Hall viscosity from elastic gauge fields in Dirac crystals
Alberto Cortijo, Yago Ferreirós, Karl Landsteiner, María A. H. Vozmediano
HHall viscosity from elastic gauge fields in Dirac crystals
Alberto Cortijo, Yago Ferreir´os, Karl Landsteiner, and Mar´ıa A. H. Vozmediano Instituto de Ciencia de Materiales de Madrid,CSIC, Cantoblanco; 28049 Madrid, Spain. Instituto de F´ısica Te´orica UAM/CSIC,Nicol´as Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain (Dated: September 26, 2018)The combination of Dirac physics and elasticity has been explored at length in graphene wherethe so–called ”elastic gauge fields” have given rise to an entire new field of research and applications:Straintronics. The fact that these elastic fields couple to fermions as the electromagnetic field, impliesthat many electromagnetic responses will have elastic counterparts not explored before. In this workwe will first show that the presence of elastic gauge fields will be the rule rather than the exceptionin most of the topologically non–trivial materials in two and three dimensions. In particular wewill extract the elastic gauge fields associated to the recently observed Weyl semimetals, the ”threedimensional graphene”. As it is known, quantum electrodynamics suffers from the chiral anomalywhose consequences have been recently explored in matter systems. We will show that, associatedto the physics of the anomalies, and as a counterpart of the Hall conductivity, elastic materialswill have a Hall viscosity in two and three dimensions with a coefficient orders of magnitude biggerthan the previously studied response. The magnitude and generality of the new effect will greatlyimprove the chances for the experimental observation of this topological, non dissipative response.
I. INTRODUCTION
In elasticity theory, Hooke’s law tells us that the stresstensor T ij applied on an elastic solid is proportional tothe strain tensor u ij = ( ∂ i u j + ∂ j u i ) / T ij = λ ijlr u lr through a four-rank tensor λ ijlr . If the system is vis-coelastic the stress tensor is also proportional to the timederivative of u ij : T ij = η ijlr ˙ u lr , (1)through another four-rank tensor η ijlr . In general, thisviscosity tensor possesses both symmetric and antisym-metric components under the permutation of pairs of in-dices. While the symmetric part is generally associatedto dissipation and vanishes at zero temperature, the an-tisymmetric part arises when time reversal symmetry isbroken . As usually happens with transport coefficientslike the gyrotropic term of the permittivity in dielectrics,the antisymmetric part of the coefficients that are oddunder time inversion are dissipation-less and they are notconstrained to vanish at zero temperatures. In two di-mensions, isotropy, the symmetry under permutations of ij and lr indices, and the aforementioned antisymmetryin permutations of pairs of indices allow for only one inde-pendent element of the antisymmetric part of the tensor η Hijlr : η ijlr = η H ( δ il (cid:15) jr + δ ir (cid:15) jl + δ jl (cid:15) ir + δ jr (cid:15) il ), where (cid:15) ij is the Levi-Civita tensor in two dimensions.The first context where this antisymmetric,dissipation-less coefficient was described was thequantum Hall effect (this is why this coefficient is calledHall viscosity) . Being non-dissipative and appearingin quantum Hall effect systems, it is not strange thatthe Hall viscosity is associated to a certain Berry phaseendowed with a topological meaning. In the case ofthe Hall conductivity, the off diagonal part of theconductivity tensor σ xy is related to a non-zero Berry phase associated to the evolution of the phase of thewave function in the Brillouin zone . In the case ofthe Hall viscosity, the parameter space is the spaceof the displacements of the atoms in the continuumlimit, u ( r ). Since these displacements are consideredadiabatic with respect to the electronic motion, one canconsider a cyclic evolution of the wave function in thisparticular parameter space and define an associatedBerry curvature of the wavefunction Φ a at some energylevel ε a : Ω ( a ) ijlr = i (cid:104) ∂ u ij Φ a | ∂ u lr Φ a (cid:105) .Also the Hall viscosity can be computed from a Kuboformula in terms of the stress-stress correlation func-tion η ijlr = (cid:82) dte i + t (cid:104) [ T ij ( t ) , T lr (0)] (cid:105) with the stress ten-sor defined as the variation of the Hamiltonian with thestrain tensor T ij = δHδu ij . After some mathematical ma-nipulations, a part of this Kubo formula appears to beproportional to the Berry curvature Ω ijlr defined above.Besides, as it happens in all transport coefficients, theHall viscosity appears in the effective action of somelow energy degrees of freedom after integrating out thefermionic degrees of freedom. The effective energy func-tional for elastic displacements (phonons after quantiz-ing) for a system in two spatial dimensions consists ontwo terms, the standard elastic terms (possibly renor-malized by the effect of electrons) and the Hall viscosityterm: U = 