Topological Properties of the Chiral Magnetic Effect in Multi-Weyl Semimetals
RRBRC 1231
Topological properties of the chiral magnetic effect in multi-Weyl semimetals
Tomoya Hayata, ∗ Yuta Kikuchi,
2, 3, † and Yuya Tanizaki ‡ Department of Physics, Chuo University, 1-13-27 Kasuga, Bunkyo, Tokyo 112-8551, Japan Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA Department of Physics, Kyoto University, Kyoto 606-8502, Japan RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000 USA (Dated: August 7, 2017)We compute the chiral magnetic effect (CME) in multi-Weyl semimetals (multi-WSMs) basedon the chiral kinetic theory. Multi-WSMs are WSMs with multiple monopole charges that havenonlinear and anisotropic dispersion relations near Weyl points, and we need to extend conventionalcomputation of CME in WSMs with linear dispersion relations. Topological properties of CME inmulti-WSMs are investigated in details for not only static magnetic fields but also time-dependent(dynamic) ones. We propose an experimental setup to measure the multiple monopole charge viathe topological nature hidden in the dynamic CME.
I. INTRODUCTION
Weyl semimetals (WSMs), a family of the topologi-cal materials, possess band touching points character-ized by the nontrivial topology (monopole charge) inmomentum space [1–6]. Excitations of electrons nearthose points share a lot of properties in common with the(3+1)-dimensional relativistic Weyl fermions in particlephysics. Henceforth, quantum field theory of relativis-tic Weyl fermions is directly applicable to WSMs and,in particular, the quantum anomaly plays an importantrole in the description of transport phenomena in WSMs.The chiral magnetic effect (CME) [7, 8] is one of the out-standing transport phenomena realized in WSMs as wellas Dirac semimetals. Experimental measurements haveconfirmed a key signal of CME in Weyl/Dirac semimet-als, that is, the negative and anisotropic magnetoresis-tance [9], over the past few years [10–16].While similarity to relativistic systems has driven thetheoretical development of WSMs, condensed matter sys-tems realizing WSMs generally do not have the Lorentzsymmetry nor continuous rotational symmetry. Such afact leads to theoretical prediction of new types of Weylfermions unique to condensed matter systems and hasdrawn further attention in recent years. One of the pos-sibilities is the WSMs with multiple monopole chargeand they are called multi-WSMs [17–20]. The multi-ple monopole charge requires nonlinear and anisotropicdispersion relations near band-touching points. It waspointed out that multi-WSMs are protected by point-group symmetries [18]. For instance, C and C discreterotational symmetries yield quadratic and cubic bandtouchings with the double and triple monopole charges,respectively. Although the existence of this type of topo-logically protected state is promising, systematic studieson transport phenomena in multi-WSMs are not yet com-pletely done. ∗ [email protected] † [email protected] ‡ [email protected] We will make a clear distinction between two differ-ent kinds of CME, static and dynamic CMEs, through-out this paper. Static CME refers CME under time-independent magnetic fields, and dynamic CME refersCME under time-dependent magnetic fields. For staticCME, there had been debates on the existence of chi-ral magnetic current in equilibirum, which is dissipation-less, and there seems to be no energy source for the in-duction of current [21–26]. It is now understood thatthe static CME necessarily requires “chiral chemical po-tential”, which is actually induced only by driving thesystem into nonequilibrium states. On the other hand,the dynamic CME is the AC current induced by time-dependent magnetic fields [27–31]. Since the dynamicalmagnetic fields drives the systems out of equilibrium, thedynamic CME can occur even in the absence of chiralchemical potential. This is expected to be observed inoptical measurement [30, 32].In this paper, we investigate the static and dynamicCMEs in multi-WSMs with focusing on their topologicalproperties, based on the chiral kinetic theory [33–35], i.e.,the kinetic theory incorporating the effect of the Berrycurvature [27, 36]. In conventional WSMs with linear dis-persion relations and unit monopole charges, CME understatic external magnetic fields B (static CME) is givenby J CME = ( e / π ) µ B , where e and µ are the unitcharge and chiral chemical potential [8] [37]. This ex-pression is robust against thermal excitations as it doesnot depend on temperature. In order to extend it tothe case with the multiple monopole charge, we need toconsider nonlinear and anisotropic band touching pointsthat cause a lot of technical complications, especially atfinite temperature. We introduce the general techniqueto compute topological quantities of multi-WSMs in anefficient way, and explicitly show that the above expres-sion is just multiplied by the monopole charge K [34]: J CME = e π Kµ B . (1)The obtained result is robust against the deformation ofdispersion relation and geometric