Higher Rank Chiral Fermions in 3D Weyl Semimetals
HHigher Rank Chiral Fermions in 3D Weyl Semimetals
Oleg Dubinkin, F. J. Burnell, and Taylor L. Hughes Department of Physics and Institute for Condensed Matter Theory,University of Illinois at Urbana-Champaign, Illinois 61801, USA Department of Physics, University of Minnesota Twin Cities, MN, 55455, USA
We report on exotic response properties in 3D time-reversal invariant Weyl semimetals with mirrorsymmetry. Despite having a vanishing anomalous Hall coefficient, we find that the momentum-spacequadrupole moment formed by four Weyl nodes determines the coefficient of a mixed electromagneticcharge-stress response, in which momentum flows perpendicular to an applied electric field, andelectric charge accumulates on certain types of lattice defects. This response is described by amixed Chern-Simons-like term in 3 spatial dimensions, which couples a rank-2 gauge field to theusual electromagnetic gauge field. On certain 2D surfaces of the bulk 3D Weyl semimetal, we findwhat we will call rank-2 chiral fermions, with ω = k x k y dispersion. The intrinsically 2D rank-2chiral fermions have a mixed charge-momentum anomaly which is cancelled by the bulk of the 3Dsystem. PACS numbers:
Chiral fermions have had a remarkable impact acrossa variety of fields of physics in the past few decades.Whether it is in the context of the weak interactions inparticle-physics[1, 2], or as low-energy edge or bulk exci-tations of topological insulators[3, 4] and semimetals[5–7], or even as heralds of non-reciprocal light transportin photonic crystal analogs[8–10], there is no denyingtheir broad relevance to a number of physical platforms.From their origin in Lorentz-invariant field theories it isknown that these massless, linearly dispersing fermionscan intrinsically appear in any odd spatial dimension.Additionally, in a condensed matter context, the famousNielsen-Ninomiya no-go theorem[11] dictates that local,time-independent lattice Hamiltonians must harbor aneven number of chiral fermions, such that the total chi-rality vanishes. Hence, these restrictions allow 1D chiralfermions to either appear as right/left-mover pairs in a1D metal, or as isolated chiral edge states of a Cherninsulator[4], while 3D chiral (Weyl) fermions appear innodal pairs in Weyl semimetal materials[5].Recent developments in the condensed matter andhigh-energy literature have opened the door to the dis-covery of new types of massless fermions and bosonsin a non-Lorentz invariant, crystalline environment[12–17]. In this work we propose a generalization of chiralfermions for 2 dimensional systems with crystalline sym-metry; we call these rank-2 chiral fermions. We presenta 2D lattice model exhibiting rank-2 chiral fermions, butwhere the total higher rank chirality vanishes. Then wepresent a 3D model in which the rank-2 chiral fermionsappear as surface states with an associated anomalous re-sponse. We find the anomalous response can be mappedonto a bulk rank-2 Chern-Simons term (in analogy withthe dipole Chern-Simons term from Ref. 18) representinga mixed charge-geometric response, and has a coefficientrelated to the Berry curvature quadrupole moment of thebulk Fermi-surface. We begin by reviewing chiral fermions in 1D. For gap-less fermions in 1D, the chirality is given by χ = sgn( v ),where v is a characteristic velocity that fixes the disper-sion relation E ( k ) = (cid:126) vk − µ . Superficially, this chiralityallows us to define two currents that are conserved by theclassical equations of motion of a general 1D system: theusual charge current j µ (where j = (cid:80) c † α c α ), and theaxial current j µχ (where j χ = (cid:80) α χ α c † α c α ), and α runsover all fermion channels. Evidently, j χ is associatedwith the conservation of the difference in the number ofright-movers (with positive chirality) and the number ofleft-movers (with negative chirality).However, it is well known that these two currents can-not be simultaneously conserved[19–21]. Indeed, in thepresence of an electric field E x , the