12 (cid:90) d r [ λ ijlr ∂ i u j ∂ l u r + η Hijlr ∂ i u j ∂ l ˙ u r ] . (2)Today there is a consensus about the meaning and theorigin of the Hall viscosity. The question at this point ishow to define the coupling between electrons and the elas-tic displacements u i or better, to the strain tensor u ij andwhere to seek for it. The quantum Hall state is a topolog-ically ordered state so it is natural to extend the search toother topologically non-trivial quantum systems. Much a r X i v : . [ c ond - m a t . m e s - h a ll ] J un work has been done in this direction and Hall viscositieshave been found in most of the topologically nontrivialcondensed matter phases apart from the quantum Hallphase like the fractional quantum Hall phase , chiralsuperfluids and superconductors , Chern insulators inpresence of torsion in two dimensions and, more re-cently, in three dimensions as well.Precisely, the step of finding topologically non trivialsystems from two to three spatial dimensions appearsto be rather challenging. To have a non-zero Hall vis-cosity in three spatial dimensions breaking time rever-sal symmetry is not enough. Spatial isotropy or rota-tional invariance in the continuum limit must be bro-ken as well . Up to now, only the torsional Hall viscos-ity defined in Weyl semimetals and three dimensionalchiral superconductors has been discussed in the litera-ture. In these systems torsion is linked to the presenceof dislocations that define a preferred direction in thespace. Also, in three dimensional topological insulators aHall viscosity is allowed when the spatial isotropy is bro-ken by the presence of sample surfaces, and time reversalsymmetry is broken by coupling magnetic elements to thetwo dimensional states appearing at these surfaces .As we will discuss in detail later, the way how theelasticity couples to the low energy degrees of freedom(electrons or phase fluctuations in the case of superfluidsor superconductors) is crucial to determine if a topo-logically nontrivial phase displays Hall viscosity. Mostof the studies available in the literature are based ona continuum formulation of the system and the elasticdegrees of freedom. The philosophy is that, since elasticdeformations can be viewed as geometrical deformationsin the medium hosting the system, the excitations feel adistorted or curved space where to propagate. We canthus define an effective metric tensor related to elasticdistortions and postulate that our system now developsits dynamics in a curved space. The general formalism offield theories in curved spaces and linear responses doesthe job . Although powerful, this approach has somelimitations in the scope of systems that can be treated.For instance, in the metric formalism (in absence of tor-sion), some non trivial topological phases with non-zeroHall conductivity have zero Hall viscosity. This is thecase of Chern insulators in two dimensions and Weylsemimetals in three dimensions. On the other hand,only a small fraction of the literature devoted to theHall viscosity in electronic systems treats the problemfrom the more conventional perspective in solid statephysics through general electron-phonon couplings .The goal in these approaches is to define an effectiveelectron-metric coupling by taking the continuum limitfrom a lattice formulation, thus reaching similar successand suffering from the same limitations as in the formercase.The present work offers a different alternative. We willshow how the standard electron-phonon coupling in aclass of electronic systems with an underlying lattice not only gives rise to a metric formulation in the continuum,but as we know from graphene, emergent vector fieldsarise in such effective description coupling to the elec-trons in a very similar way as the electromagnetic gaugefield. These vector couplings between electrons and elas-tic deformations offer new results and enlarge the kindof systems where the Hall viscosity can in principle bemeasured. II. HALL VISCOSITY FROM HALLCONDUCTIVITY IN GRAPHENE
Dirac materials is the generic name for electronic sys-tems whose low energy excitations are described by aDirac Hamiltonian, graphene being the paradigmatic ex-ample. They exist in two and three spacial dimen-sions and have been the focus of attention in last yearscondensed matter. We will first show that anomalousHall conductivity implies anomalous Hall viscosity ingraphene and we will then extend the discussion to otherDirac materials.The low energy action around one of the Fermi pointsof graphene coupled to a U(1) electromagnetic back-ground field is S = (cid:90) d x ¯ ψ (cid:16) iγ ∂ − iv F γ i ∂ i + ev F c γ i A i (cid:17) ψ. (3)A way to break time reversal invariance suggested by Hal-dane is to add complex next-to-nearest neighbours in thetight-binding on the honeycomb lattice . In the contin-uum limit, these complex hoppings lead to a mass termwith opposite sign at each Fermi point. After integratingout fermions it is easy to find the Chern Simons term asa part of the effective electromagnetic action: S