ , T µν and S µ in equilibrium and see the chiral vorticaleffect. However we have a big missing piece. In this section we have been discussing thefermion only, but we also have to include anti-fermion contributions. The total current,stress-energy tensor and entropy current are J µtot = J µ − (cid:101) J µ , T µνtot = T µν + (cid:101) T µν , S µtot = S µ + (cid:101) S µ (3.55)where, for anti-fermions, every (cid:126) / − (cid:126) /
2. The incorporation of anti-fermionsinto the collision kernel is also straightforward: now the
ABCD not only denote the particles’momenta, but also whether each particle is a fermion or anti-fermion. In equilibrium, anti-fermions should have (cid:101) U µ = U µ but (cid:101) Y = − Y , because Y is the coefficient for fermion numberconservation.Now we have all the recipes. In the (cid:82) ( p ) integration, it is natural to use the fluid velocity75s the frame vector, i.e. n µ = u µ (recall that u µ is U µ normalized). The energy in this frameis E ≡ − u · p , while the spatial momentum is (cid:126)p µ ≡ p µ − Eu µ . Using the phase space current(3.54) (and also the phase space entropy current) and performing the (cid:126)p integrals, we find( J tot ) µeq = N u µ + ξ J ω µ , ( S tot ) µeq = S u µ + ξ S ω µ , ( T tot ) µνeq = E u µ u ν + P ( η µν + u µ u ν ) + 2 ξ T u { µ ω ν } (3.56)where ω µ ≡ (cid:15) µνρσ u ν ∂ x ρ u σ / N , E , P and S are the usual density, energy density, pressure and entropy density for masslessspinless particle: N = 4 π (2 π (cid:126) ) µ + π µT , E = 4 π (2 π (cid:126) ) µ + 2 π µ T + 7 π T /
154 (3.57)and P = E / S = ∂ P /∂T . The CVE coefficients ξ J , ξ T and ξ S are given by ξ J = 4 π (2 π (cid:126) ) (cid:126) T (cid:90) ∞ dE E (cid:18) exp( E/T − Y )(exp( E/T − Y ) + 1) − exp( E/T + Y )(exp( E/T + Y ) + 1) (cid:19) = 4 π (2 π (cid:126) ) (cid:126) (cid:18) µ + π T (cid:19) , (3.58) ξ T = 4 π (2 π (cid:126) ) (cid:126) T (cid:90) ∞ dE E (cid:18) exp( E/T − Y )(exp( E/T − Y ) + 1) + exp( E/T + Y )(exp( E/T + Y ) + 1) (cid:19) = 4 π (2 π (cid:126) ) (cid:126) (cid:16) µ + π µT (cid:17) , (3.59) ξ S = 32 T ξ T − Y ξ J = (cid:126) µT . (3.60)This is in agreement with the previous literature [66, 54, 47], up to a “change of hydrodynamic76rame”, i.e. a redefinition of u µ in (3.56). (The earliest computation [76] of CVE had adifferent ξ T because the ∂ x f contribution to T µν was missing.)Usually in relativistic hydrodynamics one uses one of Eckart frame (define u so that J has no spatial component in the u frame), Landau frame (define u so that T has no mixedtemporal-spatial components in the u frame) or entropy frame (define u so that S has nospatial component in the u frame). However, our result (3.56) is expressed in a frame of fluidvelocity u that does not admit any of these three usual conditions. What is the physicalmeaning of our u frame? It is shown that our frame of fluid velocity u is the “no-dragframe” [70]: if an impurity particle interacting with the fluid is moving in the fluid with thevelocity u , it will experience no net drag force. In this chapter we clarified how Lorentz invariance is non-trivially realized in chiral kinetictheory, in the seemly frame dependent Berry curvature formalism. We provided both thephysical interpretation and the mathematical origin of the non-trivial realization. Then weproposed a formalism of collisionful chiral kinetic theory in the absence of external force,and showed how the system is can be relaxed to the chiral hydrodynamics limit, in which wecomputed the chiral vortical effect. The collisionful chiral kinetic theory can be potentiallyapplied to the study of quark matter in heavy ion collision experiment or early universe.The collisionful chiral kinetic theory is worth further study. First, our proposed formalismis restricted to the absence external electromagnetic field; currently it is unclear how to writedown a frame independent collisionful Boltzmann equation in the presence of external field.For application purpose it would be useful to make this extension. Second, we do not havea microscopic derivation for the collisionful case; we only write down the simplest possibleformalism based on Lorentz symmetry considerations. It is desired to have a derivation fromthe Kadanoff-Baym method [35] or other methods.77nother direction for further theoretical study is to couple the semi-classical Weyl fermionto curved spacetime. Within the group theoretical formalism in Section 3.3, the coupling toa background metric can be introduced [4]. For massless particles, the non-triviality is againto find quantum correction terms, this time to remedy the violation of Wigner translationgauge invariance due to the spacetime curvature. We have made certain progress in thisdirection, but there remains an unsatisfactory issue – the gravitational chiral anomaly isbeyond the reach of our current regime of validity. In this thesis we are not covering therelevant efforts. The completion towards this direction can be a future subject of study.Although our discussion focuses on the issue of Lorentz invariance, through the discussionwe gain a better understanding of the Berry curvature formalism in general. The ∂ x f terms in the current and stress-energy tensor are generic. This means at order (cid:126) the semi-classical particle’s single particle current and stress-energy tensor generically have non-point-like feature. Thus, in general semi-classical systems, we always have to carefully interpretthe “position (cid:126)x ” and “momentum (cid:126)p ” as the position of the “center” of the single particlecurrent and the momentum carried at this “center”. Moreover, if we would like to includea collision term in the Boltzmann equation, either collision between particles or collision offimpurities, it is in general invalid to assume the collision is local; there would generally besome order (cid:126) (same order as Berry curvature effects) non-locality in the collision kernel. Forchiral kinetic theory we can use Lorentz symmetry to pin down this non-locality; for generalsystems perhaps a microscopic derivation is needed.78 HAPTER 4BERRY FERMI LIQUID
From the previous chapters, we see the kinetic theory of Fermi gas with Berry curvatureis a useful approach towards a wide range of physical systems. However, for most physicalsystems in reality, the assumption of free Fermi gas does not hold. The interaction betweenthe fundamental fermions might be so strong that, if one excites a fermion with high energy,it very soon decays into low energy excitations and “dissolves” into the system; hence,the picture based on the notion “particles” becomes not so useful. In such scenarios, weshall wonder, how do we still define “Berry phase effects”? How much of those we saidabout Berry Fermi gas still survives under interactions? Moreover, even if the interaction issufficiently weak that “particles” can still be talked about and the picture of Fermi gas is agood approximation, we can still ask, does the interaction introduce any new and interestingeffects? In this chapter we explore these problems.In our discussion of Fermi gas, thanks to the single particle picture, we could discuss thephysics of far-from-equilibrium systems. Once interaction is included, far-from-equilibriumsystems become difficult to study in general. Therefore we will be less ambitious about inter-acting systems. We restrict our consideration to a large class of interacting fermionic systemscalled
Fermi liquid . A Fermi liquid is a system whose ground state has some non-zero fermiondensity specified by a Fermi surface similar to that of a non-interacting system, and moreimportantly, whose low energy (gapless) excitations behave as nearly-free fermionic particlesnear the Fermi surface, with decay rate suppressed by their low energy (while interactionsare not necessarily weak); no particular assumption is made about high energy excitations.Due to these limitations in assumptions, we study the near-ground-state behaviors of theFermi liquid under low energy, long wavelength external perturbations only.The study of Fermi liquid started with the classic works of Landau [41, 40] in 1957.Following his amazing physical intuition, Landau proposed that at low energy a Fermi liquid79an be described by collisionless kinetic theory, and the leading effect (unsuppressed by thelow energy) of interaction is a local interaction potential energy and nothing else; all othereffects such as particle decay are suppressed by the low energy, even when interaction is notweak. Using his theory of Fermi liquid, Landau successfully explained the propagation of“zero sound” in low temperature Helium-3 liquid. Landau’s Fermi liquid theory was alsosuccessful in explaining various properties of electrons in metals.Originally Landau’s theory was formulated based on physical intuition, the theory waslater derived from quantum field theory by Landau himself and others [43, 59, 49, 35, 1]. Inthe 1990s, the remarkable intuition of Landau was concretely formulated in the language oflow energy effective field theory [60, 61].Despite the remarkable success of the Landau Fermi liquid theory, the interesting anoma-lous Hall effect, chiral magnetic effect and other Berry phase physics are missed in Landau’sframework. The reason is simple: Landau focused on the behaviors that are unsuppressed bythe low energy / long wavelength, while these interesting physics are one order higher in thelow energy / long wavelength expansion. One can see this easily: for instance, the longitu-dinal current δJ L ∼ σ L E ∼ const. A where A is the electromagnetic connection, E ∼ ωA isthe electric field and σ L ∼ const./ω is the longitudinal conductivity, with ω being the smallenergy carried in A ; on the other hand, the Hall current δJ H ∼ σ H E ∼ const. ωA . Thereforewe see the longitudinal current is dominated by unsuppressed terms while the Hall currentis suppressed by ω . Thus, in order to study the anomalous Hall effect, chiral magnetic effectand other Berry phase physics in Fermi liquid, we must work to the first order in the lowenergy / long wavelength expansion. Moreover, for Berry curvature to appear, clearly thefermionic field must be multi-component. This is what we consider in this chapter, and thetheory developed here would be called “Berry Fermi liquid theory”.We start with an Invitation section, demonstrating in the simple example of (2 + 1) D Dirac fermion with a weak contact interaction that the anomalous Hall current is no longer80iven by the Berry curvature of the particles, but receives a correction due to the interaction.This will be explained by the emergent electric dipole moment as we develop the generalframework of Berry Fermi liquid theory. Between the Invitation section and the presentationof the Berry Fermi liquid theory, we briefly review the Landau Fermi liquid theory.This Chapter is mostly a reproduction of Ref. [15], with the example in Section 4.1 addedfor demonstration purpose. (2 + 1) D Dirac Fermion with Weak ContactInteraction
Let’s start with a simple example. In this simple example, interaction is so weak that thedecay or collision between the fermions can be neglected, so the very notion of particle iswell-defined. Thus, we have a physical picture that is mostly analogous to a Fermi gas,except now there is some interaction between the fermions. We want to see if there is anyinteresting new effect arising from the interactions.We consider the 2-component Dirac fermion field ψ in (2 + 1) D . We consider N identicalcopies (flavors) of the field and label the flavors as ψ a . The Lagrangian is L = − ¯ ψ a iγ µ (cid:16) − i∂ x µ − A µ − (cid:15) δ µ (cid:17) ψ a − m ¯ ψ a ψ a − σ ¯ ψ a ψ a + σ / g. (4.1)We choose the Dirac gamma matrices in the basis γ = − iσ , γ = σ , γ = − σ (whichleads to γ γ µ = − σ µ ). Here ¯ ψ ≡ ψ † ( iγ ) as usual; m is the mass parameter and (cid:15) is thechemical potential parameter. σ is an auxiliary scalar field integrating out which yields thecontact interaction − ( g/ ψ a ψ a ) . We let g to be small and keep it to first order; sinceparticle decay / collision is order g , we have the picture of stable Dirac fermions. For thisLagrangian to be invariant under large gauge transformation, N must be even, and that iswhy we did not set N = 1 for simplicity. 81e allow m to take either sign, but for definiteness let’s fix (cid:15) ≥ | m | . That is, in theground state, the Dirac sea (negative energy band) is completely filled (no anti-fermion),while we have a Fermi sea of fermions in the positive energy band.The Feynman rules are simple. The bare fermion propagator, renormalized with a filledDirac sea and no Fermi sea (i.e. renormalized at (cid:15) = 0), is given by iG ( p ) δ ab ≡ (cid:104) ψ a ( − p ) ψ † b ( p ) (cid:105) (cid:12)(cid:12)(cid:12) g =0 = iδ ab u ( (cid:126)p ) u † ( (cid:126)p ) p − ( E ( (cid:126)p ) − (cid:15) )(1 − i(cid:15) ) + iδ ab w ( (cid:126)p ) w † ( (cid:126)p ) p − ( − E ( (cid:126)p ) − (cid:15) )(1 − i(cid:15) ) (4.2)where E ( (cid:126)p ) ≡ (cid:113) (cid:126)p + m and the i(cid:15) prescription corresponds to the said ground state.The bare positive energy band spinor u satisfies the Dirac equation ( σ i p i + σ m ) u ( (cid:126)p ) = E ( (cid:126)p ) u ( (cid:126)p ) and the negative energy band spinor w satisfies the Dirac equation ( σ i p i + σ m ) w ( (cid:126)p ) = − E ( (cid:126)p ) w ( (cid:126)p ); alternatively,2 u u † = σ + m E σ + p i E σ i , w w † = σ − m E σ − p i E σ i . (4.3)The explicit solutions to the bare spinors are u ( (cid:126)p ) = cos( θ/ e iφ sin( θ/ , w ( (cid:126)p ) = − e − iφ sin( θ/ θ/ (4.4)where ( E , θ, φ ) are the spherical coordinates in the ( p , p , m ) space. Note all these arethe same as Weyl fermion but with m in place of p . The other Feynman rules are: the A µ ψ a ψ † b vertex is given by iσ µ δ ab ; the “propagator” of the auxiliary σ field is ig/N ; the σψ a ψ † b vertex is given by − iσ δ ab .If we include interaction effects and keep g to first order, the bare propagator iG is82enormalized to full propagator iG , given by the following diagrams:= + + (4.5)where the thick fermion line is the full propagator iG while the thin fermion lines arethe bare propagator iG ; the dashed lines are the σ field propagator. One can showthat after including the 1-loop diagrams, iG takes the same form as iG , but with themass parameter m renormalized to some physical mass m , and the chemical potentialparameter (cid:15) renormalized to some physical chemical potential (cid:15) F ; the renormalized en-ergy E ( (cid:126)p ) and renormalized spinors u ( (cid:126)p ) , w ( (cid:126)p ) are defined with m in place of m . More-over, p is shifted by a constant which has no physical effect and can be removed by aredefinition of p . (The details of the shifts from m to m , (cid:15) to (cid:15) F and the shift of p are unimportant for the present discussion, but we include them here for completeness.To order g the shift in m is m = m + ( g/ π ) m (cid:15) (1 − / N ). The Fermi momentum p F = (cid:113) (cid:15) − m is a physical quantity and remains unchanged, and the physical chemicalpotential is given by (cid:15) F = (cid:113) p F + m = (cid:15) + ( m /(cid:15) )( m − m ). The shift in p is given by p + (cid:15) F − (cid:15) + ( g/ πN )( (cid:15) − m ), and can be removed by a redefinition of p .)Now we are ready to get to our main point: to see whether the presence of interactionalters the relation (2.39) between anomalous Hall effect and Berry curvature. Consider thelinear response of δJ µ to an external A ν field carrying momentum q . The relevant Feynmandiagrams are qµ A ν qµ A ν qµ A ν where the fermion lines are the full propagator iG . We further require q i = 0 and q to be so83mall that can be kept to first order; the electric field is E j = iq A j and the magnetic field iszero. For the purpose of computing anomalous Hall current, we are only interested in i Π [ ij ] (at linear response iδJ µ ( q ) = i Π µν ( q ) A ν ( q )), the antisymmetric part of these diagrams.The first diagram is what would contribute when g = 0, so we expect the first diagram toreproduce the Berry Fermi gas result (2.39). Let’s demonstrate this. With q i = 0, performingthe loop integral and picking the p poles in the loop yields − iN (cid:90) d (cid:126)p (2 π ) (cid:20) θ ( (cid:15) F − E ( (cid:126)p )) − E ( (cid:126)p ) − q (cid:16) w † ( (cid:126)p ) σ [ i u ( (cid:126)p ) (cid:17) (cid:16) u † ( (cid:126)p ) σ j ] w ( (cid:126)p ) (cid:17) + θ ( (cid:15) F − E ( (cid:126)p )) − E ( (cid:126)p ) + q (cid:16) u † ( (cid:126)p ) σ [ i w ( (cid:126)p ) (cid:17) (cid:16) w † ( (cid:126)p ) σ j ] u ( (cid:126)p ) (cid:17)(cid:21) = iN (cid:90) d (cid:126)p (2 π ) (1 − θ ( (cid:15) F − E ( (cid:126)p )) q E ( (cid:126)p ) − ( q ) (cid:16) w † ( (cid:126)p ) σ [ i u ( (cid:126)p ) (cid:17) (cid:16) u † ( (cid:126)p ) σ j ] w ( (cid:126)p ) (cid:17) . (4.6)At leading order we can ignore the ( q ) in the denominator. Moreover, note that σ i = ∂ p i (cid:16) σ k p k + σ m (cid:17) , while u, w are eigenstates of σ k p k + σ m with eigenvalues ± E ( (cid:126)p ) respec-tively, therefore we have w † ( (cid:126)p ) σ i u ( (cid:126)p ) = 2 E ( (cid:126)p ) w † ( (cid:126)p ) (cid:16) ∂ ip u ( (cid:126)p ) (cid:17) = − E ( (cid:126)p ) (cid:16) ∂ ip w † ( (cid:126)p ) (cid:17) u ( (cid:126)p ) , (4.7)as well as complex conjugate version of it. Hence, using = uu † + ww † , we finally find − q N π (cid:90) d (cid:126)p π (cid:104) ( − i ) ∂ [ ip w † ∂ j ] p w + θ ( (cid:15) F − E )( − i ) ∂ [ ip u † ∂ j ] p u (cid:105) . (4.8)This is indeed the Berry Fermi gas result (2.39). The first term in the square bracket isthe Berry curvature contribution from the filled Dirac sea, yielding − (cid:15) ij sgn( m ) /
2, i.e. inthe gapped phase | (cid:15) F | ≤ | m | the Chern number would be sgn( m ) N/