axial current obeysthe anomalous conservation law ∂ µ j µχ = n eE x π (cid:126) , (1)where n is the number of channels. This anomalous re-sponse reflects the fact that during a process in whichone adiabatically shifts the vector potential A x → A x + heL , the number of right-moving (left-moving) particleschanges by δN χ =+1 = +1 ( δN χ = − = − (cid:80) α χ α (cid:54) = 0, the axial anomaly (1) also implies ananomaly in the U(1) charge current. Thus a net chiralityis impossible in a 1D lattice system that conserves theelectric charge. However, chiral fermions can appear onthe 1D edges of 2D integer quantum Hall [3, 22] or quan-tum Anomalous Hall [4, 23] systems. Moreover, a wholefamily of quasi-1D chiral edge states exists on the bound-ary of 3D, T-breaking Weyl semimetals, and forms a so-called Fermi arc having chiral dispersion in the surfaceBrillouin zone[6, 7]. In these cases, the anomalous conser-vation law on one boundary is balanced by a current flowthrough the bulk (possibly from the other boundary)[24],and is a signature of a Hall effect.We now discuss a generalization of chiral fermions to a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b E ( k ) = (cid:126) vξk x k y − µ, (2)where v is a velocity and ξ has units of length. Suchlocal Fermi-surface patches are not uncommon and in-deed are guaranteed in 2D bands by Morse theory[25].The dispersion relation and Fermi surface contours aredepicted in Fig. 1(a). To look for a charge anomaly wecan consider the current j i = ∂H/∂k i = (cid:126) vξσ ij k j where σ xy = σ yx = 1 , σ xx = σ yy = 0 . We see that upon adia-batically turning on a constant electric field E x or E y , thetotal number of charged particles below the Fermi surfacedoes not change: for every extra fermion that is addedat momentum σ ij k j in the presence of the electric field E i , there is a partner at the opposite momentum − σ ij k j that is removed. This should be contrasted with the re-sponse for a 1D chiral fermion above where inserting asingle flux of E x generates an extra particle, leading toan anomaly in the U(1) charge current; such an anomaly is not present in our rank-2 chiral system. However, wewill now show that the dispersion relation (2) does ex-hibit a 2D variant of the axial anomaly, which leads toa violation of momentum conservation in the presence ofexternal electric fields, and a violation of charge conser-vation in the presence of certain strain fields.To motivate this anomaly, let us first choose a direc-tion ˆ k in momentum space, and consider a section of ageneric 2D Fermi surface that satisfies ( v k FS · ˆ k ) (cid:54) = 0 , where v k FS is the Fermi velocity at momentum k F S onthe Fermi surface. We can associate an axial current toeach quasi-1D system at constant k (cid:107) ≡ ( k × ˆ k ) · ˆ z , via amomentum-resolved chirality χ ˆ k k (cid:107) = (cid:80) k FS sgn( v k FS · ˆ k ),where the sum runs over momenta on the relevant sectionof the Fermi surface for which ( k F S × ˆ k ) · ˆ z = k (cid:107) . For ex-ample, if we fix ˆ k = ˆ y , then at fixed k x = K x , we can de-fine an axial current j χ,y ( K x ) = (cid:80) k y,FS χ ˆ yK x ˆ n ( K x ,k y,FS ) ;choosing ˆ k = ˆ x we can define j χ,x ( K y ) similarly. Intu-itively, this definition captures the fact that if we pick adirection ˆ k that intersects the Fermi surface going fromoccupied states to unoccupied states then it has a chiraldispersion along ˆ k , and vice-versa for anti-chiral.From a global perspective, any closed Fermi surfacewill intersect each slice at fixed k (cid:107) an even number oftimes along the ˆ k direction, with equal numbers of pos-itive and negative chirality intersections. Thus for closedFermi surfaces the total chirality of each fixed k (cid:107) slicevanishes. In contrast, Eq. 2 describes open Fermi sur-faces which have well-defined, non-vanishing chiralities χ ˆ xK y = sgn( vξK y ) for slices at fixed k y = K y , and simi-larly for fixed k x = K x . These values are non-vanishingon each hyperbolic branch of the Fermi surface, sincethere is only one value of k i on the Fermi surface at whichthere is an intersection with each constant σ ij k j slice (seeFig. 1(b),(c)). Thus, each fixed momentum slice is chiral. k x k y k x k y E EE k y