deformations such as a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug strain-induced effect or temperature gradient as well asthermal excitations (Sec. IV A).Furthermore, we apply our computational method tothe dynamic CME, where the AC current is induced inthe direction of time-dependent external magnetic fields.In contrast to the static CME, the dynamic CME is not topologically protected in anisotropic systems. However,we show that the topological nature is hidden in the dy-namic CME. More specifically, we show that the trace ofthe chiral magnetic conductivity exhibits the topologicalnature even for anisotropic systems: (cid:88) i =1 , , σ ii dCME = e π b K. (2)The chiral magnetic conductivity for dynamic CME isdefined by J i dCME = σ ij dCME B j and b is an energy separa-tion between a pair of left- and right-handed Weyl nodes(see Sec. IV B for details). This quantity is again robustagainst the aforementioned perturbations as well as thechiral magnetic conductivity of static CME (1). The cru-cial difference between Eqs. (1) and (2) is that the latterdoes not depend on chiral chemical potential but dependson b . An energy separation b is an IR property of Weylcones, while chiral chemical potential strongly dependson UV properties of the entire band structures.[38] Byutilizing this advantage of dynamic CME (2), we maydetect the monopole charge without suffering from theUV structures, which cause some subtleties on the ex-perimental measurement of static CME. Based on thesearguments, we also address the experimental setup to de-tect the multiple monopole charge from measurements ofthe dynamic CME. Our proposal can be tested in multi-WSMs with broken spatial-inversion and reflection sym-metries such as SrSi . II. MONOPOLE CHARGE IN MULTI-WSM
In this section, we apply a useful technique to the com-putation of monopole charge as a demonstration. Al-though the result of this section is not new, the compu-tational method enables us to explicitly show the topo-logical properties of chiral magnetic effect in Sec. IV.We first introduce the general effective Hamiltoniandescribing the band structure near the two-band touchingpoint, H = (cid:88) i = x,y,z σ i R i ( k ) , (3)where σ i are the Pauli matrices. R i ( k ) are some realfunctions with a zero only at k = 0. Here, momentumare measured from the band-touching point. The Ein-stein convention is understood for repeated indices below.The Hamiltonian describes the linear left-handed Weylcone in R space [ R = ( R x , R y , R z )] although its shapein k space [ k = ( k x , k y , k z )] is not specified (See Fig. 1). k x k y ε ( k ) R x R y ε ( R ) FIG. 1. (Color online) The left cone is a nonlinear andanisotropic Weyl cone in k -space (Brillouin zone) with k z = 0,described by the model Hamiltonian (7). The right cone is alinear and isotropic Weyl cone in R -space with R z = 0. The only assumption is that the Jacobian det( ∂R l /∂k i )does not vanish everywhere. The eigenvalues of H are ε ± ( k ) = ±| R ( k ) | , and their corresponding normalizedeigenvectors are u ± ( k ) = 1 (cid:112) | R | ( | R | ± R z ) (cid:18) ±| R | + R z R x + iR y (cid:19) . (4)The Berry connection is defined by A ± ≡ − iu †± d u ± , andthe Berry curvature isΩ ± ≡ d A ± = ± (cid:15) lmn R l ( k )d ˆ R m ( k ) ∧ d ˆ R n ( k ) , (5)where ˆ R = R / | R | , and (cid:15) lmn are completely antisymmet-ric tensors (See Appendix A for differential forms). Thesequantities are defined in k space. The monopole chargeis defined by the Chern number, K ≡ π (cid:90) S Ω + , (6)where the integration is done over any two-sphere sur-rounding the origin k = 0. This yields the winding num-ber characterized by π ( S ) associated with a map from k space ( R \ { } ) to R space ( R \ { } ), and takes onlyan integer.Hence, the multi-WSMs are described by the Hamilto-nian (3) with an appropriate choice of R i ( k ). As an ex-ample, we consider a model Hamiltonian with anisotropicand polynomial dispersion relations [18]: H J = ( k − ) J σ + + ( k + ) J σ − + k z σ z , (7)where k ± = k x ± ik y , σ ± = ( σ x ± iσ y ) /
2, and J is aninteger. To compute the monopole charge, we introducenew coordinates ( r, θ, ϕ ) as k ± = ( r sin θ ) /J e ± iϕ , k z = r cos θ. (8)By using those coordinates, the Hamiltonian (7) reads H J = r (cid:18) cos θ sin θe − iJϕ sin θe iJϕ − cos θ (cid:19) . (9)The energy eigenvalues are (cid:15) ± = ± r , which depend onlyon r . The corresponding eigenstates are u + = (cid:18) cos( θ/ θ/ iJϕ (cid:19) , u − = (cid:18) − sin( θ/ θ/ iJϕ (cid:19) . (10)We obtain the Berry curvature in the new coordinates asΩ ± = − i d u †± ∧ d u ± = ± J sin θ d θ ∧ d ϕ. (11)The monopole charge (6) reads K = 12 π (cid:90) π J sin θ d θ (cid:90) π d ϕ = J. (12)We see that the monopole has a multiple charge, whichis determined by the power J of polynomial dispersions. III. STATIC AND DYNAMIC CME FROMCHIRAL KINETIC THEORY
In this section, we briefly review several aspects of dy-namic CME in comparison with static CME from theviewpoint of chiral kinetic theory [27–31].
