Dirac Fermions in Solids - from High Tc cuprates and Graphene to Topological Insulators and Weyl Semimetals
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Dirac Fermions in Solids — from High Tc cuprates and Grapheneto Topological Insulators and Weyl Semimetals.
Oskar Vafek
National High Magnetic Field Laboratory and Department of Physics,Florida State University, Tallahassee, Florida 32306, USA
Ashvin Vishwanath
Department of Physics, University of California, Berkeley, California 94720, USA
Abstract
Understanding Dirac-like Fermions has become an imperative in modern condensed matter sci-ences: all across its research frontier, from graphene to high T c superconductors to the topologicalinsulators and beyond, various electronic systems exhibit properties which can be well describedby the Dirac equation. Such physics is no longer the exclusive domain of quantum field theoriesand other esoteric mathematical musings; instead, real physics of real systems is governed by suchequations, and important materials science and practical implications hinge on our understandingof Dirac particles in two and three dimensions. While the physics that gives rise to the masslessDirac Fermions in each of the above mentioned materials is different, the low energy properties aregoverned by the same Dirac kinematics. The aim of this article is to review a selected cross-sectionof this vast field by highlighting the generalities, and contrasting the specifics, of several physicalsystems. ONTENTS
I. Dirac, Weyl, and Majorana 2II. When and why to expect Dirac points in condensed matter? 4A. Dirac points and Kramer’s pairs 6B. Fermion doubling: Nielsen-Nynomiya theorem and ways around it 71. Domain wall Fermions and 3D topological insulators 8III. Dirac particles subject to external perturbations 9IV. Many-body interactions 16V. Applications to various physical systems 20A. Graphene 201. Coupling to external fields 22B. Surface states of a 3D topological insulator 231. Coupling to external fields, interaction and disorder effects 25C. d x − y -wave superconductivity in copper oxides 29VI. Weyl Semimetals 33A. Topological Properties 34B. Physical Realizations 36VII. Summary 37VIII. Acknowledgments 38References 38 I. DIRAC, WEYL, AND MAJORANA
I think it is a peculiarity of myself that I like to play about with equations,just looking for beautiful mathematical relations which maybe don’t have anyphysical meaning at all. Sometimes they do. - Paul A. M. Dirac (1902 - 1984)ublished in 1928 by Paul Dirac[1], the eponymous equation is among the finest achieve-ments of human intellect. The equation, now taught in virtually every physics departmentaround the world, has brought together Einstein’s special theory of relativity and quantummechanics. It led to the prediction of antimatter, namely the positron as the electron’santi-partner. It casted the spin-1 / i ~ ∂∂t ψ = (cid:0) c α · p + βmc (cid:1) ψ, (1)where the momentum operator p = − i ~ ∇ = ( p x , p y , p z ), m is the mass of the particle, c is the speed of light in vacuum, and ψ is a 4-component object, a spinor. There are manyequivalent ways to write down the Dirac 4 × ? ], one such way is α = ( τ ⊗ σ , τ ⊗ σ , τ ⊗ σ ), and β = − τ ⊗
1. Theequation was originally intended for the electron, which is, of course, a massive, spin-1 / α matrices are block diagonal, while the term proportional tothe mass is block off-diagonal. Therefore, if we consider massless particles, the right-hand-side of the Dirac equation no longer couples the upper two components of ψ , let’s call them χ + , and the lower two components, χ − . Thus, with m = 0, it can be written in a simplerform i ~ ∂∂t χ ± = ± c σ · p χ ± . (2)This is the Weyl equation[3] and χ ’s are referred to as Weyl Fermions.Both of these equations involve real and complex numbers. Majorana noticed[4] that it ispossible to write the Dirac equation — including the mass term — entirely in terms of realnumbers[2]. This can be accomplished by choosing the α matrices to be purely real and the β matrix to be purely imaginary, because then both the right-hand-side and the left-hand-sideof the Dirac equation are purely imaginary. For example, α = ( − τ ⊗ σ , τ ⊗ , − τ ⊗ σ ),and β = τ ⊗ σ does the job. Once the equation is purely real, its solutions can also bechosen to be purely real. In quantum field theory, a real field describes a particle which isits own antiparticle.his review is about how such equations provide an accurate description of some 2- and3-dimensional non-relativistic systems, where Dirac or Weyl Fermions emerge as low energyexcitations. It is also about how these excitations behave when subjected to external fields,and how to relate the perturbing “potentials” (e.g. scalar, vector, mass etc.) appearing inthe effective Dirac equation to either externally applied fields produced in a laboratory, orto defects and impurity potentials. A few consequences of many-body interactions will alsobe reviewed. We will not discuss any of the fascinating aspects of Majorana Fermions incondensed matter; this topic has already been covered in Ref.[5] and references therein. Themain topics of this paper form a vast area of physics, and we ask the reader to keep in mindthat it is impossible to do it justice in the review with a given allotted space. II. WHEN AND WHY TO EXPECT DIRAC POINTS IN CONDENSED MAT-TER?
In a non-relativistic condensed matter setting, the time evolution of any many body state | Ψ i is governed by the Schrodinger equation i ~ ∂∂t | Ψ i = H| Ψ i (3)where H is the Hamiltonian operator. This Hamiltonian contains the kinetic energy of theelectrons and ions, as well as any interaction energy among them. Our aim is to illustratehow and when we may expect the relativistic-like Dirac dispersion to arise from H in a coldnon-relativistic solid state. We do so first by pure symmetry considerations and then in abrief survey of several physical systems realizing Dirac-like physics. We will assume thatthe heavy ions have crystallized and to the first approximation let us ignore their motion.As such, their role is solely to provide a static periodic potential which scatters the electronSchrodinger waves and, if the spin-orbit coupling is also taken into account, the electronspins. Then H → H + H int , where H includes all one body effects and H int all many-bodyelectron-electron interaction effects.According to the Bloch theorem, the energy spectrum E n ( k ) and the eigenstates | φ n, k i of H can be described by a discrete band index n as well as a continuous D -dimensional vector k , the crystalline momentum, which is defined within the first Brillouin zone. Consider nowtwo distinct but adjacent energy bands E n + ( k ) and E n − ( k ), and assume that for some rangef k the two bands approach each other, i.e. the energy difference | E n + ( k ) − E n − ( k ) | is muchsmaller than the separation to any one of the rest of the energy bands. One way to derive theeffective Hamiltonian for the two bands is to start with a pair of (orthonormal) variationalBloch states, | u k i and | v k i , consistent with, and adapted to, the symmetries of H . Thenthe effective Hamiltonian takes the form H eff = X k ψ † k H ( k ) ψ k (4)where the first component of the creation operator ψ † k adds a particle (to the N -body state)in the single particle state | u k i and antisymmetrizes the resulting N +1-body state. Similarly,the second component creates a particle in the state | v k i and H ( k ) = h u k |H | u k i h u k |H | v k ih v k |H | u k i h v k |H | v k i ≡ f ( k )1 + X j =1 g j ( k ) σ j (5)where 1 is a unit matrix and σ j are the Pauli matrices. The corresponding one particlespectrum is E ± = f ( k ) ± vuut X j =1 g j ( k ) . (6)For a general k -point and in the absence of any other symmetries, g j ( k ) = 0 for each j . Itis clear from the expression for E ± ( k ) that the two bands touch only if g j ( k ) = 0 for each j at some k .In 3D, we can vary each of the three components of k and try to find simultaneous zerosof each of the three components of g j ( k ). To see that this may be possible without fine-tuning, note that in general each one of the three equations g j ( k ) = 0 describes a 2D surfacein k -space. The first two surfaces may generally meet along lines, and such lines may thenintersect the third surface at points without additional fine-tuning. If such points exist,they generally come in pairs and the dispersion near each may be linearized. The effectiveHamiltonian near one such point k takes the form H ( k ) = E k + ~ v · ( k − k )1 + X j =1 ~ v j · ( k − k ) σ j . (7)If v = 0 and the three velocity vectors v j are mutually orthogonal this has the form ofan anisotropic Weyl Hamiltonian. Of course, far away from k both bands may dispersepwards or downwards, in which case even if the Fermi level could be set to E ( k ), therewould be additional Fermi surface(s).In 2D, only two components of k can be freely varied, and therefore it is impossible to findsimultaneous zeros of three functions g j ( k ) without additional fine-tuning. Simply stated,in general, three curves do not intersect at the same point. Therefore, in the absence ofadditional symmetries that may constrain the number of independent g j ( k )’s, the two levelswill avoid each other. A. Dirac points and Kramer’s pairs
We have intentionally refrained from any discussion of the electron spin degeneracy, ortime reversal symmetry, which were not assumed to be present in the above discussion. Fora number of physical systems considered later on, the product of the time reversal and thespace inversion leaves the crystalline Hamiltonian invariant. This symmetry implies that,at each k , every electronic level is doubly degenerate, because if φ k ( r ) is an eigenstate,then so is its orthogonal Kramers partner, iσ φ ∗ k ( − r ), where σ acts on the spin part ofthe wavefunction. Therefore, the appropriate variational quadruplet of mutually orthogonalstates describing two nearby bands can be constructed from u k ( r ) | ↑i + u k ( r ) | ↓i , itsKramers partner − u ∗ k ( − r ) | ↓i + u ∗ k ( − r ) | ↑i , and v k ( r ) | ↑i + v k ( r ) | ↓i , with its partner − v ∗ k ( − r ) | ↓i + v ∗ k ( − r ) | ↑i . In this four-dimensional subspace H ( k ) = f ( k )1 + X j =1 g j ( k )Γ j (8)where Γ = τ ⊗
1, Γ = τ ⊗