00 0 0 000 k x k y k x1 k x2 k x3 k x4 -k x4 -k x3 -k x2 -k x1 (a) (b)(c) (d) FIG. 1: (a) Dispersion relation and Fermi surface contoursfor Eq. 2. Note that when µ = 0, the the Fermi surfaceconsists of two intersecting lines. (b) Dispersion relation ofEq. 2 but with guides to illustrate the chirality χ ˆ yK x in the k y -direction for fixed values of k x . The yellow line representsthe collection of k (cid:107) Fermi points at a fixed Fermi level repre-sented by the blue plane. (c) Cross-sections of the dispersionas a function of k y for a set of fixed values of k x , colors in-dicate different χ ˆ yK x chiralities which are opposite for ± k x .(d) Fermi-level contours for the 2D tightbinding dispersion E ( k ) = sin k x sin k y . Orange and blue points are rank-2 chiralfermions having χ = ± µ = 0 Fermi surface that gain/lose par-ticles when an electric field E x is turned on. Globally there isno momentum anomaly as each fixed k x slice has a positiveand negative chirality. The same is true for E x → E y , k x → k y if we rotate the thick red and blue lines by π/ Despite this chirality, Eq. 2 has time-reversal symmetry,which implies χ ˆ k − k (cid:107) = − χ ˆ k k (cid:107) , and hence the net axialcharge of the entire Fermi surface vanishes, since eachbranch of the hyperbola has an opposite chirality. Thisconfirms our claim above about the lack of a conventionalaxial anomaly in this system.Interestingly, this failure points the way to the actualanomaly of interest since the product χ ˆ k k (cid:107) · k (cid:107) does take thesame sign on the two hyperbolic Fermi surface branches,and will also do so in general for a Fermi surface intervaland its time-reversed partner. Hence, let us specialize totime-reversal invariant systems, and focus on anomaliesin the momentum densities/currentsˆ J a = 1 A D (cid:88) k (cid:126) k a ˆ n k , (3)where A D is the area of the 2D system. More pre-cisely, given an interval of the Fermi surface with a non-vanishing chirality χ ˆ k k (cid:107) , and its time-reversed partner, wecan apply a uniform electric field in the ˆ k -direction, andconsider the change in the k (cid:107) momentum, i.e., the compo-nent of ˆ J a perpendicular to the applied field. Physicallywe expect that an electric field acting on a Fermi sur-face with a non-vanishing χ ˆ k k (cid:107) generates electrons withone sign of k (cid:107) momentum, and its time-reversed part-ner will remove electrons with the opposite sign of k (cid:107) momentum, such that the particle number stays fixed,but there is a net change in momentum. For the hy-perbolic Fermi surfaces of Eq. 2 we find that, for anyvalue of µ , turning on E x by adiabatically shifting A x by heL generates a change in the y -momentum densityequal to ∆ J y = − sgn( vξ ) (cid:126) (cid:80) Λ y k y = − Λ y | k y | , where Λ y is awavevector cutoff. If we repeat the experiment with E y we find ∆ J x = − sgn( vξ ) (cid:126) (cid:80) Λ x k x = − Λ x | k x | . In the thermo-dynamic limit, we find ∆ J a = − sgn( vξ ) a π (cid:126) Λ a when anelectric field σ ab E b is applied by inserting one flux quan-tum. There are similar anomalous responses of the Eq.2 Fermi surface for momenta orthogonal to other appliedelectric field directions, except when the field is appliedalong the ˆ x ± ˆ y directions where the anomalous responsevanishes.Naively we would like to associate the sign of the mo-mentum anomaly response sgn( vξ ) to a notion of two-dimensional chirality. However, without additional sym-metry, this chirality is not well-defined since by sim-ply rotating the coordinate system one can transform k x k y → − k x k y , hence flipping the sign of the chiral-ity. Thus, to formulate a robust notion of a rank-2chirality we need to impose symmetry. Let us imposemirror symmetry about the line x = y. This accom-plishes several things: (i) it forces k x k y and k x − k y tolie in