Dirac and Weyl fermions: from Gor'kov equations to Standard Model (in memory of Lev Petrovich Gorkov)
DDirac and Weyl fermions: from Gor’kov equations to Standard Model(in memory of Lev Petrovich Gorkov)
G.E. Volovik
1, 2, 3 Low Temperature Laboratory, Department of Applied Physics,Aalto University, PO Box 15100, FI-00076 AALTO, Finland Landau Institute for Theoretical Physics RAS, Kosygina 2, 119334 Moscow, Russia P.N. Lebedev Physical Institute, RAS, Moscow 119991, Russia (Dated: January 5, 2017)Gor’kov theory of superconductivity opened the application of the methods of quantum fieldtheory to condensed matter physics. Later the results became relevant to relativistic quantumfields. PACS numbers:
I. INTRODUCTION
Application of quantum field theory to condensed matter physics began in Soviet Union around 1956-57 . In thisapproach the Fermi sea serves as an analog of the relativistic quantum vacuum – the Dirac sea. The Gor’kov theoryof superconductivity has been the fundamental step in this direction, which in turn triggered the development ofthe relativisic theories. The composite models developed by Nambu and Jona-Lasinio and by Vaks and Larkin ,where the Higgs bosons appear as a composite states of the fermion pairs, are the direct consequences of the Gor’kovtheory. In such models the original Weyl fermions of Standard Model (such as top quarks) play the role of theelectrons in metals, while the composite Higgs bosons are analogs of the collective modes of the order parameter insuperconductors.Here we consider another consequence of the Gor’kov theory of superconductivity, where the Weyl fermions emergein superconductors as Bogoliubov quasiparticles. This in particular takes place for superconductors of the symmetryclass O ( D ), where the 4 left-handed and 4 right-handed topologically protected chiral fermions emerge , see Fig.I. Expansion of the Gor’kov Green’s function in the vicinity of each topologically protected Weyl point leads to theeffective relativistic quantum field theory with effective gauge fields and the effective gravity. This provides the hintfor possible emergent origin of the ”fundamental” Weyl fermions, gauge fields, and general relativity . II. SYMMETRY AND TOPOLOGY
The Gor’kov anomalous function F and the ordinary Green’s function G have been organized by Nambu in termsof the 2 × × G − = iω + τ (cid:15) ( p ) + τ Re ∆( p ) + τ Im ∆( p ) . (1)Here ∆( p ) is the complex gap function; τ , , are 2 × demonstrates that the majority of supercoductivity classescontains the gapless superconductors, where the symmetry supports the point nodes and/or the nodal lines in theenergy spectrum E ( p ) = (cid:112) (cid:15) ( p ) + | ∆( p ) | of Bogoliubov quasiparticles, for review see . Some of the nodal linesand points do not disappear even if the symmetry of the superconducting state is violated: such stability towardsdeformations of crystal is supported by the integer-valued topological invariant, which for the point nodes has beendiscussed in Ref. . This invariant reflects the topology of the Gor’kov-Nambu matrix Green’s function in the 3Dmomentum space or in the 4D momentum-frequency space .Since the topologically protected nodes in the energy spectrum determine the dynamics, kinetics, thermodynamicsand anomalies in superconductors, the symmetry classification of superconducting states should be supplemented bythe topological classification. The topological classification of nodes in the energy spectrum with and without thesymmetry protection has been started , but still is not completed, see the latest reviews on symmetry and topologyof superconducting and superfuid states . a r X i v : . [ c ond - m a t . s up r- c on ] J a n FIG. 1: Figure from Ref. . Arrangement of the nodes in the energy spectrum in superconductors of class O ( D ). The pointsdenote four Weyl nodes with topological charge N = +1, and crosses denote four Weyl nodes with N = −
1. The modernexpression for the topological charge in terms of the Gor’kov-Nambu Green’sfunction is in Eq.(6). In the vicinity of each Weylnode with N = +1 the chiral right-handed Weyl fermions emerge, while N = − III. WEYL AND DIRAC SUPERCONDUCTORSA. Weyl superconductors