1, Γ = τ ⊗ σ , Γ = τ ⊗ σ , and Γ = τ ⊗ σ ; the firstPauli matrix acts within the u , v space and the second within the Kramers doublets. Whilethe corresponding one particle spectrum, E ± = f ( k ) ± qP j =1 g j ( k ), exhibits a two-folddegeneracy at any k , an intersection of two Kramers pairs requires finding simultaneous zerosof five g j ( k )’s. Clearly, the bands avoid each other because, even in 3D, this condition cannotbe satisfied without additional symmetry. For example, if the spin-orbit interaction can beneglected and time reversal symmetry is preserved — based on our earlier assumptions,this also implies that space inversion is preserved — then the spin SU (2) symmetry forces g = g = g = 0. With such additional symmetry, in 3D, the accidental degeneracy mayhappen along 1D k -space curves and in 2D, at nodal points. . Fermion doubling: Nielsen-Nynomiya theorem and ways around it The Nielsen-Nynomiya theorem states that it is impossible to construct a non-interactinglattice hopping model with a net imbalance in the number of (massless) Dirac Fermions withpositive and negative chirality, provided that certain weak restrictions apply. For example,the translationally invariant hopping amplitudes are assumed to decay sufficiently fast sothat in momentum space the Hamiltonian is continuous. The full proof[6] makes use ofhomotopy theory and is beyond the scope of this review; pedagogical discussion of this “no-go” theorem can be found in [7]. Here we will illustrate the basic idea behind it in a simpleexample in two space dimensions.Consider a model with two bands which may touch, such as the one given in Eq.(5) with g ( k ) = 0. Then, g ( k ) and g ( k ) are smooth periodic functions of k x and k y . If the firstfunction vanishes along some curve in the Brillouin zone, say the one marked by red in Fig.1,and the second vanishes along another curve, blue in Fig.1, then the places where the twocurves intersect correspond to massless Dirac Fermions. Periodicity guarantees that anyintersection must occur at an even number of points, corresponding to an even number ofmassless Fermions; just touching the two curves does not produce a Dirac Fermion becauseat least one component of the velocity vanishes. Importantly, there is an equal number ofpartners with opposite chirality.One way to remove half of the massless Fermions is to bring back g ( k ) and to force itto vanish at only half of the intersections of the red and the blue curves in Fig.1. Thisgaps out the unwanted Dirac points, leaving an odd number of gapless points. Haldane’smodel for a quantum Hall effect without Landau levels is a condensed matter examplewhere such an effect occurs along the phase boundaries separating quantum Hall phases andtrivial insulating phases[8]. HgTe quantum wells are another example[9]; there such “singlevalley” massless Dirac Fermions have been experimentally realized at the phase boundaryseparating the quantum spin Hall phase[10] and a trivial insulating phase. In the latticeregularization of the relativistic high energy theory, for which the space-time points arediscrete and separated by at least a lattice constant a , a similar term corresponds to theso-called Wilson mass term: a 4-momentum dependent mass, P j =0 ∆ (1 − cos( k j a ))), whichvanishes at k = 0 and ω = 0. Adding the Wilson mass results in only one massless Fermion,but it is not chiral. Moreover, in any condensed matter setting, making the k -dependent P P + -+ -+ +- - k x k y FIG. 1: Illustration of the Fermion doubling in the 2D lattice hamiltonian. The blue andred lines correspond to the solutions of g ( k ) = 0 and g ( k ) = 0, respectively. Both g ( k )and g ( k ) are smooth and must be periodic (for illustration only 4 Brillouin zones areshown). Note that there is always an even number of intersections unless the two curvesjust touch. If we think of the two signs as points in the complex plane, we see that thegapless points have opposite chirality. Imagine displacing, say, the blue curve down,holding the red curve fixed. The two points P and P will move towards each other, andmeet when the two curves touch. In this case, one of the Dirac velocities vanishes and wedo not have a Dirac Fermion at all. Therefore, in any lattice formulation with finite rangehopping, there will always be an even number of — in general anisotropic — masslessDirac Fermions with opposite chirality.mass term vanish at an isolated k -point requires fine tuning, and therefore such gaplesspoints generally correspond to phase boundaries as opposed to phases[8][10].
1. Domain wall Fermions and 3D topological insulators
Another way of avoiding the Fermion doubling on the lattice has been well known in highenergy theory[11][12]. Kaplan’s idea has been to start with massive Fermions and to makea mass domain wall along the non-physical 4th spatial dimension, hereby labeled by w . Byass domain wall we mean that for positive w the mass is m , and for negative w it is − m .For the w = 0 lattice site the mass vanishes. To this domain wall mass term add a Wilsonmass term. There is then a range of values of m for which we have a single chiral 3+1Dmassless Dirac, i.e. Weyl, particle on the domain wall. For m <
2∆ this can be understoodas the two sides having a mass inversion at only one k -point, namely at the origin. Thiswas proposed as a method to simulate — on a lattice — chiral Fermions in odd space-timedimensions: from 4+1D to 3+1D or from 2+1D to 1+1D.Unlike the Wilson mass, its condensed matter reincarnation is frequency independent, al-though of course momentum dependent. Massless domain wall Fermions have been discussedby Volkov and Pankratov at a 2D interface between (3D) SnTe and PbTe[13]. Such mass-less Dirac Fermions are similar to those appearing at the surface of strong 3D topologicalinsulators, although there is a difference: in the former case the mass sign change occurs atan even number of points in the Brillouin zone while in the latter at an odd number[14][15]. III. DIRAC PARTICLES SUBJECT TO EXTERNAL PERTURBATIONS
For relativistic Dirac Fermions described by 4-component spinors, external perturbationstake the form of space-time dependent 4 × V ( r , t ). In theHamiltonian formalism H = Z d r ψ † ( r ) (cid:0) c α · p + mc β + V ( r , t ) (cid:1) ψ ( r ) . (9)There are 16 linearly independent 4 × V ( r , t ). In arelativistic context, their physical meaning is determined by their properties under Lorentztransformations.1. If the matrix structure of V ( r , t ) is the same as β , it clearly acts as a space-time varyingmass; because it is a scalar under the Lorentz transformation it is also sometimesreferred to as a scalar potential[16].2. Any V ( r , t ) of the form − e α · A ( r , t ) acts as the spatial component of the electro-magnetic vector potential; it enters via minimal coupling.3. If V ( r , t ) = e Φ( r , t ), then it corresponds to the time component of the electro-magneticpotential, or electrical potential.. Of the 11 remaining matrices, 6 are Lorentz tensor fields, 4 are pseudo-vectors and 1is pseudo-scalar[16].Before proceeding, it is important to stress that the appropriate V — which describeshow Dirac Fermions in a given condensed matter system react to, say, an external physical magnetic field — depends on the system itself. For example, it is not the same in grapheneand d -wave superconductors. This will be elaborated on in later sections.As mentioned earlier, for massless Dirac Fermions the kinetic energy term, α · p , canbe chosen to be block diagonal. If the external perturbation V ( r , t ) does not couple thetwo Dirac points, then such perturbation is also block diagonal. In 2D — where p is a 2-component vector — within each 2 × A = (Φ , A x , A y ). A constant mass termopens a gap in the spectrum; this gap may close at the boundaries or defects, but persistsin their absence. Simply put, for any energy − m < E < m , the equation E = c p + m forces p to be imaginary and the corresponding states can at best be evanescent. A constantelectric potential, Φ, shifts the energy eigenvalues; the constant space components, A x or A y , shift the momentum. The situation is similar in 3D, except the 2 × E = −∇ Φ, accelerates charged massless Dirac particles and leadsto non-equilibrium phenomena; it produces charge electron-positron pairs out of the filledDirac sea via the Schwinger mechanism[19]. For massless Dirac particles in 2D such rate hasbeen calculated to be ∼ ( eE ) / [19][20] and, argued to lead to electrical current increasingas E / above a finite field scale below which it is E -linear[21][22][23].The effect of a static 1D plane-wave electrical potential, Φ( x, y ) = Φ cos( qx ), on 2Dmassless Dirac Fermions was considered in Ref. [24]. Based on our discussion, we intu- x E Ñ cq à d Ε N H Ε L FIG. 2: Integrated single particle density of states for a massless Dirac Fermion in 2Dsubject to a static 1D periodic electric potential Φ cos ( qx ), blue dots, where Φ = ~ cq ;solid line is for a free massless Dirac particle. Note the buildup of the spectral weightwhich is recovered only near the cutoff energy, much larger than the scale shown.itively expect that such potential locally shifts the Fermi energy away from the Dirac pointand introduces electron-positron “stripe puddles”. The energy spectrum has a particle-hole symmetry: for every eigenstate ψ E ( x, y ) with an energy E , there is an eigenstate σ ψ E ( x + π/q, y ) with an energy − E . For this result we assumed that the kinetic energyterm is c ( p x σ + p y σ ). The full quantum mechanical solution of this problem, performednumerically using a large number of plane-wave states, shows that, while the energy spec-trum remains gapless, the spectral weight is indeed shifted towards the Dirac point. This isshown in Figure 2, where we compare the integrated density of states, starting from E = 0,in the presence and absence of the periodic potential. Clearly there is an excess number ofstates at low energy. Interestingly, the “lost” states are recovered at energies comparable tothe cutoff, which is much larger than Φ . Analogous buildup of low energy density of statesunderpins the interpretation of the measured low temperature specific heat of type-II nodald-wave superconductors in an external magnetic field, discussed in a later section.On the other hand, a uniform magnetic field directed perpendicular to the 2D plane, B = ∂A y /∂x − ∂A x /∂y , quantizes the electron orbits. The resulting spectrum consists of dis- - - - - E (cid:144) W c FIG. 3: Single particle density of states (orange) for a 2D charged massless Dirac Fermionsubject to a uniform magnetic field, the Landau levels have been broadened for easiervisualization; green line is the density of states for the free Dirac particle. The (step-like)integrated density of states shows that the spectral weight is redistributed over the energywindow given by (cid:0) √ n + 1 − √ n (cid:1) Ω c where Ω c ≡ √ ~ c/ℓ B , where ℓ B = p ~ c/eB is themagnetic length.crete Landau levels at energies E n = sgn(n) p | n | Ω c where n = 0 , ± , ± , . . . , Ω c = √ ~ c/ℓ B ,and the magnetic length ℓ B = p ~ c/eB ; this result is easily obtained by elementary meth-ods, see