A. Response to static magnetic field
We first consider the response to static (and homoge-neous) magnetic fields B = const . , and we set E = 0.The current under the external magnetic field takes thefollowing form [39], J = (cid:90) k (cid:20) e ∂∂ k ( ε − m · B ) + e B (cid:18) ∂∂ k ( ε − m · B ) · Ω (cid:19)(cid:21) n + ∇ r × (cid:90) k m n. (13)Here n is the one-body distribution function, Ω =(Ω x , Ω y , Ω z ) are given by Ω i = (cid:15) ijk Ω jk with Ω = Ω ij d k i ∧ d k j , and m is the magnetic moment, whichis decomposed into spin and orbital parts, m = − (2 g s / m e ) S + m orb . We drop the band indices for no-tational simplicity in this section. The last term is themagnetization current, which is absent in static systemsas we shall see in a moment. Under the static magneticfield, the equilibrium solution must solve the Boltzmannequation, and is given by n = f ( ε − m · B )= f ( ε ) − ( m · B ) ∂f ( ε ) ∂ε + O ( | B | ) , (14)where f is the Fermi-Dirac distribution function. Sincethis solution is uniform, the magnetization current van-ishes in the static case. The remaining current is calcu-lated to be J = e B (cid:90) k ( v · Ω ) f ( ε ) ≡ J CME , (15) where v = ∂ε∂ k . We obtain the well-known expression ofstatic CME [33–35]. B. Response to dynamic magnetic field
For dynamic magnetic field B ∝ e − iωt + i q · r , we con-sider the uniform limit | q | → / | q | . Inthis case, the electric and magnetic fields are tied viathe Faraday law, ∂∂t B + ∂∂ r × E = 0. In order to ob-tain the distribution function n , we solve the Boltzmannequation, ∂n∂t + ˙ r · ∂n∂ r + ˙ k · ∂n∂ k = − n − f (˜ ε ) τ , (16)(1 + e B · Ω ) ˙ r = ˜ v + e (cid:101) E × Ω + (˜ v · Ω ) e B , (17)(1 + e B · Ω ) ˙ k = e (cid:101) E + ˜ v × e B + ( e (cid:101) E · e B ) Ω , (18)where we defined ˜ v ≡ ∂ ˜ ε k ∂ k and (cid:101) E ≡ E − ∂ ˜ ε k ∂ r . We intro-duced ˜ ε k = ε k − m · B with m being the magnetic mo-ment for notational simplicity. We used the relaxation-time approximation in the right hand side of Eq. (16).Substituting a decomposition of the distribution function n = f ( ε − m · B ) + δf, (19)into Eq. (16), and keeping the terms up to linear orderin E and B , we obtain δf = ( iω − i q · v ) m · B + v · e E iω − i q · v − /τ ∂f ( ε ) ∂ε . (20)The distribution function becomes n = f ( ε − m · B ) + ω − q · v ω − q · v + i/τ ( m · B ) ∂f ( ε ) ∂ε − i v · e E ω − q · v + i/τ ∂f ( ε ) ∂ε = f ( ε ) − i/τω − q · v + i/τ ( m · B ) ∂f ( ε ) ∂ε − i v · e E ω − q · v + i/τ ∂f ( ε ) ∂ε . (21)Therefore, the current in the uniform limit reads [31] J = J normal + J CME + J GME + J mag , (22)with J normal = e (cid:90) k v f ( ε ) , (23) J CME = e B (cid:90) k ( v · Ω ) f ( ε ) , (24) J GME = ieωτiωτ − (cid:90) k v ( m · B ) ∂f ( ε ) ∂ε , (25) J mag = lim | q |→ ∂∂ r × (cid:90) k m (1 /τ ) m · B + v · e E iω − i q · v − /τ ∂f ( ε ) ∂ε = eiω − /τ lim | q |→ ∂∂ r × (cid:90) k m ( v · E ) ∂f ( ε ) ∂ε , (26)We use a simplified notation (cid:82) k ≡ (cid:82) d k / (2 π ) only inthis section. The first one is the normal current, and thesum of other three terms is the dynamic CME: J total = J CME + J GME + J mag . (27) J CME is a usual static CME current, which survives evenin the static limit ω → J GME is the current inducedby gyrotropic magnetic effect, which contributes to dy-namic CME. J mag also yields finite contribution to dy-namic CME.Dynamic CME itself does not show topological naturein general band structures because of J GME and J mag .Only exception is the linear and isotropic band structure(See Appendix B). Another important property of thedynamic CME is that the gyrotropic magnetic and mag-netization currents J GME and J mag can give finite contri-butions even when the chiral chemical potential vanishes.These two facts make a clear distinction of the dynamicCME from the static one. IV. CME IN MULTI-WSM
In this section, we discuss the topological propertiesof the static and dynamic chiral magnetic conductivitiesof multi-WSMs in near-equilibrium states. The necessaryformulas of the chiral kinetic theory are found in Sec. III.