2, which explains whywe said N must be even. The second term in the square bracket is the Berry curvaturecontribution from the Fermi sea, yielding (cid:15) ij (1 − | m | /(cid:15) F )sgn( m ) / g , we must also take into account the second and third Feynman diagrams.Unless those two diagrams vanish at first order in q , the extra contribution from those twodiagrams would violate the Berry Fermi gas result (2.39). It is easy to see the secondFeynman diagram vanishes when q i = 0; however, the third has a non-trivial contribution,so (2.39) is indeed violated. The 2-loop integral in the third diagram is not as hard as itmight look. Because the σ propagator is momentum independent, we can carry our themomentum integral in each loop separately (and the two loops are clearly identical up to q → − q ), so we really only need to perform a 1-loop integral. We find g m q (cid:15) ij π (cid:32) − m (cid:15) F (cid:33) . (4.9)Combining with the first diagram, we arrive at i Π [ ij ] ( q ) = N q (cid:15) ij π (cid:32) m(cid:15) F + g m πN (cid:32) − m (cid:15) F (cid:33)(cid:33) , δJ iH ( q ) = Π [ ij ] ( q ) A j ( q ) . (4.10)The first term is the expected Berry curvature contribution from both bands. However,interaction introduces the second term, and thus violates the simple relation (2.39) betweenanomalous Hall conductivity and Berry curvature in Berry Fermi gas.Through this simple example we see the violation of (2.39). What is the physical natureof the new term? Inspecting the third Feynman diagram, it seems we shall view the dashline as a quantum correction to the bare A ν ψψ † vertex, and this suggests the new term isrelated to the effective electric dipole moment arising from interactions. This is indeed thecorrect physical interpretation, as we will see along the development of the general formalismof Berry Fermi liquid.What cannot be learned from this simple example – in which the particles are stable sincedecay occurs at order g which we neglect – is whether Berry curvature effect is still worthtalking about when the interaction is so strong that particles in the Fermi sea are unstable.85he answer is positive – more particularly, only the Berry curvature near the Fermi surfaceis involved. We will also show this along the development of our general formalism. Before we present the kinetic theory of Berry Fermi liquid, let’s start with a quick review ofLandau Fermi liquid theory – in particular, the computation of linear response in the theory.We assume that there is no external field violating spacetime translational symmetry exceptfor the present external EM field. We assume the EM U (1) charge conservation is not brokenby the ground state. We do not assume the presence of any other symmetry.Let us start with a system of fermions, interacting through a finite-ranged interaction,in d spatial dimensions with d ≥ d + 1) spacetime dimensions). We assume theground state at chemical potential (cid:15) F is a Fermi liquid, with a sharp Fermi surface (FS).(We assume that the Kohn-Luttinger instability [38] occurs at a temperature much smallerthan any energy scales of interest). The low energy excitations are fermionic quasiparticles(or quasiholes) near the FS. For simplicity we assume one, non-degenerate, FS, i.e., eachmomentum (cid:126)p near the FS corresponds to only one quasiparticle.We now perturb this system by a small external EM field A µ . Physically, this causes adeformation of the FS, which can also be viewed as creating quasiparticles and quasiholes,which in Landau’s theory are described by the quasiparticle distribution function δf ( (cid:126)p ; x )with (cid:126)p near the FS. In the linear response theory we keep δf to linear order of A µ .The Landau Fermi liquid theory matches with quantum field theory at long wavelength.If we Fourier transform − i (cid:126) ∂ x µ to q µ , then in the long-wavelength limit under consideration, A µ and δf only have q modes with q (cid:28) p F and q (cid:28) (cid:126) /r int , where p F is the size scale of theFS (there is no notion of “Fermi momentum” since we do not assume rotational symmetry),and r int is the range of interaction between quasiparticles (this is why we assumed finite-ranged interactions). In practice, we keep (cid:126) ∂ x , or equivalent q , to leading order in Landau86ermi liquid theory.It can be shown that the collision (decay included) rate of quasiparticles is suppressedbeyond leading order in q , due to the limited availability of decay channels. In particular,the suppression is by an extra order of q for d ≥ q ln q for d = 2[6, 17, 18]. Thus, quasiparticle collision can be neglected in Landau Fermi liquid theory.The computation of linear response in Landau Fermi liquid theory proceeds in two steps.One first computes δf as a linear function of A by solving the Boltzmann equation, and thenexpresses (the quantum expectation of) the induced current δJ µ as a linear function of δf ,and hence of A . In Landau’s Fermi liquid theory, the energy of a single quasiparticle has theform (cid:15) ( (cid:126)p ; x ) = E ( (cid:126)p ) + (cid:90) (cid:126)k U ( (cid:126)p, (cid:126)k ) δf ( (cid:126)k ; x ) (4.11)where (cid:82) (cid:126)k ≡ (cid:82) d d k/ (2 π (cid:126) ) d . Here E ( (cid:126)p ) is the kinetic energy of the quasiparticle, and U ( (cid:126)p, (cid:126)k ),even under exchange of (cid:126)p and (cid:126)k , parameterizes the contact interaction between two quasi-particles of momenta (cid:126)p and (cid:126)k . (If the system has rotational symmetry, the Landau Fermiliquid parameters are obtained by putting (cid:126)p and (cid:126)k on the Fermi surface and expanding U in angular harmonics in the angle between (cid:126)p and (cid:126)k .) Both E and U are microscopic inputsinto Landau’s theory. Landau’s Fermi liquid theory postulates the collisionless Boltzmannequation (2.30): ∂f ( (cid:126)p ; x ) ∂t + ∂(cid:15) ( (cid:126)p ; x ) ∂p i ∂f ( (cid:126)p ; x ) ∂x i + (cid:18) F i ( x ) + F ij ( x ) ∂(cid:15) ( (cid:126)p ; x ) ∂p j − ∂(cid:15) ( (cid:126)p ; x ) ∂x i (cid:19) ∂f ( (cid:126)p ; x ) ∂p i = 0 . (4.12)where F i = ∂ x i A − ∂ t A i is the electric field, F ij = ∂ x i A j − ∂ x j A i is the magnetic field,and we have absorbed the electric charge into the field potential A . Writing f ( (cid:126)p ; x ) =87 ( (cid:15) F − E ( (cid:126)p )) + δf ( (cid:126)p ; x ) and linearizing over δf and A , one finds v µ ( (cid:126)p ) ∂ x µ δf ( (cid:126)p ; x ) = δ ( (cid:15) F − E ( (cid:126)p )) v i ( (cid:126)p ) (cid:0) F i ( x ) − ∂ x i (cid:15) ( (cid:126)p ; x ) (cid:1) (4.13)where v ≡ v i ( (cid:126)p ) ≡ ∂ ip E ( (cid:126)p ), and x ≡ t . Notice that, due to the delta function on the righthand side, Eq. (4.13) involves only the FS, but not, say, the whole Fermi sea. Performingthe Fourier transformation − i∂ x µ → q µ , where, in our convention, − q = q is the energy,while q i = q i is the momentum, the Boltzmann equation then reads δf ( (cid:126)p ; q ) = δ ( (cid:15) F − E ) v i v µ q µ − i(cid:15) (cid:18) − iF i ( q ) − q i (cid:90) (cid:126)k U ( (cid:126)p, (cid:126)k ) δf ( (cid:126)k ; q ) (cid:19) (4.14)where F µν ( q ) = 2 iq [ µ A ν ] . This is an integral equation from which one can find δf in termsof A . It follows from Eq. (4.14) that the the coefficient of linear dependence between δf and A is finite in the limit q → q / | q | fixed. In this thesis we count this as zeroth order(leading order) in q . Note that we placed an i(cid:15) prescription in the denominator; its sign issuch that q appears as q + i(cid:15) . This corresponds to the retarded boundary condition thatat infinite past the system is in its ground state.Now suppose we have solved for δf as a linear function of A from (4.14). Then theinduced current in Landau Fermi liquid theory is given by δJ µ ( x ) = (cid:90) (cid:126)p (cid:16) v µ ( (cid:126)p ) δf ( (cid:126)p ; x ) + δ ( (cid:15) F − E ( (cid:126)p )) δ µi v i ( (cid:126)p ) ( (cid:15) ( (cid:126)p ; x ) − E ( (cid:126)p )) (cid:17) . (4.15)The first term is simply the current created by the quasiparticles that were excited. Thesecond term, by recognizing δ ( (cid:15) F − E ) v i = − ∂ ip θ ( (cid:15) F − E ) and integrating by parts over (cid:126)p , is the current due to quasiparticles in the Fermi sea having their velocity perturbed byinteractions with the excited quasiparticles ∂ ip ( (cid:15) − E ). (Although “quasiparticles in the Fermisea” are generally not well-defined far from the FS, from the expression (4.15) we clearly88ee only those quasiparticles near the FS are involved.) This is the procedure of computinglinear response in Landau Fermi liquid theory. As introduced at the beginning of this chapter, we are to develop a kinetic theory of Fermiliquid incorporating Berry curvature. The most outstanding questions are how to defineBerry curvature effects when high energy quasiparticles are unstable, which properties ofBerry Fermi gas survive in the presence of interactions, and whether any new effects arisefrom interactions. The assumptions about the Fermi liquid is mostly the same as in Landau’stheory. The differences are that here the fermionic field must be multi-component, andthat we work to one order higher in low energy / long wavelength expansion (consequently,collisions cannot be neglected even for the meta-stable quasiparticles near the FS).The kinetic formalism of Berry Fermi liquid theory, similar to Landau Fermi liquid theory,consists of two parts: the Boltzmann equation, and the expression of the current in termsof the distribution function. We will also find that the consistency of the theory requirescertain relationships between the chemical potential dependence of the Fermi velocity andthe Landau interaction potential, and between the chemical potential dependence of theHall conductivity tensor (to be defined later) and the Berry curvature of the fermionicquasiparticle. We will present this formalism in this Section. In the next Section, we willshow this kinetic formalism exactly matches with quantum field theory (QFT) computationto all orders in diagrammatic expansion, for a large class of QFTs.We make a final remark. For simplicity, we again assume only one band crosses theFermi level. However, the formalism below can be easily generalized to the cases of either i)multiple degenerate bands crossing the Fermi level, or ii) multiple bands crossing the Fermilevel with disjoint FS; the generalizations are obvious in the QFT derivation. Therefore,with straightforward generalization, our formalism encompasses the example in Section 4.189n which N degenerate bands cross the Fermi level. In a Berry Fermi liquid, as in the usual Fermi liquid theory, the energy of a quasiparticlewith momentum (cid:126)p near the FS depends on the occupation at other momenta. To first orderin A and first order in ∂ x , the energy is (cid:15) ( (cid:126)p ; x ) = E ( (cid:126)p ) − µ µν ( (cid:126)p ) F µν ( x )2 + (cid:90) (cid:126)k (cid:16) U ( (cid:126)p, (cid:126)k ) δf ( (cid:126)k ; x ) + V ν ( (cid:126)p, (cid:126)k ) ∂ x ν δf ( (cid:126)k ; x ) (cid:17) . (4.16)Compared to (4.11), here µ µν , antisymmetric in µν , is the EM dipole moment of the quasi-particles (the purely spatial components µ ij correspond to the magnetic dipole moment andthe mixed components µ i to the electric dipole moment), and V ν ( (cid:126)p, (cid:126)k ), odd under exchangeof (cid:126)p and (cid:126)k , is the gradient interaction potential between quasiparticles. The function V µ ( (cid:126)p, (cid:126)k )is the additional function parametrizing the dependence of the energy of the quasiparticlewith momentum (cid:126)p on the gradient of the distribution function at (cid:126)k . Since we are perform-ing a gradient expansion of the interaction between two quasiparticles, our assumption ofinteraction being finite-ranged is needed.Extended to sub-leading order in spacetime derivative, the linearized (in δf and A )Boltzmann equation now includes collision term. Although we need to include collision forcompleteness, we emphasize it is “uninteresting” towards the focus of this paper as it doesnot contribute to interesting physics such as the anomalous Hall effect, as we will show laterin Section 4.4.6.The collision term is different from that in classical Boltzmann equation, and must beobtained quantum mechanically. The collisionful Boltzmann equation we find is to modify(4.13) by the replacement v i ( (cid:126)p ) ∂ x i → v i ( (cid:126)p ) ∂ x i − δ ( (cid:15) F − E ( (cid:126)p )) (cid:82) (cid:126)k C ( (cid:126)p, (cid:126)k ) ∂ t on both sides,90ielding v µ ( (cid:126)p ) ∂ x µ δf ( (cid:126)p ; x ) − δ ( (cid:15) F − E ( (cid:126)p )) (cid:90) (cid:126)k C ( (cid:126)p, (cid:126)k ) ∂ t δf ( (cid:126)k ; x )= δ ( (cid:15) F − E ( (cid:126)p )) (cid:18) v i ( (cid:126)p ) F i ( x ) − v i ( (cid:126)p ) ∂ x i (cid:15) ( (cid:126)p ; x )+ (cid:90) (cid:126)k C ( (cid:126)p, (cid:126)k ) δ ( (cid:15) F − E ( (cid:126)k )) ∂ t (cid:15) ( (cid:126)k ; x ) (cid:19) . (4.17)Here C ( (cid:126)p, (cid:126)k ), symmetric under exchange of (cid:126)p and (cid:126)k , is the effective collision kernel definedon the FS. It has the following properties (which we will show when we perform the QFTderivation in the next Section): • Collisions do not change the total number of fermionic excitations, i.e. (cid:90) (cid:126)p δ ( (cid:15) F − E ( (cid:126)p )) C ( (cid:126)p, (cid:126)k ) = 0 . (4.18) • C ( (cid:126)p, (cid:126)k ) is not regular over the FS. It can be separated into a positive “quasiparticledecay” piece that is non-vanishing only when (cid:126)p = (cid:126)k on the FS, plus a piece that isnon-vanishing for general values of (cid:126)p and (cid:126)k .The ∂ t in the collision term has been long known. Recall that in the “thermal regime” wheretemperature T (cid:29) ∂ t , linearizing the classical Boltzmann collision term yields the scaling of T . But here we are in the “quantum regime” where temperature is negligible, T (cid:28) ∂ t ;according to Landau’s semi-classical argument [40], in this regime the scaling should bereplaced by ∂ t . Luttinger also has a field theory power counting argument [48]; we willadopt this method in Section 4.4.6.We have to emphasize that such parametrization of the collision term is only valid for d ≥
3. In d = 2 the collision term cannot be parametrized in any simple form [17, 18], aswe will discuss in Section 4.4.6. Fortunately, our main focus – the computation of the Hall91urrent – is not undermined by this failure of parametrizing collisions in d = 2. In particular,we will show in Section 4.4.6 that at order q and q ln q , collisions only contributes to thelongitudinal current but not the Hall current.The Boltzmann equation can be solved in principle, order by order in q . First in (4.17)we Fourier transform − i∂ x µ into q µ . Let us separate δf = δf + δf , where the subscriptlabels the order in q . Then the Boltzmann equation (4.17) reads (the collision term onlyholds for d ≥ δf ( (cid:126)p ; q ) = δ ( (cid:15) F − E ) v i v µ q µ − i(cid:15) (cid:18) − iF i ( q ) − q i (cid:90) (cid:126)k U ( (cid:126)p, (cid:126)k ) δf ( (cid:126)k ; q ) (cid:19) , (4.19) δf ( (cid:126)p ; q )= δ ( (cid:15) F − E ) v i q i v µ q µ − i(cid:15) (cid:18) µ µν F µν ( q )2 − (cid:90) (cid:126)k (cid:16) U ( (cid:126)p, (cid:126)k ) δf ( (cid:126)k ; q ) + V ν ( (cid:126)p, (cid:126)k ) iq ν δf ( (cid:126)k ; q ) (cid:17)(cid:19) + δ ( (cid:15) F − E ) i ( q ) v µ q µ − i(cid:15) (cid:90) (cid:126)k C ( (cid:126)p, (cid:126)k ) (cid:18) δf ( (cid:126)k ; q ) + δ ( (cid:15) F − E ( (cid:126)k )) (cid:90) l U ( (cid:126)k, l ) δf ( l ; q ) (cid:19) (4.20)at zeroth and first order in q respectively. We will prove these two equations from QFT inSection 4.4.4. Note that the zeroth order Boltzmann equation (4.19) is that in Landau Fermiliquid theory.One may have noticed that there is no reference to Berry curvature in the Boltzmannequation. Notably, the Berry curvature b ij should induce an anomalous velocity b ij F jν v ν [72].However, since (4.17) is of order A , the effect of anomalous velocity will be order A , whichwe assumed to be negligible. (If one works beyond linear response, and assume stable quasi-particle, the anomalous velocity term would be present [62].) Other effects of Berry curvatureare negligible in the Boltzmann equation for the same reason.92 .3.2 Current At equilibrium there is some equilibrium current J µeq. . In most systems at equilibrium onlythe charge density J eq. is non-zero, while J ieq. = 0. As we perturb the system, extra current δJ µ of order A will be induced. In our formalism, we propose δJ µ ( x ) = (cid:90) (cid:126)p (cid:16) v µ ( (cid:126)p ) δf ( (cid:126)p ; x ) + µ µν ( (cid:126)p ) ∂ x ν δf ( (cid:126)p ; x ) + δ ( (cid:15) F − E ( (cid:126)p )) δ µi v i ( (cid:126)p ) ( (cid:15) ( (cid:126)p ; x ) − E ( (cid:126)p )) (cid:17) + σ µνλ F νλ ( x )2 . (4.21)Inside the integral on the right-hand side of (4.21) are three terms. The first term is thecurrent due to the velocity of the deformation of the FS. The second term is the magnetization/ electric polarization current [14] due to the deformation of the FS. The third term, as inLandau Fermi liquid theory, is the current due to quasiparticles in the Fermi sea gettingextra velocity ∂ ip ( (cid:15) − E ) (rewritten by integrating (cid:126)p by parts). All these three terms involveonly (cid:126)p near the FS, as desired.In the last term of (4.21), σ µνλ , totally antisymmetric in µνλ , is the Hall conductivitytensor. As we will discuss in Section 4.3.4, it has very interesting relation to the FS, andthat is how Berry curvature enters the formalism.Now in (4.21) we Fourier transform − i∂ x µ into q µ . At zeroth and first order in q respec-tively, the current reads δJ µ ( q ) = (cid:90) (cid:126)p (cid:18) v µ δf ( (cid:126)p ; q ) + δ ( (cid:15) F − E ) δ µi v i (cid:90) (cid:126)k U ( (cid:126)p, (cid:126)k ) δf ( (cid:126)k ; q ) (cid:19) , (4.22)93 J µ ( q ) = (cid:90) (cid:126)p (cid:20) v µ δf ( (cid:126)p ; q ) + µ µν iq ν δf ( (cid:126)p ; q )+ δ ( (cid:15) F − E ) δ µi v i (cid:18) − µ νλ F νλ ( q )2+ (cid:90) (cid:126)k (cid:16) U ( (cid:126)p, (cid:126)k ) δf ( (cid:126)k ; q ) + V ν ( (cid:126)p, (cid:126)k ) iq ν δf ( (cid:126)k ; q ) (cid:17)(cid:19)(cid:21) + σ µνλ F νλ ( q )2 . (4.23)Notice δJ µ is that in Landau Fermi liquid theory. We will prove these two equations fromQFT in Section 4.4.5. In the proof, we will also discuss the microscopic contributions to µ µν . The magnetic dipole moment is generally non-zero; in the presence of interactions [62],the electric dipole moment will also be non-zero in general, as we will see in the proof.Although we call σ µνλ the Hall conductivity tensor, it is not the full Hall conductivity asmeasured in linear response. The full Hall conductivity also receives contributions from the (cid:126)p integral, and depends on the ratio | (cid:126)q | /q . For example, in order to find the Hall conductivityfor spatially homogeneous electric field, we set q j = 0 and choose the gauge A = 0, so F j = − iq A j . From the Boltzmann equation we have δf = − δ ( (cid:15) F − E ) v j A j , which leadsto the anomalous Hall current δJ iH = (cid:18) σ ij − (cid:90) (cid:126)p δ ( (cid:15) F − E ) 2 v [ i µ j ]0 (cid:19) F j ( q j = 0 , q small) (4.24)(although δf is non-zero due to collisions, we already mentioned that collisions do notcontribute to the Hall current, as shown in Section 4.4.6). Thus, the full Hall conductivity,in the limit of taking q j = 0 first and then taking q small, receives contribution from boththe σ tensor and the electric dipole moment µ j of the quasiparticles – the latter is generallynon-zero in the presence of interaction [62], as we will show in the QFT derivation. Similarly,if we take the other order of limits, q = 0 first and q j small so that F j = iq j A , we will94nd δJ iH = (cid:18) σ ij + (cid:90) (cid:126)p δ ( (cid:15) F − E ) µ ij (cid:19) F j ( q = 0 , q j small) (4.25)where µ ij is related to the magnetic dipole moment µ ij via the recursion relation µ ij ( (cid:126)p ) = µ ij ( (cid:126)p ) − (cid:90) (cid:126)k δ ( (cid:15) F − E ( (cid:126)k )) U ( (cid:126)p, (cid:126)k ) µ ij ( (cid:126)k ) . (4.26)So again the σ tensor does not give the measured Hall conductivity. We have separated the quasiparticle distribution into an equilibrium part θ ( (cid:15) F − E ) andan excitation part δf . Within Fermi liquid theory, such a separation is ambiguous: thesame state may equally well be described either by starting with a slightly lower chemicalpotential and exciting some quasiparticles above the FS, or by starting with a slightly higherchemical potential and exciting some quasiholes below the FS. Clearly, for the theory to beself-consistent, all these different descriptions of the same state must be equivalent. For this,the following relationship between E at different chemical potentials must hold: ∂E ( (cid:126)p ) ∂(cid:15) F = (cid:90) (cid:126)k U ( (cid:126)p, (cid:126)k ) ∂∂(cid:15) F θ ( (cid:15) F − E ( (cid:126)k )) = (cid:90) (cid:126)k U ( (cid:126)p, (cid:126)k ) (cid:32) − ∂E ( (cid:126)k ) ∂(cid:15) F (cid:33) δ ( (cid:15) F − E ( (cid:126)k )) . (4.27)This can be physically understood from (4.16), setting F µν = 0 and ∂ x δf = 0. Furthermore,we will prove it from QFT in Section (4.4.2). Taking ∂ ip of (4.27), we obtain the chemicalpotential dependence of v i ( (cid:126)p ) on the FS.Strictly speaking, the reasoning above only applies when the FS changes continuouslywith the chemical potential. If the system undergoes a quantum phase transition at some (cid:15) F , around which the FS develops new disconnected components, as illustrated in the Figure95 F increases −−−−−−−−−−−→ Figure 4.1: Around some discrete values of chemical potential, the Fermi surface may developnew disconnected components, which may lead to a quantum phase transition. The behaviorof the interacting system around such values of chemical potential remains unknown.4.1, then the formula (4.27) not necessarily holds.