In order to make connection to the vacuum of quantum field theory, we consider the superconducting state in thecrystals with the maximal possible symmetry – the cubic symmetry (note that as has been found by Gor’kov , evenin the cubic crystals the upper critical field H c can be anisotropic). The state, where the gap nodes obtained fromthe symmetry consideration appear to be topologically stable, belongs to the superconducting class O ( D ). This statecontains 8 point nodes in the energy spectrum in Fig. I, which are situated on the four C symmetry axes in theBrillouin zone (the more general situation with the pockets of the Fermi surfaces see in Ref. ). Let us consider the gapfunction ∆( p ) for the momentum p along one of these threefold axes. The important property of the unconventionalsuperconductivity discussed in , is that the symmetry group of the order parameter (or of the gap function ∆( p ))contains the elements with combined symmetry. In particular, the symmetry group of the O ( D ) state contains thecombined symmetry: ˜ C = C exp (cid:0) πi (cid:1) . This combined symmetry means that the gap function does not change ifthe C rotation of the momentum p around one of the threefold axis is accompanied by the U (1) gauge transformationby the phase π/
3. Applying this to the gap function ∆( p ) with momentum along the C rotation axis one obtains :∆( p ) = ˜ C ∆( p ) = C e πi ∆( p ) = e πi ∆( C p ) = e πi ∆( p ) . (2)This gives ∆( p ) = 0 for the momentum p on the threefold axis.This can be visualized using the simple model Hamiltonian for fermions in the spin-singlet d -wave superconductorin cubic crystal, which belongs to O ( D ) symmetry class. The gap function has the form∆( p ) = a ( p x + (cid:15)p y + (cid:15) p z ) , (cid:15) = exp (cid:18) πi (cid:19) . (3)The corresponding Gor’kov-Nambu Hamiltonian for fermions is H = (cid:15) ( p ) τ + a
12 (2 p x − p y − p z ) τ + a √
32 ( p y − p z ) τ . (4)The intersections of the lines of zeroes in the gap function, ∆( p ) = 0, with the Fermi surface, (cid:15) ( p ) = 0, give the pointnodes in the energy spectrum. In the considered case of cubic symmetry of the crystal there are 8 point nodes at thevertices of cube in the momentum space in Fig. I: p ( n ) = p F √ ± ˆ x ± ˆ y ± ˆ z ) , n = 1 , . . . , . (5)It appears that these zeroes in the energy spectrum do not disappear if the crystal is deformed and the cubicsymmetry is violated. The reason for that is that these point nodes are topologically stable: they are supported bytopological invariant, which takes values N = ± . This invariant is related to the winding number of the phase ofthe gap function ∆( p ) = | ∆( p ) | e i Φ( p ) around the nodal line in the gap function . It also represents the charge ofthe Berry phase monopoles in momentum space , and can be analytically expressed in terms of the Gor’kov-Green’sfunction : N = e αβµν π tr (cid:20)(cid:90) σ dS α ˆ G∂ p β ˆ G − ˆ G∂ p µ ˆ G − ˆ G∂ p ν ˆ G − (cid:21) . (6)Here σ is the 3D surface around the point nodes of the inverse Green’s function in the 4D frequency-momentum space p µ = ( ω, p ); and tr denotes the trace over all the indices of the matrix ˆ G ( p µ ) (in addition to Bogoliubov-Nambu-Gor’kov spin index, the Green’s function contains also the ordinary spin index, band index in crystals, and indicesof the fermionic species in particle physics). In a different form the topological stability of the Weyl nodes has beendiscussed for the neutrino sector in Refs. .The important property of the point nodes with N = ± H ( n ) = e ( n ) iα τ α (cid:16) p i − p ( n ) i (cid:17) , n = 1 , . . . , . (7)Under space and time dependent deformations the expansion parameters e ( n ) iα ( r , t ) become the fields. These areanalogous to the tetrad field in general relativity acting on Weyl fermion. In the same way p ( n ) i ( r , t ) play the role ofthe effective gauge fields.Altogether the O ( D ) superconducting state contains 4 left-handed fermions ( N = −