A. Static CME
As we saw in the last section, the chiral magnetic cur-rent induced by a static magnetic field is provided by J CME = − e (cid:88) s = ± (cid:90) d k (2 π ) (cid:18) Ω s · ∂f ( ε s ) ∂ k (cid:19) ε s B ≡ σ CME B , (28)where s = ± refers to the contributions from the upperand lower Weyl cones. Ω ± = (Ω x ± , Ω y ± , Ω z ± ) are given byΩ i ± = (cid:15) ijk Ω jk ± with Ω ± = Ω ij ± d k i ∧ d k j , and f ( ε ) =(1 + e β ( ε − µ ) ) − is the Fermi-Dirac distribution function.For the Weyl cones described by the model Hamiltonian(7), the chiral magnetic conductivity σ CME at the zerotemperature with positive doping is written as σ CME = e µ (2 π ) (cid:90) d k δ ( ε + ( k ) − µ ) ∂ε + ( k ) ∂ k · Ω + . (29)The explicit and direct evaluation of the right-hand-sidewould require some tedious computation for J >
1, so wehere just mention its physical meaning and the result. Weshall compute this quantity for more general situationsin the moment. (cid:82) d k δ ( ε + ( k ) − µ ) is an integration overthe entire Fermi surface, and ∂ε + ( k ) ∂ k is a vector normalto the Fermi surface. Since ∂ε + ( k ) ∂ k · Ω + is the monopole flux penetrating the Fermi surface per unit area, the in-tegration in Eq. (29) measures the total monopole fluxpenetrating the Fermi surface, which should be nothingbut the monopole charge K given in Eq. (6). As a result,we should get J CME = ( e / (2 π ) ) Kµ B for the staticCME in a left-handed multi-Weyl fermion.In order to calculate the chiral magnetic conductiv-ity for the general two-band Hamiltonian (3) at finitetemperatures in an efficient manner, we rewrite σ CME inEq. (29), by using differential forms, as (see Appendix A) σ CME = − e (cid:88) s = ± (cid:90) d k (2 π ) (cid:15) ijk jks ∂f ( ε s ) ∂k i ε s = − e (2 π ) (cid:88) s = ± (cid:90) Ω s ∧ d f ( ε s ) ε s . (30)The energy eigenvalues of the Hamiltonian (3) is ε ± = ±| R | , and the equilibrium distribution function f ( ε s ( k ))depends on k only through ±| R ( k ) | . Also, Ω ± ( k ) de-pends only on ˆ R . Then by using the expression (30) thechiral magnetic conductivity can be readily computed as σ CME = − e (2 π ) (cid:88) s = ± (cid:90) Ω s ∧ d f ( ε s ) ε s = − e (2 π ) (cid:90) ε + d( f + ( ε + ) − f − ( ε + )) (cid:90) Ω + = e (2 π ) µK. (31)Here, we introduced the Fermi-Dirac distributions f + ( ε ) = f ( ε ) and f − ( ε ) = 1 − f ( − ε ) = (1 + e β ( ε + µ ) ) − for the upper and lower Weyl cones, and we used theidentity (cid:82) r d r ∂∂r ( f + ( r ) − f − ( r )) = − µ . Equation (31)represents contribution only from the left-handed Weylcones. Provided both of the left- and right-handed Weylcones have the same monopole charge, the total chiralmagnetic current is given by J CME = e π Kµ B , (32)with µ ≡ ( µ L − µ R ) /
2. This extends the universal for-mula of the static CME in the linear WSMs to multi-WSMs by direct computation. We find that the chi-ral magnetic conductivity is independent of tempera-tures and the energy of Weyl points for not only the lin-early dispersive WSMs but also multi-WSMs. AlthoughEq. (32) was derived in Ref. [34] we do not assume that ∂n/∂ k = 0 at the monopole singularity, which was as-sumed in [34]. This extension enables us to use Eq. (32)in the case where Fermi level is close to monopole sin-gularities. Furthermore, one can explicitly see that thechiral magnetic conductivity (30) does not depend onmetric in curved spacetime, meaning that it is robustagainst geometric deformations. B. Dynamic CME