for instance [25]. Therefore, unlike for a Schrodinger electron, the energy differencebetween the Landau levels of a massless 2D Dirac electron decreases with increasing en-ergy. Each Landau level is N -fold degenerate, where N = Area/ (2 πℓ B ); the degeneracy,being proportional to the sample area, is macroscopically large. As shown in Figure 3,the uniform magnetic field causes redistribution of spectral weight over the energy interval (cid:0) √ n + 1 − √ n (cid:1) Ω c ; the number of states which are ‘moved’ to the Landau levels equals tothe total number of states which would be present between the Landau levels in the absenceof the external B-field.The effects of a perpendicular magnetic field and an in-plane electric field have beenstudied in the context of proving the absence of the relativistic correction to quantum Halleffect in ordinary 2D electron gas[26]. The eigenfunctions and eigenvalues can be determined E Ñ cq à d Ε N H Ε L FIG. 4: Massless Dirac Fermion in 2D subject to the 1D periodic mass, m ( x, y ) = m cos( qx ) with m = cq . Note the suppression of the spectral weight, which isrecovered only near the cutoff energy, again, much larger than the scale shown.analytically, either directly[26], or, if B > E , by first Lorentz boosting the space-timecoordinates and the Dirac spinors into a frame in which the electric field effectively disappearsand only the Lorentz contracted magnetic field enters[27] [we discussed this simpler problemabove] and then ‘inverse’ Lorentz boosting the wavefunctions and eigenenergies.Effects of non-uniform Dirac mass are quite fascinating, particularly when the mass pro-file is topologically non-trivial and can lead to fractionalization of Fermion’s quantum num-bers. We will illustrate the effect for 1D Dirac particles, first published in 1976 by Ro-man Jackiw and Claudio Rebbi[28]. The kinetic energy and the mass term together give H JR = cσ p + σ m ( x ), where m ( x ) is fixed to approach ± m as x → ±∞ , vanishing oncesomewhere in between. One such kink configuration is, for example, m ( x ) = m tanh ( x/ξ ).The spectrum of H JR is particle-hole symmetric, because for any state ψ E ( x ) with en-ergy E , there is a state σ ψ E ( x ) with energy − E . As we argued earlier, any “midgap”state with − m < E < m must be localized. Let us therefore seek states at E = 0;they must satisfy i ~ cσ ψ ′ ( x ) = m ( x ) σ ψ ( x ). If we write ψ ( x ) = σ χ ( x ) and sub-stitute, then we find ~ cχ ′ ( x ) = m ( x ) σ χ ( x ). The solution now follows immediately: χ ( x ) = N exp (cid:2) ~ c R x dx ′ m ( x ′ ) σ (cid:3) χ (0). Since any χ (0) can be decomposed into a lin-ar combination of the +1 and − σ , we see that because the term in theintegral is positive, χ must be purely the − − i , otherwise the solutionis not normalizable. There is therefore a single isolated energy level at E = 0. For a generalsingle kink mass profile, there may be other mid-gap states, but they must come in pairs atnon-zero energies ± E .The remarkable consequence of this isolation is that if the E = 0 midgap state is empty,while all the negative energy states are occupied with charge e Fermions, then the resultingstate carries an excess localized charge of − e/ e/
2. This follows fromthe fact that a symmetric configuration of a widely separated kink and an anti-kink leadsto a pair of essentially zero energy states. In effect, one level has been “drawn” from the“conduction band” and one from the “valence band”, each of which are missing one state.If the zero energy doublet is unoccupied, then the total charge of this state differs from theconstant mass state by − e . Because the two localized states at the kink and the anti-kinkare perfectly symmetric, we must find that the total amount of charge in the vicinity ofeach kink is the same, namely, − e/ H JR asmall constant term proportional to σ , then the localized states carry irrational charge[29].Such ideas have fascinating applications to the physics of conducting polymers[30][31] andthere is an extensive literature on the subject reviewed in Ref.[32].In higher dimensions, the topologically non-trivial configurations also lead to zeromodes[28][33]. Just as in 1D, such results are insensitive to the details of the mass configu-ration, only the overall topology matters[34].As an illustration of an effect a non-topological configuration of the mass has on a 2Dmassless Dirac Fermion, we consider a 1D plane wave m ( x, y ) = m cos( qx ). The resultingHamiltonian, c ( p x σ + p y σ ) + m ( x, y ) σ , has a particle hole symmetry, in that for everyeigenfunction ψ E ( x, y ) with energy E , there is an eigenfunction σ ψ E ( x + π/q, y ) with energy − E . The momentum along the y -axis, k y , is conserved due to the translational symmetryn the y -direction. The momentum in the x -direction, k x , is conserved only modulo thereciprocal lattice vector. At k x = k y = 0 we can construct the E = 0 state explicitly, just aswe did for the Jackiw-Rebbi problem, but now both choices for χ lead to Bloch normalizablewavefunctions. There is therefore a doublet of states at k = 0 and E = 0. Away from k = 0,there is a new anisotropic Dirac cone, with renormalized velocities. Interestingly, at k = 0,the spectrum consists only of doublets at any energy because for every ψ E ( x, y ) there is σ ψ ∗ E ( x + π/q, y ) which is also at k = 0, has the same energy, and is orthogonal to ψ E ( x, y ).The overall effect on the integrated density of states is shown in Figure 4 for m = ~ cq .The minimum of the 2nd band is at E ≈ . ~ cq and is responsible for the change of slope.Overall, there is a suppression of the number of states at low energy — an opposite effectcompared to the electric potential case. Similarly, the “lost” states are recovered only atenergies comparable to the cutoff, which is much larger than m .To conclude this section, we briefly mention the chiral anomaly associated with the mass-less Dirac equation[35][36]. The anomalies in quantum field theory are a rich subject[37] andplay a very important role in elementary particle physics[38]. In order to illustrate the effect,note that the massless Dirac Hamiltonian in 3D and in the presence of an arbitrary externalelectro-magnetic field, R d r ψ † ( r ) (cid:0) c α · (cid:0) p − ec A ( r , t ) (cid:1) + e Φ( r , t ) (cid:1) ψ ( r ), formally commuteswith both the total particle number operator — or equivalently, the total charge operator — R d r ψ † ( r ) ψ ( r ), and the total “chiral” charge operator R d r ψ † ( r ) τ ⊗ ψ ( r ). Here we usedthe representation for α used in Eq.(1). The equation of motion for an operator O ( t ) in theHeisenberg picture is d O ( t ) /dt = [ O ( t ) , H H ( t )] /i ~ , where H H ( t ) is the Dirac Hamiltonian inthe Heisenberg representation. Because the commutator vanishes for both the total chargeand the total “chiral” charge, they should both be constants of motion. However, closerinspection reveals that in explicit calculations[35][36][38] an ultra-violet regularization mustbe adopted in order to obtain finite results. What’s more, if the regularization is chosen insuch a way as to maintain the conservation of charge — a physically desirable consequenceof a useful theory — then for some configurations of electromagnetic fields, the chiral chargeis not conserved and changes in time. As an illustration, one such configuration consists ofa uniform magnetic field along the z -direction and a parallel weak electric field[38]. Thiscan be described by Φ = 0 and A ( t ) = ( − By, , A z ( t )) where the electric field is given by − c ddt A z ( t ); the time variation of A z ( t ) is therefore slow. For a system with size L and peri-odic boundary conditions, the momentum is quantized in units of 2 π/L and the separationetween the adjacent energy levels is non-zero. If the rate of change of A z ( t ) is much smallerthan the separation of the energy levels, then we can use the adiabatic theorem, solve forthe eigen-energies using the instantaneous A z ( t ), and then monitor the energy spectrumin time. Such an energy spectrum is easily constructed once we notice that we are effec-tively dealing with ± σ · ( c p − e A ). These are just two copies — with opposite sign of theHamiltonian — of the Landau level problem of a massive Dirac particle in 2D, with themass set by c ~ k z − eA z ( t ). The spectrum for each is given by ± q ( c ~ k z − eA z ( t )) + n Ω c ,where n = 1 , , , . . . , together with the two anomalous levels, one for each chirality, at ± ( c ~ k z − eA z ( t )). If, at t = 0, we start with the many-body state where all negative en-ergy single-particle states are occupied and all positive energy ones are empty, and thenadiabatically increase A z from 0 to hc/eL , then, while their energy is changing, none ofthe anomalous single-particle states change because their phase is locked by the periodicboundary condition. Once A z reaches hc/eL , we can perform the gauge transformation thatremoves A z from the Hamiltonian and that is consistent with the periodic boundary condi-tions, and find that we end up with the many-body state which appears to differ from theinitial many-body state by the occupation of one additional negative chirality anomalousLandau level at energy hc/L and one fewer positive chirality Landau level at energy − hc/L .Note that the infinitely deep negative energy Dirac sea plays a key role in this argument.Since the degeneracy of each Landau level is L / πℓ B , we change the difference in the num-ber of the positive and negative chirality states, δN + − δN − , by − L / πℓ B ) ( eL/hc ) δA z .Relating δA z to the electric field, we find∆ N + − ∆ N − = 12 π e ~ c Z dt Z d r E · B . (10)This expression for the non-conservation of the total “chiral” charge is a direct consequenceof the Adler-Bell-Jackiw anomaly. IV. MANY-BODY INTERACTIONS