The Hall conductivity tensor in (4.21) seem to have no reference to the FS. But in fact theHall conductivity tensor is related to the FS via the Berry curvature in a very interestingmanner. We will distinguish two cases. In the first case, either d = 2, or d > d , but not the anomalous Hall effect in (2 + 1) d .) Then we turn to the case in d > Without Anomaly-Related Transport
Let’s review the story in Berry Fermi gas. In Fermi gas, particles are stable, so one candefine the Berry connection a j and Berry curvature b ij for all particles in the Fermi sea: a j ( (cid:126)p ) ≡ ( − i (cid:126) ) u † α ( (cid:126)p ) ∂ jp u α ( (cid:126)p ) , (4.28) b ij ( (cid:126)p ) ≡ ∂ [ ip a j ] ( (cid:126)p ) = ( − i (cid:126) ) ∂ [ ip u † α ( (cid:126)p ) ∂ j ] p u α ( (cid:126)p ) (4.29)96here u α ( (cid:126)p ) is the spinor or Bloch state of the fermion. The Berry curvature induces ananomalous velocity [72] and a change of the classical phase space measure [84, 26], leadingto Hall conductivity tensor of the form σ µνλ = σ µνλo + 3 (cid:90) (cid:126)p θ ( (cid:15) F − E ( (cid:126)p )) v [ µ ( (cid:126)p ) b νλ ] ( (cid:126)p ) (4.30)where a = 0 , b µ = 0. Here σ µνλo is the contribution from valence bands / Dirac sea, andis independent of (cid:15) F . The second term seems like a Fermi sea property, but as observed byHaldane [32], one can integrate (cid:126)p by parts and get σ µνλ = σ µνλo + 6 (cid:90) (cid:126)p δ ( (cid:15) F − E ( (cid:126)p )) δ [ µ v ν ( (cid:126)p ) a λ ] ( (cid:126)p ) , (4.31)so that the kinetic part of the Hall conductivity tensor is actually a FS property; notice thekinetic part has no ijk components, but only ij d > d is the numberof spatial dimensions), there is another way to integrate (4.30) by parts, also promoted byHaldane [32]. Using v ≡ ∂ kp p k /d , we have σ ij = σ ij o + 3 d − (cid:90) (cid:126)p θ ( (cid:15) F − E ( (cid:126)p )) b [ ij ( (cid:126)p ) ∂ k ] p p k = σ ij o + 3 d − (cid:90) (cid:126)p δ ( (cid:15) F − E ( (cid:126)p )) b [ ij ( (cid:126)p ) v k ] ( (cid:126)p ) p k + 6 d − (cid:90) (cid:126)p ∂ [ kp (cid:16) δ ( (cid:15) F − E ( (cid:126)p )) v i ( (cid:126)p ) a j ] ( (cid:126)p ) p k (cid:17) (4.32)and σ ijk = σ ijko . The last line is a boundary term that is non-vanishing if the fermion is in alattice and the FS intersects the boundary of our choice of first Brillouin zone [32] (because p k is not continuous when we identify the opposite boundaries of the first Brillouin zone).The advantage of (4.32) over (4.31) is that it involves b ij instead of the gauge dependent a i (except for the boundary term); as we will see later, this makes (4.32) is more convenient97or generalization to include anomaly-related transport effects.Now we turn to the (cid:15) F dependence of σ µνλ in Berry Fermi liquid. In the presenceof interaction, the picture of quasiparticles is only valid near the FS, so whether the (cid:15) F dependence of σ µνλ can be expressed as a FS property becomes important at conceptuallevel: It determines, in order to study linear response to EM field at long wavelength,whether knowing the system is a Fermi liquid at low energy is enough, or we have to knowmore beyond the low energy behaviors. Our conclusion is, the former is true – the fact thatthe system is a Fermi liquid is enough. More exactly, we will show in Section 4.4.5 that ifthe FS changes continuously with the chemical potential, then dσ µνλ d(cid:15) F = 3 (cid:90) (cid:126)p δ ( (cid:15) F − E ( (cid:126)p )) (cid:18) v [ µ ( (cid:126)p ) b νλ ] ( (cid:126)p ) − ∂E ( (cid:126)p ) ∂(cid:15) F δ [ µ b νλ ] ( (cid:126)p ) − δ [ µ v ν ( (cid:126)p ) b λ ] F ( (cid:126)p ) (cid:19) . (4.33)Here b λF is the mixed Berry curvature of momentum and chemical potential: b kF ( (cid:126)p ) ≡ ( − i (cid:126) ) (cid:32) ∂ kp u † α ( (cid:126)p ) ∂ u α ( (cid:126)p ) ∂(cid:15) F − ∂ u † α ( (cid:126)p ) ∂(cid:15) F ∂ kp u α ( (cid:126)p ) (cid:33) (4.34)and b F = 0; it satisfies the Bianchi identity ∂b νλ /∂(cid:15) F = 2 ∂ [ λp b ν ] F . The spinor / Bloch state u ( (cid:126)p ) is understood as that of an on-shell quasiparticle near the FS. We will also show (4.33)is equivalent to dσ µνλ d(cid:15) F = dd(cid:15) F (cid:90) (cid:126)p δ ( (cid:15) F − E ( (cid:126)p )) δ [ µ v ν ( (cid:126)p ) a λ ] ( (cid:126)p ) , (4.35)and, for d >
2, also equivalent to dσ ij d(cid:15) F = dd(cid:15) F d − (cid:90) (cid:126)p δ ( (cid:15) F − E ( (cid:126)p )) b [ ij ( (cid:126)p ) v k ] ( (cid:126)p ) p k + dd(cid:15) F d − (cid:90) (cid:126)p ∂ [ kp (cid:16) δ ( (cid:15) F − E ( (cid:126)p )) v i ( (cid:126)p ) a j ] ( (cid:126)p ) p k (cid:17) , (4.36)98 σ ijk d(cid:15) F = 0 . (4.37)Thus, we conclude that in Berry Fermi liquid, (4.31) and (4.32) still hold as in Berry Fermigas. Although we demonstrated in Section 4.3.2 that σ µνλ is not the full Hall conductivity,those remaining contributions are nevertheless always FS integrals. Therefore the full con-ductivity is always equal to a chemical potential independent part (as long as the FS changescontinuously) plus a FS integral.In Berry Fermi gas in d = 2, σ µνλo is topological [34, 32]. It would be interesting to studyif it is still topological in Berry Fermi liquid. In particular, it is unknown whether σ µνλo can have a jump when the FS develops new disconnected components, as in the example ofFigure 4.1. With Anomaly-Related Transport in d > For d >
2, when the Berry curvature is not an exact 2-form on the FS, the system hasanomaly-related transport effects.Let’s first review the effects in Berry Fermi gas. The expression (4.30) still holds, and westart from there. Now we have to take extra care when rewriting it via integration by parts.More precisely, a i cannot be continuously defined over the entire FS, so the expression (4.31)is not so useful. The alternative expression (4.32) promoted by [32] is still useful as longas we take into account the “Berry curvature defects” where ∂ [ kp b ij ] (cid:54) = 0 (e.g. monopoles in99 = 3): σ ij = σ ij o + 3 d − (cid:90) (cid:126)p θ ( (cid:15) F − E ( (cid:126)p )) b [ ij ( (cid:126)p ) ∂ k ] p p k = σ ij o − d − (cid:90) (cid:126)p θ ( (cid:15) F − E ( (cid:126)p )) ∂ [ kp b ij ] ( (cid:126)p ) p k + 3 d − (cid:90) (cid:126)p δ ( (cid:15) F − E ( (cid:126)p )) b [ ij ( (cid:126)p ) v k ] ( (cid:126)p ) p k + 6 d − (cid:90) (cid:126)p ∂ [ kp (cid:16) δ ( (cid:15) F − E ( (cid:126)p )) v i ( (cid:126)p ) a j ] ( (cid:126)p ) p k (cid:17) . (4.38)The boundary term in the last line is explained below (4.32); although a i is not continuouslydefined over the FS, it can be continuously defined around where the FS intersects theboundary of the first Brillouin zone. The defects lie along where ∂ [ kp b ij ] (cid:54) = 0, and they aregenerically d − (cid:15) F .In this spirit, we can combine the σ ij o term and the ∂ [ kp b ij ] term and call their sum σ ij a . Asimilar integration by parts can be carried out in the spatial components [67, 68]: σ ijk = σ ijko + 3 (cid:90) (cid:126)p θ ( (cid:15) F − E ( (cid:126)p )) b [ ij ( (cid:126)p ) ∂ k ] p E ( (cid:126)p )= σ ijko − (cid:90) (cid:126)p θ ( (cid:15) F − E ( (cid:126)p )) ∂ [ kp b ij ] ( (cid:126)p ) E ( (cid:126)p )+ 3 (cid:90) (cid:126)p δ ( (cid:15) F − E ( (cid:126)p )) b [ ij ( (cid:126)p ) v k ] ( (cid:126)p ) (cid:15) F (4.39)(the second and third term separately vanish in the absence of Berry curvature defect),whose ∂ [ kp b ij ] term is again independent of small continuous variation of (cid:15) F , and again wecan combine the σ ijko term and the ∂ [ kp b ij ] term and call their sum σ ijka . The simplest exampleof (4.38) with Berry curvature defect is the anomalous Hall effect in Weyl metals [86]; thesimplest example of (4.39) is the chiral magnetic effect [67, 69, 68].100or Berry Fermi liquid, (4.33) still holds when the Berry curvature is not exact on theFS, and we start from there. In Section 4.4.5 we will show (4.33) is equivalent to dσ ij d(cid:15) F = dd(cid:15) F d − (cid:90) (cid:126)p δ ( (cid:15) F − E ( (cid:126)p )) b [ ij ( (cid:126)p ) v k ] ( (cid:126)p ) p k + dd(cid:15) F d − (cid:90) (cid:126)p ∂ [ kp (cid:16) δ ( (cid:15) F − E ( (cid:126)p )) v i ( (cid:126)p ) a j ] ( (cid:126)p ) p k (cid:17) , (4.40) dσ ijk d(cid:15) F = dd(cid:15) F (cid:90) (cid:126)p δ ( (cid:15) F − E ( (cid:126)p )) b [ ij ( (cid:126)p ) v k ] ( (cid:126)p ) (cid:15) F (4.41)as long as there is no Berry curvature defect near the FS; they reduce to (4.36)(4.37) if theBerry curvature is exact on the FS. Thus, for Berry Fermi liquid we can write σ ijλ = σ ijλa + 3 (cid:90) (cid:126)p δ ( (cid:15) F − E ( (cid:126)p )) b [ ij ( (cid:126)p ) v k ] ( (cid:126)p ) P λk ( (cid:126)p )+ 6 (cid:90) (cid:126)p ∂ [ kp (cid:16) δ ( (cid:15) F − E ( (cid:126)p )) v i ( (cid:126)p ) a j ] ( (cid:126)p ) P λk (cid:17) , (4.42)where P k ≡ p k / ( d −
2) and P lk ≡ (cid:15) F δ lk (the second line vanishes if λ is spatial). Here σ µνλa is independent of (cid:15) F for generic values of (cid:15) F ; but it depends on (cid:15) F at special values of (cid:15) F where some Berry curvature defect is brought across the Fermi level. Moreover, as before, itis unknown whether σ µνλa can have a jump in situations like Figure 4.1. In (4.38) and (4.39)for Fermi gas, we are able to separate σ µνλa into σ µνλo plus a Berry curvature defect terminside the Fermi sea. Such separation is generally impossible for Fermi liquid.A final subtlety needs to be addressed. If we shift the definition of (cid:126)p by a constant vector,or shift the definitions of E ( (cid:126)p ) and (cid:15) F together by a constant value, no physics should change.However, the kinetic term in (4.42), due to its P λk factor, does not necessarily satisfy thisproperty in the presence of anomaly-related transport effects. There is no inconsistency here,as our starting point (4.33) does not have this problem. This just implies that, if we perform101uch shifts, σ µνλa also needs to be shifted such that σ µνλ remains unchanged. In Berry Fermigas, this can be verified explicitly in (4.38) and (4.39). In the next Section we will provide a diagrammatic derivation of the Berry Fermi liquidtheory. Here we summarize the identification of the various quantities appearing the BerryFermi liquid theory and objects in the resummed perturbation theory. • δf ( (cid:126)p ; x ), the quasiparticle distribution, is given by the FS singular part of the perturbedWigner function, as introduced in Section 4.4.4. • δJ µ ( x ), the induced current, is the quantum expectation (4.128). • E ( (cid:126)p ), the kinetic energy of a quasiparticle, is defined by the full propagator (4.44) nearthe FS. Its chemical potential dependence is given by (4.87). • u α ( (cid:126)p ), the spinor / Bloch state of an quasiparticle, needed to define the Berry curvature,is defined in (4.43) and above (4.45). Its chemical potential dependence is given below(4.88). • µ µν ( (cid:126)p ), the EM dipole moment of a quasiparticle, is given by (4.118), and discussed indetail in Section 4.4.4. • U ( (cid:126)p, (cid:126)k ), the contact interaction energy between two quasiparticles, is given by (4.87). • V µ ( (cid:126)p, (cid:126)k ), the gradient interaction energy between two quasiparticles, is given by (4.117). • C ( (cid:126)p, (cid:126)k ), the near-FS effective collision kernel between two quasiparticles (in d ≥ • σ µνλ , the Hall conductivity tensor, is defined in (4.142). Its chemical potential depen-dence is given by the Berry curvature around the FS, as shown in Section 4.4.5.102 ( (cid:126)p ) and U ( (cid:126)p, (cid:126)k ) are familiar parameters in Landau Fermi liquid theory, while the otherparameters µ µν ( (cid:126)p ) , V µ ( (cid:126)p, (cid:126)k ) , C ( (cid:126)p, (cid:126)k ) and σ µνλ are new. In this Section we will prove the kinetic formalism presented above by analyzing the quantumfield theory (QFT) to all orders in perturbation theory. Before we go into any details, wesummarize the idea behind our proof as the following. Our goal is to compute linear response,i.e. induced current δJ as a linear function of electromagnetic (EM) connection A , to firstorder in the external momentum q carried in A . The evaluation of δJ µ can be separated,technically, into two parts: • The first part is related to Cutkosky cut, and corresponds to the quasiparticle contri-butions to δJ , that is, the first line of (4.21). This part includes the excitation andcollision of quasiparticles, described by the Boltzmann equation (4.17). • The second part is the remaining contributions that are unrelated to Cutkosky cut, i.e.non-quasiparticle contributions. This second part gives rise to the Hall conductivitytensor in (4.21), whose chemical potential dependence is given by the Berry curvatureon the Fermi surface (FS).The idea is simple and clear. Now we devote into the technical details.First we state the assumptions about our QFT and its ground state and low energyspectrum.Our QFT consists of a multi-component fermionic field ψ α , charged under EM. Theindex α can be a spinor index if ψ is a Dirac spinor, or in general labels different bands.The fermionic field may interact via massive fields (generically denoted as φ ) or/and viafinite-ranged self-interactions. But we assume any field other than ψ to be EM neutral, andthe EM couplings to ψ only take place in the non-interacting terms of ψ in the Lagrangian,103ut not in any interacting terms. Thus, there can be EM couplings such as Aψ † ψ (including F ψ † ψ ) and AAψ † ψ , but there is no EM coupling like Aφψ † ψ or Aψ † ψψ † ψ .We assume the system is under chemical potential (cid:15) F for the fermionic field and atnegligible temperature (but high enough to avoid to Kohn-Luttinger instability [38]). Weassume the EM U (1) gauge invariance is not broken by the ground state. We assume there isno band degeneracy near the FS, and for simplicity, we assume the Fermi level crosses onlyone band of the spectrum of the fermionic field. We assume the only low energy excitationsare quasiparticles of this band. Thereby the system is said to be a Fermi liquid at low energy.We assume spacetime translational symmetry is not broken by anything except for thepresent external EM field. We do not assume any symmetry otherwise. In this Section weset (cid:126) to 1.The proof is organized as the following. We first discuss the properties of a single fullpropagator and a pair of full propagators. Then we introduce the irreducible 2-particle in-teraction vertex, and discuss its relation to the chemical potential dependence of the fullpropagator (from which the chemical potential dependence (4.27) of the kinetic energy fol-lows). Then we introduce the properties of the bare EM coupling vertex. Next we presentthe recursion relation satisfied by the full EM coupling vertex; we will also extract the impli-cations of the Ward-Takahashi identity. Having had all these preliminaries, we are ready toprove the main results. We first show the Boltzmann equation (4.19)(4.20) follows from therecursion relation satisfied by the full EM coupling vertex. Then we compute the quantumexpectation of the current and show it takes the form (4.22)(4.23) given in the kinetic theory.Finally we study the chemical potential dependence (4.33) of the Hall conductivity tensor.Along the way, we will also discuss the microscopic ingredients of the EM dipole moment.As a bonus, we obtain an alternative diagrammatic proof to the Coleman-Hill theorem [19]for QFTs restricted to our assumptions. 104 .4.1 Propagator Single Propagator pα β
The full propagator (all QFT quantities are time ordered unless otherwise specified) iG αβ ( p ) is a matrix in the components of the fermionic field. We let the energy p = − p = 0on the FS. The assumption of Fermi liquid amounts to the assumptions of the form of iG at small p . By general analytic properties of fermionic propagators [48, 1] (in particular,the property that G must be Hermitian at p = 0), and the specific requirement that atlow energy there is one species of stable quasiparticle, as p → iG αβ ( p ) (cid:39) iu α ( p ) u † β ( p ) χ u ( p ) + (cid:88) w iw α ( p ) w † β ( p ) χ w ( p ) . (4.43)The eigenvector u is the band that crosses the Fermi level, with singular eigenvalue whoseinverse is of the form χ u ( p ) = p − ξ ( (cid:126)p ) + i(cid:15) sgn ξ ( (cid:126)p ) Z ( (cid:126)p ) + · · · (4.44)where ξ ( (cid:126)p ) ≡ E ( (cid:126)p ) − (cid:15) F and ( · · · ) are terms of higher suppression in p ; the quasiparticlerenormalization factor Z ( (cid:126)p ) should be understood as the inverse of the coefficient of p in χ u (with rotational invariance, Z can depend on | (cid:126)p | ; without rotational invariance, it maydepend on all components of (cid:126)p ). The w ’s are all other eigenvectors, and their eigenvalues1 /χ w are regular and nearly real.For Landau Fermi liquid theory, (4.44) is enough, but for Berry Fermi liquid we needto know one order higher in p , i.e. work up to ( p ) order in the ( · · · ) terms. This will105e handled later in Section 4.4.2 and Section 4.4.6. One should also worry about whetherthe diagonalization (4.43) fails as we consider one order higher in p ; using the method inSection 4.4.6 one can easily see this problem occurs only at two orders higher in p , so inthis paper we do not need to worry about this.In Section 4.3.4, we used the quasiparticle spinor / Bloch state u ( (cid:126)p ); it refers to u ( p )with p on-shell and near the FS, i.e. p = ξ ( (cid:126)p ) →