We calculate the dynamic chiral magnetic current (22)for WSMs with general dispersion relations. The dy-namic CME is composed of the three contributions asin Eq. (27). The contribution from J CME was alreadycalculated and given by Eq. (32). We next discuss gy-rotropic magnetic current J GME (25) and magnetizationcurrent J mag (26) by paying particular attention to theirhidden topological properties.In the Weyl semimetals, since the spin degrees of free-dom are decoupled, the magnetic moment m s is given bythe orbital magnetic moment [27], m s ≡ i e (cid:28) ∂u s ∂ k (cid:12)(cid:12)(cid:12)(cid:12) × ( H − ε s ) (cid:12)(cid:12)(cid:12)(cid:12) ∂u s ∂ k (cid:29) , (33)Using differential forms, we can show that the magneticmoment 2-form is related to the Berry curvature 2-formas m s ≡ i e (cid:2) d u † s ∧ ( H − ε s )d u s (cid:3) = i e ε − s − ε s ) (cid:2) d u † s ∧ (1 − u s u † s )d u s (cid:3) = − e ε − s − ε s ) [d A s + i A s ∧ A s ]= e ε s − ε − s )Ω s . (34)Here, we used the spectral decomposition of the Hamilto-nian, H = (cid:80) s = ± ε s u s u † s , and 1 = (cid:80) s = ± u s u † s . Since wedid not specify the energy dispersion relations, the rela-tion between the magnetic moment and Berry curvatureobtained here holds for general dispersion relations.Although the GME and magnetization currents them-selves are interesting, they are not topologically pro-tected except for isotropic systems. Despite this fact,we now extract the topological nature hidden in thosecurrents. First we consider the conductivity tensor ofthe GME (25): J i GME = (cid:88) j = x,y,z σ ij GME B j . (35)By taking the trace of the conductivity tensor, we obtain (cid:88) i = x,y,z σ ii GME = e (2 π ) (cid:88) s = ± (cid:90) m s ∧ d f ( ε s )= e (2 π ) (cid:88) s = ± (cid:90) Ω s ∧ d f ( ε s ) ε s − ε − s , (36)which elucidates the topological property hidden inGME. This yields the same expression with Eq. (31) forthe general Weyl Hamiltonian (3).Next, we consider the magnetization current (26) andextract the topological part of the conductivity. Supposewe have the time-dependent external magnetic field inthe z direction, B = B e − iωt + iqx ˆ z . Then the correspond-ing electric field is E = B e − iωt + iqx ˆ y , which satisfies the Maxwell equation. The induced current along the z di-rection can be calculated as J z mag = eiω − /τ ∂E y ∂x (cid:88) s = ± (cid:90) d k (2 π ) m ys ∂f ( ε s ) ∂k y = eiω − /τ ( iωB z ) (cid:88) s = ± (cid:90) d k (2 π ) m ys ∂f ( ε s ) ∂k y ≡ σ z mag B z . (37)The similar expression holds for the current in the x or y direction when a parallel external magnetic field is ap-plied along that direction. Summing up the conductiv-ities in these three cases, we obtain the exactly sameexpression as that of the GME: (cid:88) i = x,y,z σ i mag = e (2 π ) (cid:88) s = ± (cid:90) Ω s ∧ d f ( ε s ) ε s − ε − s , (38)in the clean limit. We note that in general σ x mag (cid:54) = σ y mag (cid:54) = σ z mag in anisotropic systems. V. DETECTION OF MONOPOLE CHARGE
Here we discuss how to detect the monopole charge inmulti-WSMs by experimental measurement of the chiralmagnetic current. Both static and dynamic CMEs yieldthe currents in the direction of the magnetic field at finitechiral chemical potential. However, the chiral chemicalpotential µ itself highly depends on the UV property ofthe band structure, which causes the chirality relaxationmechanism, even though Eq. (32) is universal [8]. Hence,it is nontrivial how to extract the signal of the monopolecharge from the static CME measurements.For µ = 0 the static CME clearly vanishes, but thedynamic CME is not necessarily zero. Let us think ofthe following Hamiltonian for the left- and right-handedWeyl fermions, H L/R = ± (cid:88) i =1 , , R i ( k ) σ i + b , (39)where ± corresponds to the left- and right-handed Weylfermions, respectively. The Hamiltonian describes a pairof Weyl cones whose Weyl points are shifted in energy by b . Such a model Hamiltonian can be realized in WSMswith broken spatial-inversion symmetry [40, 41]. Theenergy eigenvalues are ε L, ± = ±| R | + b / ε R, ± = ±| R | − b /
