Chiral Anomaly in Dirac Semimetals Due to Dislocations M

Chiral anomaly in Dirac semimetals due to dislocations M. N. Chernodub, Mikhail Zubkov

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M. N. Chernodub, Mikhail Zubkov. Chiral anomaly in Dirac semimetals due to dislocations. Physical Review D, American Physical Society, 2017, 95, pp.115410. hal-01182539v2

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Distributed under a Creative Commons Attribution - NonCommercial| 4.0 International License Chiral anomaly in Dirac semimetals due to dislocations

M. N. Chernodub1, 2, ∗ and M.A. Zubkov3, 1, 4, 5, † 1Laboratoire de Mathématiqueset Physique Théorique,Universitéde Tours, 37200, France 2Far Eastern Federal University, School of Biomedicine, 690950 Vladivostok, Russia 3LE STUDIUM, Loire Valley Institute for Advanced Studies, Tours and Orleans, 45000, France 4National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe highway 31, 115409 Moscow, Russia 5Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, Moscow, 117259, Russia (Dated: June 5, 2016) The dislocation in Dirac semimetal carries an emergent magnetic flux parallel to the dislocation axis. We show that due to the emergent magnetic field the dislocation accommodates a single fermion massless mode of the corresponding low-energy one-particle Hamiltonian. The mode is propagating along the dislocation with its spin directed parallel to the dislocation axis. In agreement with the chiral anomaly observed in Dirac semimetals, an external electric field results in the spectral flow of the one-particle Hamiltonian, in pumping of the fermionic quasiparticles out from vacuum, and in creating a nonzero axial (chiral) charge in the vicinity of the dislocation.

PACS numbers: 75.47.-m,03.65.Vf,73.43.-f

I. INTRODUCTION AND MOTIVATION reported in different Dirac and Weyl semimetals (see [27] and references therein). The Dirac semimetals are novel materials that have Similarly to graphene [28–35] the fermionic quasipar- been discovered recently (Na3Bi and Cd3As2 [1–3]). A ticles in Dirac and Weyl semimetals experience emergent possible appearance of Dirac semimetals in the other sys- gauge field and emergent gravity in the presence of elas- tems (for example, ZrTe5 [4, 5], and Bi2Se3 [6]) was also tic deformations of the atomic lattices (see, for example, discussed. In Dirac semimetals the fermionic quasipar- [36–39] and references therein). In this paper we will ticles propagate according to the low energy action that concentrate on dislocations in the crystalline order of the has an emergent relativistic symmetry. Both in Na3Bi atomic lattice, which are particularly interesting cases of (0) and Cd3As2 there exist two Fermi points ±K . At the elastic deformations of the ion crystal lattice [40, 41]. each Fermi point the pair of left-handed and right-handed The dislocation is a line-like defect characterized by the fermions appears. The Dirac semimetals represent an Burgers vector b which determines the physical displace- arena for the observation of various effects specific for ment of the atomic lattices along the dislocation. The the high energy physics. In particular, the effects of chi- vector b is a global characteristic of the dislocation be- ral anomaly play an important role in physics of these cause it is a constant quantity over the entire length of materials [3, 7–11]. the dislocation. In rough terms, one may imagine the dis- In the Weyl semimetals, which were also discov- location as a vortex which possesses a fixed “vorticity” ered recently (in particular, TaAs [12]) one of the given by the Burgers vector b. The extreme examples two Fermi points hosts a right-handed Weyl fermion of the dislocations are the screw dislocation (shown in while another Fermi point hosts a left-handed Weyl Fig. 1) and the edge dislocation (illustrated in Fig. 2) fermion. Various relativistic effects were discussed in for which the corresponding Burgers vectors are parallel Weyl and Dirac semimetals already before their exper- and, respectively, perpendicular to dislocations’ axes n. imental discovery[13–25]. There are other types of the dislocations lying in between In [4] the experimental observation of the chiral these two extreme cases. anomaly in ZrTe5 was reported as measured through In [39] the effect of the dislocation on the geometry ex- their contributions to the conductance of the sample. It perienced by fermionic quasiparticles in Dirac semimetals has been shown, that in the presence of parallel exter- was considered for the first time. Aharonov-Bohm effect nal magnetic field and external electric field the chiral and Stodolsky effect (the latter effect describes a correc- anomaly leads to the appearance of nonzero chiral den- tion to the Aharonov-Bohm effect due to torsion) were sity and, correspondingly, a nonzero chiral chemical po- investigated for the scattering of the quasiparticles on dis- tential. This work was followed be a number of papers, locations. Besides, basing on an obvious analogy with the where the experimental detection of chiral anomaly was results of [4] it was proposed, that the dislocation (that carries an emergent magnetic flux) becomes the source of chiral anomaly and chiral magnetic effect. This occurs because the dislocation carries emergent magnetic field. ∗ [email protected] Therefore, it was argued, that the chiral anomaly and chi- † [email protected] ral magnetic effect occur without any external magnetic 2