CS = e πc sign ( m ) (cid:90) d x(cid:15) ijk A i ∂ j A k , (4)after considering all the degeneracies (and in units of h = 1). The term (4) is a consequence of the parityanomaly. Coupling the effective action (4) to a back-ground electromagnetic source J i A i allows to derive ananomalous quantum Hall effect: J i ≡ δSδA i = 2 e πc sign ( m ) (cid:15) ijk ∂ j A k . (5)A very interesting aspect of graphene shared by manyother 2D materials is the tight connection between latticestructure and electronic excitations. Within the simplesttight binding-elasticity approach, lattice deformationscouple to the Dirac fermion as elastic U(1) vector fieldsthat depend linearly on the strain tensor u ij = ∂ i u j + ∂ j u i as (see section Methods V A) A el = βa ( u − u ) ,A el = − βa u , (6)where β is the dimensionless Gr¨uneisen parameter esti-mated to be of order 2 in graphene, and a is the latticeconstant. Since lattice deformations do not break timereversal symmetry, the elastic vector field couples withopposite signs to the two Dirac points and behaves as anaxial vector field in a four dimensional representation.If we consider this elastic axial vector coupling to elec-trons instead of the electromagnetic field we have S g = (cid:90) d x ¯ ψ (cid:0) iγ ∂ − iv F γ i ∂ i + v F γ i A eli (cid:1) ψ + m ¯ ψψ, (7)It is obvious that the same line of arguments as before willgive rise to a Chern-Simons action like (4) in terms of theelastic instead of the electromagnetic field. In particularwe will get S CS = 2 π sign ( m ) (cid:90) d x (cid:16) A el ˙ A el − A el ˙ A el (cid:17) , (8)substituting from eq. (6) we get in the Chern Simonsaction a term of the form S CS [ u ] = 4 β πa sign ( m ) (cid:90) d x [ u ˙ u − u ˙ u ++ u ˙ u − u ˙ u ] . (9)To this effective action we can now couple a source termfor a time-dependent elastic deformation S u = u ij T ij and compute the averaged value of the 11 component ofthe stress tensor (cid:104) T (cid:105) ≡ δSδu = 4 β πa sign ( m ) ˙ u . (10)Hence we get the quantized Hall viscosity η H = 4 β πa sign ( m ) . (11)The coefficient is determined by the parity anomaly in(2+1) dimensions, and the characteristic length associ-ated to the viscosity is given by the lattice spacing a .This new response is a genuine viscoelastic response:Elastic gauge fields inducing a Hall viscosity.The Hall viscosity described here is an intrinsic prop-erty of the Haldane model of graphene in the same senseas the quantum anomalous Hall effect (QAHE) of theoriginal proposal . It has been argued that the intrin-sic Hall viscosity in the topological phase of the Haldanemodel is η H = ¯ hρ/
4, where ρ is the electronic density.When the Fermi level lies within the gap, no Hall viscos-ity appears in the Haldane model.Although the Haldane model has been realised experi-mentally in optical lattices it seems challenging to gen-erate it in graphene. The Haldane model is the modelof the QAHE in graphene and the QAHE by magneticproximity effect has been measured , making graphenea suitable platform to potentially measure effects inducedby the presence of the Hall viscosity. The same reasoning as done in the Haldane model canbe applied to real graphene in a perpendicular magneticfield breaking time reversal symmetry. The Hall conduc-tivity in graphene is σ H = 4 e ( n +1 / /π, n = 0 , , .. .By the same argument as before in the quantum limitwhere only the lowest Landau level is filled ( n = 0), thereis a contribution to the Hall viscosity coming from theelastic vector fields: η H ∼ β a . (12)The complete low energy action for deformed graphenein the elastic limit has been derived in . In the presenceof a magnetic field other terms coupling the elasticity tothe Dirac fermions contribute to the Hall viscosity. Inparticular the following term H = i βu ij (cid:0) ψ + σ i ∂ j ψ + ∂ j ψ + σ i ψ (cid:1) , (13)gives a contribution to the Hall viscosity that has beencomputed to be η ( B ) H = 4 β [ | n | ( | n | + 1) / / eB/ π. (14)In the quantum limit ( n = 0) we thus have η ( B ) H ∼ β /l B ,where l B is the magnetic length. This result might seema little bit puzzling at a first glance. We have stressedthat in the absence of torsional couplings and magneticfields, the Haldane model has no Hall viscosity unless oneconsiders the elastic gauge fields. The reason is that thecoupling in (13) is a derivative coupling. When one con-sider both terms (7) and (13) and find the effective theoryfor elasticity, the term coming from (13) has more thanone space-time derivatives and it does not contribute tothe Hall viscosity. When a magnetic field is taken intoaccount, the partial derivative ∂ j transforms into a co-variant derivative ∂ j + ieA j (with (cid:126)B = ∇ × (cid:126)A ) and themagnetic length l B = (cid:112) c/eB permeates through the cal-culations allowing for the result (14). Taking grapheneas an example, the ratio between the two contributionsis η H η BH ∼ l B a ∼ B , (15)where a is the lattice constant and B is the magnitudeof the magnetic field in Tesla. For a standard magneticfield of 10T, the contribution from the elastic gauge fieldsis three orders of magnitude larger than the one comingfrom (13). It means that if the Hall viscosity is measuredin graphene under quantizing magnetic fields, it is thecomponent discussed in the present work the one thatwill be measured. III. GENERALITY OF THE EFFECT