In all condensed matter applications, the velocity of the massless Dirac particles, v F ,is much smaller than the speed of light in vacuum, c . This difference is important whenmany-body interactions are considered, and therefore, from now on, we shall intentionallydistinguish between v F and c .n a 2D semi-metal such as graphene, we can imagine integrating out all high-energyelectronic modes outside of a finite energy interval about the Dirac point. The Fermi level isassumed to be close to the energy of the Dirac point. Since none of the gapless modes havebeen integrated out, there can be no non-analytic terms generated at long wavelengths,and in particular no screening of the 1 /r electron-electron interaction whose 2D Fouriertransform is, of course, non-analytic in momentum. Indeed, the long distance tail of thebare electron-electron interactions falls off as e / (4 πǫ d r ), where ǫ d is the dielectric constantof the 3D medium in which the graphene sheet has been embedded. At long distances, ǫ d is independent of the screening within the graphene sheet coming from the core carbonelectrons. This can be shown by solving an elementary electrostatic problem of a pointcharge inserted in the middle of an infinite dielectric slab of finite thickness placed in a3D medium with a dielectric constant ǫ d [39][40][41]. At distances much greater than thethickness of the slab, the Coulomb field within the slab is entirely determined by ǫ d . A finiteon-site Hubbard-like interaction is usually taken to model the very short distance repulsion.What then are the consequences of such electron-electron interactions if the Dirac pointcoincides with the Fermi level? The importance of each of the terms can be determinedby dimensional analysis: in 2D, the Dirac field scales as an inverse length and thereforethe short distance (contact) coupling g , multiplying four Dirac fields, has dimensions oflength. In any perturbative series expansion, each power of g must be accompanied by apower of an inverse length to maintain the correct dimensions of a physical quantity thatis being computed. Since it is critical, the only lengthscales in the problem are associatedwith finite temperature, i.e. the thermal length ~ v F /k B T , or the wavelength (frequency)of the external perturbation. As such length scales become very long, each term in theperturbative series in g becomes small and we expect the series to converge. In the parlanceof critical phenomena, the short range interaction is perturbatively irrelevant at the non-interacting (Gaussian) fixed point (see e.g. Ref.[42]). Therefore, while there can be finitemodifications of the Fermi velocity or of the overlap of the true (dressed) quasiparticle withthe free electron wave function, the asymptotic infrared properties of the model must beidentical to the non-interacting Dirac problem[43, 44].Using a similar analysis for the 1 /r tail of the non-retarded Coulomb interaction, onefinds that e / ( ǫ d ~ v F ) is dimensionless. Despite the superficial similarity with the 3+1DQED fine structure constant e / ~ c , the physics here is different. First of all, the charge,eing a coefficient of a non-analytic term in the Hamiltonian, does not renormalize whenhigh energy modes are progressively integrated out[45][46]. Any renormalization group flowof the dimensionless coupling e / ( ǫ d ~ v F ) must therefore originate in the flow of v F , whichis no longer fixed by the Lorentz invariance because such symmetry is violated by theinstantaneous Coulomb interaction. Detailed perturbative calculations reveal[47] that v F grows to infinity logarithmically at long distances thereby shrinking e / ( ǫ d ~ v F ). Physically,however, v F cannot exceed the speed of light c . Instead, once the retarded form of theelectron-electron interaction is properly included via an exchange of a (3D) photon, the flowof v F saturates at c . The resulting theory is quite fascinating, in that the 2D massless DiracFermions and the 3D photons propagate with the speed of light and, unlike in 3+1D QED,the coupling e / ~ c remains finite in the infra-red[47]. Unfortunately, since the flow of v F isonly logarithmic, and since initially there is a large disparity in the values of v F and c , sucha fixed point is practically unobservable. Instead, in practice, the physics is at best givenby the crossover regime in which v F increases, but never to values comparable to c .The 1 /r Coulomb interaction induced enhancement of the Fermi velocity is expected tolead to a suppression of the low temperature specific heat below its non-interacting value[48], as well as other thermodynamic quantities [49]. Interestingly, the suppression of thesingle particle density of states does not lead to a suppression of the ac conductivity; in thenon-interacting limit it takes a (frequency independent) value σ = N e / ~ where N is thenumber of the 2-component “flavors”. Again, the reason is the enhancement of the velocity:loosely speaking, while there are fewer excitations at low energy, those that are left have ahigher velocity and therefore carry a larger electrical current. The expression [50] for thelow frequency ac conductivity has the form σ ( ω ) = σ (cid:16) Ce / ( ~ v F + e log v F Λ ω ) (cid:17) , whereΛ is a large momentum cutoff. In the limit ω →
0, the correction to the non-interactingvalue is seen to vanish[49][50][51]. The value of the (positive) constant C in this expressionhas been a subject of debate as it seems to depend on the details of the UV regularizationprocedure[50][51][52] [53][54][55]. Recently, the calculation of C within a honeycomb tight-binding model [56], which provides a physical regularization of the short distance physics,found C = 11 / − π/ ≈ .
26; this value was also obtained within a continuum Diracformulation using dimensional regularization [53] by working in 2 − ǫ space dimensions, andeventually setting ǫ = 0.Increasing the strength of the electron-electron interactions, while holding the kineticnergy fixed, is expected to cause a quantum phase transition into an insulating state with aspontaneously generated mass for the Dirac Fermions [57][58]. Since, as we just argued, weakinteractions are irrelevant at long distances, such transition must happen at strong coupling,making it hard to control within a purely Fermionic theory. The full phase diagram alsodepends on the details of the interaction and is difficult to determine reliably using analyticalmethods. However, if one assumes that there is a direct continuous quantum phase transitionbetween the semi-metallic phase at weak coupling and a known broken-symmetry strongcoupling phase, say an anti-ferromagnetic insulator, then the critical theory can be arguedto take the form of massless Dirac Fermions Yukawa-like coupled to the self-interacting orderparameter bosonic field [59]. The advantage of this formulation is that the upper critical(spatial) dimension is 3, and therefore such theory can be studied in 3 − ǫ space dimensionswithin a controlled ǫ -expansion, eventually extrapolating to 2 space dimensions by setting ǫ = 1. The transition thus found is indeed continuous and governed by a fixed point at finiteYukawa and quartic bosonic couplings. To leading order in ǫ , the critical exponents havebeen determined[59]; for the semi-metal to the antiferromagnetic insulator quantum phasetransition, the correlation length exponent ν = 0 .
882 and the bosonic anomalous dimension η b = 0 .
8. Since the dynamical critical exponent has been found to be z = 1, these valuesimply that the order parameter vanishes at the transition as | u − u c | β with the exponent β = 0 . u c is a critical interaction. The 1 /r Coulomb interaction has been found tobe irrelevant at this fixed point.Given that at half-filling the theory does not suffer from the Fermion sign problem, avery promising theoretical approach in this regard is numerical. The Hubbard model onthe honeycomb lattice, with the nearest neighbor hopping energy t and the repulsive on-site interaction U , has been studied using quantum Monte Carlo methods [60][61][62][63].Recent simulations on cluster sizes of up to 2592 sites show strong indications of a directcontinuous phase transition at U/t ≈ . ± .
013 between the (Dirac) semi-metal and theanti-ferromagnetic insulator[63], disfavouring earlier claims[62] on the existence of a spinliquid phase for intermediate values of couplings 3 . . U/t . . β = 0 . ± .
04 extracted in Ref.[63] is in excellentagreement with the value obtained using the analytic Yukawa-like theory[59]. In subsequentnumerical simulations, the anti-ferromagnetic order parameter has been pinned by intro-ducing a local symmetry breaking field[64]. The resulting induced local order parameter farrom the pinning center was then ‘measured’. This procedure resulted in an improved res-olution, confirming a continuous quantum phase transition between the semi-metallic andthe insulating anti-ferromagnetic states. The single particle gap was found to track thestaggered magnetization, while the critical exponents obtained from finite size scaling agreewith those obtained to leading order in ǫ -expansion [59].The 1 /r Coulomb interaction can also be simulated efficiently without the Fermionsign problem using a hybrid Monte Carlo algorithm [65] using either staggered Fermions[65][66][67] or, preferentially, directly on a honeycomb tight-binding lattice[68][69][70][71][72].The critical strength of the interaction necessary to achieve a quantum phase transition intoan insulating state seems to depend on the details of the short distance part of the re-pulsion. Moreover, the system sizes studied numerically [72] may be too small to explorethe unscreened long distance tail of the 1 /r interactions and to therefore unambiguouslyestablish theoretically whether suspended monolayer graphene should be insulating. It isworth pointing out here that experiments on the suspended high purity monolayer graphenesamples show no sign of spontaneous symmetry breaking and would thus place it on thesemi-metallic side. V. APPLICATIONS TO VARIOUS PHYSICAL SYSTEMSA. Graphene