2. Therefore, for µ = 0 the trace part ofthe dynamic chiral magnetic conductivity for the totalcurrent consists of Eqs. (36) and (38): (cid:88) i =1 , , σ ii dCME = e π b K. (40)We should emphasize here that this expression is robustagainst thermal excitations, deformation of Weyl conesand geometric deformation, which is certainly not truewithout taking trace. Furthermore, the quantity doesnot depend on UV property due to the absence of µ incontrast to the static CME as mentioned above. Pro-vided the energy shift of the Weyl points, the monopolecharge can be extracted by measuring the dynamic chi-ral magnetic currents parallel to the external magneticfields in the x , y , and z directions and summing up eachconductivity.To extract the topological property from the observa-tion of the chiral magnetic current, we need to requirethe absence of the anomalous Hall current, J AHE = E × (cid:88) s = ± (cid:90) d k (2 π ) Ω s f ( ε s ) , (41)which may give an unwanted contribution in the dynam-ical case for anisotropic systems. We raise three possibil-ities to eliminate the anomalous Hall contribution frommeasurements of the dynamic CME: (i) The anomalousHall current vanishes in the presence of the time-reversalsymmetry. (ii) The anomalous Hall conductivity van-ishes under discrete rotational symmetries about two dif-ferent directions. The point group symmetries ( C or C ) protecting multi-WSMs can be important also foreliminating the anomalous Hall current. (iii) Externalelectric fields may be tuned to be small around the sam-ple by using the standing electromagnetic wave. An ex-ample of standing wave is B z = B cos( qx ) cos( ωt ) and E y = E sin( qx ) sin( ωt ), then one can put the sample at x = 0 when the sample size is negligible compared with1 /q .A candidate material that fits to our proposal is SrSi ,and it is numerically predicted to have the following threeproperties [42] (SrSi was pointed out in previous works[30, 41, 43]): (A) The band structure possesses the Weylpoints with double monopole charges. (B) The crystallacks both inversion and reflection symmetries, whichmakes it possible to observe the dynamic CME even with-out chiral chemical potential. The numerical band com-putation indeed shows the finite energy shift between theleft and right double-Weyl points ( b ∼ × [meV]).(C) The system is time-reversal symmetric. Therefore,the anomalous Hall current should be excluded as men-tioned above (i). VI. CONCLUSIONS
We considered the general effective Hamiltonian de-scribing multi-WSMs, and analyzed topological proper-ties of the static and dynamic CMEs. In particular,the monopole charge of each Weyl point is character-ized by the winding number of the map from k spaceto R space. We rewrite the expression of the static chi-ral magnetic conductivity using differential forms, whichmakes the topological feature of the static CME man-ifest. The obtained formula for multi-WSMs gives the straightforward extension of that for conventional WSMswith unit monopole charge. On the other hand, the dy-namic CME in multi-WSMs is not manifestly topological,but we found the topological feature hidden there. Weshowed the general relation for multi-WSMs between theBerry curvature and orbital magnetic moment. Using therelation, the topological nature hidden in the dynamicCME is extracted by taking the trace of the gyrotropicmagnetic and magnetization conductivities.We proposed an idea to experimentally observe themultiple monopole charge through the measurement ofthe dynamic CME. The multi-WSMs must have the en-ergy shift of a pair of Weyl points, i.e., the spatial inver-sion symmetry must be broken. In addition, the absenceof reflection symmetry is necessary for the non-vanishingdynamic CME. We also mentioned the anomalous Halleffect that can mask the signal of the dynamic CME, andseveral possibilities were discussed to evade the problem.SrSi is the WSMs satisfying all the criteria necessary forthe detection of the multi-monopole charge. ACKNOWLEDGMENTS
The authors thank D. Kharzeev for reading themanuscript and making useful comments, particularly,on the experimental realization. Authors also thank theorganizers of the workshop “Chiral Matter 2016”, wherethis work started. Y. K. is supported by the Grants-in-Aid for JSPS fellows (Grant No.15J01626). T. H. issupported by JSPS Grant-in-Aid for Scientific Research(Grant No. JP16J02240). Y. T. is supported by the Spe-cial Postdoctoral Researchers Program of RIKEN.