different symmetry representations such that onecan no longer continuously deform k x k y → − k x k y with-out breaking symmetry[45], (ii) it establishes a naturalpair of directions (up to a sign) in which to consider theanomalous momentum response; these directions are ex-changed by the mirror symmetry, and orthogonal to eachother, e.g., ˆ x, ˆ y for this mirror symmetry, (iii) it requiresΛ x = Λ y ≡ Λ . Hence, with mirror symmetry the anoma-lous momentum response is characterized by: ∂ µ J µa = χ σ ab e Λ π E b a, b = x, y, (4)where χ is the rank-2 chirality, which is equal to sgn( vξ )for the dispersion in Eq. 2.With mirror symmetry we now have a well-defined no-tion of a chirality, but we still have not indicated a linkto a rank-2 structure. The rank-2 nature is more clearlyexpressed at this level through a reciprocal anomalousresponse where a gauge field e aµ , that couples canonicallyto the momentum current J µa , can produce an anomalyin the ordinary charge current. Eq. (3) implies that each electron couples to e aµ with a charge equal to its a -momentum; thus adiabatically shifting e aj is heuristicallylike turning on an electric field in the − j (+ j )-directionfor electrons with negative (positive) k a momentum, witha magnitude proportional to | k a | . For the rank-2 chi-ral fermions, the opposite electric fields are applied toopposite chirality Fermi surface branches, hence we ex-pect such a shift will generate an excess of charge for therank-2 chiral fermions. We note in passing that while itis tempting to identify the fields e aµ as frame fields, weonly consider fields that couple to momenta along trans-lationally invariant lattice directions, which has the effectof limiting the gauge transformations to δ λ e aµ = ∂ µ λ a . To see the anomaly explicitly, let us shift e yx by L y /L x in a system with periodic boundary conditions. Physi-cally, this describes a process in which a dislocation withBurgers vector L y ˆ y is threaded adiabatically through thehole spanned by the periodic x -direction[26] (the finalresult is akin to a twisted carbon nanotube). Mathe-matically, for an electron having a fixed k y , this shiftis equivalent to a shift in A x by (cid:126) e k y e yx [26–28]. Hencean electron with momentum k y = 2 πs y /L y ( s y ∈ Z )will experience a shift k x → k x + 2 πs y /L x . If the statesat this k y have χ ˆ xk y,FS > χ ˆ xk y,FS <
0) then thiswill generate s y ( − s y ) particles at k y . The states at − k y have the opposite chirality, but also the oppositeshift, and thus also contribute s y ( − s y ) particles. Intotal the change in the charge for positive chirality is∆ Q = e (cid:80) Λ y k y = − Λ y L y π k y = e (cid:16) L y Λ y π (cid:17) . In the presence ofmirror symmetry there is a symmetric effect for a shift in e xy → e xy + L x /L y that will produce a ∆ Q = e (cid:0) L x Λ x π (cid:1) , and these effects can be summarized by an anomalousconservation law ∂ µ j µ = χ e Λ π E xy , (5)where mirror fixed L x = L y = L, Λ x = Λ y = Λ , and E xy = ∂ x e y + ∂ y e x − ∂ t e xy is an effective rank-2 electricfield (with a vector charge Gauss’ law[29–31]). Here, themirror symmetry allows us to combine e xy ≡ e yx + e xy toform a symmetric rank-2 gauge field with gauge trans-formations e xy → e xy + ∂ x λ y + ∂ y λ x which match thoseof a vector-charge rank-2 gauge field where the vec-tor charge is crystal momentum. Hence, our mirror-symmetric rank-2 chiral fermions have an anomalousrank-2 response, which is the origin of their name.Before moving on to lattice models with rank-2 chi-ral fermions, a few remarks in order. First, for moregeneric dispersion relations E = g ij k i k j , where g ij is a Lorentzian metric, a rank-2 chirality can be defined bychoosing a mirror line M through the origin of momen-tum space, i.e., M γ ,γ : γ x k x = γ y k y , that exchangesthe principal axes ˆ x , ˆ x of the quadratic form g ij . Theseaxes are unique up to a sign, which will preserve the rel-ative distinction between the two chiralities. The result-ing rank-2 chiral fermions exhibit anomalies described byEqs. 