It is interesting to consider the massless Dirac Fermions in graphene[73] within the per-spective outlined above. Pure symmetry arguments are a powerful tool in this regard; ourgoal is to carry out such arguments in full detail in this section in order to illustrate theirutility. Assuming a perfectly flat, sp hybridized carbon sheet, the relevant atomic orbitalsforming both the conduction and the valence bands are the carbon 2 p z orbitals[73][25]. Agood variational ansatz for u k ( r ) would be P R e i k · R φ p z ( r − R − δ ), where φ p z ( r ) is aL¨owdin orbital[ ? ] with the same symmetry as the atomic p z orbital[74]. The exact formof the L¨owdin orbital is unimportant for us now, its symmetry is what matters. In an ide-alized situation, without externally imposed strains or any other lattice distortions, the setof vectors R could be chosen to span the triangular sublattice of the graphene honeycomblattice: m R + n R with R = √ x , R = R + a ˆ y , and m, n are integers. The basisector δ = √ a ˆ x + a ˆ y . Note that this Bloch state is manifestly periodic in k . Similarly,we can choose v k ( r ) as P R e i k · R φ p z ( r − R + δ ). This physically motivated choice, alongwith u k ( r ) = v k ( r ) = 0, defines our four basis states used to construct the Eq.(8).A flat graphene sheet is invariant under the mirror reflection about the plane of thelattice which further constrains H ( k ). Such operation reverses the in-plane components ofthe electron spin — an axial vector — and leaves the perpendicular component unchanged,thus acting on the spin state as a π -rotation about the axis perpendicular to the graphenesheet. Additionally, the p z orbitals are odd under the mirror reflection. Therefore, theeffective Hamiltonian in Eq.8 is constrained to satisfy 1 ⊗ σ H ( k ) 1 ⊗ σ = H ( k ) for anyin-plane k . This forces g = g = 0 in the Eq.(8). Because the remaining three g j ’s are ingeneral non-zero, we see that with only two components of k we cannot find simultaneouszeros of three independent functions. Therefore, in the absence of any other symmetry, weshould expect level repulsion.We can find the location of the Dirac points by taking into account additional symmetries.The space inversion symmetry, say about the center of the honeycomb plaquette, requires τ ⊗ H ( − k ) τ ⊗ H ( k ). This forces g ( k ) and g ( k ) to be odd under k → − k and g ( k )to be even. If the lattice also has a threefold symmetry axis perpendicular to the sheet andpassing through the plaquette center, then g and g must vanish at the two inequivalentpoints k = ± K = ± π √ a ˆ x , as well as, of course, all points equivalent to ± K by periodicityin the momentum space. This follows from our formalism when we note that the effect ofthe π rotation, induced on our wavefunctions by the operator e − i π ~ ˆ L z e − i π σ , affects ourfour basis states as e iφτ ⊗ σ e − i π ⊗ σ , where φ = k ′ · R and k ′ is the result of rotating k counter-clockwise by 120 ◦ . Then, the identity e iφτ ⊗ σ H ( k ′ ) e − iφτ ⊗ σ = H ( k ) evaluated at k = ± K immediately leads to g ( ± K ) = g ( ± K ) = 0. Interestingly, g is finite at ± K withvanishing derivatives, although if we also assumed spin SU (2) symmetry, which allows us toflip the spins using τ ⊗ σ , then g would vanish as well. In such case, irrespective of themicroscopic details of the full Hamiltonian, the two bands must touch at ± K .The Dirac particles of graphene therefore live at ± K . Strictly speaking, they are notquite massless because of non-zero spin-orbit coupling which makes g ( k ) finite. Such aterm has been introduced by Kane and Mele[75]. However, this term is very small in planargraphene structures, because the carbon atom is light and because graphene has a reflectionsymmetry about the vertical plane passing through the nearest neighbor bond[76][77]. Theres therefore only a negligibly small Dirac mass at K of order 10 − meV.Expanding H ( ± K + δ k ) to first order in δ k we find H eff = ± m QSH τ ⊗ ± ~ v F δk k τ ⊗ ~ v F δk ⊥ τ ⊗ σ , (11)where the 3-fold rotational symmetry guarantees that the δk k and δk ⊥ are two mutuallyorthogonal projections of δ k . In the coordinate system we have adopted, the mirror reflectionsymmetry about the x − z plane forces δk k = δk x + √ δk y and δk ⊥ = − √ δk x + δk y . Atenergy scales much smaller that m QSH , this Hamiltonian describes the quantum spin Hallstate: a gapped phase with counter-propagating edge states[75]. Due to the smallnessof m QSH in graphene, for all practical purposes we can set it to zero. The particle holeasymmetry, which arises from the δ k dependence of g , is also small in that it guaranteesthat the Fermi level can in principle be tuned to the Dirac point without the appearance ofadditional Fermi surfaces. The value for the Fermi velocity, v F ≈ m/s , can be obtainedfrom approximate first principle calculations or from experiments.
1. Coupling to external fields
Perhaps the greatest utility of the Dirac-like equation (11) is its ability to capture both the kinematics of the low energy excitations and their dynamics when subjected to external,or internal, fields. The former are of course the experimental tool of choice in studying thesystem.In our theoretical description, we are tempted to minimally couple the external vectorpotential A ( r ), associated with the perpendicular magnetic field B ( r ) = ∇ × A ( r ), andscalar potential associated with either an applied electric field or to the field induced byimpurities. While some care must be applied since we are working with a Bloch basis whoseperiodic part changes with k , to the order in δ k that the Eq.11 has been written, we areactually allowed to perform such minimal substitution[78][79]. Therefore, as long as thefields are sufficiently weakly varying in space, or for the uniform magnetic field as long asthe magnetic length p ~ c/eB is much longer than the lattice spacing, we have H eff = ± v F (cid:16) p k − ec A k ( r ) (cid:17) τ ⊗ v F (cid:16) p ⊥ − ec A ⊥ ( r ) (cid:17) τ ⊗ σ + U ( r )1 + H Z . (12)where the Zeeman term is H Z = gµ B ( B x τ ⊗ σ + B y τ ⊗ σ + B z ⊗ σ ). The aboveHamiltonian governs the behavior of graphene in an external magnetic field. The resultingandau level structure has been directly observed in scanning tunneling spectroscopy[80][81][82].Its utility in understanding the experiments on graphene hetero-junctions has been reviewedin Ref.[18]. The Schwinger mechanism, discussed in Section III, has been experimentallytested in Ref.[83]. H eff can also accommodate a time dependence of external poten-tials, important for interpreting the optical[84] or infra-red spectroscopy measurements ofgraphene[85]. The enhancement of the Fermi velocity, which, as discussed in Section IV, isa signature of electron-electron interactions, have been reported in Ref.[86], with no signs ofgap opening at the Dirac point. The effects of strain, as an effective potential in H eff , arediscussed in Refs.[87][88][89]. By and large, realistic impurity potentials in graphene cannotbe treated in linear response theory[79][90]; the review of transport effects can be found inRef.[91]. B. Surface states of a 3D topological insulator
An example of a 3D topological insulator[14][15][92][93] is Bi Se [94][95][96]. Its excita-tion spectrum is gapped in the 3D bulk, but its 2D surfaces accommodate gapless excitationswhich carry electrical charge, conduct electricity, and the dispersion of the surface excita-tions obeys massless Dirac equation. Unfortunately, presently the actual material suffersfrom imperfections causing finite bulk conductivity, a complication which we will largelyoverlook in this review.The electronic configuration of Bi is 6 s p and of Se is 4 s p . Since the p -shellsof Se lie ∼ . eV below Bi [97], a naive valence count would suggest that the two Bi atoms donate six of their valence p -electrons to fill the p -shell of Se . We would thereforeincorrectly conclude that the system is a simple, or trivial, insulator with a fully filled Se -like p -band and empty Bi -like conduction band, perhaps with an appreciable band gap.Interestingly, the strong spin-orbit coupling causes a “band inversion”[94][95] near the Γ-point (the origin of the Brillouin zone), where the Bi -like states lie below the Se -like states.Because the rhombohedral crystal structure of Bi Se has a center of inversion, the exactBloch eigenstates must be either even or odd under space inversion at the crystal momentawhich map onto themselves under time reversal, modulo a reciprocal lattice vector, i.e., k = − k + G . Clearly, Γ is such a point. As shown by Fu and Kane [15], a sufficientcondition for a band insulator with a center of inversion to be a 3D topological insulator isf such band inversion happens at an odd number of time reversal invariant points. Moreprecisely, the system is a 3D topological insulator if the product of the parity eigenvalues ofthe occupied bands at the time reversal invariant k -points is odd, with the understandingthat we count the parity eigenvalue of only one of the members of the Kramers pair. This isindeed what happens within a more realistic band structure calculation[94] [95] of Bi Se .At the Γ point — but not at the other time reversal invariant k -points — the parity evencombination of the p z -like Bi states are spin-orbit coupled to the more energetic p x ± ip y -like Bi states, and get pushed below the parity odd combination of the Se p z -like and p x ± ip y -likestates.The Eq.8 must describe the dispersion near the Γ point inside the bulk of the 3D system.This can be seen explicitly if we choose u k ( r ) to be predominantly made of the parityeven combination of Bi p z -like orbitals and − u k ( r ) of the Bi p x + ip y -like orbitals; i.e.,the states which are mixed due to the spin-orbit interaction. Similarly, for the proximateband, we should have v k ( r ) made predominantly of the parity odd combination of the Sep z -like orbitals, and − v k ( r ) of Se p x + ip y -like orbitals[95]. Then, up to the quadraticorder in deviation from the Γ point, g ( k ) = M + M k z + M (cid:0) k x + k y (cid:1) with M < M , >
0. No k -odd terms are allowed here because the states are of definite parity. Notethat because M is negative, in the immediate vicinity of the Γ point the Bi -like states liebelow the Se -like states. At higher k , we revert to the expected band ordering. For theother terms in the Eq.8, g ( k ) = 0 to linear order in k , due to additional 3-fold rotationalsymmetry; it is non-zero when we include terms up to order k , since the k -cubic invariantexists. The remaining terms must be k odd, because they couple opposite parity states: tolinear order then, g ( k ) = B k z , g ( k ) = − A k x , and g ( k ) = − A k y , where A & B > f ( k ) is also finite, but since its presence leads toqualitatively same conclusions, it will be ignored[95].Since g ( k ) is finite at Γ, which in this approximation is the only place where g , g , and g vanish, the spectrum in the bulk is of course gapped. However, the surface is gapless.To see this explicitly[94, 95], consider a semi-infinite interface in the x − y plane, set k x = k y = 0, and construct evanescent zero energy states along the z -direction. There are alwaystwo such normalizable states, which can be used as a basis for the low energy subspace.The effective surface Hamiltonian for small k x and k y can be obtained by sandwiching thebulk Hamiltonian between these two states. For macroscopically thick material, we cangnore the exponentially small overlap between the surface states, and we find H surf = ± A ( k x σ y − k y σ x ), where the top sign is for the top surface, z = L , and the bottom sign forthe bottom surface z = − L . A similar procedure along the right, y = L , and left, y = − L ,surfaces leads to H surf = ± ( B k z σ x + A k x σ z ); the effective Hamiltonians are simply relatedto each other by space inversion. In general, H surf = ˆ n ′ · ( ~σ × k ′ ) (13)where ˆ n ′ is obtained by rotating the normal to the surface, ˆ n , by 180 ◦ about the z-axis,and k ′ = ( − A k x , − A k y , B k z ). We thus arrive at an equation for massless, anisotropic,Dirac particles. However, unlike in graphene which has four “flavors”, the surface of the 3Dtopological insulator can support a single flavor.