tion core of radius ξ ∼ |b|, where b is the Burgers vector of the dislocation. In [39] the simple model of the dislocation was used, in which it is represented as a tube of size ξ with the emergent magnetic field inside it. The further examination of the mentioned above problem has led us to the conclusion, that the naive appli- cation of the pattern of chiral anomaly discussed in [4] to the case, when the magnetic field is emergent and is caused by dislocations, has certain restrictions. Strictly speaking, the mentioned above model of the fermionic excitations and chiral anomaly within the dislocation may be applied to the investigation of real materials only if the emergent magnetic flux of the low energy field theory is distributed within the area of size ξ essentially larger, than the interatomic distance a while the emer- FIG. 1. Illustration of the screw dislocation of the atomic lattice with the Burgers vector b parallel to the axis n of the gent magnetic flux of the dislocation is essentially larger, dislocation (the green line). The semitransparent plane points than 2π. In this situation we are formally able to use the out to the region where the atomic planes experience a shift. low energy field theory for the description of fermionic excitations inside the dislocation core. This occurs for the strong dislocations with sufficiently large values of the Burgers vector b, when the crystal lattice is distorted considerably (or in the case, when many parallel dislocations with small values of the Burgers vectors are located close to each other). In this case the dislocation core size ξ ∼ |b| a is much larger than the interatomic distance a. In the present paper we consider the opposite situation, when in Dirac semimetals the values of Burgers vector are relatively small, so that the magnetic flux at the dislocation is smaller than 2π or around 2π. In this situation the crystal structure is not violated strongly, so that the dislocation core size is, presumably, of the order of the interatomic distance ξ ∼ a. The low energy theory is developed for the states with the typical values of momenta much smaller, that 1/a. Therefore, in this case the states localized within the dislocation core cannot be described by the field theory. In order to describe such FIG. 2. Illustration of the edge dislocation with the Burgers states the microscopic theory is to be applied. vector b perpendicular to the dislocation axis n (the blue It appears, that the anomaly in the right- and left- line). The semitransparent plane shows the extra half-plane handed quasiparticle currents is given by of ions introduced in the crystal. 1 1 h∂ jµ i = ± EB or h∂ jµi = EB , (2) µ R,L 4π2 µ 5 2π2 field. According to [39] the contribution of topology to where the upper and lower signs in the first equation cor- magnetic flux Φ is equal to the scalar product K(0)b, respond to the right-handed and the left-handed quasi- where b is the Burgers vector. There may also appear particles, respectively. The important feature of Eq. (2) the contribution to the flux Φ proportional to the tensor is that the effective magnetic field B – contributing to of elastic deformations caused by the dislocation with the the anomaly (2) differs from the emergent magnetic field coefficients of proportionality that are analogous to the at the dislocation H as given in Eq. (1). Gruneisen parameter of graphene. The emergent mag- The basic reason for the difference between the emer- netic flux is associated with emergent magnetic field. gent magnetic field H and the effective magnetic field Z B is that the emergent magnetic field H of the disloca- i i (3) H (x) = Φ dy δ (x − y), (1) tion has a small (of the order of unity or even smaller) magnetic flux Φ. In this case the contribution to both where the integral is taken along the dislocation. The mentioned effects is given by the single fermionic mode appearance of the delta-function in Eq. (1) in the low- (related to the zero mode of the one-particle Hamilto- temperature theory corresponds to the fact that the nian) propagating along the dislocation rather than by a emergent magnetic flux is localized within the disloca- large ensemble of the lowest Landau modes with a huge 3 degeneracy factor. To be more precise, the effective mag- the right- and left-handed fermions at one Dirac cone netic field B is expressed through the probability density and the magnetic field −B(x) acting on the right- and corresponding to the zero mode of the one-particle Hamil- left-handed fermions at another cone. These fields enter tonian in the background of the emergent magnetic field expression for the anomaly Eq. (2). H due to the dislocation. The appearance of the propagating (zero) mode at the dislocation is a natural effect, which is known to exist in topological insulators with If now one applies an external static electric field E lattice dislocations [46]. along the axis of the dislocation, then the quantum The contribution of the individual zero mode to the anomaly will generate the chiral charge at a rate pro- chiral anomaly (2) can be described with