0. The p derivatives of u ( p ) and u ( (cid:126)p ) arerelated by ( ∂ µp + v µ ∂ p ) u ( p ) (cid:12)(cid:12)(cid:12) p on − shell = ∂ µp u ( (cid:126)p ) . (4.45)Here ∂ p ≡ ∂/∂p = − ∂/∂p = ∂ p = − ∂ p while ∂ ip ≡ ∂/∂p i = ∂/∂p i = ∂ ip = ∂ p i , and recallthat v ≡ v i ≡ ∂ ip E as introduced in Section 4.3.The following identity, which follows from the product rule of derivative, is useful in thispaper: ∂ µp ( G − ) αβ u β = − ( G − ) αβ ∂ µp u β + ∂ µp χ u u α + χ u ∂ µp u α (4.46)for p near the FS; note that − Z∂ µp χ u = − Z u † α ∂ µp ( G − ) αβ u β = v µ + (terms vanish on the FS) . (4.47)There is a similar identity for u † α ∂ µp ( G − ) αβ .Now we look at the momentum derivative of the full propagator: ∂ νp iG αβ ( p ) = iG αα (cid:48) ∂ νp ( iG − ) α (cid:48) β (cid:48) iG β (cid:48) β − iZu α u † β iπδ ( p − ξ ) ∂ νp sgn ξ = iG αα (cid:48) ∂ νp ( iG − ) α (cid:48) β (cid:48) iG β (cid:48) β − iZu α u † β iδ F S δ νi v i , (4.48)106here δ F S ( p ) ≡ πδ ( p ) δ ( ξ ( (cid:126)p )) . (4.49)The presence of the second term in (4.48) is because as p i varies across the FS, the p polein 1 /χ u moves across the real axis. This abrupt change is not captured by the first term.The expression of the second term can be obtained by principle function decomposition( x ± i(cid:15) ) − = P x − ∓ iπδ ( x ).To avoid having too many fermion component indices in equations in this thesis, we in-troduce two notations: single fermion linear space and double fermion linear space. Consider G αβ . In single fermion linear space, αβ are viewed as two indices, so G is viewed as a matrixin single fermion linear space. In double fermion linear space, αβ together is viewed as oneindex, so G is viewed as a vector in the the double fermion linear space. In our proof, only afew index contractions are to be understood in single fermion linear space, most are under-stood in double fermion linear space. To distinguish them, we will enclose objects contractedin single fermion linear space by curly brackets { } , while do not enclose objects contractedin double fermion linear space by anything. For example, according to (4.47), v µ can beexpressed as − v µ Z = u † α ∂ µp ( G − ) αβ u β (cid:12)(cid:12)(cid:12) p on F S = (cid:110) u † ∂ µp G − u (cid:111)(cid:12)(cid:12)(cid:12) p on F S = ( uu † ) T ∂ µp G − (cid:12)(cid:12)(cid:12) p on F S (4.50)in explicit index notation, single fermion notation, and double fermion notation respectively.We will introduce more about double fermion notation in Section 4.4.2.In the double fermion notation introduced above, (4.48) can be expressed compactly as ∂ νp iG = i ∆ ∂ νp iG − + ( Zuu † ) δ F S δ νi v i , (4.51)107here i (∆ ) α γδ, β ( p ) ≡ iG αβ ( p ) iG γδ ( p ) (4.52)is a matrix in double fermion notation. Since the momentum argument in both iG ’s is thesame, i ∆ has a double pole in p when all of its four indices are in the u band. Double Propagator
An important step towards the QFT foundation of the Landau Fermi liquid theory is theobservation that, the semiclassical notion of “deformation δf of the FS” originates from thepole structure of the double propagator iG ( p + q/ iG ( p − q/ q small and p near theFS [43, 1]. Now we make a similar analysis, but with non-trivial u α ( p ), and work to firstorder in q . p − q/ p + q/ αδ βγ Consider the product of two full fermionic propagators, as drawn above, with arbitrary p and small q ; more exactly, q (cid:28) p F where p F is the size scale of the FS. To first order in q , we express the product in the form iG αβ ( p + q/ iG γδ ( p − q/ (cid:0) i ∆ ( p ) + i ∆ (cid:48) ( p ; q ) + i ∆ r ( p ; q ) + i ∆ s ( p ; q ) + i ∆ (cid:48) ( p ; q ) (cid:1) α γδ, β − D α γδ, β ( p ; q ) , (4.53)where the subscripts 0 or 1 denote the order in q . Here ∆ , ∆ r , ∆ s are regular as q →
0. Inparticular, ∆ has been introduced in (4.51), and ∆ r and ∆ s follow from the two terms of1084.48) when expanding G ( p ± q/
2) in q :(∆ r ) α γδ, β ( p ; q ) ≡ − i q λ (cid:18)(cid:110) G ∂ νp G − G (cid:111) αβ G γδ − G αβ (cid:110) G ∂ νp G − G (cid:111) γδ (cid:19) , (4.54)(∆ s ) α γδ, β ( p ; q ) ≡ q λ δ λk v k Zδ F S (cid:16) u α u † β G γδ − G αβ u γ u † δ (cid:17) . (4.55)Clearly both ∆ r and ∆ s vanish when all indices are projected onto one band.When there is no FS, the expansion of the double propagator as i ∆ + i ∆ r + i ∆ s islegitimate. When FS is present, such naive expansion in q misses contributions that arerelated to the pole structure difference across the FS. These extra contributions are denotedby ∆ (cid:48) and ∆ (cid:48) , which are singular as q →
0. Explicitly, they are given by∆ (cid:48) = ∆ (cid:48) ( uu † )( uu † ) T , (4.56)∆ (cid:48) = ∆ (cid:48) iq µ A µ , (4.57)where we defined ∆ (cid:48) ( p ; q ) ≡ Z ( (cid:126)p ) δ F S ( p ) v i ( (cid:126)p ) q i v µ ( (cid:126)p ) q µ − i(cid:15) sgn q , (4.58)which is familiar from Landau Fermi liquid theory [43, 1] when u α is one-component (i.e. u = 1 trivially). Also, we introduced the abbreviation( A µ ) α γδ, β ≡ − i (cid:16) ∂ µp u α u † δ − u α ∂ µp u † δ (cid:17) ( u γ u † β ) − − i u α u † δ ) (cid:16) ∂ µp u γ u † β − u γ ∂ µp u † β (cid:17) . (4.59)Below we present the derivation for ∆ (cid:48) and ∆ (cid:48) .109et’s focus on the double u -band term in the double propagator: i (cid:16) Zu α u † β (cid:17) ( p + q/ p + q / − ξ ( (cid:126)p + (cid:126)q/
2) + i(cid:15) sgn ξ ( (cid:126)p + (cid:126)q/ i (cid:16) Zu γ u † δ (cid:17) ( p − q/ p − q / − ξ ( (cid:126)p − (cid:126)q/
2) + i(cid:15) sgn ξ ( (cid:126)p − (cid:126)q/ . (4.60)What are missing in the naive expansion over q are the contributions when the two i(cid:15) prescriptions in the denominators take opposite signs. To extract these missing pieces, weperform principle function decomposition ( x ± i(cid:15) ) − = P x − ∓ iπδ ( x ) and keep those termswhich are non-vanishing only when sgn ξ ( (cid:126)p ± (cid:126)q/
2) are opposite. Such terms are (cid:20) iπ sgn ξ ( (cid:126)p − (cid:126)q/ − sgn ξ ( (cid:126)p + (cid:126)q/ − q + ξ ( (cid:126)p + (cid:126)q/ − ξ ( (cid:126)p − (cid:126)q/ × δ ( p + q / − ξ ( (cid:126)p + (cid:126)q/ δ ( p − q / − ξ ( (cid:126)p − (cid:126)q/ − ( iπ ) θ ( − sgn ξ ( (cid:126)p − (cid:126)q/ ξ ( (cid:126)p + (cid:126)q/ × δ ( p + q / − ξ ( (cid:126)p + (cid:126)q/ δ ( p − q / − ξ ( (cid:126)p − (cid:126)q/ (cid:21) i (cid:16) Zu α u † β (cid:17) ( p + q/ i (cid:16) Zu γ u † δ (cid:17) ( p − q/ . (4.61)Expanding the generalized functions in the square bracket in q , we have (cid:34) iπ − δ ( ξ ( (cid:126)p )) v i q i v µ ( (cid:126)p ) q µ δ ( p − ξ ( (cid:126)p )) − ( iπ ) | v i ( (cid:126)p ) q i | δ ( ξ ( (cid:126)p )) δ ( p − ξ ( (cid:126)p )) δ ( v µ ( (cid:126)p ) q µ ) + O ( q ) (cid:35) i (cid:16) Zu α u † β (cid:17) ( p + q/ i (cid:16) Zu γ u † δ (cid:17) ( p − q/ . (4.62)Now we recognize the square bracket is nothing but − i ∆ (cid:48) /Z expressed in principle functiondecomposition. Finally we expand the two ( Zuu † )’s to zeroth and first order in q , we obtainthe expression for i ∆ (cid:48) + i ∆ (cid:48) presented above.Developing along this line of thinking, one is led to the formalism of Cutkosky cut [22],which we will discuss in Section 4.4.6. In particular, see Eq. (4.162) for the derivation of ∆ (cid:48) D in (4.53)? The step (4.60) is not quite right, for it completely ignored the( · · · ) terms in (4.44). While it is legitimate to do so at leading order in q (in Landau’s theory),at first order in q there are missed contributions, which we call D . We will postpone itsdiscussion to Section 4.4.2, when we discuss the quasiparticle decay term along with otherquasiparticle collision terms. q -2PI Interaction Vertex Let i (cid:101) V α γδ, β ( p, k ; q ) be the full q -2PI (defined below) interaction vertex, with two incomingfermions of momentum and index ( p − q/ , δ ) and ( k + q/ , β ), and two outgoing fermionswith momentum and index ( k − q/ , γ ) and ( p + q/ , α ), as drawn below. i (cid:101) V p − q/ k + q/ k − q/ p + q/ αδ βγ (In this thesis, external propagators without a solid dot at the end are always stripped off.)Here q -2PI means that i (cid:101) V is a sum of connected, 1PI (with respect to the fermion only)interaction diagrams, such that in each diagram there does not exist two internal fermionpropagators that are dictated by momentum conservation to have momenta differing by q .Equivalently, for each diagram, one cannot find two internal fermion propagators cuttingwhich will disconnect the diagram into two parts, such that the external lines of ( p − q/ , δ )and ( p + q/ , α ) are on one part, while the external lines of ( k + q/ , β ) and ( k − q/ , γ ) areon the other part. For example, in the four diagrams below (fermionic propagators alwaysmean full propagators), the two on the left are q -2PI, while the two on the right are not.111ore about i (cid:101) V is said in Section 4.4.7.For Fermi liquid, the limit (cid:101) V ( p, k ; 0) is generally regular and analytic in q . Keeping zerothand first order in q , we write (cid:101) V ( p, k ; q ) = (cid:101) V ( p, k ) + (cid:101) V ( p, k ; q ).The q -2PI interaction vertex is the building block of full interaction vertex iV : The latteris a geometric series given by the recursion relation iV = i (cid:101) V + i (cid:101) V iV (4.63)The full interaction vertex is singular in the q → (cid:48) in thedouble propagators, as well as collision factors to be discussed in Section 4.4.2.Before we proceed, we say a bit more about the double fermion notation. Consider anobject, perhaps with spacetime indices, (cid:16) X α γδ, β (cid:17) µνρ... ( p, k ; q ). This object is a matrix inthe double fermion linear space. We now introduce its transpose: (cid:16) X α γδ, β (cid:17) µνρ... ( p, k ; q ) = (cid:16)(cid:16) X γ αβ, δ (cid:17) µνρ... ( k, p ; − q ) (cid:17) T . (4.64)Diagrammatically, the transpose in double fermion linear space corresponds to “turningthe diagram 180 degrees”; note that the spacetime indices are unaffected by the transpose.Finally, we introduce the convention that, for objects like X which involve two momenta p and k , the contraction with another object implies a momentum integral, for example( X µνρ... Y ) αδ ( p ; q ) ≡ (cid:90) k (cid:16) X α γδ, β (cid:17) µνρ... ( p, k ; q ) Y βγ ( k ; q ) (4.65)where (cid:82) k ≡ (cid:82) d d +1 k/ (2 π ) d +1 . 112ow, by definition of i (cid:101) V , we see it satisfies i (cid:101) V = ( i (cid:101) V ) T , and similarly for all the ∆’s and D . We will need these transpose properties when we derive the current in Section 4.4.5. Full Interaction Vertex
Using the double fermion notation introduced above, we expand the recursion relation (4.63)to zeroth and first order in q . At zeroth order, iV = i (cid:101) V + i (cid:101) V (cid:0) i ∆ + i ∆ (cid:48) (cid:1) iV = i ¯ V + i ¯ V i ∆ (cid:48) iV (4.66)where we defined the geometric series i ¯ V via the recursion relation i ¯ V = i (cid:101) V + i (cid:101) V i ∆ i ¯ V , (4.67) i ¯ V can be understood as iV in the limit q → q i /q → (cid:48) vanishes in the q i /q → U [43, 1],as we will see later.At first order, iV = i (cid:101) V + i (cid:101) V (cid:0) i ∆ + i ∆ (cid:48) (cid:1) iV + i (cid:101) V (cid:0) i ∆ r + i ∆ s + i ∆ (cid:48) (cid:1) iV + i (cid:101) V ( − C ) iV + i (cid:101) V (cid:0) i ∆ + i ∆ (cid:48) (cid:1) iV . (4.68)Here C α γδ, β ( p, k ; q ) is the quasiparticle collision term. We explain this term below. Quasiparticle Decay and Collision Vertices
Let us emphasize our comment in Section 4.3.1 again: Collision is a single band ( u band)effect that is “uninteresting”, as our main interest is multi-component effects such as Berrycurvature. More particularly, in Section 4.4.6 it is shown that collision has no contribution to113he antisymmetric part of the current-current correlation (which include interesting physicssuch as anomalous Hall effect and chiral magnetic effect). Here we are including collisionjust for completeness.The quasiparticle collision term C α γδ, β ( p, k ; q ) is defined as − C ( p, k ; q ) ≡ − D ( p ; q ) (2 π ) d +1 δ d +1 ( p − k ) − C ph ( p, k ; q ) − C pp ( p, k ; q ) . (4.69)The decay term D in C is from (4.53) but left unexplained there. Where do D , C ph and C pp come from? Recall that in (4.53) we could not naively expand the two propagators in q individually; there are terms non-analytic in q to be carefully taken care of. Similarly, here inthe recursion relation for iV , we cannot naively expand the i (cid:101) V ’s and the double propagatorsindividually. The non-analytic contributions that are missed from such naive expansion are D , C ph and C ph .Formally, the three terms in the definition of − C ( p, k ; q ) correspond to the followingthree pairs of Cutkosky-cut sub-diagrams: p − q/ p + q/ p − q/ k + lp + lk − q/ p − q/ p + q/ p + lk + q/ p + q/ k + lp − q/ k + q/ k − q/ p + lp + q/ k + l p − q/ k + q/ k − q/ p + lp + q/ k + l − q/ k + q/ k − l p + lk − q/ p + q/ p − q/ k + q/ k − l p + lk − q/ p + q/ iV . The two cut sub-diagrams for − D involve a quasiparticle decaying into two quasiparticles and a quasihole (or a hole decayinginto two holes and a particle). The cut sub-diagrams for − C ph involve the exchange of anon-shell particle-hole pair, while the cut sub-diagrams for − C pp involve the exchange of anon-shell particle-particle (or hole-hole) pair. The computation of cut diagrams is explainedin Section 4.4.6; there, we will also argue that these three pairs are the only cut sub-diagramsthat contribute at order q .For d ≥ C should scale as ∼ ( q ) . This can be seen by countingthe availability of collision channels constraint by energy and momentum conservation in thepresence of FS [43, 1]. As we show in Section 4.4.6, in d ≥ − D ( p ; q ) = − γ ( (cid:126)p ) δ F S ( p ) Z ( (cid:126)p ) ( uu † )( p ) ( uu † ) T ( p ) | q | ( q ) ( v µ ( (cid:126)p ) q µ ) , (4.70) − C ph ( p, k ; q ) = 2 λ ph ( (cid:126)p, (cid:126)k ) δ F S ( p ) δ F S ( k ) ( Z uu † )( p )( Z uu † ) T ( k ) | q | ( q ) ( v ( (cid:126)p ) µ q µ )( v ( (cid:126)k ) µ q µ ) , (4.71) − C pp ( p, k ; q ) = − λ pp ( (cid:126)p, (cid:126)k ) δ F S ( p ) δ F S ( k ) ( Z uu † )( p )( Z uu † ) T ( k ) | q | ( q ) ( v ( (cid:126)p ) µ q µ )( v ( (cid:126)k ) µ q µ ) . (4.72)(We have omitted the i(cid:15) prescription accompanying v µ q µ in the denominator; in time-orderedcorrelation its sign should be − sgn( q ), i.e. sgn( q ), as usual.) In particular, the parameter115 ( (cid:126)p ), defined near the FS, is positive and regular, and is related to the imaginary part of thefermion self-energy via χ u ( p ) = p − ξ ( (cid:126)p ) + i(cid:15) sgn ξ ( (cid:126)p ) Z ( (cid:126)p ) + i γ ( (cid:126)p ) p | p | + (higher orders in p ) (4.73)as explained in Section 4.4.6. The other two parameters, λ ph ( (cid:126)p, (cid:126)k ) and λ pp ( (cid:126)p, (cid:126)k ), defined nearthe FS, are both positive and regular, and symmetric under exchange of (cid:126)p and (cid:126)k . Moreover,from the computation in Section 4.4.6, we have the relation (cid:90) k − v µ ( (cid:126)k ) q µ Z ( (cid:126)k ) ( C phR ) ( p, k ; q ) = 2 v µ ( (cid:126)p ) q µ Z ( (cid:126)p ) ( D R ) ( p ; q ) = 2 (cid:90) k v µ ( (cid:126)k ) q µ Z ( (cid:126)k ) ( C ppR ) ( p, k ; q ) . (4.74)In terms of the parameters γ , λ ph and λ pp , this reads (cid:90) k δ F S ( k ) Z ( (cid:126)k ) λ ph ( (cid:126)p, (cid:126)k ) = γ ( (cid:126)p ) = (cid:90) k δ F S ( k ) Z ( (cid:126)k ) λ pp ( (cid:126)p, (cid:126)k ) . (4.75)In particular, this relation implies (cid:90) k C ( p, k ; q ) uu † ( k ) Z ( (cid:126)k ) v µ ( (cid:126)k ) q µ = 0 . (4.76)This is related to the Ward-Takahashi identity, as we will see in Section 4.4.3. More phys-ically, it is related to the fact that collisions do not change the total number of fermionicexcitations, as discussed above (4.18).Piecing up the above, the collision factor C can be written as − C ( p, k ; q ) = −| q | ( Zuu † )( p ) q v µ ( (cid:126)p ) q µ δ F S ( p ) C ( (cid:126)p, (cid:126)k ) δ F S ( k ) ( Zuu † ) T ( k ) q v µ ( (cid:126)k ) q µ (4.77)where C ( (cid:126)p, (cid:126)k ), symmetric in (cid:126)p, (cid:126)k , is microscopically defined when both (cid:126)p and (cid:126)k are on the116S: δ F S ( p ) C ( (cid:126)p, (cid:126)k ) δ F S ( k ) ≡ δ F S ( p ) Z ( (cid:126)p ) γ ( (cid:126)p ) (2 π ) d +1 δ d +1 ( p − k )+ δ F S ( p ) Z ( (cid:126)p ) (cid:16) − λ ph ( (cid:126)p, (cid:126)k ) + λ pp ( (cid:126)p, (cid:126)k ) (cid:17) δ F S ( k ) Z ( (cid:126)k )(4.78)(since C is defined only when (cid:126)p, (cid:126)k are on the FS, we can “remove” δ F S ( p ) δ F S ( k ) from thefirst term on the right-hand-side unambiguously) and it satisfies (cid:90) k C ( (cid:126)p, (cid:126)k ) δ F S ( k ) = 0 . (4.79)This is the collision effect C appearing in the kinetic theory in Section 4.3.1.For d = 2 spatial dimensions, D , C ph and C pp cannot be parametrized in any simpleform, as discussed in Section 4.4.6. Moreover, they are not order q ; they also involve order q ln q terms, which are less suppressed than order q . The failure of the parametrizationraises problem in the computation for e.g. the longitudinal current in d = 2. But as shownin Section 4.4.6, collisions do not contribute to the anomalous Hall current and the chiralmagnetic current, so our main discussion about them is not undermined. Also, despite thatthere is no simple parametrization in d = 2, (4.76) must still hold as it is dictated by theWard-Takahashi identity. Chemical Potential dependence of Propagator
Having defined the q -2PI vertex i (cid:101) V , we are ready to find the chemical potential dependenceof the propagator. The procedure below is analogous to [59], but allowing multi-component u α .We define the notation ∂ F ≡ ∂/∂(cid:15) F − ∂ p . (4.80)117he subtraction of ∂ p is because our p is defined such that p = 0 at the FS, and we want ∂ F to extract the effects of physically shifting the FS; for example, ∂ F ( p − ( E − (cid:15) F )) = ∂ F E = ∂E/∂(cid:15) F . The FS dependence of the propagator can be derived in analogy to (4.51),but with ∂ νp replaced with ∂ F : ∂ F iG = i ∆ ∂ F iG − − ( Zuu † ) (cid:16) − ∂ F E (cid:17) δ F S (4.81)where the expression of ∂ F G − , and hence ∂ F E , are to be derived below. The second termin (4.81) relied on the assumption that when the chemical potential changes, the FS changescontinuously, so (4.81) (and hence all discussions below) does not apply to discrete values of (cid:15) F around which the FS develops new disconnected components, as in the example Figure4.1.Let G bare be the bare fermion propagator and G − bare is its inverse ignoring the i(cid:15) polestructure. In the kinetic energy sector of the bare Lagrangian, (cid:15) F always appears as i∂ x + (cid:15) F ,therefore ∂ F G − bare = 0. Because G − = G − bare − Σ where Σ is the self-energy, we get ∂ F G − = − ∂ F Σ. Diagrammatically, one can see in the presence of interaction, when thepropagator is varied, the self-energy varies as − δi Σ = i (cid:101) V δiG . Therefore ∂ F iG − = − ∂ F i Σ = i (cid:101) V ∂ F iG. (4.82)Substituting (4.81) into the above yields ∂ F iG − = − ∂ F i Σ = − i ¯ V ( Zuu † ) (cid:16) − ∂ F E (cid:17) δ F S . (4.83)Recall that ¯ V is defined by the recursion relation i ¯ V = i (cid:101) V + i (cid:101) V i ∆ i ¯ V .Let’s focus on the change of the u -band eigenvalue of G − , given by ∂ F χ u = ( uu † ) T ∂ F G − . (4.84)118ow take p near the FS. We can expand this in powers of p . In particular, by comparisonwith (4.44), we shall identify the coefficients at zeroth and first order in p as ∂ F χ u = (cid:32) − ∂ F EZ − ( E − (cid:15) F ) ∂ F Z (cid:33) + p ∂ F Z + O (( p ) ) . (4.85)To make this parallel with (4.47), we shall define v F ≡ ∂ F E . Notice v F has nothing todo with “Fermi velocity” (in this thesis there is no notion of Fermi velocity, as we did notassume rotational symmetry).Now, for p near the FS, we can read-off: ∂ F Z ( (cid:126)p ) = ∂ p (cid:16) − Z ( uu † ) T ∂ F G − (cid:17)(cid:12)(cid:12)(cid:12) p on − shell = Z ( (cid:126)p ) ∂ p (cid:90) k ( Zuu † ) T ( p ) ¯ V ( p, k )( Zuu † )( k ) (cid:16) − ∂ F E ( (cid:126)k ) (cid:17) δ F S ( k ) (cid:12)(cid:12)(cid:12)(cid:12) p on − shell (4.86)and ∂ F E ( (cid:126)p ) = − ( Zuu † ) T ∂ F G − (cid:12)(cid:12)(cid:12) p on − shell = (cid:90) k U ( (cid:126)p, (cid:126)k ) (cid:16) − ∂ F E ( (cid:126)k ) (cid:17) δ F S ( k ) , U ( (cid:126)p, (cid:126)k ) ≡ ( Zuu † ) T ( p ) ¯ V ( p, k ) ( Zuu † )( k ) (cid:12)(cid:12)(cid:12) p,k on − shell . (4.87)Thus we have proven (4.27). At the same time we found the microscopic expression for U ,which is the same as that in [1] except here we need to contract with the four u ’s. U is evenunder the exchange of (cid:126)p, (cid:126)k , because ¯ V = ( ¯ V ) T .We can also find the change of the eigenvector u for p near the FS. Up to an unimportantcomplex phase, we have ∂ F u α ( p ) = (cid:88) w w α − χ w (cid:110) w † ∂ F G − u (cid:111) , ∂ F u † α ( p ) = (cid:88) w (cid:110) u † ∂ F G − w (cid:111) w † α − χ w , (4.88)119here ∂ F G − is given by (4.83). ∂ F u is related to ∂ F u in a way similar to (4.45), with v µ replaced by v F = ∂ F E . It appears in the kinetic formalism through (4.33), which we willprove in Section 4.4.5. Bare Electromagnetic Vertices