1. Coupling to external fields, interaction and disorder effects
The existence of a single Dirac flavor on the surface of the 3D topological insulator hasimportant consequences for robustness of the surface states towards impurity disorder. Thestates at k and at − k have opposite spin, leading to the suppression of back scattering[98][99]and absence of localization for weak (scalar potential) disorder[100][101][102]. Theoretically,such a (non-interacting) system is always expected to display electrical conductivity whichincreases towards infinity as a logarithm of the system size. Recall that in graphene witha pair of Dirac cones at K and − K , such back scattering is always present and thereforeweak localization is expected to eventually set in[103][104], although for smooth impuritypotentials, it may be very small[105][106].Recent numerical study[107] of a topologically non-trivial 3D lattice model — with ran-dom on-site energy intentionally placed only on the surface of the 3D system — indicates,that the effective continuum description with Dirac particles scattered by a scalar potentialholds if the disorder strength is much weaker than the bulk gap ( ∼ . eV in Bi Se ). Theassertion is based on identification of Dirac-like features in a momentum resolved spectralfunction, even when the translational symmetry of the lattice is broken by disorder. As thetypical disorder strength increases beyond the 3D bulk gap value, the surface states appeardiffusive. For even larger disorder strength, the outermost surface states are localized, but weakly disordered Dirac-like states reappear directly beneath it. Apparently, for large sur-face disorder, an interface between a strongly localized Anderson insulator and a topologicalnsulator is formed[107]. As such calculations were performed on finite size systems, whichare too small to detect an Anderson localization transition, it is presently impossible toconclude whether there is a true phase transition at zero temperature separating the weak,the moderate, and the strong disorder regimes. The combined effects of scalar disorder andelectron-electron (Coulomb) repulsion have been studied in Ref.[108] using the continuumDirac approximation. The authors argue that 3D topological insulators are different fromgraphene, and that the single Dirac flavor makes the system metallic with finite conductivityat zero temperature. Transport properties of topological insulators have been reviewed inRef.[109].Because the electron spin is strongly coupled to its momentum, unlike in graphene, theZeeman coupling to the external magnetic field does not lead to simple spin splitting. Rather,it opens up a gap, turning massless Dirac particles massive. To further illustrate the differ-ence between the Dirac particles in a 3D topological insulator and graphene, consider nowthe situation in which the external uniform magnetic field is applied along the z -axis, andthe field is sufficiently strong to quantize the orbital motion of the surface electrons. Theequation describing the states on the top and the bottom surfaces is then h ± v F (cid:16)(cid:16) p x + ec By (cid:17) σ y − p y σ x (cid:17) + g z µ B Bσ z i ψ ( x, y, ± L ) = Eψ ( x, y, ± L ) , (14)where ~ v F = A and g z is the effective Lande g-factor. Indeed, the Zeeman coupling acts asa Dirac mass and does not lead to the usual splitting of the spin degenerate energy levels.It is straightforward to find the eigenvalues of this operator provided we are sufficiently farfrom any edge. The resulting Landau level spectrum is E n = ± s A (cid:18) eB ~ c (cid:19) n + ( g z µ B B ) , n = 1 , , , . . . (15) E = g z µ B B. (16)The physics in a quantizing magnetic field differs from graphene near the edge in anotherimportant way: the top and the bottom surfaces are coupled through the side surfaces.The applied magnetic field is parallel to the side surfaces and therefore there is no Landauquantization along this surface; even the Zeeman term does not open up a gap on the sidesurfaces, it merely shifts the momentum by a constant. Therefore, as the guiding center ofthe Landau levels approaches the edge, they start mixing into the continuum of the statesn the side surfaces. Fig.5 shows the electronic spectrum of a 3D topological insulator semi-infinite slab of finite thickness vs. the “guiding center” coordinate. Far away from anyedges, the spectrum exhibits the usual Dirac Landau level quantization, E = √ n √ ~ v F /ℓ B ,where ℓ B = p ~ c/eB and for Bi Se , v F = A . Every such Landau level is doubly degeneratebecause the top and the bottom surfaces are assumed to be identical. Such degeneracy wouldbe lifted if the inversion symmetry is broken by, say, a constant chemical potential differencebetween the top and the bottom surfaces. As the guiding center coordinate approaches theright edge — or the outer edge for the “Corbino” geometry — the Landau level states mergewith the plane-wave states from the vertical side surface. In the limit of very large thicknesssuch plane-wave states form a Dirac continuum.This poses interesting questions: how robust is the quantum Hall effect and how tomeasure it[110]? If the Fermi energy lies between the two Landau levels, the spectrumcontains M = 2 n + 1 chiral edge modes in addition to 2 N non-chiral ones. Clearly, inany Hall bar geometry the leads necessarily couple to the continuum of the states in theside surfaces, which present additional (unwanted) channels of conduction. Assuming thatthe side modes equilibrate with each other and result in a finite conductivity, the chemicalpotential will drop smoothly between µ R and µ L along each edge, and no quantizationof Hall conductance is expected[110][111][112]. Interestingly, quantization of σ xy has beenreported in a strained 70-nm-thick HgTe layer[113], with a well developed plateau at ν = 2and plateau-like features at ν = 3 and 4. At the same time, the longitudinal resistance R xx measured at 50 mK shows a suppression by few tens of percents, but it does not reachzero. While this observation awaits a complete theoretical treatment, if the sample is thinthen there are only a few non-chiral modes along the side surfaces which may get Andersonlocalized with sufficient side surface roughness, leaving only chiral modes at the edges.On the other hand, measurement of σ xy in the Corbino geometry is expected to leadto quantization[110][111]. The idea[111] is to perform the analog of the Laughlin thoughtexperiment, experimentally realized in 2D electron gas heterostructures in Ref.[114]. Onemeasures the amount of charge ∆ Q transferred from the inner surface to the outer surfacein response to the induced EMF produced in the azimuthal direction by a slow change in themagnetic flux ∆ ϕ threading the sample. Then σ xy = − c ∆ Q/ ∆ ϕ . For σ xy = n e h , half of thecharge travels through the top surface and the other half through the bottom surface. Anadditional advantage of the Corbino setup is that any interaction-driven fractional quantum Ñ v F l B Ñ v F l B Ñ v F l B kl B - L y l B - - H N+M N+M N+M N+M N N N (cid:80) R (cid:22) (cid:21) (cid:24) (cid:25) (cid:80) L N+M N N N+M N (b) H (cid:73) (t) (c)(a) FIG. 5: (a) Electronic spectrum of a 3D topological insulator semi-infinite slab of finitethickness vs the ”guiding center” coordinate. Far away from any edges, the spectrumexhibits the usual Dirac Landau level quantization, E = √ n √ ~ v F /ℓ B , where themagnetic length is ℓ B = p ~ c/eB , and ~ v F = A ≈ . eV ˚ A for Bi Se . Every such Landaulevel is doubly degenerate. If the Fermi level lies between the two Dirac Landau levels, theedge spectrum contains M = 2 n + 1 chiral modes in addition to 2 N non-chiral ones. (b)Schematic of a Hall bar geometry in a 3D topological insulator. (c) Corbino geometrysetup for measurements of quantum Hall conductivity.Hall states formed by the surface electrons can in principle also be detected[114].If the external electro-magnetic potentials are weak, the linear response theory is applica-ble. Naively, for a non-interacting system with a gap, we expect that at long wavelength andow frequency the response functions simply change, or renormalize, the dielectric constantand the magnetic permeability; after all, the system is a dielectric insulator. Interestingly,a 3D topological insulator gives rise to additional terms in the electro-magnetic response,some of which are analogous to axion electrodynamics[115][116][117][118]. C. d x − y -wave superconductivity in copper oxides Low energy quasiparticles obeying the Dirac equation may also emerge as a consequenceof a phase transition associated with the condensation of Cooper pairs. The specific examplewhich we consider here is the so called d x − y pairing which occurs in cuprate high tempera-ture superconductors [119][120]. In these layered, quasi 2D, materials, one may focus on theelectronic structure of a single CuO layer. A simple effective Hamiltonian for this systemis H = X k ,σ ( ǫ k − µ ) c † σ ( k ) c σ ( k ) + X k (cid:16) ∆ k c †↑ ( k ) c †↓ ( − k ) + h.c. (cid:17) , (17)where k = ( k x , k y ). The normal state dispersion, given by ǫ k , describes a closed Fermisurface, centered around ( π, π ), and equivalent points in momentum space. The anomalousself-energy, ∆ k , must in principle be determined from a microscopic theory; since such theoryis currently missing, one proceeds phenomenologically. Assuming time reversal symmetry, ǫ k = ǫ − k , and ∆ k can be chosen real. Since it transforms as x − y , it must change signunder a 90 ◦ rotation and vanish along the Brillouin zone diagonals, where it intersects withthe Fermi surface at four inequivalent points. Weak orthorhombic distortions, such as inYBCO, move the points of intersection slightly away from the zone diagonals[121], but donot change the low energy physics in an important way.The energy spectrum of the Fermionic quasiparticles can be obtained by solving theHeisenberg equation of motion for c ↑ ( k ) and c †↓ ( − k ): i ~ ∂∂t c ↑ ( k ) c †↓ ( − k ) = ǫ k − µ ∆ k ∆ k − ǫ k + µ c ↑ ( k ) c †↓ ( − k ) , (18)finding E ( k ) = q ( ǫ k − µ ) + ∆ k . Near the points of intersection between the Fermi surfaceand the zeros of ∆ k , we may expand ǫ k − µ ≈ ~ v F k ⊥ and ∆ k ≈ ~ v ∆ k k , where k ⊥ and k k arethe deviation perpendicular and parallel to the Fermi surface respectively. In the vicinity ofsuch points, the above has the form of an anisotropic massless Dirac equation.nterestingly, the Dirac node remains at zero energy even as the chemical potential, µ ,is varied. This is unlike in the previous examples, which involved Dirac particles in semi-conductors, where µ must be fine tuned to coincide with the Dirac node, otherwise we haveFermi circles with finite density of states at zero energy. Furthermore, given that the systemis a superconductor, the long range Coulomb interaction is screened. Since the discovery ofcuprates being d x − y superconductors, there has been a tremendous effort in trying to un-derstand the role of various perturbations. Here we focus on the question ‘How does such asystem behave in an external magnetic field?’