1) and 4 right-handed fermions( N = +1). In case of Fermi pockets discussed in Ref. the number of Weyl fermions may increase. If the pocketsare on the three-fold axes, one would have 8 left-handed and 8 right-handed fermions. This can be compared with 8left-handed and 8 right-handed Weyl fermions in each generation of fermions in Standard Model of particle physics.The natural arrangement of the 8 left and 8 right Weyl fermions also occurs on the vertices of the cube in the four-dimensional ( ω, p x , p y , p z ) space . This is one of many examples when the topologically protected nodes in thespectrum serve as an inspiration for the construction of the relativisitc quantum field theories. B. Dirac superconductors
Topological stability of nodes in the spectrum is important, because it does not depend on the microscopic mecha-nism of superconductivity. On the other hand, the symmetry arguments are rather subtle, especially in case of com-plicated many-band structure of the Fermi surface and of the gap function, and also the symmetry can be violated.An example of state with unstable Fermi points is the triplet state of the symmetry class O ( T ) with Gor’kov-NambuGreen’s function ˆ G − = iω + τ (cid:15) ( p ) + τ σ i d i ( p ) , (8)where the vector gap function is d ( p ) ∝ ˆ x p x ( p y − p z ) + ˆ y p y ( p z − p x ) + ˆ z p z ( p x − p y ) . (9)Here σ i are Pauli matrices for ordinary spin, and the energy spectrum E ( p ) = (cid:112) (cid:15) ( p ) + d ( p ) in case of real vector d ( p ).This state has 14 = 8 + 6 point nodes . The zeroes are at the 8 points, where the Fermi surface intersects the fourthreefold axes, and in addition there are 6 points at the intersections of the Fermi surface with three fourfold axes,see Fig. III A. This state is time reversal invariant (the vector d ( p ) is real), and thus the nodes do not represent theWeyl points. These are the Dirac points, which are massless (gapless) only due the symmetry of the state, and canbe destroyed by deformations of the crystal lattice. Depending on the type of the violated symmetry, the fermionsmay acquire the gap (mass), or instead the Dirac points split into the pairs of Weyl points . The first scenario FIG. 2: Unstable point nodes in the triplet state of the symmetry class O ( T ). This state has 14 = 8 + 6 point nodes . Thezeroes are at the 8 points, where the Fermi surface intersects the four threefold axes, and at 6 points at the intersections ofthe Fermi surface with three fourfold axes. These points may split into pairs of Weyl points if the time reversal symmetry isviolated. The extension to the 4D space ( ω, p x , p y , p z ) may give 24 = 16 + 8 point nodes, and up to 48 Weyl fermions, if thetime reversal symmetry is violated. takes place in Standard Model, with the possible exception in the neutrino sector, where the second scenario is stillnot excluded.The extension to the four-dimensional ( ω, p x , p y , p z ) space with cubic symmetry gives 16 + 8 = 24 Dirac points,which can be compared with 24 Dirac fermions in 3 generations of Standard Model. The crystal symmetry ofsuperconductors does not allow us to consider the point groups of higher symmetry, such as the icosahedral groupand its 4D extensions. However, this can be extended by considering the symmetry groups in the internal isospinspace . IV. CUPRATES AND DIRAC NODAL LINES
Though the mechanism of high temperature superconductivity (HTSC) in cuprates remains unknown , the d -waveorder parameter symmetry and existence of the nodal lines are well established. Let us start with cuprates whichhave tetragonal symmetry. The superconducting state there belongs to the class D ( D ). The relevant combinedsymmetry, which gives rise to the nodes in the spectrum, is U e πi . It is the π rotation about the horizontal axiscombined with the U (1) gauge transformation by phase π/ . Under this combined transformation the gap function∆( p x , p y , p z ) = U e πi ∆( p x , p y , p z ) = − ∆( p y , p x , − p z ) = − ∆( − p y , − p x , − p z ) = − ∆( p y , p x , p z ) . (10)Here we used the symmetry element C of the group D ( D ) in the third equality and the space inversion symmetryin the last one. The equation (10) gives ∆( p ) = 0 for momenta on the vertical plane p x = p y . The same considerationshows that the gap function is zero also on another vertical plane, p x = − p y . As a result one obtains 4 nodal lines inthe energy spectrum at the intersections of the Fermi surface (cid:15) ( p ) = 0 with the two vertical symmetry planes, where∆( p ) = 0. The corresponding Gor’kov-Nambu model Hamiltonian is H = (cid:15) ( p ) τ + a ( p x − p y ) τ . (11)These nodal lines survive if the tetragonal symmetry is distorted, but the time reversal symmetry is obeyed. Thereason for that is again the topological stability. The nodal lines are protected by another topological invariantexpressed via the general Gor’kov-Nambu Hamiltonian N = 14 πi tr (cid:73) C dl τ H − ∇ l H . (12)Here the integral is now along the loop C around the nodal line in 3D momentum space.Recently the polar phase of superfluid He has been discovered, which has the spin degenerate nodal line in theenergy spectrum . V. PHYSICAL CONSEQUENCES OF GAP NODES IN SUPERCONDUCTORS