By our assumptions about the QFT, the only bare EM coupling vertices take the form A n ψ † ψ for integer n ≥
1. Due to the smallness of A , we only need to concern about n = 1 , i (cid:101) Γ αδ ) µ ( p ; q ) the bare Aψ † ψ EM vertex with incoming fermionwith momentum p − q/ δ , and outgoing fermion with momentum p + q/ α . We denote by ( i (cid:101) Ξ αδ ) µν ( p ; q, q (cid:48) ) the bare AAψ † ψ vertex with incoming fermion withmomentum p − ( q + q (cid:48) ) / δ , outgoing fermion with momentum p + ( q + q (cid:48) ) / α , and photons with incoming momenta q and q (cid:48) . qαp + q/ p − q/ δ µ i (cid:101) Γ q (cid:48) qαp + ( q + q (cid:48) ) / p − ( q + q (cid:48) ) / δ νµ i (cid:101) Ξ Since they are bare quantities, both of them are regular and analytic as q → q (cid:48) → (cid:101) Ξ). To first order in q (and q (cid:48) together for (cid:101) Ξ) we separate them as (cid:101)
Γ = (cid:101) Γ + (cid:101) Γ and (cid:101) Ξ = (cid:101) Ξ + (cid:101) Ξ .The EM U (1) gauge invariance of the bare Lagrangian requires q µ i (cid:101) Γ µ ( p ) = iG − bare ( p − q/ − iG − bare ( p + q/ , (4.89)120 µ i (cid:101) Ξ µν ( p ; q, q (cid:48) ) = i (cid:101) Γ ν ( p − q/ q (cid:48) ) − i (cid:101) Γ ν ( p + q/ q (cid:48) ) ,q (cid:48) ν i (cid:101) Ξ µν ( p ; q, q (cid:48) ) = i (cid:101) Γ µ ( p − q (cid:48) / q ) − i (cid:101) Γ µ ( p + q (cid:48) / q ) . (4.90)These lead to i (cid:101) Γ µ = − i∂ µp G − bare , i (cid:101) Γ µ = i ˆ µ µν iq ν , (4.91) i (cid:101) Ξ µν ( p ) = − ∂ νp i (cid:101) Γ µ ( p ) = − ∂ µp i (cid:101) Γ ν ( p ) , i (cid:101) Ξ µν ( p ; q, q (cid:48) ) = − ∂ νp i (cid:101) Γ µ ( p ; q ) − ∂ µp i (cid:101) Γ ν ( p ; q (cid:48) ) . (4.92)Here (ˆ µ αδ ) µν ( p ) is the bare EM dipole matrix (such as that in the Pauli term) that isantisymmetric in µν and Hermitian in αδ . Full Electromagnetic Vertex i Γ = + i (cid:101) V i
ΓDiagrammatically, one can see the full Aψ † ψ EM vertex is given by the recursion relation i Γ µ = i (cid:101) Γ µ + iV (cid:8) iG i (cid:101) Γ µ iG (cid:9) = i (cid:101) Γ µ + i (cid:101) V { iG i Γ µ iG } (4.93)as drawn above. The recursion relation at zeroth order in q is i Γ ν = i (cid:101) Γ ν + i (cid:101) V (cid:0) i ∆ + i ∆ (cid:48) (cid:1) i Γ ν = i ¯Γ ν + i ¯ V i ∆ (cid:48) i Γ ν , (4.94)121here we defined i ¯Γ ν ≡ (cid:0) + i ¯ V i ∆ (cid:1) i (cid:101) Γ ν . (4.95)The purpose of the second equality of (4.94) is that, now the effect of ∆ (cid:48) , to be related tothe deformation of the FS later, is singled out, and i ¯Γ is independent of q .The recursion relation at first order in q is i Γ ν − i ¯ V i ∆ (cid:48) i Γ ν = (cid:0) + i ¯ V i ∆ (cid:1) i (cid:101) Γ ν + i ¯ V (cid:0) i ∆ (cid:48) + i ∆ r + i ∆ s − C (cid:1) i Γ ν + (cid:0) + i ¯ V i ∆ (cid:1) i (cid:101) V (cid:0) i ∆ + i ∆ (cid:48) (cid:1) i Γ ν . (4.96)Of course the recursion (4.96) can be expressed in many equivalent ways; we have chosen toexpress it such that on the right-hand-side there is no ∆ (cid:48) (including those hidden in Γ ν ) tothe left of any quantity of order q . For the purpose of deriving the Boltzmann equation, wewant to further rewrite (4.96) so that each Γ ν has ∆ (cid:48) or ∆ (cid:48) or C on its immediate left. Wecan achieve so by substituting (4.94) for those Γ ’s in (4.96) whose immediate left are notyet ∆ (cid:48) or ∆ (cid:48) or C . The result is i Γ ν = i ¯Γ ν + i ¯ V i ∆ (cid:48) i Γ ν + i ¯ V i ∆ (cid:48) i Γ ν + + i ¯ V ( − C ) i Γ ν + i ¯ V i ∆ (cid:48) i Γ ν , (4.97)where i ¯ V ≡ (cid:0) + i ¯ V i ∆ (cid:1) i (cid:101) V (cid:0) + i ¯ V i ∆ (cid:1) T + i ¯ V ( i ∆ r + i ∆ s ) i ¯ V , (4.98) i ¯Γ ν ≡ (cid:0) + i ¯ V i ∆ (cid:1) i (cid:101) Γ ν + i ¯ V i ∆ i (cid:101) Γ ν + i ¯ V ( i ∆ r + i ∆ s ) i (cid:101) Γ ν = (cid:0) + i ¯ V i ∆ (cid:1) (cid:0) i (cid:101) Γ ν + i (cid:101) V i ∆ i ¯Γ ν (cid:1) + i ¯ V ( i ∆ r + i ∆ s ) i ¯Γ ν (4.99)122re partial sums at first order in q that involve no factor of ∆ (cid:48) or C . By construction, i ¯ V and i ¯Γ are analytic in q as q →
0. Thus, in (4.97) we singled out the ∆ (cid:48) , ∆ (cid:48) and C effects,which are to be related to the quasiparticle excitations.We do not need to consider the “full AAψ † ψ vertex”. In fact, the only place (cid:101) Ξ showsup in our proof is the expression of the current, in which we will immediately use (4.92) toeliminate (cid:101) Ξ. Ward-Takahashi Identity
Later in Section 4.4.4 we will show how the Boltzmann equation (4.19)(4.20) follow exactlyfrom (4.94) and (4.97). Before that, we need to answer a question: In QFT, it is the matrixΓ ν governing the coupling to A , while in the kinetic formalism, it is the velocity v ν (plusorder q couplings such as EM dipole). How to relate Γ ν to v ν ? The answer is the generalizedWard-Takahashi identity [73]: { iG ( p + q/ i Γ ν ( p ; q ) iG ( p − q/ } q ν = iG ( p − q/ − iG ( p + q/ . (4.100)We want to extract its implications at leading and sub-leading orders in q in the presence ofFS.At leading order in q , the Ward-Takahashi identity reads (cid:0) i ∆ + i ∆ (cid:48) (cid:1) Γ ν q ν = − i ∆ ∂ νp G − q ν + i ( Zuu † ) δ F S δ νi v i q ν (4.101)using (4.51). This is equivalent toΓ ν q ν = − ∂ νp G − q ν . (4.102)One can easily verify the equivalence by contracting i ∆ + i ∆ (cid:48) on the left of (4.102), with123he aid of (4.50), to recover (4.101). We will see the result (4.102) is related to the gaugeinvariance of δf .We can extract more detailed information from (4.102) – we gain an identity similar tothe original Ward identity [79], but in the presence of FS. For this purpose let’s treat | (cid:126)q | /q as an independent small expansion parameter, and expand (4.102) to its zeroth and firstorder. This gives us two equations, about ¯Γ and ¯Γ i respectively. Solving them with thehelp of (4.94) and the explicit expression for ∆ (cid:48) , we find the Ward identity in the presenceof FS: ¯Γ ν = − ∂ νp G − + ¯ V ( Zuu † ) δ F S δ νi v i . (4.103)We can equivalently express (4.103) as i ∆ i ¯Γ ν = − ∂ νp iG + (cid:0) + i ∆ i ¯ V (cid:1) ( Zuu † ) δ F S δ νi v i (4.104)using (4.51). As we will see later, this result will help us relate the EM vertex in QFT tothe velocity in the kinetic formalism.At sub-leading order in q , the Ward-Takahashi identity reads (cid:0) ∆ + ∆ (cid:48) (cid:1) Γ ν q ν + (cid:0) ∆ r + ∆ s + ∆ (cid:48) + iC (cid:1) Γ ν q ν = 0 . (4.105)The C Γ ν q ν term vanishes on its own, due to (4.102), (4.50) and (4.76). For the remainingterms, we can conclude∆ Γ ν q ν + ∆ r Γ ν q ν = 0 = ∆ (cid:48) Γ ν q ν + (cid:0) ∆ s + ∆ (cid:48) (cid:1) Γ ν q ν . (4.106)The two sides must vanish separately because the right-hand-side involves the singular factor δ ( ξ ( (cid:126)p )), while the left-hand-side does not. Later we will see (4.106) is related to the gauge124nvariance of δf . Having extracted (4.103) from the Ward-Takahashi identity, we are ready to prove the Boltz-mann equation (4.19)(4.20) from the recursion relations (4.94) and (4.97). The distributionof excitations δf will be defined in terms of QFT quantities, and as one should expect,our definition agrees with the Wigner function approach. Our derivation also provides themicroscopic expressions for V ν and µ µν . Zeroth Order in q When an external EM field of small q is present, the propagation of a quasiparticle is nolonger translationally invariant – the two-point propagator now depends on both p and q .More precisely, iG ( p ) −→ iG ( p ) + { iG ( p + q/ i Γ ν ( p ; q ) iG ( p − q/ } A ν ( q ) (4.107)at linear response. We will focus on the shifted piece.At zeroth order in q , using the identity (4.103), we can express the recursion relation(4.94) as i Γ ν A ν = − i∂ νp G − A ν − i ¯ V ( Zuu † ) δW , (4.108)where δW is a quantity restricted on the FS: δW ≡ ( uu † ) T Z ∆ (cid:48) Γ ν A ν − δ F S v i A i . (4.109)We can see δW is gauge invariant from (4.102) and (4.50). Substituting (4.108) into (4.109),125e find the recursion relation for δW : δW = δ F S v i v µ q µ − i(cid:15) sgn( q ) ( − iF i − q i U δW ) . (4.110)This proves the Boltzmann equation (4.19) at zeroth order in q , if we make the identification2 πδ ( p − ( E ( (cid:126)p ) − (cid:15) F )) δf ( (cid:126)p ; q ) ≡ δW ( p ; q ) (4.111)to factor out the on-shell condition. Note that the computation above is time-ordered,therefore the i(cid:15) prescription depends on sgn( q ); when computing the physical quasiparticledistribution in kinetic theory, retarded boundary condition should be used, which corre-sponds to removing the sgn( q ) factor in the i(cid:15) prescription. This proof is a generalizationto that in [1], with multi-component spinor / Bloch state u α and the presence of externalEM field, and without rotational symmetry.The definition (4.109) of δW agrees with the quasiparticle Wigner function to firstorder in A and zeroth order in q . The first term of (4.109) corresponds to the singularpart of (4.107) projected onto the u band (at zeroth order in q ), which we identify as thedistribution of excited quasiparticles; the factor of Z difference is the quasiparticle wavefunction renormalization. The second term of (4.109) is due to the Peierl’s substitution inthe equilibrium part θ ( (cid:15) F − E ) of the Wigner function; Fourier transforming to the positionspace, it corresponds to the Wilson loop at first order in A in the Wigner function. First Order in q At first order in q , we assert we should define δW ≡ ( uu † ) T Z (cid:0) ∆ (cid:48) Γ ν A ν + ∆ (cid:48) Γ ν A ν + iC Γ ν A ν (cid:1) . (4.112)126ts gauge invariance follows from (4.106) and (4.76). It also agrees with the order q singularpart of the Wigner function – as can be seen from (4.107) – projected onto the u band. Inparticular, the projection onto the u band should be done by the momentum space Wilsonline lim n →∞ (cid:16) u α u † α (cid:17) ( p + q/ (cid:16) u α u † α (cid:17) ( p + q ( n − / n ) · · · (cid:16) u α n u † β (cid:17) ( p − q/ . (4.113)It equals u α ( p ) u † β ( p ) + O ( q ), so we can just use ( uu † ) T ( p ) at first order in q .Now we derive the kinetic recursion relation for δW . Substituting (4.94) and (4.97) into(4.112), we have δW = ( uu † ) T Z (cid:2) − ∆ (cid:48) ¯ V ∆ (cid:48) Γ ν − ∆ (cid:48) ¯ V ∆ (cid:48) Γ ν + iC Γ ν − ∆ (cid:48) ¯ V iC Γ ν − (cid:0) ∆ (cid:48) ¯ V ∆ (cid:48) + ∆ (cid:48) ¯ V ∆ (cid:48) (cid:1) Γ ν + (cid:0) ∆ (cid:48) ¯Γ ν + ∆ (cid:48) ¯Γ ν (cid:1)(cid:3) A ν . (4.114)We use the identity∆ (cid:48) = ( uu † )( uu † ) T ∆ (cid:48) + ∆ (cid:48) ( uu † )( uu † ) T = ( uu † )( uu † ) T ∆ (cid:48) + iq λ A λ ∆ (cid:48) = ∆ (cid:48) iq λ A λ + ∆ (cid:48) ( uu † )( uu † ) T (4.115)and the fact ( uu † ) T ∆ (cid:48) ( uu † ) = 0 to rewrite δW as δW = ∆ (cid:48) ( uu † ) T Z (cid:104) − (cid:16) ¯ V + iq λ A λ ¯ V + ¯ V iq λ A λ (cid:17) ∆ (cid:48) Γ ν + (cid:16) ¯Γ ν + iq λ A λ ¯Γ ν (cid:17) − ¯ V ( uu † )( uu † ) T (cid:0) ∆ (cid:48) Γ ν + ∆ (cid:48) Γ ν + iC Γ ν (cid:1)(cid:105) A ν + ( uu † ) T Z iC Γ ν A ν (4.116)The second line can be easily identified as (1 /Z )∆ (cid:48) U δW . In the first line, we substitute1274.109) for ∆ (cid:48) Γ ν A ν . Then we define the gradient interaction potential via iq µ V µ ( (cid:126)p, (cid:126)k ) ≡ ( Zuu † ) T ( p ) (cid:2) ¯ V ( p, k ; q ) + iq µ (cid:0) A µ ( p ) ¯ V ( p, k ) + ¯ V ( p, k ) A µ ( k ) (cid:1)(cid:3) ( Zuu † )( k ) (cid:12)(cid:12)(cid:12) p,k on F S (4.117)(note that even if the microscopic interaction is contact interaction, in kinetic theory V µ isstill non-zero) and define the EM dipole moment via iq µ µ µν ( (cid:126)p ) ≡ ( Zuu † ) T (cid:0) ¯Γ ν + iq µ A µ ¯Γ ν (cid:1)(cid:12)(cid:12)(cid:12) p on F S − iq µ V µ δ F S δ νi v i . (4.118)As we will show explicitly below, µ µν is antisymmetric in µν . With these definitions, therecursion relation for δW becomes δW = δ F S v i q i v µ q µ − i(cid:15) sgn( q ) (cid:18) µ νλ F νλ − U δW − V ν iq ν δW (cid:19) + ( uu † ) T Z iC Γ ν A ν . (4.119)(The gauge invariance of δW also implicitly requires the antisymmetry of µ µν .)The last step is to rewrite the C term:( uu † ) T Z iC Γ ν A ν = i | q | q v µ q µ δ F S C δ F S (cid:32) v i q i v µ q µ − (cid:33) ( Zuu † ) T Γ ν A ν = i | q | q v µ q µ δ F S C (cid:104) ( δW + δ F S v i A i ) − δ F S ( Zuu † ) T (cid:16) − ∂ νp G − A ν − ¯ V ( Zuu † ) δW (cid:17)(cid:105) = i | q | q v µ q µ δ F S C ( δW + δ F S U δW ) (4.120)where in the second equality we used (4.109) and (4.108), and in the third equality we used1284.50) and (4.76).Now we have δW = δ F S v µ q µ − i(cid:15) sgn( q ) (cid:20) v i q i (cid:18) µ νλ F νλ − U δW − V ν iq ν δW (cid:19) + i | q | q C ( δW + δ F S U δW ) (cid:21) . (4.121)The computation done here is time-ordered. When computing physical quasiparticle distri-bution, we should use retarded boundary condition, which corresponds to using the retardedversions of ∆ (cid:48) and C – that is, to remove the sgn( q ) on the i(cid:15) prescription, and remove theabsolute value on | q | in the collision term. This proves (4.20). Electromagnetic Dipole Moment
The definition (4.118) of µ µν is unusual, and its antisymmetry in µν is not manifest. Nowwe present it in a more familiar form that is explicitly antisymmetric. Using (4.103), (4.104)and the explicit expressions of ¯Γ and ¯ V , we can express the EM dipole moment as µ µν = µ µνbare + µ µνband + µ µνanom. (4.122)which we explain term by term below.The bare EM dipole moment is due to the bare EM dipole matrix (e.g. the Pauli term): µ µνbare ≡ ( Zuu † ) T (cid:0) − ¯ V ∆ (cid:1) ˆ µ µν (cid:12)(cid:12)(cid:12) p on F S (4.123)where ˆ µ µν has been introduced in (4.91) and is antisymmetric in µν .The band EM dipole moment, due to the p dependence of u , is µ µνband ≡ − ( Zuu † ) T A µ ∂ νp G − (cid:12)(cid:12)(cid:12) p on F S = − iZ (cid:110) ∂ [ µp u † G − ∂ ν ] p u (cid:111)(cid:12)(cid:12)(cid:12) p on F S . (4.124)129n the second equality we used the trick (4.46). It is explicitly antisymmetric in µν . Innon-interacting theory, u depends only on (cid:126)p but not p , so µ band would be purely magnetic(e.g. the g = 2 magnetic dipole of free Dirac fermion). In interacting theory, u may or maynot depend on p , so µ band may or may not have electric dipole components.The anomalous EM dipole moment, due to interactions, is defined via iq µ µ µνanom. ≡ − ( Zuu † ) T ¯ V (cid:16) − (∆ r + ∆ s ) ∂ νk G − + iq µ A µ ( Zuu † ) δ F S δ νi v i (cid:17)(cid:12)(cid:12)(cid:12) p on F S − ( Zuu † ) T (cid:0) − ¯ V ∆ (cid:1) (cid:101) V ∂ νk iG (cid:12)(cid:12)(cid:12) p on F S . (4.125)To get a better understanding of µ µνanom. , we do the following. For the term with ∆ r , we usethe explicit expression of ∆ r . For the term with ∆ s , we use the identity∆ s ∂ νk G − = A ν ( Zuu † ) δ F S δ µi v i iq µ (4.126)which again follows from the trick (4.46). Now, the anomalous EM dipole moment reads µ µνanom. = − ( Zuu † ) T ¯ V (cid:16)(cid:110) G ( ∂ [ µp G − ) G ( ∂ ν ] p G − ) G (cid:111) + 2 A [ µ ( Zuu † ) δ F S δ ν ] i v i (cid:17)(cid:12)(cid:12)(cid:12) p on F S + ( Zuu † ) T (cid:0) − ¯ V ∆ (cid:1) ∂ µq (cid:0) i (cid:101) V ∂ νk iG (cid:1)(cid:12)(cid:12)(cid:12) p on F S . (4.127)The antisymmetry in µν is manifest in the first line. Gauge invariance of (4.119) requires thesecond line above to be antisymmetric in µν too; more explicitly we show this from diagramsin Section 4.4.7.In general, µ i anom. (cid:54) = 0, so even when there is no bare electric dipole matrix, the quasi-particle will still acquire an electric dipole moment due to interactions. This gives rise tothe second term in (4.24) which is absent in usual Fermi gas.130 .4.5 Current We now prove the expression of the current (4.22)(4.23). Previously we have defined δW , U , V ν and µ µν from QFT, but we have not shown they are real in the position space. But theseimmediately follow once we have (4.21), because in position space the quantum expectationof the current must be real for arbitrary A , q , interaction strength and initial / boundaryconditions of δW . (However, we note that µ µνband and µ µνanom. are not separately real in general– due to interactions, the χ w ( p )’s are generally complex near the FS, and hence is µ µνband .) qi Γ µ A ν qµ A ν As drawn above, the expectation of the current induced by A at linear response is givenby iδJ µ ( q ) = − (cid:90) p tr (cid:8) i (cid:101) Γ µ ( p ; − q ) iG ( p + q/ i Γ ν ( p ; q ) iG ( p − q/ (cid:9) A ν ( q ) . − (cid:90) p tr (cid:8) i (cid:101) Ξ µν ( p ; − q, q ) iG ( p ) (cid:9) A ν ( q ) (4.128)where the negative sign is due to the fermion loop. In the second line, we use (4.92) andintegrate p by parts to eliminate (cid:101) Ξ. Below we work in double fermion notation, at zerothand first order in q separately.We emphasize that here we are computing the time-ordered correlation of δJ and A ,while in linear response we should compute the retarded correlation. This difference onlyshows up in the recursion relation that δW satifies, i.e. the Boltzmann equation, and therewe have already handled this difference. The expression of δJ in terms of δW is the samefor time-ordered and retarded correlation. 