The novel effect described in graphene is based on twomain ingredients: The existence of elastic gauge fields,FIG. 1:
Two dimensional Dirac materials . Themost representative example of the so-called Diracmaterials in two spatial dimensions is graphene. Ingraphene, carbon atoms are arranged in a honeycomblattice, (a). The highest occupied and the lowest emptybands touch at the corners of the Brillouin zone and thelow energy band dispersion is well described by twospecies of massless Dirac fermions (c). MoS is therepresentative of the family of transition metaldichalcogenides, (b). MoS is a semiconductor withbands that admit a low energy effective description interms of two species of massive Dirac fermions (d).and the non–trivial topology of the electronic bands ofthe system. This last ingredient is what allows to writedown a Chern Simons effective action in 2D given in eq.(4) which is the key issue of the approach (D will de-note the spacial dimensions of the (D+1) system). Thenon trivial topology can arise directly as a Berry phaseassociated to the Bloch bands of the crystal as in theHaldane model , or from the topology of a general 2Delectron gas in a perpendicular magnetic field throughthe Hofstadter model . In all cases the Hall conductiv-ity is associated to the topological Chern–Simons actiondepicted in eq. (4). As we have demonstrated, a Hallconductivity and elastic gauge fields inmediately impliesthe new Hall viscosity. We will now discuss the general-ity of the presence of elastic gauge fields in real systemsand give some concrete examples.Although first derived in a tight binding formulationon the lattice, the generation of pseudogauge fields cou-pling shape deformations to electronic degrees of freedomin the low energy effective Hamiltonian of electron sys-tems, is a very general phenomenon. It also arises in ageneral construction of low energy effective actions basedon a symmetry analysis as described in . The generalconditions for a 2D lattice to support elastic gauge fieldshave been discussed in . The two essential ingredientsare: The presence of (at least) two atoms per unit cell(this can arise from geometry as in graphene, from or-bital degrees of freedom, or by other mechanisms). Lack of inversion symmetry in the little group leaving a Fermipoint invariant, ensures that the Fermi points sit at non-equivalent high symmetry points of the Brillouin zone. Itis easy to see that the same structure giving rise to theDirac Hamiltonian causes the minimal coupling to thevector fields associated to lattice deformations implyingthat the phenomenon can be generalized to other crystalsin two and three dimensions. As it is clear from the anal-yses in the symmetry group of the lattice dictates theprecise form of the elastic vector fields.In 3D, an analysis of all crystallographic groups thatsustain Weyl points in its low energy action in com-plete analogy to the case of graphene was first presentedin ref. . Since the Dirac structure in these crystalscomes from the lattice, these having Weyl points in non–equivalent high symmetry points of the Brillouin zonewill probably generate elastic vector field couplings un-der strain although a general analysis as the one done in2D is still lacking. A. 2D and Van der Waals systems
New two dimensional materials of scientific and tech-nological interest are being synthesized in the “post–graphene” era . They share many of the properties ofgraphene and often have large gaps making them moresuitable for electronic applications. Single-layer and mul-tilayer transition metal dichalcogenides M X ( M = Mo,W and X = S, Se) do support elastic gauge fields.Although having a noticeable band gap and a compli-cated orbital structure, M oS and the related compoundsbased on the honeycomb lattice, support at low energy,the same elastic gauge fields as graphene given in eq.(6) . These are also found in bilayer graphene whichhas a quadratic dispersion and a Hall conductivity twotimes that of graphene. They will also occur in rombohe-dral multilayer graphene where the effective low energymodel of the n layer compound has a Lifschitz dispersionrelation ω ∼ k n . Our mechanism for Hall viscosity willdirectly apply in these 2D (or effectively 2D) compounds.In these multilayer systems formed by a family of iso-lated metallic 2D lattice planes indexed by a primitivereciprocal lattice vector G , the 3D Hall conductivity is σ abH = 12 π σ DH (cid:15) abc G c . (16)Any 2D system having a quantum Hall effect (QHE) willexhibit, when stacked to form a 3D material, a 3DQHEif the inter–plane coupling is sufficiently weak, as it isthe case of the so called Van der Waals (VdW) systems.When the 2D constituents support elastic gauge fields,a 3D Hall viscosity will arise whose coefficient is not re-lated to a quantum anomaly. The obvious example inthis class is graphite whose 3D Hall conductivity has beendescribed in . In a homogeneous magnetic field perpen-dicular to the layers and when the Fermi level is in the3D gap, the Hall conductance is found to be (in unitsFIG. 