[122][123][124][125][126] The first step towardsanswering this question is to recognize that the upper and the lower components of the‘spinor’ in Eq.18 acquire an opposite phase under a U(1) charge gauge transformation, andtherefore, an external magnetic field cannot couple minimally[126][127][128][129][130][131].Moreover, the pair potential must also be modified. In a mean-field calculation, it is com-puted self-consistently, with the solution depending on the value of the external magneticfield[123][125]. But even in the absence of a microscopic theory — which may justify aself-consistent mean-field calculation — we can establish this fact by noting, that near thetransition temperature, the existence of the Ginzburg-Landau functional follows quite gen-erally from the order parameter having the charge 2 e and the transition being continuous.Given that in cuprates the magnetic penetration depth is much longer than the coherencelength, for most of the magnetic field range the field penetrates in the form flux tubes andthe order parameter phase winds by 2 π near the core of each vortex. Therefore, in thepresence of the external magnetic field, the equation which generalizes Eq.18 is i ~ ∂∂t c r ↑ c † r ↓ = X r ′ t rr ′ − µ ↑ δ rr ′ ∆ rr ′ ∆ ∗ rr ′ − t ∗ rr ′ + µ ↓ δ rr ′ c r ′ ↑ c † r ′ ↓ , (19)where we assumed that the electrons hop on a square lattice given by r , with a complexamplitude t rr ′ . The phase of the complex singlet pair potential ∆ rr ′ winds by 2 π whenits center of mass coordinate encircles a vortex sufficiently far from the vortex core; itsdependence on the relative coordinate has d x − y symmetry.When the typical separation between vortices, set by p hc/eB , is much smaller than thepenetration depth, the magnetic field inside is almost uniform. Clearly, in such a case, theplane waves with the wave-number k are no longer eigenstates of the kinetic energy operator.One may attempt to proceed by working with Landau levels, which, in the continuum limitof the above lattice model, are eigenstates of the kinetic energy operator for a uniformagnetic field[125]. However, the number of the Landau levels below the Fermi energy,as determined from the quantum oscillations experiments on the overdoped side of thephase diagram[132][133], is of order 10 at magnetic fields of 1Tesla, this number decreasingwith 1 /B . The energy scale associated with the pair potential is approximately given by( v ∆ /v F ) E F , decreasing the number of Landau levels mixed by ∆ rr ′ by only one order ofmagnitude. Moreover, the resulting Hamiltonian matrix is dense, prohibiting the use ofefficient algorithms for determining the eigenvalues of sparse matrices.In the relevant magnetic field range H c ≪ H ≪ H c a different approach was proposedby Franz and Tesanovic[126], circumventing the use of the Landau level basis. The idea isto map the problem onto an equivalent one but at zero average magnetic field, in whichcase the plane wave basis may be used. This can be accomplished by performing a singulargauge transformation, familiar in the context of the fractional quantum Hall effect. Theythen argued that the relevant low energy excitations reside in the vicinity of the Diracnodal points, and that, in the continuum limit, the vortices together with the magnetic fieldact as an effective potential scattering the Dirac particles. As the magnetic field decreasesso does the strength of the effective potential, making a natural connection with the zerofield problem. For each of the four massless Dirac particles, which were assumed to bedecoupled[124], the combination v F · (cid:0) ~ ∇ φ − ec A (cid:1) entered the Dirac equation as an effectiveelectrical potential, Φ [126]. Here ∇× A = B and ∇×∇ φ = 2 π ˆ z P j δ ( r − R j ). The additionalminus signs acquired by the quasiparticles upon encircling an odd number of vortices wasencoded using a statistical U (1) field, minimally coupled to the Dirac particles[126]. Suchan approach provided an explicit method to (numerically) compute the scaling functions,whose existence was proposed earlier by Simon and Lee[124], as well as to test the validityof the semiclassical approach advanced by Volovik[122].In the vicinity of each vortex, the effective potential ~ ∇ φ − ec A grows with the inverse ofthe distance to the vortex. Since the kinetic energy of a massless Dirac particle also scaleswith inverse length, the vortices constitute a singular potential. It is therefore not obviousthat the long wavelength expansion, which led to the effective Dirac description in the firstplace, can be directly applied. Indeed, in the continuum limit, one must carefully specifythe boundary conditions at the vortex core by requiring that the effective Hamiltonian isa self-adjoint operator[134]. A choice of such, so called, self-adjoint extensions should bedetermined by matching to a well regularized lattice theory. Unfortunately, so far, it hasot been possible to determine their form. Since the choice is not unique, and since differentphysically allowable choices appear to lead to a qualitative difference in the low energyspectra (e.g. gapped or gapless), one is led to work with the lattice theory[128][130][135][136].The usual choice is to set t rr ′ = − te − iA rr ′ where the magnetic flux, ϕ , through an elementaryplaquette enters the Peierls factor via A rr +ˆ x = − πyeϕ/hc and A rr +ˆ y = πxeϕ/hc . The ansatzfor the pairing term is ∆ rr + δ = ∆ η δ e iθ rr + δ , where the d x − y -wave symmetry is encoded by η δ = +( − ) for δ k ˆ x (ˆ y ), and the vortex phase factor e iθ rr ′ = (cid:0) e iφ r + e iφ r ′ (cid:1) / | e iφ r + e iφ r ′ | . Thischoice is motivated by its behavior in the long distance limit[128][137].For a periodic vortex arrangement, and after the appropriate lattice version of the singulargauge transformation, one can take advantage of the Bloch theorem. The quasiparticlespectrum is then a function of a “vortex crystal” momentum q . It can be shown[130] thatif the vortex lattice has a center of inversion and if the Zeeman term is ignored, then foreach eigenstate with an eigenvalue E at q , there is a corresponding eigenstate with aneigenvalue − E at the same q . Therefore, any zero energy state at a fixed q must be at leasttwo-fold degenerate. However, because our problem breaks time reversal symmetry, sucha degeneracy can only be achieved by fine tuning an additional parameter besides the twocomponents of q . Therefore, the quasiparticle spectrum of an inversion symmetric vortexlattice is in general gapped. The Zeeman term corresponds to a simple overall shift of thequasiparticle energy and does not destroy the avoided crossing, it simply moves it to a non-zero energy. Some further non-perturbative aspects of this problem have been discussed inRef.[135].In Figure 6 we show the quasiparticle contribution to the specific heat obtained by thenumerical diagonalization of the resulting (sparse) Hamiltonian matrix for different values ofmagnetic field. The result is re-scaled according to the Simon and Lee scaling[124][137]. Wesee that in the mixed state of a d x − y superconductor, for v F /v ∆ = 7 and 14, increasing themagnetic field indeed increases the specific heat in an intermediate temperature window,in accord with the semiclassical prediction by Volovik[122]. At the lowest temperatures,however, there is a crossover into the quantum regime where the interference effects set inand the finite spectral gap rapidly decreases the specific heat. Note that the entropy at low T , i.e. R T C ( T ′ ) /T ′ dT ′ , increases with an increasing magnetic field. Entropy must of coursebe conserved and independent of the magnetic field when T → ∞ ; the effect comes from thetransfer of the spectral weight from energies above ∼ ∆ . It is similar to the effect discussedn the context of the Dirac particle in a periodic electrical potential whose average vanished,see Fig.2.We see then, that despite being described by similar kinematics, there is a very importantdifference in the way the d x − y -wave Dirac particles couple to the physical external magneticfield from the way the graphene or the 3D topological quasiparticles couple. In the lattercase, the specific heat may oscillate with the field, but when averaged over few oscillations,its value is field independent. In the former case, it is the average value that increases withthe external field. VI. WEYL SEMIMETALS
My work always tried to unite the truth with the beautiful, but when Ihad to choose one or the other, I usually chose the beautiful. - Hermann Weyl(1885-1955)It has long been known that band touchings in three dimensions are very stable[144, 145], asdescribed in Section 2. When the chemical potential lines up with the band touching points,and no other Fermi surfaces intersect it, a semimetal results. The low energy dispersion ofelectrons then closely resembles the Weyl equation of particle physics, hence these semimetalshave been termed Weyl semimetals[146]. The generic form is shown in Equation 4. Initially,the Weyl equation was believed to describe neutrinos, which however had to be given up withwith the discovery of neutrino mass. Thus, an experimental realization of a Weyl semimetalwould be the first physical realization of this fundamental equation. Here we will brieflyreview topological aspects of Weyl semimetals and their possible realizations in solids. Forsimplicity, consider the following simplified form of Equation 4: H ± = ± v F ( p x σ + p y σ + p z σ ) (20)where we have expanded about a pair of band touchings located at k ± and have denoted p = ~ ( k − k + ) (for example). The Pauli matrices σ j act in the space of the pair of bandsthat approach each other and touch at the Weyl nodes. The energy spectrum then is E ( p ) = v F | p | for both nodes. At each node we can associate a chirality, which measuresthe relative handedness of the three momenta and the Pauli matrices associated in the Weylequation. The chirality is ± H ± . This is a general property of Weylermions realized in band structures - their net chirality must cancel. A simple physical proofof this Fermion doubling theorem is pointed out below. In a clean system, where crystalmomentum is well defined, one can focus at one or the other node and hence effectivelyrealize the Weyl equation. Note, we have assumed that the bands are individually non-degenerate. This requires that either the time reversal symmetry, or the inversion symmetry(parity), is broken. In order to realize the minimal case of just a pair of opposite chiralityWeyl nodes, time reversal symmetry must be broken[146]. In practice this is achieved bymagnetic order in the crystal. Alternately, one may consider systems with broken inversionsymmetry[147], where a minimum of four Weyl nodes are present.It is useful to describe a toy lattice model where the above dispersion is simply realized[148,149]. Consider electrons hopping on a cubic lattice, where on every site the electron can bespin up or down. Now, assume a spin-orbit type hopping in the y and z directions whichproceeds by flipping spin, while along the x direction the sign of hopping depends on thespin projection. The coresponding Hamiltonian is H ( k ) = ~ v F a ([cos( k x a ) + m (2 − cos( k y a ) − cos( k z a ))] σ + sin( k y a ) σ + sin( k z a ) σ ) . (21)This Hamiltonian has Weyl nodes located at ( ± π/ a, , H = − ( ~ v F /a ) h Z σ shifts the nodes to ( k ± , ,