Appendix A: Some formulas of differential forms
We here briefly summarize some useful formulas, whichare necessary for the calculations in the main text (seee.g., [44, 45] for more details). We only consider threedimensional flat space, which corresponds to k space inthe main text.The differential line element d x i is called (differential)1-form. The higher-forms are defined by antisymmetriz-ing the corresponding tensor fields. For instance, by us-ing 1-forms d x i,j ( i, j = 1 , , x i ∧ d x j = 12 (d x i ⊗ d x j − d x j ⊗ d x i )= − d x j ∧ d x i , (A1)where ⊗ is the tensor product and ∧ is called wedge prod-uct . The 3-form d x i ∧ d x j ∧ d x k is obtained by the samemanner with antisymmetrizing the three indices. For p -form ω p and q -form η q , ω p ∧ η q = ( − pq η q ∧ ω p issatisfied. Under the coordinate transformation ˜ x a,b =˜ x a,b ( x ) , ( a, b = 1 , , x a ∧ d˜ x b = 12 (cid:18) ∂ ˜ x a ∂x i ∂ ˜ x b ∂x j − ∂ ˜ x a ∂x j ∂ ˜ x b ∂x i (cid:19) d x i ∧ d x j , (A2)where the quantity inside the brackets is the Jacobianassociated with the coordinate transformation.The exterior derivative operation on differential forms,which turns p -forms into ( p + 1)-forms, is defined byd ω = ∂ω∂x i d x i , (A3)d( ω j d x j ) = ∂ω j ∂x i d x i ∧ d x j , (A4)d( ω jk d x j ∧ d x k ) = ∂ω jk ∂x i d x i ∧ d x j ∧ d x k , (A5)where ω , ω j , and ω jk are scalar functions of x . The exter-nal derivative of a 3-form is identically zero in three di-mensions. The following property of the exterior deriva-tive dd ω p = 0 , (A6)for an arbitrary p -form ω p can be easily checked from theabove definition.The hodge star operation on differential forms in three-dimensional flat space, which turns p -forms into (3 − p )-forms, is defined by ∗ ω = (cid:15) ijk ω d x i ∧ d x j ∧ d x k , (A7) ∗ ( ω i d x i ) = (cid:15) ijk ω i d x j ∧ d x k , (A8) ∗ ( ω ij d x i ∧ d x j ) = (cid:15) ijk ω ij d x k , (A9) ∗ ( ω ijk d x i ∧ d x j ∧ d x k ) = (cid:15) ijk ω ijk . (A10)With the use of a differential p-form ω p = ω i ...i p d x i ∧· · · ∧ d x i p , the integration over the p-dimensional space M p is defined by (cid:90) M p ω p = (cid:90) M p (cid:15) i ...i p ω i ...i p d x . . . d x p , (A11)where we used d x i ∧ · · · ∧ d x i p = (cid:15) i ...i p d x ∧ · · · ∧ d x p and suppressed the wedge products in the right hand side.The integral of the inner product is expressed by usingthe hodge star operation. For instance, the integrationof two 1-forms ω = ω i d x i and η = η i d x i over the three-dimensional space M is given as (cid:90) M ω ∧ ∗ η = (cid:90) M ( ω i d x i ) ∧ (cid:16) (cid:15) jkl η j d x k ∧ d x l (cid:17) = (cid:90) M ω i η j (cid:15) jkl (cid:15) ikl d x d x d x = (cid:90) M ω i η i d x d x d x . (A12)As an example, we see how the expression of CMEin Eq. (28) is rewritten, by using the differential forms, as Eq. (30). First, the Berry connection 1-form A andcurvature 2-form Ω defined in Eq. (5) are expressed as A = A i d k i , (A13)Ω = d A = 12 (cid:18) ∂A j ∂k i − ∂A i ∂k j (cid:19) d k i ∧ d k j = 12 Ω ij d k i ∧ d k j . (A14)It is noted that Ω i = (cid:15) ijk Ω jk , in other words, the 1-formΩ i d k i is the hodge dual of the Berry curvature 2-formΩ = Ω ij d k i ∧ d k j :Ω i d k i = ∗ (cid:18)
12 Ω ij d k i ∧ d k j (cid:19) = ∗ Ω . (A15)Similarly, Ω = ∗ (Ω i d k i ) . (A16)Therefore, the integration in Eq. (28) is written as (cid:90) Ω is · ∂f∂k i ε s d k = (cid:90) ∗ (Ω js d k j ) ∧ ε s ∂f∂k i d k i = (cid:90) Ω s ∧ d f ε s , (A17)where we used Eq. (A12) in the first equality andEq. (A16) in the second equality. Appendix B: Dynamic CME in isotropic two-bandmodel