131 eroth Order in q At zeroth order in q , iδJ µ = − ( i (cid:101) Γ µ ) T (cid:0) i ∆ i Γ ν + i ∆ (cid:48) i Γ ν + ∂ νp iG (cid:1) A ν (4.129)where the integration over p is understood. For the i Γ ν in the first term, whose immediateleft is not ∆ (cid:48) , we apply the recursion relation (4.94), and get iδJ µ = − ( i (cid:101) Γ µ ) T (cid:0) + i ∆ i ¯ V (cid:1) i ∆ (cid:48) i Γ ν A ν − ( i (cid:101) Γ µ ) T (cid:0) i ∆ i ¯Γ ν + ∂ νp iG (cid:1) A ν (4.130)Due to (4.104) and the facts ( ¯ V ) T = ¯ V , (∆ ) T = ∆ , the above reduces to δJ µ = (¯Γ µ ) T (cid:16) ∆ (cid:48) Γ ν A ν − ( Zuu † ) δ F S δ νi v i (cid:17) A ν = (¯Γ µ ) T ( Zuu † ) δW . (4.131)Finally, applying (4.103), we obtain δJ µ = ( v µ ) T δW + (cid:16) δ µi v i δ F S (cid:17) T U δW . (4.132)The transpose on the left implies integration over p . This is (4.22). First Order in q At first order in q , iδJ µ = − ( i (cid:101) Γ µ ( − q )) T (cid:0) i ∆ + i ∆ (cid:48) (cid:1) i Γ ν A ν − ( i (cid:101) Γ µ ) T (cid:0) i ∆ + i ∆ (cid:48) (cid:1) i Γ ν ( q ) A ν − ( i (cid:101) Γ µ ) T (cid:0) i ∆ (cid:48) + i ∆ r + i ∆ s − C (cid:1) ( q ) i Γ ν A ν − ( i (cid:101) Γ µ ( − q )) T ∂ νp iG A ν − ( i (cid:101) Γ ν ( q )) T ∂ µp iG A ν . (4.133)132e rewrite this according to the following: If the immediate left of an i Γ ν is not ∆ (cid:48) or ∆ (cid:48) or C , we apply the recursion relations (4.94) to it; similarly, if the immediate left of an i Γ ν is not ∆ (cid:48) , we apply (4.97) to it. We find iδJ µ = − ( i ¯Γ µ ) T (cid:0) i ∆ (cid:48) i Γ ν ( q ) + i ∆ (cid:48) ( q ) i Γ ν − C ( q ) i Γ ν (cid:1) A ν − ( i ¯Γ µ ( − q )) T i ∆ (cid:48) i Γ ν A ν + (terms regular in q ) . (4.134)We will take care of the terms in the second line of (4.134) later. To terms in the first line,we apply the identity (4.115), and get( i ¯Γ µ ) T ( Zuu † ) δW + (cid:16) ( i ¯Γ µ ( − q )) T + ( i ¯Γ µ ) T iq λ A λ (cid:17) ( Zuu † ) (cid:16) δW + δ F S v j A j (cid:17) (4.135)Now use (4.103) in the first term, and (4.118) and the facts A λ = − ( A λ ) T , V ν ( (cid:126)p, (cid:126)k ) = −V ν ( (cid:126)k, (cid:126)p ) in the second term, the first line of (4.134) becomes i ( v µ ) T δW + (cid:16) δ µi v i δ F S (cid:17) T i U δW + (cid:16) ( iµ µν ) T + ( δ F S δ µi v i ) T i V ν (cid:17) iq ν (cid:16) δW + δ F S v j A j (cid:17) . (4.136)Notice that the δW dependence agrees with (4.23).The second line of (4.134) – terms regular in q – can be read-off diagrammatically: − ( i (cid:101) Γ µ ( − q )) T (cid:0) i ∆ i ¯Γ ν + i∂ νp G (cid:1) A ν − (cid:16) ( i ∆ i ¯Γ µ ) T + ( i∂ µp G ) T (cid:17) i (cid:101) Γ ν ( q ) A ν − ( i ¯Γ µ ) T ( i ∆ r + i ∆ s ) ( q ) i ¯Γ ν A ν − (cid:0) i ∆ i ¯Γ µ (cid:1) T i (cid:101) V ( q ) (cid:0) i ∆ i ¯Γ ν (cid:1) A ν (4.137)where the ∂ p G terms follow from the (cid:101) Ξ terms in (4.128). There are many equivalent ex-pressions; we have chosen to express it so that it appears “symmetric” to read from left toright and from right to left. Now, substitute (4.103) into the (∆ r + ∆ s ) term, and substitute1334.104) into the rest; next, for each of the two terms in the second line above, we expandlike − a c b = − a c b − a c b + a c b − a c b for a = a + a , b = b + b . The result is − ( i ¯Γ µ ( − q )) T ( Zuu † ) δ F S δ νj v j A ν − ( δ F S δ µi v i ) T ( Zuu † ) T i ¯Γ ν ( q ) A ν + ( δ F S δ µi v i ) T ( Zuu † ) T i ¯ V ( q ) ( Zuu † ) δ F S δ νj v j A ν − ( i∂ µp G − ) T ( i ∆ r + i ∆ s ) ( q ) i∂ νp G − A ν − ( i∂ µp G ) T i (cid:101) V ( q ) i∂ νk G A ν . (4.138)The last term is to be expressed using (4.176); it vanishes, as we show diagrammatically andcombinatorially in Section 4.4.7. The remaining terms, inspecting the definitions of µ µν and V ν , can be expressed line by line as − (cid:16) ( iµ µλ ) T + ( δ F S δ µi v i ) T i V λ − ( i ¯Γ µ ) T A λ ( Zuu † ) (cid:17) iq λ δ F S δ νj v j A ν − ( δ F S δ µi v i ) T (cid:16) iµ λν − ( Zuu † ) T A λ i ¯Γ ν + ( Zuu † ) T (cid:16) A λ i ¯ V − i ¯ V A λ (cid:17) ( Zuu † ) δ F S δ νj v j (cid:17) iq λ A ν − ( i∂ µp G − ) T ( i ∆ r + i ∆ s ) ( q ) i∂ νp G − A ν . (4.139)Substituting (4.103) for ¯Γ , and using the definition of µ µνband , we find − (cid:16) ( iµ µλ ) T + ( δ F S δ µi v i ) T i V λ (cid:17) iq λ δ F S δ νj v j A ν − ( δ F S δ µi v i ) T iµ λν iq λ A ν + ( iµ µλband ) T δ F S δ νj v j iq λ A ν + ( δ F S δ µi v i ) T iµ λνband iq λ A ν − ( i∂ µp G − ) T ( i ∆ r + i ∆ s ) ( q ) i∂ νp G − A ν . (4.140)Finally, for the ∆ r term, use its explicit expression, and for the ∆ s term, use (4.126) and134he definition of µ µνband . We arrive at − (cid:16) ( iµ µν ) T + ( δ F S δ µi v i ) T i V ν (cid:17) iq ν δ F S v j A j − ( δ F S δ µi v i ) T iµ νλ F νλ / iσ µνλ F νλ / , (4.141)where the Hall conductivity tensor σ µνλ , totally antisymmetric in µλν , is defined by σ µνλ ≡ σ µνλr + σ µνλs , with σ µνλr ≡ (cid:90) p tr (cid:110) ( ∂ [ µp iG − ) iG ( ∂ νp iG − ) iG ( ∂ λ ] p iG − ) iG (cid:111) ,σ µνλs ≡ (cid:90) p δ F S v i δ [ µi µ νλ ] band . (4.142)One may notice the similarity between the definitions of σ µνλ and µ νλanom. (except in σ µνλ ,the (cid:101) V term vanishes due to the proof in Section 4.4.7).After combining (4.136) and (4.141) into (4.134), we arrive at δJ µ = ( v µ ) T δW + ( µ µν ) T iq ν δW + ( δ F S δ µi v i ) T (cid:16) − µ νλ F νλ / U δW + V ν iq ν δW (cid:17) + σ µνλ F νλ / . (4.143)This is (4.23). Detour: Coleman-Hill Theorem
Interestingly, our derivation for δJ µ above, most crucially the cancellation in Section 4.4.7,provides an alternative diagrammatic proof to the Coleman-Hill theorem, for QFTs restrictedto our assumptions (which are less general than in the original proof). The theorem statesthat in a gapped fermionic system, the Hall conductivity is unaffected by the interactions.When the system is gapped, i.e. in the absence of FS, our result reduces to δJ µ = σ µνλr F νλ / σ µνλr . Let g be some interaction strength, wehave (denoting ∂ g ≡ ∂/∂g ) ∂ g σ µνλr = − (cid:90) p ∂ g tr (cid:110) ( ∂ [ µp G − ) G ( ∂ νp G − ) G ( ∂ λ ] p G − ) G (cid:111) = − (cid:90) p ∂ [ g tr (cid:110) ( ∂ µp G − ) G ( ∂ νp G − ) G ( ∂ λ ] p G − ) G (cid:111) . (4.144)In the second equality we added some total derivative terms so to antisymmetrize the ∂ g altogether with the three ∂ p ’s. But because of the antisymmetrization, the integrand actuallyvanishes. This means the full Hall conductivity is independent of interaction strength. Thisproves the Coleman-Hill theorem, for QFTs restricted to our assumptions. In asserting“the integrand vanishes”, we implicitly made use of the fact that ∂iG = { iG ∂iG − iG } inthe absence of FS, the fact that G − bare by definition is independent of g , and the physicalassumption that the dependence of the self-energy Σ on g is non-singular. Chemical Potential Dependence of the Hall Conductivity Tensor
Now we prove (4.33), the important result relating the Hall conductivity to the Berry cur-vature on the FS.We first consider the (cid:15) F dependence of σ µνλr : ∂ F σ µνλr = − (cid:90) p ∂ [ F tr (cid:110) ( ∂ µp G − ) G ( ∂ νp G − ) G ( ∂ λ ] p G − ) G (cid:111) . (4.145)The integrand is non-vanishing because in the presence of FS, the derivative outside thetrace acts on the p pole structure of the iG ’s, and the pole structure depends on p i and (cid:15) F .136n fact, by similar reasoning that led to the FS term in (4.48), here we are led to ∂ F σ µνλr = 12 (cid:90) p iπδ ( p − ξ ) ∂ [ F sgn ξ × (cid:88) w (cid:88) w (cid:48) Zχ w χ w (cid:48) (cid:110) u † ( ∂ µp G − ) ww † ( ∂ νp G − ) w (cid:48) w (cid:48)† ( ∂ λ ] p G − ) u (cid:111) + 12 (cid:90) p iπ (cid:16) − ∂ p δ ( p − ξ ) (cid:17) ∂ [ F sgn ξ × (cid:88) w Z χ w (cid:110) w † ( ∂ µp G − ) uu † ( ∂ νp G − ) uu † ( ∂ λ ] p G − ) w (cid:111) . (4.146)The first term arises from the single pole (appearing as iπδ ( p − ξ ) sgn ξ in principle functiondecomposition) when one of the three G ’s is in the u band; the second term arises from thedouble pole (appearing as iπ ( − ∂ p δ ( p − ξ )) sgn ξ ) when two of the G ’s are in the u band.The triple pole contribution when all three G ’s are in the u band vanishes under the totalantisymmetrization. We can evaluate the above using the trick (4.46). We find ∂ F σ µνλr =12 (cid:90) p iπδ ( p ) ∂ [ F sgn ξ (cid:110) ∂ µp u † ( − uu † ) Z ( ∂ νp G − ) ( − uu † ) ∂ λ ] p u (cid:111) − (cid:90) p iπδ ( p ) ∂ [ F sgn ξ ∂ p (cid:40) ∂ µp u † Z ∂ νp χ u (cid:88) w ( χ u − χ w ) ww † χ w ∂ λ ] p u (cid:41) . (4.147)In the first line, − uu † can be further replaced by thanks to (4.46) and the antisym-metrization. In the second line, we need the following relevant terms, according to (4.73): − Z ∂ νp χ u ( χ u − χ w ) (cid:12)(cid:12)(cid:12) p on F S = Zv ν χ w (4.148)137 p (cid:16) − Z ∂ νp χ u ( χ u − χ w ) (cid:17)(cid:12)(cid:12)(cid:12) p on F S = Zv ν ∂ p χ w + (cid:16) ∂ νp Z + 2 iZγ ∂ νp | p | (cid:17) χ w − v ν χ w . (4.149)Similar results hold when ∂ νp χ u is replaced with ∂ F χ u ; recall that v F ≡ ∂ F E . The remainingproblem is, how to understand δ ( p ) ∂ p | p | ? We should understand it as 0, because inthis thesis, the generalized function δ ( p ) always arises as the approximation to a narrowrectangular function over an interval centered at p = 0 (more precisely, the rectangularfunction is θ ( p − q / − θ ( p + q / δ ( p ) ∂ p | p | correspondsto taking the difference of | p | between the two sides of the interval, which is obviously 0.(Note that in d = 2, the parametrization (4.73) does not apply; however, general analyticproperties of δ Σ still requires it to be odd in p [48, 1], and therefore the correspondingcontribution here must still vanish.) Thus, at the end, we have ∂ F σ µνλr = 12 (cid:90) p i πδ ( p ) ∂ [ F θ ( (cid:15) F − E ) × (cid:16) − v µ (cid:110) ∂ νp u † ∂ λ ] p u (cid:111) + (cid:16) ∂ µp + v µ ∂ p (cid:17) (cid:110) ∂ νp u † ZG − ∂ λ ] p u (cid:111)(cid:17) . (4.150)Note that the derivatives of − θ ( (cid:15) − E ) are always the same as those of sgn ξ ; we choose toexpress as the former because it admits the intuition as the “Fermi sea”, at least near theFS.Next, from the expression of µ νλband , we observe σ µνλs can be expressed as σ µνλs = 3 (cid:90) p πδ ( p − ( E − (cid:15) F )) ∂ [ µp θ ( (cid:15) F − E ) i (cid:110) ∂ νp u † ZG − ∂ λ ] p u (cid:111) . (4.151)138ow we take ∂ F and find ∂ F σ µνλs = 12 (cid:90) p ∂ [ F (cid:16) πδ ( p − ( E − (cid:15) F )) ∂ µp θ ( (cid:15) F − E ) i (cid:110) ∂ νp u † ZG − ∂ λ ] p u (cid:111)(cid:17) = − (cid:90) p i π ∂ [ F θ ( (cid:15) F − E ) (cid:16) δ ( p ) ∂ µp − v µ ∂ p δ ( p ) (cid:17) (cid:110) ∂ νp u † ZG − ∂ λ ] p u (cid:111) = − (cid:90) p i πδ ( p ) ∂ [ F θ ( (cid:15) F − E ) (cid:16) ∂ µp + v µ ∂ p (cid:17) (cid:110) ∂ νp u † ZG − ∂ λ ] p u (cid:111) (4.152)In the second equality we used ∂ µp δ ( p − ( E − (cid:15) F )) = − v µ ∂ p δ ( p − ( E − (cid:15) F )) (4.153)and likewise for ∂ F .Finally we combine (4.150) and (4.152) and obtain ∂ F σ µνλ = 12 (cid:90) p πδ ( p ) ∂ [ F θ ( (cid:15) F − E ) v µ ( − i ) (cid:110) ∂ νp u † ∂ λ ] p u (cid:111) . (4.154)Due to (4.45) (and the similar version for ∂ F ) and the antisymmetrization, we can replace u with u , and perform the p integral to obtain ∂ F σ µνλ = 12 (cid:90) (cid:126)p ∂ [ F θ ( (cid:15) F − E ) v µ b νλ ] . (4.155)Expanding the antisymmetrization explicitly, this is (4.33).If d = 2, or if d > u ( (cid:126)p ) over the FS, and derive (4.35) via integration by parts: ∂ F σ µνλ = 24 (cid:90) (cid:126)p ∂ [ F (cid:16) ∂ µp θ ( (cid:15) F − E ) v ν a λ ] (cid:17) = ∂ F (cid:90) (cid:126)p δ ( (cid:15) F − E ) δ [ µ v ν a λ ] (4.156)139here in the first equality we used the fact that v µ and v F are respectively − ∂ µp and − ∂ F acted on p − ( E − (cid:15) F ), and in the second equality, total (cid:126)p derivatives vanish due to the (cid:126)p integral, while total p derivative vanishes trivially as the integrand has no p dependence.If d > P λk introduced below (4.42): ∂ F σ ijλ = 12 (cid:90) (cid:126)p ∂ [ F θ ( (cid:15) F − E ) b ij ∂ k ] p P λk (4.157)where the total antisymmetrization is in indices [ F ijk ]. Using the assumption that there isno band degeneracy near the FS and hence ∂ [ kp b ij ] = 0 near the FS, we have ∂ F σ ijλ = 12 (cid:90) (cid:126)p ∂ [ F (cid:16) − ∂ kp θ ( (cid:15) F − E ) b ij ] P λk (cid:17) . (4.158)When we proceed further, note that if the fermions are in a lattice and if the FS intersectsthe boundary of our choice of first Brillouin zone, then for λ = 0 we cannot drop the total p -derivative terms because p k is not continuous when we identify the opposite boundaries ofthe first Brillouin zone [32]. We have ∂ F σ ijλ = ∂ F (cid:18) (cid:90) (cid:126)p δ ( (cid:15) F − E ) b [ ij v k ] P λk + 6 (cid:90) (cid:126)p ∂ [ kp (cid:16) δ ( (cid:15) F − E ) v i a j ] P λk (cid:17)(cid:19) (4.159)(we used the Bianchi identity ∂ F ∂b νλ = 2 ∂ [ λp b ν ] F ). This proves (4.40) and (4.41), and hence(4.42).We remind that the derivation above fails at discrete values of (cid:15) F around which the FSdevelops new disconnected components (e.g. Figure 4.1). Across those values of (cid:15) F it isunclear whether σ µνλ may have a jump, as commented in Section 4.3.4.140 .4.6 Cutkosky Cut and Quasiparticle Collision In the proof present above, we have left behind the technical discussion of the Cutkoskyformalism and the computation of quasiparticle collisions. In this section we complete therelevant discussions. More exactly, we present the followings: • First we present the Cutkosky cutting rule for our fermionic system. • Then we introduce how to count the power of q in a cut diagram, and show D , C ph and C pp are the only cut sub-diagrams that contribute at order q . Then restrict to d ≥ • Then we discuss the difficulty of parametrizing quasiparticle collision in d = 2, andexplain why (4.76) still holds in d = 2 despite the difficulties of parametrization. • Finally, we show (regardless of d ≥ d = 2) that collisions have no contributionto the antisymmetric part Π [ µν ] of the current-current correlation. So collision are“uninteresting” to the main focus of this thesis. Review of Cutkosky Cut
The formalism of Cutkosky cut is a formal procedure to compute discontinuities (with respectto external momenta) in Feynman diagrams. A Feynman diagram has a discontinuity if, inan intermediate step of the process described the diagram, all intermediate particles go on-shell. We have seen an example of this in the derivation for ∆ (cid:48) following (4.60). From thederivation for ∆ (cid:48) , we have the sense that all it matters is the i(cid:15) prescription whose effectscan be extracted by principle function decomposition; other details of the propagator areirrelevant. This suggests the method can be generalized to other diagrams. Indeed, Landaumade pioneering contribution in this direction [42], and the full formalism was established by141utkosky [22]. The Cutkosky cutting rule was originally developed for bosons. Here, basedon our needs, we present the rule for fermions; the modifications have clear physical meaningsas long as one is familiar with some basic analytic properties of fermion propagators [48, 1].Consider a two point correlation Π with momentum q (for definiteness, we let q run fromthe right to the left of the diagram). In this section, unless otherwise specified, Π refers tothe current-current correlation Π µν ( q ), and ReΠ and ImΠ really mean the Hermitian andanti-Hermitian parts of Π µν . The Cutkosky cutting rule gives the difference between theretarded correlation and the advanced correlation:Π cut ≡ − i (Π R − Π A ) = ImΠ R − ImΠ A = 2ImΠ R = − A = Π cut − − Π cut + . (4.160)The three equalities in the first line follow from general analytic properties of two-pointcorrelations [48, 1]. In the second line, Π cut + is defined as the following. Consider a certainFeynman diagram in Π, with a certain Cutkosky cut – a cut through a number of internalfermion propagators such that the Feynman diagram is disconnected into two parts, withone current insertion (or other operators, depending on what Π is) contained in each part.Clearly the total momentum running from right to left across the cut is q . For those fermionpropagators that are being cut, we place the fermions on-shell, which, according to theCutkosky cutting rule, means to replace each cut propagator by iG ( p ) −→ πZ ( (cid:126)p ) δ ( p − ξ (cid:126)p ) sgn( p ) θ ( ∓ p ) , (4.161)where θ ( ∓ p ) is taken when the fermion runs across the cut from right to left / from leftto right (given we have chosen q to run from right to left). Then we sum over all possibleways of cutting over all Feynman diagrams, and the result is defined as Π cut − . And Π cut + is defined in a similar manner, but with θ ( ± p ) taken when the fermion runs across the cut142rom right to left / from left to right.An important consequence is, by energy conservation, all these cut propagators musthave energies between ±| q | . This is because, those step functions require the on-shellquasiparticles’ energies to appear in energy conservation in the form “the sum of positiveenergies minus the sum of negative energies is equal to ∓ q ” respectively in Π cut ∓ (so Π cut ∓ is non-vanishing only for negative / positive q respectively, hence our ∓ subscript). Inretrospect, this justifies why we could restrict to the u band in a multi-band system, andwhy we could ignore the possibility of cutting through an interaction mediator: Because bythe assumptions about our QFT, neither the other bands of the fermion nor the short-rangedinteraction mediator(s) have any low energy on-shell excitation. Computation of Quasiparticle Collision d ≥ Below we discuss how to count the power of q in a cut sub-diagram. We have to note thatthe power counting introduced below has missing piece. In d = 2 the missing piece is order q ln q (less suppressed than order q when q is small) and lead to complications to be discussedlater. For now we work with d ≥