0) where k ± a = ± cos − h Z . Essentially, this stability to perturbations arises from the fact that thereis no ‘fourth’ Pauli matrix available to gap out the node. Only when the field h Z is largeenough | h Z | ≥ A. Topological Properties
The stability of the Weyl nodes is tied to a topological protection inherent to this bandstructure. Away from the band touching points, there is a clear demarcation between filledand empty bands. Consider the state obtained at a particular crystal momentum by fillingthe negative energy states (below the chemical potential). By studying how this state evolveson varying the crystal momenta one can extract a Berry phase, from which a Berry flux ( k ) = ∇ k × A ( k ) can be defined. The Weyl nodes are sources, or monopoles, of Berry flux- thus ∇ · B ( k ) = ± δ ( k − k ± ), hence their stability. They can only disappear by annihilatinga monopole of the opposite charge - which is a Weyl node of opposite chirality[150].This band topology of Weyl semimetals has two direct physical consequences. The firstis an unusual type of surface state, unique to Weyl semimetals - called Fermi arcs [146].Consider a 3D slab of Weyl semimetal with a surface in the x-y plane. Translation invariancealong these directions allow us to label single electron states by crystal momenta in this plane.Let us assume we have a single pair of Weyl nodes in the bulk as in the model in Eq. 21.At this same energy we can ask what are the surface states in the system. Surface statesare well defined at this energy at all momenta away from the Weyl nodes, because thereare no bulk excitations with the same energy and momenta. It is easily seen that surfacestates should form a Fermi arc. The arc terminates at the crystal momenta correspondingto the bulk Weyl nodes (see Figure 7). This result follows from the fact that Weyl nodesare monopoles of Berry flux. Therefore, the 2D Brillouin zones that lie between the pairof Weyl nodes will have a different Chern number than the planes outside (see Figure 7).These planes may be interpreted as 2D Quantum Hall states associated with a chiral edgestate which is guaranteed to cross the chemical potential. The locus of these crossings givesthe Fermi Arc surface state.If one considers both top and bottom surfaces of a Weyl semimetal one should recover aclosed Fermi surface as one would expect for a 2D system. Indeed the two Fermi arc stateson opposite surfaces, taken together, form a closed 2D Fermi surface. Thus a thin slab ofsemimetal may be viewed as a 2D system with a closed Fermi surface. As the thickness isincreased, two halves of this Fermi surface are spatially separated to opposite sides of thesample. Probing these surface states in surface sensitive probes such as ARPES and STMshould provide smoking gun evidence for this unusual phase of matter.A second physical consequence of the topology of Weyl nodes is their response to anapplied electric and magnetic field. As discussed in Section III, a single Weyl node possessesa Chiral Anomaly: the net number of charged particles would not be conserved if a singleWeyl node was present[36][35]. Rather the continuity equation is modified ∂n∂t + ∇ · J = ± π e ~ c E · B , where the sign is determined by the chirality of the Weyl node. Thus charge conservationrovides a rationale for why Weyl nodes always must occur in a band structure with zeronet chirality. Although the net charge is then conserved, the chiral anomaly does lead to aninteresting effect. Consider for example the case of a pair of nodes with opposite chiralityas in Equation 21. Then the difference in density between excitations near the two nodes(the valley polarization) is governed by d ( n + − n − ) dt = 12 π e ~ c E · B (22)thus, applying parallel electric and magnetic fields can be used to control the valley polariza-tion - which will lead to new transport phenomena and possibly even applications for Weylsemimetals. There are close connections between this phenomena and the chiral hydrody-namics recently described in the high energy literature[151]. A related physical effect is agiant anomalous Hall effect expected for the case of a pair of Weyl nodes which is proportionalto the separation between the Weyl nodes in momentum space. Thus σ yz = e πh ( k + − k − ).If combined with an independent measurement of the momentum separation ( k + − k − ) be-tween Weyl nodes, obtained for example via ARPES, leads to a quantized ratio. In Weylsemimetals with higher symmetry, such as cubic symmetry, the anomalous Hall conductancevanishes. However, under a uniaxial strain that lowers symmetry, a large anomalous Halleffect is expected[148].We note that the two topological properties mentioned above required that the Weylnodes be separated in crystal momentum. In the presence of breaking of crystalline transla-tion symmetry, such a distinction may be lost, which would obstruct defining a sharp physicalproperty that reflects the underlying topology. Thus it appears that while semimetals likethe Weyl semimetal may be topological states, the topology associated with them is sharplydefined in the presence of translation symmetry, in contrast to insulating topological phaseswhich do not require any such assumption. However, in practice disorder is rarely strongenough to completely destroy well separated nodal points, as evidenced in the example ofgraphene. Thus realistic systems should display the novel features we mentioned above. B. Physical Realizations
Despite being a very natural band structure, currently there are no clearly established ma-terials with Weyl nodes near the chemical potential, although several promising candidatesxist. It has been proposed that members of the family of material A Ir O (pyrochloreiridates), where A = Y or a rare earth such as A = Eu, N d, Sm may be in or proximateto the Weyl semimetal phase[146]. This is currently an active area of experimental work[152][153][154][155]. Spinels based on osmium[156] and HgCr Se [157] have also been pro-posed as candidates. Another route has been to try to engineer Weyl semimetals usingheterostructures of topological insulators[158, 159]. Interestingly, a proposal to realize Weylpoints in a photonics band structure has recently appeared[160]. A general symmetry anal-ysis of crystal structures that may host Weyl semimetals appeared in [161]. Further detailson this topic may be found in the longer review [162]. VII. SUMMARY
We reviewed general conditions under which one may expect gapless Dirac points to occurin solids. Their appearance may be a consequence of band-structure effects, of symmetrybreaking due to many-body effects such as superconductivity or as a surface state of a bulktopological phase. If a Dirac point exists, additional fine-tuning of the chemical potentialis necessary in order for the Dirac point to coincide with the Fermi level, unless the Diracpoint appears as a consequence of the condensation of Cooper pairs. Then, the Dirac point“rides” along with the chemical potential.We also reviewed how the Dirac Fermions respond to externally applied perturbations andwhy the response differs in the case of graphene, topological insulators, Weyl semimetals,and d-wave superconductors. External potentials cause a redistribution of the quasiparticlespectral weight: space-dependent electrical potential tends to transfer the spectral weightfrom large energies towards the Dirac point, while the Dirac mass term tends to removethe states from the vicinity of the Dirac point, pushing them towards the large energies.Uniform magnetic field redistributes the states over the energy scale set by the cyclotronfrequency. In this context, the magnetic field induced enhancement of the low temperaturespecific heat in the vortex state of d-wave superconductors is also reviewed.When weak, finite range electron-electron interactions result in only finite renormalizationof the Dirac particle dispersion, without leading to any qualitative changes. As the strengthof the interactions increases, a quantum phase transition occurs into an insulating state. Inthe case of the half-filled repulsive Hubbard model on the honeycomb lattice, the transitionppears to be into a Neel anti-ferromagnetic state. Among its attractive features is thepossibility to study the transition either using a quantum Monte Carlo method without theFermion sign problem, or analytically using the ǫ -expansion around 3 + 1 dimensions for thecontinuum field theory description with the massless Dirac particles Yukawa coupled to aself-interacting O (3) bosonic field. Understanding why an interacting system may undergoa symmetry breaking transition into a state with massless Dirac Fermions, such as in thecuprate superconductors, rather than avail of a fully gapped state, remains a fascinatingopen problem.Finally we discussed recent developments of three dimensional Weyl Fermions, includingtheir robust topological properties in the form of unusual surface states and magneto-electricresponses, and possible physical realizations. VIII. ACKNOWLEDGMENTS
OV was supported by the NSF CAREER award under Grant No. DMR-0955561, NSFCooperative Agreement No. DMR-0654118, and the State of Florida. AV was supported byARO MURI grant W911NF-12-0461. [1] Dirac PAM. 1928
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Physical Review B TH @ K (cid:144) T (cid:144) D CT H @ mJ (cid:144)H mol K T (cid:144) LD Α D = (a) TH @ K (cid:144) T (cid:144) D CT H @ mJ (cid:144)H mol K T (cid:144) LD Α D = (b) FIG. 6: Electronic contribution to the low temperature specific heat of a d x − y superconductor in the vortex state[138], scaled according to the Simon and Leescaling[124]. The thick lines are with Zeeman term included, the thin lines are without it.The electrons hop with the nearest neighbor amplitude t on a tight-binding lattice with alattice spacing a = 3 . A . The chemical potential was set to µ = 0 . t corresponding to15% doping. The Fermi velocity v F = 2 . ∗ m/s was taken to agree with thephotoemission experiments on YBCO[139] by setting t = 132meV; the Dirac coneanisotropy α D = v F /v ∆ = 7 in panel (a) and α D = 14 in panel (b). Insets show the squarevortex lattice used. The dashed lines correspond to the values extracted experimentally:(a) ∼ . mJ/molK √ T at 10% doping by Riggs et.al. [140]; (b) ∼ . mJ/molK √ T at15% doping by Moler et.al. [141] (lower dashed line) and ∼ . mJ/molK √ T at 15%doping by Wang et.al. [142] (higher dashed line); see also [143]. " !" &&:F=&:FK& !"!"