2: Band structure of representative Weylsemimetals . Typical band structure of a Weylsemimetal with two nodal points at different points inthe momentum space. The vector connecting the nodalpoints is a key ingredient in the axial anomaly.of the quantum of conductance) σ xy = 2 (2 n +1) c where c is the c-axis coupling constant of graphite estimated tobe c = 6 . A . The corresponding 3D Hall viscosity willhave an in-plane component η H = 2 (2 n + 1) β πa c , (17)where a = 2 . will also have theproposed elastic response. B. Anomalous Hall viscosity in a Weyl semimetal
The derivation of the Hall viscosity from Hall con-ductivity can be extended to Weyl crystals in threespacial dimensions . For electronic states in a three-dimensional periodic potential in a uniform magneticfield and when the Fermi level lies in an energy gap, theHall conductivity adopts the form σ ij = e πc ε ijk G k , where G is a vector in the reciprocal lattice of the peri-odic potential. The Hall current is given by J = e π E × G .The presence of elastic gauge fields will give rise in these3D crystals, to an anomaly related Hall viscosity coeffi- cient η H obtained from the 3D Chern Simons term S CS = (cid:90) d xG i (cid:15) ijkl A elj ∂ k A ell . (18)For a general 3D crystal breaking time reversal symme-try, the anomalous hall effect is characterized by a mo-mentum space vector (cid:126)ν called the Chern vector. Theanomalous hall conductivity is given by the same expres-sion: σ ij = e πc (cid:15) ijk ν k , (19)with the lattice vector replaced by (cid:126)ν . Our mechanism forthe Hall viscosity will also apply in the three dimensionalcase by replacing the gauge fields in eq. (18) by the elasticgauge fields. The elastic Hall viscosity coefficient will be η H ∼ β a l , (20)where l is a characteristic length associated to the Chernvector. This is the basic mechanism at work. In (3+1)dimensions there are some subtleties concerning the coef-ficient of the Chern Simons action that will be discussedin detail in the Methods sec. V B. In what follows wewill concrete these general expressions working out theanomalous Hall viscosity of Weyl semimetals.Weyl semimetals (WSM) are the “3D graphene”:They break time reversal symmetry and have pairs ofWeyl nodes of opposite chirality connected by Fermiarcs. Their low energy description in terms of 3D Weylfermions has given rise to an enormous amount of worksrelated to quantum field theory physics: experimentalrealization of the axial anomaly, chiral magnetic effect,etc. (see and the references there in). To enhancethe excitement, WSM physics has been reported recentlyin several compounds, particularly TaAs . WSM whoseFermi surface crosses a given set i of Weyl nodes of chi-ralities ξ i located at positions (cid:126)P i have an anomalous Hallconductivity given by eq. (19) with a Chern vector (cid:126)ν = Σ i ( − ξ i (cid:126)P i . Nevertheless the Hall viscosity of the in-trinsic material at zero temperature and chemical poten-tial is zero . By our proposed mechanism Weyl semimet-als will have an anomalous Hall viscosity at zero temper-ature and zero chemical potential given by the coefficientdictated precisely by the chiral anomaly supplementedwith the elastic constant of the material.A detailed derivation of the elastic gauge fields for alattice model of a Weyl semimetal is given in sec. Meth-ods V B. The low energy effective action in the continuumlimit around the Weyl points ( ± λ ) is given by ( v = at ) H W ( k ) = ψ + ± , k (cid:0) σ ( v k ⊥ ± A el ⊥ ) ∓ ( v k ± A el ) σ (cid:1) ψ ± , k . (21)The elastic gauge fields are proportional to the strain ten-sor u ij with coefficients that depend on the parametersof the model: A el = β (cid:113) b − m u ,A el = β (cid:113) b − m u ,A el = β r b − m t u . (22)Integrating out the fermions in the four componentsformulation where the field λ couples as an axial field,we get through the electromagnetic chiral anomaly, theeffective actionΓ eff [ u ] = 148 π (cid:90) d x(cid:15) µνρσ λ µ A elν ∂ ρ A elσ == β π (cid:18) b − m v (cid:19) (cid:90) d x ( u ˙ u − u ˙ u ) , (23)The numerical coefficient in (23) is explained in sec.Methods V B). The anomalous Hall viscosity (in the ab-sence of a magnetic field) is η H = β π a (cid:16) b − m t (cid:17) . (24)As it can be read from eq. (23) the new Hall viscos-ity obtained from the elastic gauge coupling is a coeffi-cient η that points along the direction of the vector λ breaking time reversal and rotation symmetries. Sincein the perpendicular plane between the two nodal pointsthe system is a Chern insulator (see Fig. 2), there willalso be an in–plane component η coming from theelectron-phonon mechanism described in ref. . IV. DISCUSSION AND WRAPPING UP