The nodes in the spectrum lead to the finite density of states (DoS) in the presence of the superfluid velocity, thisis because the nodes expand due to the Doppler shift and form the Fermi surface pockets around the former nodes.Due to the superflow around Abrikosov vortices, the Fermi pockets produce the singular dependence of the DoS onmagnetic field. The point nodes lead to the DoS, which is proportional to v s . Integration over the cross section of theAbrikosov vortices gives the B ln B dependence of DoS in the vortex lattice . In case of the line nodes the DoS haslinear dependence on | v s | , see Ref. , and correspondingly one obtains the √ B dependence of DoS in the Abrikosovlattice . This square root dependence of DoS has been observed in cuprates .The systems with Weyl fermions should experience the chiral anomaly effect in the external fields or in theeffective gauge fields produced by deformations. The latter has been observed in chiral superfluid He-A as an extraforce acting on vortex-skyrmions . This is the spectral flow force or the Kopnin force, see e.g. Ref. . It can beexpressed in terms of the effective ”magnetic” B and ”electric” E fields acting on Weyl fermions. These fields areproduced by deformation in superconductors, or by the moving skyrmion texture in He-A . The Kopnin forceis proportional to B · E , and the measured coefficient in front of B · E was found to be in agreement with theAdler-Bell-Jackiw equation , desribing production of the fermionic charge generated by the chiral anomaly.The topologicaly protected point and line nodes in the energy spectrum lead to the fermion zero modes in the coreof vortices and to the topologically protected gapless edge states. This is the result of the so-called bulk-surface andbulk-vortex topological correspondence. The Weyl fermions in bulk produce the dispersionless flat band of fermionswith zero energy living in the vortex core and the Fermi arc of the edge states . The nodal lines in bulksuperconductors lead to the condensation of the Caroli-de Gennes-Matricon levels in the vortex and to the flat bandof the edge states .The dimensional reduction of the 3D superconducting states with Weyl fermions to the 2D or quasi-3D films leadsto the two-dimensional fully gapped superconductivity without the gap nodes, which however retains the nontrivialtopological character. The topological invariant for these 2D superconductors is obtained by dimensional reductionof Eq.(6) for the Weyl point. It is expressed in terms of Gor’kov-Nambu matrix Green’s function, but the integrationnow is over the frequency (on the imaginary axis) and over the 2D Brillouin zone, see Refs. : N = e βµν π tr (cid:20)(cid:90) dωd p ˆ G∂ p β ˆ G − ˆ G∂ p µ ˆ G − ˆ G∂ p ν ˆ G − (cid:21) . (13)Here p µ = ( p x , p y , ω ). The Abrikosov vortex in such 2D superconductors contains in its core the Majorana fermionwith zero energy . VI. CONCLUSION
The majority of the topological classes of superconductivity contain the topologically protected Weyl points. Ex-pansion of the Gor’kov-Nambu matrix Green’s function in the vicinity of each Weyl point leads to the effectiverelativistic quantum field theory with effective gauge fields and effective gravity. In this analogy different Weyl pointscorrespond to different fermionic species of Standard Model - quarks and lepton. The effective quantum field theoryin the vicinity of the nodal line is still waiting for its development, as well as in the vicinity of nodes of arbitraryco-dimension. Probably this may have relation to branes in string theory . ACKNOWLEDGMENTS
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