3, where this missing piece is neglected as they are beyondorder q . After introducing the power counting, we will argue the only cut sub-diagram thatcontributes at zeroth order in q is a cut through double propagator, leading to ∆ (cid:48) , and theonly ones that contribute at first order in q are those three pairs of cut sub-diagrams for D , C ph and C pp .Consider a cut through n > | q | , hencethe integration over the n internal energies yields a suppression of order ( q ) n ; on the otherhand, the argument of the delta function of energy conservation is of order q . Therefore,the contribution of the n cut propagators is of order ( q ) n − .The case of n = 2 has a difference. For n = 2, when one on-shell fermion is low143nergy, and hence near the FS, the other one, due to spatial momentum conservation andthe smallness of (cid:126)q , is automatically near the FS too, and hence low energy too (because ofon-shell). This means the smallness of their energies provides only one constraint, insteadof two independent constraints. This lowers the power counting of q by one. Thus, the cutthrough two propagator is not first order but zeroth order in q . In particular, let’s computethe cut through the double propagator of momenta p ± q/
2. According to the cutting rule,the cut sub-diagram is equal to2Im∆ (cid:48) R ( q )= 2 πZ ( (cid:126)p + (cid:126)q/ δ ( p + q / − ξ ( (cid:126)p + (cid:126)q/ πZ ( (cid:126)p − (cid:126)q/ δ ( p − q / − ξ ( (cid:126)p − (cid:126)q/ − (cid:16) θ ( − ( p + q / θ ( p − q / − θ ( p + q / θ ( − ( p − q / (cid:17) = (2 πZ ( (cid:126)p )) q δ ( p ) δ ( p − ξ ( (cid:126)p )) δ ( v µ ( (cid:126)p ) q µ ) + O ( q ) . (4.162)To relate this to the time-ordered ∆ (cid:48) , we use the Kramers-Kronig dispersion relation of ageneral two-point correlation [48, 1]: i Π( q ) = i π (cid:90) dω Π cut ( ω, (cid:126)q ) − q + ω − i(cid:15) sgn q + i (real terms unrelated to Cutkosky cut) . (4.163)(The Π here is time-ordered; if retarded or advanced, the sgn q should be replaced with ± i ∆ (cid:48) ( q ) in (4.56), as desired. (Theintegration generally involves Π cut at non-small values of ω . But Im∆ (cid:48) in particular isnon-vanishing only when ω equals the small value v i q i .)For n >
2, the n cut propagators contribute order ( q ) n − , and for current-currentcorrelation n must be even (with n/ n/ (cid:48) (beyond n = 2) are at least of order q . (This justifies our analytic expansion144f the q -2PI interaction vertex to zeroth and first order in q .) How can there be order q sub-diagrams? Consider the following situation. Given that all cut propagators are on-shell,if there is a pair of propagators, one cut and one uncut, whose momenta are dictated bymomentum conservation to differ by q , then that uncut propagator will be nearly-on-shell(due to the smallness of q ), and contributes a factor of order 1 /q . There can be at mostone such nearly-on-shell propagator on either the left or the right of the cut, so there can beat most two of them in total. Therefore, there exist cut sub-diagrams at order q : Such cutsub-diagrams have four cut propagators, and two nearly-on-shell propagators, one on eachside of the cut. These propagators can be organized in six different ways, which are the threepairs of cut sub-diagrams for D , C ph and C pp respectively, presented in Section 4.4.2.Now we evaluate the sub-diagrams for D , C ph and C pp according to the cutting rule.First, 2( D R ) ( p ; q )= (cid:90) k,l (cid:18) − (cid:19) (cid:18) iZ ( (cid:126)p − (cid:126)q/ − q − ξ ( (cid:126)p − (cid:126)q/
2) + ξ ( (cid:126)p + (cid:126)q/ (cid:19) i | V ( p − q/ , k + l → k − q/ , p + l ) | (2 π ) Z ( (cid:126)p + (cid:126)q/ Z ( (cid:126)k − (cid:126)q/ Z ( (cid:126)p + l ) Z ( (cid:126)k + l ) δ ( p + q / − ξ ( (cid:126)p + (cid:126)q/ δ ( k − q / − ξ ( (cid:126)k − (cid:126)q/ δ ( p + l − ξ ( (cid:126)p + l )) δ ( k + l − ξ ( (cid:126)k + l )) (cid:104) θ ( − ( p + q / θ ( k − q / θ ( − ( k + l )) θ ( p + l ) − θ ( p + q / θ ( − ( k − q / θ ( k + l ) θ ( − ( p + l )) (cid:105) − (with q ↔ − q ) . (4.164)The − / , k and l to order q ; for example, the first product of step functions restricts q / 2. We already argued that the q suppression in the cut sub-diagramsis dictated by the step functions, so in the Z ’s, the V ’s and the on-shell delta functions, wecan neglect the q -dependences, as well as the p , k , l dependences. We are then led to2( D R ) ( p ; q )= − (cid:90) k,l i Z ( (cid:126)p ) Z ( (cid:126)k ) Z ( (cid:126)p + l ) Z ( (cid:126)k + l )( v µ ( (cid:126)p ) q µ ) i | V ( p, k + l → k, p + l ) | (2 π ) δ ( ξ ( (cid:126)p )) δ ( ξ ( (cid:126)k )) δ ( ξ ( (cid:126)p + l )) δ ( ξ ( (cid:126)k + l )) (cid:104) θ ( − ( p + q / θ ( k − q / θ ( − ( k + l )) θ ( p + l ) − θ ( p + q / θ ( − ( k − q / θ ( k + l ) θ ( − ( p + l ))+ θ ( − ( k + q / θ ( p − q / θ ( − ( p + l )) θ ( k + l ) − θ ( k + q / θ ( − ( p − q / θ ( p + l ) θ ( − ( k + l )) (cid:105) . (4.165)Inspecting the p, q dependence, together with power counting, we justify the parametrization(4.70) with non-negative γ (what is remained to be shown is that the γ here is the same γ that appears in ImΣ). In particular, the sign of ( D R ) is given by the sign of q , and so the146ime-ordered D = ( D R ) sgn q is non-negative. Next,2( C phR ) ( p, k ; q )= (cid:90) l ( − iZ ( (cid:126)p − (cid:126)q/ − q − ξ ( (cid:126)p − (cid:126)q/ 2) + ξ ( (cid:126)p + (cid:126)q/ iZ ( (cid:126)k + (cid:126)q/ q − ξ ( (cid:126)k + (cid:126)q/ 2) + ξ ( (cid:126)k − (cid:126)q/ iV ( p − q/ , k + l → k − q/ , p + l ) iV ( k + q/ , p + l → p + q/ , k + l )(2 π ) Z ( (cid:126)p + (cid:126)q/ Z ( (cid:126)k − (cid:126)q/ Z ( (cid:126)p + l ) Z ( (cid:126)k + l ) δ ( p + q / − ξ ( (cid:126)p + (cid:126)q/ δ ( k − q / − ξ ( (cid:126)k − (cid:126)q/ δ ( p + l − ξ ( (cid:126)p + l )) δ ( k + l − ξ ( (cid:126)k + l )) (cid:104) θ ( − ( p + q / θ ( k − q / θ ( − ( k + l )) θ ( p + l ) − θ ( p + q / θ ( − ( k − q / θ ( k + l ) θ ( − ( p + l )) (cid:105) + (with p ↔ k, except the arguments of the V ’s kept unchanged) . (4.166)Making the small q approximations we made for ( D R ) , we are led to2( C phR ) ( p, k ; q )= − (cid:90) l i Z ( (cid:126)p ) Z ( (cid:126)k ) Z ( (cid:126)p + l ) Z ( (cid:126)k + l ) − ( v µ ( (cid:126)p ) q µ )( v µ ( (cid:126)k ) q µ ) i | V ( p, k + l → k, p + l ) | (2 π ) δ ( ξ ( (cid:126)p )) δ ( ξ ( (cid:126)k )) δ ( ξ ( (cid:126)p + l )) δ ( ξ ( (cid:126)k + l )) (cid:104) θ ( − ( p + q / θ ( k − q / θ ( − ( k + l )) θ ( p + l ) − θ ( p + q / θ ( − ( k − q / θ ( k + l ) θ ( − ( p + l ))+ θ ( − ( k + q / θ ( p − q / θ ( − ( p + l )) θ ( k + l ) − θ ( k + q / θ ( − ( p − q / θ ( p + l ) θ ( − ( k + l )) (cid:105) . (4.167)147his justifies the parametrization (4.71) with non-negative λ ph . And last,2( C ppR ) ( p, k ; q )= (cid:90) l (cid:18) − (cid:19) iZ ( (cid:126)p − (cid:126)q/ − q − ξ ( (cid:126)p − (cid:126)q/ 2) + ξ ( (cid:126)p + (cid:126)q/ iZ ( (cid:126)k − (cid:126)q/ − q − ξ ( (cid:126)k − (cid:126)q/ 2) + ξ ( (cid:126)k + (cid:126)q/ iV ( p − q/ , k + q/ → k − l, p + l ) iV ( k − l, p + l → p + q/ , k − q/ π ) Z ( (cid:126)p + (cid:126)q/ Z ( (cid:126)k + (cid:126)q/ Z ( (cid:126)p + l ) Z ( (cid:126)k − l ) δ ( p + q / − ξ ( (cid:126)p + (cid:126)q/ δ ( k + q / − ξ ( (cid:126)k + (cid:126)q/ δ ( p + l − ξ ( (cid:126)p + l )) δ ( k − l − ξ ( (cid:126)k − l )) (cid:104) θ ( − ( p + q / θ ( − ( k + q / θ ( k − l ) θ ( p + l ) − θ ( p + q / θ ( k + q / θ ( − ( k − l )) θ ( − ( p + l )) (cid:105) − (with q ↔ − q, except the arguments of the V ’s kept unchanged) . (4.168)Making the small q approximations we made for ( D R ) , we are led to2( C ppR ) ( p, k ; q )= − (cid:90) l i Z ( (cid:126)p ) Z ( (cid:126)k ) Z ( (cid:126)p + l ) Z ( (cid:126)k − l )( v µ ( (cid:126)p ) q µ )( v µ ( (cid:126)k ) q µ ) i | V ( p, k → k − l, p + l ) | (2 π ) δ ( ξ ( (cid:126)p )) δ ( ξ ( (cid:126)k )) δ ( ξ ( (cid:126)p + l )) δ ( ξ ( (cid:126)k − l )) (cid:104) θ ( − ( p + q / θ ( − ( k + q / θ ( k − l )) θ ( p + l ) − θ ( p + q / θ ( k + q / θ ( − ( k − l )) θ ( − ( p + l ))+ θ ( k − q / θ ( p − q / θ ( − ( p + l )) θ ( − ( k − l )) − θ ( − ( k − q / θ ( − ( p − q / θ ( p + l ) θ ( k − l ) (cid:105) . (4.169)148his justifies the parametrization (4.72) with non-negative λ pp .Inspecting the exact expressions (4.164), (4.166) and (4.168) (these expressions are beforewe apply power counting, and hence hold in d = 2 as well), we can observe the importantrelation (4.74). In particular, to see the second equality in (4.74), we shift k → k + l ∓ q/ C ppR ) . This leads to (4.76), which is requiredby the Ward-Takahashi identity.We have computed the order q contributions to ImΠ R . Is there an associated part in ReΠthrough the Kramers-Kronig dispersion relation (4.163)? For ImΣ R , and hence the ( D ) R contribution to ImΠ R , it is known that the associated real part amounts to a correction to Z ( (cid:126)p ); but Z ( (cid:126)p ) itself appeared in our cutting rule to start with, so this just means we mustuse the self-consistent, i.e. physical, value of Z ( (cid:126)p ). As long as we have done so, there is nofurther contribution to ReΠ from ( D ) R . From Ward identity we know ( C ph ) R and ( C pp ) R must have no further contribution to ReΠ either. This differs from the scenario in the zerothorder contribution, where ∆ (cid:48) in (4.56) has both real and imaginary parts.It remains to show the γ in the parametrization of D is the same γ that appears in ImΣ.Now we apply the Cutkosky cutting rule to − Σ( p ) with small p . The leading cut diagramis order ( p ) , involving three cut propagators [48]. More explicitly, at this order we have2Im( − Σ R )( p ) = (cid:90) k,l (cid:18) − (cid:19) i | V ( p, k + l → k, p + l ) | (2 π ) Z ( (cid:126)k ) Z ( (cid:126)k + l ) Z ( (cid:126)p + l ) δ ( k − ξ ( (cid:126)k )) δ ( k + l − ξ ( (cid:126)k + l )) δ ( p + l − ξ ( (cid:126)p + l )) (cid:104) ( − θ ( − k ) θ ( k + l ) θ ( − ( p + l )) − ( − θ ( k ) θ ( − ( k + l )) θ ( p + l ) (cid:105) . (4.170)We can see − ImΣ R is explicitly positive, as it should [48, 1]. Combined with power counting,we justify the parametrization − ImΣ = − ImΣ R sgn p = γ (cid:48) p | p | at small p with positive γ (cid:48) . To verify γ (cid:48) = (3 / γ , we compare the evaluations of − ImΣ R and ( D R ) and restrict to149rder q , we find2( D R ) ( p ; q )= i Z ( v µ q µ ) πZδ ( ξ ) (cid:16) θ ( − ( p + q / θ ( p − q / 2) + θ ( p + q / θ ( − ( p − q / (cid:17)(cid:104) sgn( p − q / − R ( p + q/ − sgn( p + q / − R ( p − q/ (cid:105) = 2 Z πδ ( ξ )( v µ q µ ) sgn( q ) θ ( | q | / − | p | ) ( − ImΣ R ( p − q/ − ImΣ R ( p + q/ . (4.171)The last factor is equal to γ (cid:48) ( (cid:126)p ) (cid:0) p ) + ( q ) / (cid:1) to leading order. Now, we average p between ±| q | / δ ( p ) q ; this is equivalent to makingthe approximationsgn( q ) θ ( | q | / − | p | ) = 2 δ ( p )1! q ∂ p δ ( p )3! (cid:18) q (cid:19) + · · · , (4.172)where the second term is needed to take into account the ( p ) from ImΣ R . This leads to(4.70), and in particular, verifies γ (cid:48) = (3 / γ . Difficulties in d = 2 As we mentioned previously, the power counting of q presented there has missing piece. Hereit is: The power counting relied on the assumption that the internal momenta carried bythe cut propagators are generically not close to each other. However, in the integration overthe internal momenta, there must be some region where this assumption does not hold –the internal momenta, restricted near the FS, can appear collinear with one another. Toestimate the scale of the the contribution from the collinear regime, one views this regime asa quasi-one-dimensional system [6]. Take the self-energy Σ for example. In d = 1 dimension,the self-energy Σ ∼ p ln p – this leads to the well-known non-Fermi liquid behavior. In150igher dimensions, quasi-one-dimensional power counting estimates the contribution fromthe collinear regime to be ∼ ( p ) d ln p . For d = 2 this is less suppressed than the usual( p ) .Denote the collinear regime contribution to Σ as δ Σ; the leading contribution has threeintermediate collinear on-shell fermions. Based on the quasi-one-dimensional power count-ing, along with the general analyticity requirements that − Σ R ( ω, (cid:126)p ) is analytic in the ω upper-half-plane and − ImΣ R > − δ Σ R ( ω, (cid:126)p ) = ia ( (cid:126)p ) ω ln( − iω/p F ) + (higher order contributions) (4.173)for small ω in the upper-half-plane. Here p F is the size scale of the FS, and a is positive; thebranch cut of ln is placed along the negative real axis. Taking ω = p + i(cid:15) gives − δ Σ R ( p ).Unfortunately, when ξ ( (cid:126)p ) is of the same order as p , such parametrization is wrong. It isknown [17, 18] that − δ Σ R has complicated dependence on p and p − ξ ( (cid:126)p ), but still scalesas ( p ) ln p when ξ ( (cid:126)p ) and p are of the same order.Now that there is no simple way to parametrize − δ ImΣ, there is no simple way toparametrize the decay factor − D (despite the 1 subscript, here it also involves terms oforder q ln q ), because the latter can be expressed in terms of the former, with p being order q and p − ξ being order v µ q µ . The collinear regime contributions to − C ph and − C pp areterms of order q ln q proportional to δ d +1 ( p ∓ k ) (respectively corresponding to forward andback scattering); this terms have no simple parametrization either.Although the collision term in d = 2 is not parametrized, we know (4.74) must still hold,as it is required by the Ward-Takahashi identity.151 ollision being Uninteresting Finally we want to show that collisions are “uninteresting” to the focus of this thesis – thecollision term C has no contribution to the antisymmetric part Π [ µν ] of the current-currentcorrelation, and therefore has no contribution to the anomalous Hall effect or the chiralmagnetic effect. First, we note that C (despite the 1 subscript, in d = 2 it also involvesterms of order q ln q ) has a special property C ( p, k ; q ) = C ( k, p ; q ) . (4.174)In particular, the D term in C has this property simply because it is proportional to δ d +1 ( p − k ). On the other hand, inspecting the exact expressions (4.166) and (4.168), wesee C ph and C pp are symmetric under p ↔ k up to the q -dependences in the V ’s. But the q -dependences in the V ’s can be neglected, for theirs effects would be further suppressed byorder q , i.e. contribute to order q (and order q ln q in d = 2) which are beyond order q .Therefore (4.174) holds to order q – including in d = 2, since the argument here did not relyon power counting. The collision contribution to i ( i ImΠ µν ) is given by − (cid:90) p,k i ReΓ µ ( p ; − q ) ( − C ( p, k ; q )) i ReΓ ν ( k ; q ) − (cid:90) p,k i (cid:0) i ImΓ µ (cid:1) ( p ; − q ) ( − C ( p, k ; q )) i ( i ImΓ ν ) ( k ; q ) . (4.175)The q -dependence in Γ µ comes from ∆ (cid:48) which is even in q , so Γ µ is also even in q . Thus,due to (4.174), the expression above is symmetric in µν . Now that C does not contributeto ImΠ [ µν ] , by the Kramers-Kronig dispersion relation it has no associated contribution toReΠ [ µν ] either, and thus our claim is proven.152 .4.7 Some Cancellation of Diagrams In the presentation of the proof, there are two technical diagrammatic cancellations that areleft unproven. One is (4.176) in the EM dipole moment, and the other is the vanishing ofthe (cid:101) V term in (4.138) in the Hall conductivity tensor. Here we complete these technicalsteps. In Electromagnetic Dipole Moment We explicitly show from Feynman diagrams the second line of (4.127) is antisymmetric in µν . Consider diagrams in the q -2PI sum i (cid:101) V α γδ, β ( p, k ; q ). We can separate these diagramsinto two types: • Type I: The q -2PI diagram has a fermion line at the top, running in with momentumand index ( p − q/ , δ ) and running out with ( k − q/ , γ ), and a fermion line at thebottom, running in with momentum and index ( k + q/ , β ) and running out with( p + q/ , α ). Type I diagrams are summed in (cid:101) V with plus sign. Some examples areshown below.For Type I diagrams, we can assign momenta on the internal propagators so that − q/ q/ q . • Type II: The q -2PI diagram has a fermion line on the left, running in with momentumand index ( p − q/ , δ ) and running out with ( p + q/ , α ), and a fermion line on the right,running in with momentum and index ( k + q/ , β ) and running out with ( k − q/ , γ ).153ype II diagrams are summed in (cid:101) V with minus sign, because of fermionic statistics.Some examples are shown below.For Type II diagrams, however we assign the internal momenta, q will in generalappear in some propagator(s) on both the left and right fermion lines, as well as onsome internal propagators in the middle.We want to show (cid:82) k (cid:0) i (cid:101) V (cid:1) α γδ, β ( p, k ; q ) ∂ νk iG βγ ( k ) is equal to q µ times a quantity antisym-metric in µν . We consider the Type I and Type II contributions separately.For Type I diagrams in i (cid:101) V ( p, k ; q ), when expanded to linear order in q , we pick onepropagator iG ( l ± q/ 2) on the bottom (top) fermion line and replace it with ( ± q µ / ∂ µl iG ( l ),and in all other propagators set q to zero; we sum up all possible ways of such expansion,and sum up all possible Type I diagrams. (One may wonder why the interaction vertices areindependent of q . We explain this at the end of this section.) As a result, Type I contributionto (cid:82) k (cid:0) i (cid:101) V (cid:1) α γδ, β ( p, k ; q ) ∂ νk iG βγ ( k ) can be summarized as q µ (cid:90) k (cid:90) l (cid:0) i (cid:101) Y (cid:1) α ξ γδ, ζ, β ( p, l, k ) ∂ µl iG ζξ ( l ) ∂ νk iG βγ ( k ) , (4.176)where i (cid:101) Y is a sum of connected diagrams: • Diagrams satisfying the following conditions are summed in i (cid:101) Y with plus sign:There is a fermion line running in with momentum and index ( p, δ ) and running outwith ( k, γ ), a fermion line running in with momentum and index ( k, β ) and runningout with ( l, ξ ), and a fermion line running in with momentum and index ( l, ζ ) andrunning out with ( p, α ). 