In the present work, we have identified a new mech-anism to generate a Hall viscosity and a large numberof new materials and systems that will display such ef-fect. As it can be understood from the previous generaldiscussion , it is natural to measure changes in thephonon structure due to the presence of η H . In-planephonon dispersion measurements can be performed byX-ray scattering , Brillouin scattering , and electronenergy loss spectroscopy , all already done in the caseof graphene and VdW materials. Also, since now we arealso open to three dimensional electronic systems, threedimensional probes like neutron scattering can in prin-ciple be also used. The Hall viscosity can be related tochanges in the phonon dispersions in presence of an ex-ternal magnetic field. In order to get some numbers wecan take graphene as the paradigmatic example. Usingstandard values for the Lam´e coefficients λ = 2 eV˚A − and µ = 10 eV˚A − a characteristic frequency associatedto the Hall viscosity can be defined, ω H = πa µβ ¯ h (cid:39)
95 eV, which is a rather large frequency scale compared with thein-plane acoustic phonon frequencies in most systems ( ∼ hundreds of meV). It intuitively means that the changesin the acoustic phonons will be hard to observe withincurrent experimental resolutions at least in two dimen-sions. In spite of that, we believe that increasing thenumber of systems that display a Hall viscosity throughelastic gauge fields, it is a matter of time to find an ex-perimental probe that could detect and measure η H .We end up summarizing the main findings of this workand their physical implications: • The elastic gauge fields encoding the interactionof the electronic properties o with lattice deforma-tions of crystals are a very general phenomenon.First described in graphene, they have also beenobtained in the low energy effective models of other2D systems as MoS but there were no examples in3D. We have deduced the presence and structureof the elastic gauge fields in a 3D model of Weylsemimetals (Sec. Methods). • In any compound supporting elastic gauge fieldsHall conductivity implies Hall viscosity. This isparticularly important in 3D where the precise formof the gravitational and chiral anomalies excludesthe possibility of standard Hall viscosity in the ab-sence of torsion. For these systems with finite den-sity or temperature where a Hall viscosity exists,the contribution to the Hall viscosity from elasticgauge fields is several orders of magnitude biggerthan that coming from phonons or metric deforma-tions.The lattice aspects of the viscoelastic response of topo-logical crystals have been recently analyzed in . Ournew contribution to the Hall viscosity, although not ex-plicitly discussed, could certainly be worked out as a partof their general analysis. What we emphasize here is theanalogy of the elastic gauge fields with the standard elec-tromagnetic partners which makes the connection withthe anomaly related topological aspects very transpar-ent. Acknowledgments
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Phys. Rev. B , 134409 (2014). V. METHODSA. Elastic gauge fields in graphene
The fact that the lattice deformations couple to thelow energy electronic excitations of graphene in the formof elastic gauge fields has been explained in a number ofarticles and reviews. Nevertheless, being central to ourwork, we will repeat the derivation here taken mostlyfrom ref. . Elastic deformations can be introduced ina tight binding calculation simply changing the hoppingparameters. In the case of non equal hopping parameters γ i = t + t i , one gets, near the a K point, the effectivelow energy Hamiltonian: H = v F ψ + σ i (cid:0) i∂ i − A eli (cid:1) ψ, (25)where v F = (3 / ta , t is the hoping parameter of theundeformed lattice and a is the lattice constant. Thevector field (cid:126)A is A el = √ v F ( t − t ) ,A el = 12 v F ( t + t − t ) , (26)In the weakly deformed lattice, assuming that the atomicdisplacements (cid:126)u are small in comparison with the latticeconstant a one has t i = t + βta (cid:126)δ i · ( (cid:126)u i − (cid:126)u ) , (27)where (cid:126)δ i are the nearest-neighbor vectors, (cid:126)u is the dis-placement vector for the central atom, and β = − ∂ ln t∂ ln a (cid:39) (cid:126)u ( (cid:126)r )is performed by making the substitution (cid:126)u i − (cid:126)u ∝ (cid:16) (cid:126)δ i · ∇ (cid:17) (cid:126)u ( (cid:126)r ) , (29)from where we obtain the effective gauge field : A el = βa ( u − u ) ,A el = − βa u . (30)For the other valley, K (cid:48) , the sign of the vector potential(30) is opposite, in agreement with the requirement oftime-reversal invariance.The gauge field (30) is proportional to the deformationtensor which is directly involved in the density of elasticenergy. This means that, although the coupling to thefermions is gauge invariant, the problem as a whole isnot. A pure gauge rotation by the gradient of a scalarfunction will change the kinetic term of the elastic part. B. Emergent elastic vector fields in a Weylsemimetal.