One may wonder whether dynamic CME shows thetopological properties as well as static CME, which iswell known to be proportional to the monopole charge forisotropic and linear Hamiltonian. It turned out that thechiral magnetic conductivity of dynamic CME has thetopological property for an isotropic band as we will seebelow. However, this is generally not true for anisotropicmodels such as multi-Weyl semimetals. In order to figureout the difference between static and dynamic CME, wegive the computation of dynamic CME for an isotropictwo-band model with the linear dispersion relation.If the system is isotropic, the gyrotropic magnetic cur-rent (25) and magnetization current (26) are further re-duced to J GME = ieωτiωτ − B (cid:90) k m · ∂f ( ε ) ∂ k , (B1) J mag = eiω − /τ
13 lim | q |→ ∂∂ r × E (cid:90) k m · ∂f ( ε ) ∂ k = ieωτiωτ − B (cid:90) k m · ∂f ( ε ) ∂ k (B2)In the clean limit ωτ (cid:29)
1, we have J GME = J mag = e B (cid:90) k m · ∂f ( ε ) ∂ k . (B3)In this derivation, the isotropy of the GME or magneti-zation current is assumed, and thus the expression doesnot hold for anisotropic systems.We apply the expressions given in Eqs. (24) and (B3)to the two-band model with linear band structure aroundeach Weyl nodes for the purpose of illustrating the sim-ilarity and difference between the static and dynamicCMEs. Consider the following Hamiltonian H = χv F k · σ , (B4)near the Weyl point with the chirality χ = ±
1. In thiscase, the energy eigenvalues are ε ± = ± χv F | k | corre-sponding to the upper and lower Weyl cones. Also, theBerry curvature is calculated to be Ω ± = ± χ k / (2 | k | ).Hence, static chiral magnetic current takes the followingform: J CME = − e B (cid:90) k ,s (cid:18) sχ k | k | · ∂f ( ε s ) ∂ k (cid:19) ε s = e (2 π ) χµ B , (B5)where we used the notation (cid:82) k ,s ≡ (cid:80) s = ± (cid:82) d k / (2 π ) and ε s = sχv F | k | with s = ±
1. Next we evaluateEq. (B3).In this model, the orbital magnetic moment is calcu-lated to be m ± = m orb ± = ± eχv F k | k | . (B6) Substituting it into Eq. (B3), we have J GME = J mag = e B (cid:90) k ,s sχv F k | k | · ∂f ( ε s ) ∂ k = − e π ) χv F k F B . (B7)Therefore, collecting the contributions to the dynamicCME, we obtain J total = J CME + J GME + J mag = e (2 π ) χ (cid:18) µ − v F k F (cid:19) B . (B8)We should emphasize two points on this computation.Firstly, the dynamic chiral magnetic conductivity is topo-logical in the sense that it is proportional to the monopolecharge χ . This is because we assumed the isotropic bandHamiltonian, and not necessarily true for generic bandstructure. Nevertheless, we claim that the topological na-ture is hidden in (25) and (26) even in anisotropic cases aswe will discuss in the main text. Secondly, it is noted thatthe second term does not necessarily vanish even if thechiral chemical potential is zero ( µ = ( µ L − µ R ) / [1] S. Murakami, New Journal of Physics , 356 (2007).[2] X. Wan, A. M. Turner, A. Vishwanath, and S. Y.Savrasov, Phys. Rev. B , 205101 (2011).[3] A. A. Burkov and L. Balents, Phys. Rev. Lett. ,127205 (2011).[4] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane,G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez,B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin,S. Jia, and M. Z. Hasan, Science , 613 (2015).[5] L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos,and M. Soljaˇci´c, Science , 622 (2015).[6] B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao,J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen,Z. Fang, X. Dai, T. Qian, and H. Ding, Phys. Rev. X ,031013 (2015).[7] H. B. Nielsen and M. Ninomiya, Phys. Lett. B130 , 389(1983).[8] K. Fukushima, D. E. Kharzeev, and H. J. Warringa,Phys. Rev.
D78 , 074033 (2008).[9] D. T. Son and B. Z. Spivak, Phys. Rev.