154oreover, among the internal fermion propagators on these three fermion lines, none ofthem is dictated by momentum conservation to have momentum p, k or l , and no pairof them is dictated by momentum conservation to have same momenta. Equivalently,among those fermion propagators, one cannot cut any one or two of them to disconnectthe diagram. kpl kp l βγδα ζ ξ For example, the diagram on the left contributes to i (cid:101) Y , while the three on the rightdo not. • Diagrams satisfying the conditions above, but with ( k, γβ ) and ( l, ξζ ) switched, aresummed in i (cid:101) Y with minus sign.Notice that i (cid:101) Y is a totally antisymmetric 3-tensor in the double fermion linear space, i.e. itis antisymmetric under the exchange of any two of ( p, αδ ), ( k, γβ ) and ( l, ξζ ). Thus, Type Icontribution is antisymmetric in µν .For Type II diagram contribution, we use a diagrammatic technique developed by Ward [ ? ].Consider, for instance, the diagram below.Let us call this a “prototype diagram”. Let us call the loop on the right the k -loop, whoseloop momentum is assigned k ; it consists of five fermion propagators. A Type II diagram155ontribution to (cid:82) k i (cid:101) V ( p, k ; q ) ∂ νk iG ( k ) is obtained by picking one propagator iG on the k -loop – in the example above there are five ways to do so – and replacing it with ∂ νk iG , andthen letting the external momentum q flow-in through it. The sum of all these five resultingdiagrams forms a “prototype class” associated with the above prototype diagram. For eachprototype class, we can fix one interaction propagator (it maybe an auxiliary propagator),through which q in the k -loop flows out towards the left; for instance, in the example above,we may assign internal momenta so that q always flows out the k -loop along the dashed lineon the top. Now, for each Type II diagram in this prototype class, we expand q at firstorder (since we are looking at (cid:101) V ). This corresponds to picking one internal propagator(fermion or interaction) that has q in its argument, replacing it with q µ times its momentumderivative, and then in all other internal propagators set q to zero. Now: • If our picked q -dependent propagator is on the k -loop, then we have a ∂ νk iG ( k ) and a ∂ µk iG ( k ) on the k -loop, and summing up all such possibilities in the prototype classyields a quantity antisymmetric in µν , in a manner similar to the Type I diagramcontribution. • If our picked q -dependent propagator is not on the k -loop, then there is only one ∂ νk iG ( k ) on the k -loop, and summing up all such possibilities is equivalent to taking atotal k -derivative on the k -loop (and k is integrated over later). So the sum of suchpossibilities vanishes.Thus, Type II diagram contribution to (cid:82) k ( ∂ µq i (cid:101) V ( p, k ; q )) ∂ νk iG ( k ) is also antisymmetric in µν . (There is a small caveat in the use of prototype diagrams. For example the prototypebelow 156as a symmetry of exchanging the two propagators on the k -loop. So when relating it toType II contribution by replacing one iG on the k -loop with ∂ νk iG , we need an extra factorof 1 / 2. Clearly this does not affect the final antisymmetry in µν .)A left-over subtlety has to be addressed: When we were expanding the momentum run-ning along a fermion line, we did not have contribution from the interaction vertices on thefermion line. Why is that? Recall our assumption (for simplicity, not for principle) about theQFT that any bare interaction vertex has no coupling to A , i.e. there is no e.g. Aφψ † ψ barevertex or Aψ † ψψ † ψ bare vertex. By EM U (1) gauge invariance, this also means the bare in-teraction vertices cannot depend on the momentum running along the charged fermion line.So a bare interaction vertex at most depends on the momentum running along the neutralinteraction lines (which maybe auxiliary), for example in ( ψ † ψ ) ∂ x φ or in ( ψ † ψ ) ∂ x ( ψ † ψ ). In Hall Conductivity Tensor In this section we show σ µνλY ≡ ( i∂ µp G ) T ∂ νq i (cid:101) V ( q ) i∂ λk G, (4.177)which appears in (4.138), vanishes. According to the previous discussion, we separate thecontributions of Type I diagrams and Type II diagrams in i (cid:101) V . In σ Y , since ∂iG is contractedon both sides, it is easy to see Type II contribution vanishes using Ward’s method presentedpreviously – at least one of the k -loop and the p -loop involves a total derivative. We are leftwith Type I contribution to σ Y . By (4.176) we can express it as σ µνλY = 12 (cid:90) k (cid:90) l (cid:90) p (cid:0) i (cid:101) Y (cid:1) α ξ γδ, ζ, β ( p, l, k ) ∂ µp iG δα ( p ) ∂ νl iG ζξ ( l ) ∂ λk iG βγ ( k ) . (4.178)We can describe diagrams in σ µνλY as the following: • The diagram has a fermion loop, which we call the outer loop (formed by connecting the157hree fermion lines in i (cid:101) Y with the three differentiated propagators). We redefine p sothat it is now the loop momentum running around the outer loop. Interaction verticesseparate the outer loop into n ≥ ∂ µp iG , ∂ νp iG and ∂ λp iG , the remaining n − iG .Moreover, the interaction lines inside make the outer loop 2PI; that is, among allsegments on the outer loop, no pair of them are dictated by momentum conservationto have the same momentum.If ∂ µp , ∂ νp , ∂ λp appear on the outer loop in the cyclic order against the fermion arrow,then the diagram is summed in σ µνλY with coefficient − / (cid:101) Y ). If they appearon the outer loop in the cyclic order along the fermion arrow, then the diagram issummed with coefficient +1 / σ µνλY = 0, below we introduce four notions.First, let us be blind between iG and ∂ p iG on the outer loop. Then we are led to considerprototype diagrams like this one .A prototype diagram defines a prototype class: Diagrams contributing to σ Y are in the sameprototype class if, after ignoring the distinction between ∂ p iG and iG on the outer loop, theyreduce to the same prototype diagram. In fact, the sum S µνλ (we drop the µνλ indices fromhere on) of diagrams (with coefficients ± / σ Y = 0, because clearly diagrams in σ Y arepartitioned into prototype classes. 158econd, let us fix a prototype class, and consider the placement of the three ∂ p iG ’s onthe outer loop. For simplicity, in the below we will restrict to prototype classes with nosymmetry factor (there will be a symmetry factor of 1 /n s if the prototype diagram has a Z n s cyclic symmetry with respect to the outer loop, where n s divides n ); we will return tothe case with symmetry factor later. Now consider for example the diagram.This diagram represents the sum (with coefficients ± / ∂ p iG ’s appear on the three thickened segments.This sum is manifestly antisymmetric in µνλ . Let us call the set of diagrams contributing tosuch sum an “antisymmetrization class”. Obviously a prototype class can be partitioned intoantisymmetrization classes. The purpose of introducing this notion is only for introducingthe next notion.The third notion to introduce is a partitioning finer than a prototype class (still, werestrict to those without symmetry factor) but coarser than an antisymmetrization class.In an antisymmetrization class, the three ∂ p iG ’s are separated by a number of iG ’s, forexample, in the previous antisymmetrization diagram, the three ∂ p iG ’s are separated by 0 , iG ’s. But there are other antisymmetrization diagrams whose three ∂ p iG ’s are alsoseparated by 0 , iG ’s. Let us introduce the notation (012), which represents the sumof them: .In general, we call the set of diagrams contributing to ( abc ) a “cyclic class”. The name isbecause ( abc ) is by definition the same object as ( bca ) and ( cab ). Clearly, a prototype class159s partitioned into cyclic classes, and a cyclic class is partitioned into antisymmetrizationclasses. It is easy to see the sum S of diagrams in the prototype class can be expressed as S = (cid:48) (cid:88) a + b + c = n − n · s abc · ( abc ) , s abc = / a = b = c a, b, c are non-negative integers, and the prime on the sum means we only count( abc ), ( bca ) and ( cab ) once because they are the same object. We will see soon that byintroducing the notion of cyclic class, we boil the Feynman diagram cancellation problem toa combinatorial problem.In general ( abc ), the sum of diagrams in a cyclic class, is not equal to zero. We need tointroduce the fourth notion that bridges between prototype class and cyclic class; but unlikethe three notions above, this fourth notion is not a partitioning. Given the prototype class,consider the following diagrams with two ∂ p iG ’s: along the fermion arrow on the outer loop,we have ∂ µp iG , then m iG ’s, then ∂ νp iG , and then the remaining ( n − − m ) iG ’s. We assume m ≤ n − − m . Sum up all such diagrams. But before we integrate over the outer loopmomentum p , we take a total ∂ λp derivative. Then we totally antisymmetrize between µνλ and multiply by 3. We denote the result by (cid:104) m (cid:105) . By construction, (cid:104) m (cid:105) = 0 due to the total160erivative. But on the other hand, it is easy to see (cid:104) m (cid:105) is a sum of cyclic classes: (cid:104) m (cid:105) = (cid:48) (cid:88) a + b + c = n − s mabc · ( abc ) , (4.180) s mabc = a, b, c is m ,and the other two do not sum up to m − 11 + 1 = 2 if two of a, b, c are m , the other is not m a = b = c = m − a, b, c sum up to m − 1, and the other is not m − a, b, c is m , and the other two sum up to m − 10 otherwise . We used the fact that the vertices on the outer loop are independent of p , whose reason isexplained at the end of the previous section. Diagrams contributing to a fixed (cid:104) m (cid:105) do notform an equivalence class, because clearly a given ( abc ) can appear in several different (cid:104) m (cid:105) ’s.Now that every (cid:104) m (cid:105) is equal to 0, it would be desirable to express S as a linear combinationof (cid:104) m (cid:105) ’s. Indeed, now we shall show that S = m ≤ ( n − / (cid:88) m =0 ( n − − m ) · (cid:104) m (cid:105) = 0 . (4.181)This is a simple combinatorial problem that can be shown by matching the coefficient ofeach ( abc ) on both sides, using equations (4.179) and (4.180). Due to the cyclic property of( abc ), we can assume a ≤ b , a ≤ c . Then we discuss over 8 possibilities:1. a < b < c < ( n − / 2: On the right-hand-side, ( abc ) appears in three different (cid:104) m (cid:105) ’s:161 m = a : The coefficient of ( abc ) is ( n − − a ) · • m = b : The coefficient of ( abc ) is ( n − − b ) · • m = c : The coefficient of ( abc ) is ( n − − c ) · n , matches with the coefficient on the left-hand-side.The case a < c < b < ( n − / a < b < c = ( n − / 2: On the right-hand-side, ( abc ) appears in three different (cid:104) m (cid:105) ’s: • m = a : The coefficient of ( abc ) is ( n − − a ) · • m = b : The coefficient of ( abc ) is ( n − − b ) · • m = c : The coefficient of ( abc ) is 0.The sum of these coefficients is n , matches with the coefficient on the left-hand-side.The case a < c < b = ( n − / a < b < ( n − / < c : On the right-hand-side, ( abc ) appears in three different (cid:104) m (cid:105) ’s: • m = a : The coefficient of ( abc ) is ( n − − a ) · • m = b : The coefficient of ( abc ) is ( n − − b ) · • m − a + b : The coefficient of ( abc ) is ( n − − a + b + 1)) · ( − n , matches with the coefficient on the left-hand-side.The case a < c < ( n − / < b works in the same manner.4. a = b < c < ( n − / 2: On the right-hand-side, ( abc ) appears in two different (cid:104) m (cid:105) ’s: • m = a = b : The coefficient of ( abc ) is ( n − − a − b ) · • m = c : The coefficient of ( abc ) is ( n − − c ) · n , matches with the coefficient on the left-hand-side.The case a = c < b < ( n − / a = b < c = ( n − / 2: On the right-hand-side, ( abc ) appears in two different (cid:104) m (cid:105) ’s: • m = a = b : The coefficient of ( abc ) is ( n − − a − b ) · • m = c : The coefficient of ( abc ) is 0.The sum of these coefficients is n , matches with the coefficient on the left-hand-side.The case a = c < b = ( n − / a = b < ( n − / < c : On the right-hand-side, ( abc ) appears in two different (cid:104) m (cid:105) ’s: • m = a = b : The coefficient of ( abc ) is ( n − − a − b ) · • m − a + b : The coefficient of ( abc ) is ( n − − a + b + 1)) · ( − n , matches with the coefficient on the left-hand-side.The case a = c < ( n − / < b works in the same manner.7. a < b = c < ( n − / 2: On the right-hand-side, ( abc ) appears in two different (cid:104) m (cid:105) ’s: • m = a : The coefficient of ( abc ) is ( n − − a ) · • m = b = c : The coefficient of ( abc ) is ( n − − b − c ) · n , matches with the coefficient on the left-hand-side.8. a = b = c : On the right-hand-side, ( abc ) appears only when m = a = b = c , withcoefficient ( n − − n − / · n/ 3. This matches with the coefficient on theleft-hand-side.This completes our proof, for prototype classes that have no symmetry factor.163or a prototype class whose prototype diagram has a Z n s symmetry with respect to theouter loop, we can pick one segment on the outer loop to be “the special segment”; in thesum of diagrams, this leads to over-counting by a factor of n s . But now that there is no Z n s symmetry any more, we can show the sum is zero as before. The factor of n s has no effecton the zero. In this chapter, we have extended Landau’s Fermi liquid theory to incorporate Berry cur-vature effects. Among other effects, we showed the anomalous Hall conductivity receivestwo contributions: the non-quasiparticle Hall conductivity tensor, plus the contribution ofelectric dipole moment of the excited quasiparticles. The latter can be viewed a new effectdue to interactions, because a non-interacting fermion usually has no intrinsic electric dipole.As for the former, it can be further separated into a chemical potential dependent part anda chemical potential independent part. Remarkably, we showed the chemical potential de-pendent part is a Fermi surface property given by the Berry curvature around the Fermisurface, as in non-interacting Fermi gas. On the other hand, some puzzles about chemicalpotential independent part remain, as we will comment about below.One can see several directions to extend our theory. First, we assumed the Fermi levelcrosses only one band; one can generalize this to multiple bands. Two scenarios are ofparticular physical interest: either that the multiple bands crossing the Fermi level arecompletely degenerate [62], or that these multiple bands have completely disjoint Fermisurfaces. The generalization of our theory to both scenarios is straightforward. Second,our discussion is limited to linear response. It would be interesting to extend the scope ofthe kinetic theory to include also nonlinear response, so that important effects such as the(3 + 1) d chiral anomaly can be captured. Third, we assumed that the quantum field theorydescribing the fermions does not have couplings of the type Aφψ † ψ . It would be interesting164o see if the kinetic theory can be extended to include couplings of this type in the QFT.Also, one may try to understand if long-ranged interactions can be included, to the extentthat these interactions do not destroy the Fermi liquid ground state.Some interesting questions are raised in the context of our Berry Fermi liquid theory. Wehave found that, beside a (cid:15) F -dependent piece, the Hall conductivity contains a constant piece σ µνλo in Section 4.3.4. Is this contribution topological and not renormalized by interactions?Does it receive jumps at discrete values of (cid:15) F around which the FS develops new disconnectedcomponents (e.g. in Figure 4.1)? In gapped system, σ µνλo = σ µνλr is topological [34].More broadly, one may ask: Is it possible to have a notion of topologically equivalent/ distinct Fermi liquids? For example, are the Fermi liquids in normal metal and in Weylmetal topologically distinct under some notion? In this thesis, we see both a puzzle and ahint regarding such notion. The puzzle is the possible jump of σ µνλ mentioned previously.The hint is the manifestation of anomaly-related transport effects in the distinction between(4.41) and (4.37). It would be interesting to study if such problems can be covered under acoherent framework.We note also that the matching between the microscopic theory and the Fermi liquidtheory is done here at the level of dynamical equations. If there is a way to do the matchingat the level of action and path integral measure, like that in Berry Fermi gas [14] (see Section2.4), it would provide a much more transparent derivation of the Berry Fermi liquid theory.It may also help to extend the kinetic theory beyond linear response.Finally, given the generality of the assumptions, the formalism should have broad appli-cations in physical systems. It would be interesting if predictions of the Berry Fermi liquidtheory can be directly compared to experiments.165 HAPTER 5CONCLUSION In this thesis we first studied general Berry Fermi gas theory and its microscopic justifica-tion. Then we studied the chiral kinetic theory of Weyl fermions as a specific example, anddiscussed how Lorentz invariance is non-trivially realized in chiral kinetic theory. Finallywe studied general Berry Fermi liquid theory and its microscopic justification. The Berryphase physics being non-trivial and interesting in these systems should now be evident. Atthe end of each chapter, we have a “Summary and Outlook” section, summarizing the im-portant ideas and results in that chapter, and more importantly, discussing future directionsof study relevant to that chapter. We are not going to repeat those summarizations anddiscussions here. Here we discuss some more general possibilities.From our study of Berry Fermi liquid, it is clear that to define momentum space Berrycurvature, all it needs is the eigenvector u of the full propagator; the eigenvalue part of thepropagator is irrelevant to the definition. This suggests we should be able to define themomentum space Berry curvature even for non-Fermi liquids. Then, we should wonder whatwe can say about Berry curvature in such systems. For example, at least for non-Fermiliquids that are “sufficiently similar” to a normal Fermi liquid (for instance those whose p in the denominator of the propagator is renormalized to the form p ln( p )), can we stillexplore the relation between anomalous Hall effect and Berry phase? If something can besaid, it will not only be theoretically satisfactory, but may also have potential applicationvalue to important systems such as the composite Fermi liquid at half-filled Landau level.Another problem worth thinking about is a more theoretical one. In d spatial dimensionsthe Berry curvature defect where ∂ [ ip b jk ] (cid:54) = 0 is generally d − d > EFERENCES [1] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski. Methods of quantum field theoryin statistical physics . Dover, 1975.[2] Stephen L. Adler. Axial vector vertex in spinor electrodynamics. Phys.Rev. , 177:2426–2438, 1969.[3] Stephen L Adler and William A Bardeen. 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