To illustrate how emergent vector fields associated toelasticity appear in a Weyl semimetal phase we can con-sider the following simple model of s-, and p-like elec-trons hopping in a cubic lattice and chirally coupled to anon-site constant vector field b . The parameters t, r, andm, represent, in a tight-binding description, the hoppingmatrix elements between s and p states, hopping betweenthe same kind of states, and the difference of on-site en-ergies between s and p states, respectively. The vectorfield b breaks time reversal symmetry. Without loss ofgenerality, we will choose the vector field b to point alongthe OZ direction. The tight binding Hamiltonian is: H = (cid:88) i,j c + i ( itα j − rβ ) c i + j + ( m + 3 r ) (cid:88) i c + i βc i + h.c, (31)where i labels the position R i and j labels the six nextnearest neighbors a j of length a in the cubic lattice. Thematrices α i and β are the standard Dirac matrices. Inthe unstrained situation we will set all the hopping terms t equal for simplicity.We will focus on the parameter regime 0 > m > − r corresponding to a topological insulating phase. The longwavelength limit of this model around the Γ point ( k = 0)is an isotropic massive Dirac system. Since H describesa massive Dirac model in the continuum limit in threespatial dimensions, in order to find a non vanishing Hallviscosity we need to break both time reversal symmetryand the effective rotational symmetry in the continuum.In the lattice model we can do both operations by includ-ing an on-site axial coupling between the fermion fieldsand a constant vector field b . Without loss of general-ity we will choose b to point along the third axis, with b > m : H b = (cid:88) i b c + i α γ c i . (32)With the choice b > m the spectrum of the total Hamil-tonian (31)+(32) consists in two bands crossing at twoFermi points in the Brillouin Zone and two bands athigher and lower energies, as it can be seen in Fig.(3).To find the low energy effective model around these twoFermi points, we shall proceed in two steps: we will ex-pand the lattice model for small momenta compared with1 /a , so after Fourier transforming (31)+(32) and intro-ducing the following approximations sin( k j a ) (cid:39) k j a andcos( k j a ) (cid:39) − k j a , one gets ( v = at ) H ( k ) = (cid:88) j c + k ( vα j k j + mβ + b α γ ) c k ≡≡ (cid:88) j c + k H ( k ) c k . (33)And now we project out the high energy bands. Aftera unitary transformation, the Hamiltonian matrix H ( k ) can be written as H ( k ) = (cid:18) v σ · k ⊥ + ( b + m ) σ vk σ vk σ v σ · k ⊥ + ( b − m ) σ (cid:19) , (34)with k ⊥ = ( k , k ). The four-component wavefunctioncan be written in two-component blocks ( φ k , ψ k ). Forenergies E (cid:28) m + b and momenta k j (cid:28) ( m + b ) /v we can write φ k (cid:39) − k m + b ψ k and the effective two-bandmodel takes the form: H eff ( k ) = ψ + k (cid:18) v σ · k ⊥ + 1 m + b ( b − m − v k ) σ (cid:19) ψ k . (35)From this effective Hamiltonian it is easy to see thatthe momenta where the two bands cross are λ ± =(0 , , ± (cid:113) b − m v ). Expanding now around these twopoints k (cid:39) λ ± + δ k , the final Hamiltonian takes theform of two massless three dimensional Dirac fermions ψ ± separated 2 λ in the momentum space: H W ( δ k ) = ψ + ± , k ( v σ · δ k ⊥ ∓ v δk σ ) ψ ± , k , (36)with v = 2 v (cid:113) b − mb + m .Now let us focus on the part of the original latticeHamiltonian that depends on the strain. As discussedin the case of graphene, the strain tensor u ij enters inthe tight binding approach through the change of thehopping parameters t when the lattice is distorted. Thegeneral recipe is (see also ref. ) for t , tα → t (1 − βu ) α + tβu α + tβu α , (37)and the same for t and t , and r → r j (cid:39) r (1 − βu jj ) , (38)being β the corresponding Gr¨uneisen parameter of themodel. Inserting these hopping changes in the originalHamiltonian (31), we can define the strained Hamiltonianas the sum of the original Hamiltonian H and the straindependent part H [ u ij ]. For our purposes we will firstfocus on these two parts of H [ u ij ]: δH [ u ij ] = tβ (cid:88) k c + k [( u α + u α − u α ) sin( k a )] c k , (39)and δH [ u ij ] = rβ (cid:88) k u (1 − cos( k a )) c + k βc k . (40)Projecting out the high energy sector and expandingaround the two nodal points λ ± , the strain dependentHamiltonian part takes the form δH [ u ij ] = ± λβv (cid:88) k (cid:88) j =1 , u j ψ + ± , k σ j ψ ± , k −− ra β λ u ψ + ± , k σ ψ ± , k . (41)0 (a) k E ( k ) (b) k (c) k b < m b > mb = 0 FIG. 3:
Evolution from a topologogical insulator to a Weyl semimetal . For b = 0 the spectrum consists intwo pairs of degenerate bands due to time reversal symmetry, (a). When 0 < b < m the band degeneracy breaksdown and a high energy sector differentiates from a low energy sector, but the system is still gapfull, (b). When b > m , the low energy bands cross each other at two definite points in the Brillouin zone. At sufficiently lowenergies, the system consists in two pairs of Weyl fermions with opposite chirality, (c).We have found that, around the two nodal points, straincouples to the low energy electronic sector as a vectorfield: A el = β (cid:113) b − m u ,A el = β (cid:113) b − m u ,A el = β r b − m t u . (42)We also note that this coupling is chiral, that is, theelastic vector field A el couples with opposite signs to theelectronic excitations around the two Weyl nodes. Whatwe have found is that, when the cubic symmetry in theoriginal lattice model, or the continuous rotation symme-try in the effective model is broken by a constant vector b , strain couples to electrons as a chiral vector field, sim-ilarly to what happens in graphene or other two dimen-sional systems.Integrating out fermions to get an effective theory forthe elastic distortions will give rise, through the electro-magnetic chiral anomaly, to a term of the formΓ[ u ij ] = 13 β λ π (cid:90) d x (cid:16) A el ˙ A el − A el ˙ A el (cid:17) == 13 β λ π (cid:90) d x ( u ˙ u − u ˙ u ) , (43)and the Hall viscosity associated to the chiral anomalymechanism is (in units ¯ h = 1) η H = β π a (cid:16) b − m t (cid:17) . (44) C. Chiral anomaly and Hall viscosity in Weylsemimetals
Let us consider he action for a massless Dirac fermioncoupled to a U(1) vector A µ and to a constant axial vec-tor b µ in (3+1) dimensions: S = (cid:90) d x ¯ ψγ µ ( i∂ µ + eA µ + b µ γ ) ψ. (45)At the classical level the action is invariant separatelyunder vector and axial transformations: ψ → exp[ iα ] ψ , ψ → exp[ iθγ ] ψ what ensures the separate conservationof the vector J µ = ¯ ψγ µ ψ and axial J µ = ¯ ψγ µ γ ψ cur-rents: ∂ µ J µ = 0, ∂ µ J µ = 0. Also, since we have chosen b µ to be constant, classically we can make a change in thechiral phase of the fermion fields to eliminate it from theaction (45). As we know well, we cannot remove com-pletely the vector b µ from our theory at the quantumlevel due to the chiral anomaly , that translates intothe presence in the effective action for the electromag-netic field of the following term:Γ a [ A ] = e π (cid:90) d x ( b σ x σ ) (cid:15) µναβ F µν F αβ . (46)Integrating by parts and neglecting boundary terms, onecan recognize in (46) the so-called Bardeen counter-term . From the action (46), one easily obtains theanomalous current conservation laws at the quantumlevel: ∂ µ J µ = 0 and ∂ µ J µ = π (cid:15) µναβ F µν F αβ .Things become more involved when, besides the field A µ ( x ) and the vector b µ , we consider adding to (45) anaxial vector field A µ that couples to the chiral current1 (a) A VV (b)
A AA
FIG. 4:
Anomalous Feynman diagrams.
When both axial and vector fields are coupled to Weyl fermions, thereare two diagrams that contribute to the axial anomaly. Imposing that the electromagnetic current must beconserved forces these two amplitudes to not be equal being the proportionality factor . J µ . In this case the chiral current is not only coupled toa constant vector that leads to (46) after a chiral gaugetransformation. In this case, the new effective action canbe obtained by computing the amplitudes diagrammati-cally depicted in Fig.(4) . Up to quadratic terms in A µ and A µ , the effective action now has an extra term:Γ a [ A ] = 13 116 π (cid:90) d x ( b σ x σ ) (cid:15) µναβ f µν ( x ) f αβ ( x ) , (47)where f µν is the field strength associated to A µ . The rel-ative factor of 1 / ∂ µ J µ = 0.We are now ready to obtain the term in the effectiveaction for elasticity corresponding to the Hall viscosityfrom (47) by identifying the vector field b µ with (0 , λ )and A µ as the elastic vector field in (42). According toour discussion on the elastic gauge fields, since they areaxial vector fields, and so is the vector (cid:126)b(cid:126)b