4 and 5, but with the replacement E xy → E x x etc.A second comment concerns the cutoff dependence of theresponse action. We emphasize that this is a momentumcutoff, not an energy cutoff: it describes the range of k values over which we have included states on the Fermisurface.[46] Below, we consider a 3D model where thesecutoffs have a natural physical origin.In 2D, where the Fermi surface must be closed, thenet anomalous response necessarily vanishes when we ex-tend the momentum cutoff to include the entire Brillouinzone. To illustrate this, consider a 1-band, 2D latticemodel on a square lattice with only next-nearest neigh-bor hopping such that the dispersion relation is given by E ( k ) = sin k x sin k y . This model has M , mirror symme-try, hence we evaluate the k x and k y momentum anoma-lies. If one expands around ( k x , k y ) = (0 ,
0) the dis-persion is E ( k ) ∼ χ k x k y with χ = +1, while around( k x , k y ) = ( π,
0) the dispersion is E ( k ) ∼ χ (cid:48) k x k y , with χ (cid:48) = − k x (and to k y ), which make opposite contributions in Eq. (4).A net rank-2 chirality can be realized, however, at thesurface of a 3D system. Let us consider a 2-band BlochHamiltonian for a Weyl semimetal: H ( k ) = sin k x sin k y Γ x + sin k z Γ y ++ ( m + t (cos k x + cos k y + cos k z )) Γ z , (6)where if Γ a = τ a Pauli matrices the model has time-reversal symmetry with T = K , C z = I , and M , = I , or if Γ x = τ y ⊗ σ y , Γ y = τ x ⊗ σ y , Γ z = τ z ⊗ I themodel has spinful time-reversal symmetry T = i I ⊗ σ y K. We focus on the two-band model for simplicity, as thefour-band model behaves just as two copies of the for-mer. In this model we find several Weyl semimetalregimes summarized in Fig. 2(a),(b),(c). In particu-lar, let us focus on the range − t < m < − t , wherethe system has four gapless Weyl points in the k z = 0plane located at: k = ( ± arccos( − m/t − , , T and(0 , ± arccos( − m/t − , T . One can establish the exis-tence of rank-2 chiral fermions on the z -surfaces via nu-merical diagonalization, or with analytical lattice meth-ods to solve for surface states shown in, for example, Refs.32–34. As indicated by the surface BZ projections in Fig.2, we find surface states that have a dispersion relation E ( k ) = ± sin k x sin k y centered around the Γ-point, withzero-energy lines that terminate at the four Weyl nodes,and an overall sign that flips when the surface normal vec-tor is ± ˆ z . On the side surfaces we find Fermi arcs thatexhibit an anisotropic momentum anomaly, but whichare not rank-2 chiral fermions.From our continuum calculations we expect the rank-2surface states to be anomalous, with the momentum lo-cations of the Weyl nodes serving as natural momentumcutoffs in the x and y -directions. The remaining ques-tion is: can we describe the anomalous surface response FIG. 2: (a)–(c): Weyl node positions in the 3D bulk Bril-louin zone and surface state structure on the (0,0,1) surfacefor different values of the parameters t and m . Red (blue)dots depict Weyl nodes with chirality χ = +1( − . Red linesin the 2D surface BZ depict the surface zero modes. (d): Dis-tribution of the k x momentum density J x weighted by theWeyl quadrupole moment Q xx = π / e/ π fora lattice with N y × N z = 40 ×
40 and m = − t in the pres-ence of two opposite magnetic flux lines carrying Φ x = ± / y, z ) = (20 ,
10) and ( y, z ) = (20 , y, z ) = (20 ,
10) on the amount of magnetic fluxΦ x threaded through this plaquette. Red line is a fit of thenumerical data (blue dots) with the slope ≈ . e/ (8 π ). We find thesame dependence for the charge density j localized on a tor-sional magnetic flux B xx (see Supplement for Figure). as a bulk response in analogy to how the anomalous re-sponse of chiral Fermi arcs encode the bulk anomalousHall effect[6, 35–37]. To this end, let us consider the mo-mentum space locations of the Weyl nodes K ( α ) . We findthat the pair of nodes lying on the k x axis have Weyl chi-rality +1, while the pair of nodes on the k y axis both haveWeyl chirality − . Hence, the total chirality and Weylmomentum dipole moment (i.e., the anomalous Hall co-efficients) vanish: (cid:80) α =1 χ ( α ) = 0 , (cid:80) α =1 K ( α ) χ ( α ) = 0 . However, we note that the Weyl momentum quadrupolemoment is non-vanishing: Q xx = (cid:88) α =1 χ ( α ) ( K ( α ) x ) = − (cid:88) α =1 χ ( α ) ( K ( α ) y ) = Q yy . (7)Indeed since the mirror symmetry M (1 , also flips thesign of the Weyl chiralties it enforces Q xx = − Q yy , and Q xy = 0 . We will see below that the anomalous rank-2 surface response is determined by exactly this Weylquadrupole moment.To determine the bulk response we calculate a charge-current (ˆ j i = ∂ k i E ( k ))– momentum-current ( ˆ J ia = k a ˆ j i )correlation function using the Kubo formula (see Supple-ment). We find a bulk linear response: J µa = e π (cid:15) µνρσ Q νa ∂ ρ A σ ,j µ = e π (cid:15) µνρσ Q νa ∂ ρ e aσ , (8)where (when µ = 0) the response coefficients are: Q ia = − π ˆ d k (cid:15) ijk k a F jk , (9)where F ij is the Berry curvature. We show in the Sup-plement that for a gapless system, Q ia can be reducedto an integral over the Fermi surface analogous to thearguments in Ref. 38: Q ia = 12 π ˆ F S k a k i F µν ds µ ∧ ds ν , (10)where { s , s } are the coordinates that parametrize theFermi surface. For a Weyl semimetal, the Fermi surfacesplits into disjoint Fermi surfaces F S α , each enclosingan individual Weyl node carrying a Weyl chirality χ ( α ) . Hence, the integral over the Fermi surface simplifies to asum over a discrete set of Weyl nodes: Q ia = N Nodes (cid:88) α =1 χ ( α ) K ( α ) a K ( α ) i , (11)and is exactly the Weyl momentum space quadrupolemoment mentioned above. We note that recent work hasshown that the Hall viscosity can be determined by themomentum space quadrupole of the occupied states[39],while our work shows that a similar quantity, calculatedon only the Fermi surface, describes a mixed charge-momentum response.The bulk responses in Eq. (8) are described by theaction S [ A, e ] = − e π ˆ d x(cid:15) iνρσ Q ia A ν ∂ ρ e aσ . (12)For our model, and any Weyl quadrupole model havingtime-reversal and M (1 , mirror symmetry, we find a sin-gle non-vanishing response coefficient: ¯ Q = Q xx = − Q yy (note Q xy = 0 by symmetry), and we can simplify theresponse theory to find: S = − e ¯ Q π ˆ d x (cid:2) A z E xy − A y E xz − A x E yz − A ( B yy − B xx ) (cid:3) , (13)where E az = ∂ z e a − ∂ t e zz , and B ai = (cid:15) ijk ∂ j e ak . We writethe action in this suggestive form to make a direct com-parison to the rank-2 scalar charge dipole Chern-Simons theory of Ref. 18. Indeed we see Eq. 13 is just a vector-charge version of that Chern-Simons theory, i.e, insteadof a rank-2 scalar dipole charge we have a rank-2 vectormomentum charge. Interestingly, unlike the scalar chargecase which was shown to be a pure boundary term[18],this theory gives the bulk responses in Eq. 8. This actionshould be contrasted with previous work on geometricresponse in Weyl semimetals which focus on the termsinduced by a Weyl dipole[40–43].Let us now try to understand the physical meaning ofthe bulk response. First, consider J a = e ¯ Q π B a ( δ ax − δ ay ) , (14)which indicates a momentum density attached to mag-netic flux. Second, consider j = e ¯ Q π B ai ( δ ix δ ax − δ iy δ ay ) , (15)which is a charge density attached to a torsional magneticfield B ai parallel to the i -th direction and having Burgersvector in the a -th direction. For our model, the secondresponse requires the torsional flux and Burgers vectorto be parallel, which is naturally represented by a screwdislocation. Hence, on screw dislocations in the x or y -directions we should find a bound charge density. We nu-merically confirm both of these responses by inserting lo-calized magnetic (torsional) flux tubes in the x -directionand calculating the localized momentum (charge) den-sity. Our numerical results exactly match the responseequations as shown in Fig. 2(c),(d).To see how this bulk response is connected to the sur-face states, we observe that the response action Eq. 13is not gauge invariant in the presence of a boundary. Asa model for the surface we can let ¯ Q ( z ) be a domain wallin the z -direction[37]. If we put such a domain wall con-figuration into Eq. 13 and integrate by parts, then theaction localized on the domain wall is S ∂z = e ¯ Q π ˆ dxdydt [ A e xy − A y e x − A x e y ] . (16)If we treat the Weyl points as momentum cutoffs in the x and y directions then we can make the replacement ¯ Q =2Λ . The end result has response equations that exactlymatch the anomalous conservation laws in Eqs. (4) and(5). 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In this section we present a detailed derivation of theresponse equations (8). For clarity, we will limit thederivation here to a 2-band free-fermion model with theHamiltonian of the general form H ( k ) = d ( k ) · σ . Wewill eventually specialize to a case with a band structurein which Weyl nodes are arranged in a quadrupolar pat-tern. Let us consider the mixed momentum current J µa – electromagnetic current j λ response and calculate thecorresponding linear response coefficient: t µ,λa = lim ω → iω Π µ,λa ( ω + iδ ) , (17)Π µ,λa ( iν m ) = 1 V β (cid:88) k ,n tr (cid:2) J µa ( k ) G ( k , i ( ω n + ν m )) j λ ( k ) G ( k , i ( ω n )) (cid:3) . (18)The electromagnetic current carried by a single-particlestates with momentum k is defined in the usual way: j λ ( k ) = i e (cid:126) (cid:104) [ ˆ H, ˆ x λ ] (cid:105) k = e (cid:126) ∂H ( k ) ∂k λ . (19)Assuming that the internal degrees of freedom transformtrivially under rotations we define unsymmetrized straingenerators as ˆ J µa = ˆ p a ˆ x µ which lead to the following ex-pression for the momentum currents tensor: J µa ( k ) = − i (cid:126) (cid:104) [ ˆ H, ˆ J µa ] (cid:105) k = k a ∂H ( k ) ∂k µ ≡ (cid:126) e k a j µ ( k ) . (20)From this we recognize that Eq. 18 is a conventionalcurrent-current response function, but with an additionalfactor of k a in the summand. Following the same steps as in the derivation of the Hall conductivity for a 2-bandfree-fermion model in Ref. 44, we arrive at the followingequation for our linear response coefficient: t µ,λa = 12 V (cid:88) k k a (cid:34) (cid:15) αβγ ∂ ˆ d α ( k ) ∂k µ ∂ ˆ d β ( k ) ∂k λ ˆ d γ ( k ) (cid:35) × ( n + ( k ) − n − ( k )) . (21)where n + ( k ) ( n − ( k )) is the occupation of states at mo-mentum k above (below) the Fermi level. Setting theFermi level precisely at the Weyl nodes, which fills upcompletely the lower band and leaves the upper bandempty, and translating from the sum over momentumspace to a continuous integral, we find: t µ,λa = − π ) ˆ F V d k k a F µλ (22)where F µλ is the Berry curvature and the integral istaken over the Fermi Volume. The combination in theintegrand has a very intuitive interpretation: for partic-ular choices of response coefficients one can rewrite thisintegral as an integral over the Fermi surface. For exam-ple, we can show that: t y,zx = − π ) ˆ F V d k k x F yz = − π ) ˆ F V d ( k x A z dk z ∧ dk x − k x A y dk x ∧ dk y ) . (23)Since the integrand in this expression is a total derivative,we can reduce the expression for t y,zx to the integral overthe Fermi Surface. Furthermore, introducing a pair ofcoordinates { s , s } parametrizing the Fermi Surface weget: t y,zx = − π ) ˆ F S k x A z dk z ∧ dk x − k x A y dk x ∧ dk y = − π ) ˆ F S k x ( s ) F ij ( s ) ds i ∧ ds j (24)where F ij is the Berry curvature expressed in the coor-dinates on the Fermi Surface, and we have integrated byparts in passing from the first line to the second, giving anextra factor of k x . Setting the Fermi energy to match thelocation of the Weyl nodes, the integral over the Fermisurface turns into a sum over the total Berry curvatureslocalized near each of the N W eyl nodes weighted by thesquared k x position of the node inside the Brillouin zone: t y,zx = − π N Weyl (cid:88) α =1 χ ( α ) ( K ( α ) x ) = − Q xx π , (25)where α is the Weyl node’s index and K ( α ) x is the positionof that node along the k x axis. Similarly, we can lookat the other coefficients for the { ˆ x, ˆ y } plane momentumcurrent responses to the electric E z field to find: t x,zy = Q yy π , t x,zx − t y,zy = − π ) Q xy , (26)where we always define the quadrupole moment compo-nents for the set of nodes with chiralities χ ( α ) located atthe set of N W eyl
Weyl points as: Q µa = N Weyl (cid:88) α =1 k µ k a χ ( α ) . (27)Finally, to reconstruct the response action we will usethe fact that the coefficient t µ,λa describes the linear re-sponse of the k a momentum current J µa to an appliedelectric field t µ,λa = ∂ J µa ∂E λ = ∂∂E λ δSδ e aµ . (28)More specifically, from the k x momentum response to the E z electric field we find: − Q xx π = t y,zx = ∂∂ ( ∂ t A z ) δδ e xy α ia ˆ d x(cid:15) iνρσ A ν ∂ ρ e aσ = α xx . (29)Repeating this analysis for other response coefficients al-lows us to confirm that the matrix α ia in front of thelinear response action is given by: α ia = − Q ia π . (30) Numerics
In this section we present the results of our numericalsimulations which allow us to confirm the structure of theeffective response action (12). To do this we will insert apair of spatially separated lines carrying opposite fluxes(either magnetic or torsional). We study a tight-bindingmodel for a 3D Weyl semimetal with four Weyl nodes ar-ranged in a quadrupolar pattern. The particular modelwe use is given in Eq. 6. We simulate the model on a com-pletely periodic N x × N y × N z = 40 × ×
40 lattice. Weintroduce a pair of opposite flux lines (either magneticor torsional) stretching along the ˆ x -direction, inserted through the plaquettes with coordinates ( y, z ) = (20 , y, z ) = (20 , y and z -directions in real space and thetranslationally invariant x -direction in momentum space.We investigate the electric charge and k x -momentumdensities as a function of the y and z coordinates. Asshown in Fig. 3, we find a momentum (electric) chargedensity localized in the vicinity of the (torsional) mag-netic flux line.We introduce the pair of magnetic flux lines by shift-ing the Peierls’ phases α y ( l ) = e (cid:126) ´ l A y dy by a constantamount for a set of links l with coordinates y = 20 and10 ≤ z <
30. To implement the torsional magnetic flux,we introduce a lattice gauge field e ai which couples to themomentum k a , and then we shift e xy by a constant amounton the set of links l described above. This results in amomentum-dependent Peierls’ phase β y ( l ) = k x ´ l e xy dy .The torsional magnetic flux obtained from shifting e xy byone lattice constant a is equivalent to introducing a screwdislocation to our system. An electron with fixed mo-mentum k x in the ˆ x direction that travels around a tor-sional flux line in the { ˆ y, ˆ z } plane results in an effectiveAharonov-Bohm phase factor e ik x a arising from translat-ing the electron by one lattice constant in the ˆ x direction.To investigate the structure of the effective responseaction we perform the set four tests: (1) we fix thevalue of the Weyl quadrupole moment to be Q xx = − Q yy = π / t = 1, m = − t = − T x (Fig. 3(c)); (2) for the sameconfiguration of the Weyl nodes, we study the depen-dence of the k x -momentum density bound to a magneticflux line as a function of magnetic flux Φ x (Fig. 3(d)); (3)we tune the value of the quadrupole moment and trackthe dependence of the charge density bound to a fixedtorsional magnetic field with Φ T x = − Q xx whichcan be varied by tuning the parameters m and t of thetight-binding model (Fig. 3(e)); (4) similarly, we trackthe k x -momentum density bound to a magnetic flux linewith Φ x = 1 /N x magnetic flux quanta as a function ofthe quadrupole moment Q xx (Fig. 3(f)). In all four caseswe observe linear dependencies with the slopes of all fourplots closely approximating the coefficient of 1 / (8 π ) infront of the action. FIG. 3: (a): Distribution of charge density in units of the quadrupole moment Q xx = π / T x = ± y, z ) = (20 ,
10) and ( y, z ) = (20 ,
30) respectively. (b): Distribution of the k x momentum density in units of Q xx in the presence of two opposite magnetic flux lines carrying Φ x = ± //