the help of portional to the scalar product EB. The generated chi- the effective magnetic field B, which carries a unit ele- ral charge will dissipate, both due to the chiral-changing mentary magnetic flux contrary to the original emergent processes inside the region of size ξ0 around the disloca- magnetic field H, which may have an arbitrary (but still tion and due to the spatial diffusion of the chiral charge. small) value of the total flux Φ. The effective field B is Next we notice that the equilibrium distribution of the localized in the wide area of linear size ξ0, where 1/ξ0 chiral charge – which can effectively be described by a is the infrared cutoff of the considered field theoretical spatially nonconstant but otherwise static chiral chemi- low energy approximation (below we argue that ξ may 0 cal potential µ5 – is subjected to the intrinsic magnetic be identified with the mean free path of the quasipar- field of the dislocation itself. The chemical potential µ5 ticles. For example, in Cd3As2 the mean free path is is distributed around the dislocation with the character- ξ ∼ 200 µm [8]). 0 istic length LV (for example, in Cd3As2 this length is In this paper we demonstrate that for the straight of the order of LV ∼ 2 µm.) The chirally imbalanced screw dislocation directed, for example, along the sym- matter in the presence of magnetic field generates dis- metry axis of the crystal the emergent magnetic flux as- sipationless electric current directed along the disloca- sociated with the emergent field H is given by tion and concentrated in the spatial vicinity around it. Therefore, the intrinsic magnetic field of the dislocation (0) β Φ = K · b + (n · b) , (3) would lead to a spatially-dependent (negative) magne- 2a toresistance around the dislocation. Similar arguments where the first term is of the topological origin [39] while were used in Ref. [4] to experimentally investigate effects the second term is not topological (here β is an ana- of the chiral anomaly in ZrTe5 in the presence of external logue of the Gruneisen parameter of graphene [33]). The magnetic field. magnetic field associated with the flux (3) is localized within the dislocation core of a typical size ξ ∼ a, where a ∼ 1 nm is a typical interatomic distance. In Eq. (3) (0) The paper is organized as follows. In Sect. II we re- the vector K encodes positions of the Fermi points call briefly general theory of quasi-relativistic fermions in (0) k = ±K in momentum space, and n is the direction Dirac semimetals in the presence of elastic deformations of the dislocation axis. For the straight screw disloca- which leads both to the emergent gauge field and to the tion vectors K(0), b and n in Eq. (3) are parallel to each emergent gravity (the latter is described by an emergent other. vielbein [42]). In Section III we discuss these effects fo- We will discuss effects, which appear due to the in- cusing on dislocations, partially following Ref. [39]. In terplay between quantum anomaly and dislocations in Sect. IV we consider the zero modes of the one-particle the crystal structure of Dirac semimetals. A fermion Hamiltonian and demonstrate, that there always exists excitation is affected by the dislocation, in particular, a single mode with the definite spin directed along the via the mentioned above intrinsic magnetic field, which emergent magnetic flux, which is localized in a wide area is localised in a spacial vicinity of the dislocation and around the dislocation. In Sect. V we show that the spec- is directed along the axis of the dislocation. In princi- tral flow along the branch of spectrum (that crosses zero ple, the emergent magnetic fields corresponding to differ- at the mentioned zero mode) gives rise to the anomalies ent Weyl fermions (that belong to different Fermi points in quasiparticle currents: in a Dirac semimetal the chiral and/or have different chiralities) differ from each other. anomaly appears. For the sake of simplicity, these results However, there exists an approximation, in which those are discussed first for the strait dislocation, is directed emergent fields H have the same absolute values, but along the symmetry axis z of the crystal that coincides opposite directions for the two Fermi points ±K(0). If with the direction of the Fermi point K(0) in momentum this approximation is not violated strongly (which is the space. We extend our results to the case of strait dis- general case) the signs of the emergent magnetic fluxes locations with arbitrary direction in Sect. VI. Then in experienced by the quasiparticles living near to the Fermi Sect. VII we discuss the generation of the chiral charge points ±K(0) are opposite. In the Dirac semimetal both via the chiral anomaly (2) due to the interplay between right- and left-handed fermion excitations are present in an external electric field and the internal magnetic field of each (of the two) Dirac cone, therefore in this case we the dislocation. The last section is devoted to discussions have a standard effective magnetic field B(x) acting on and to our conclusions. 4

II. RELATIVISTIC FERMIONS IN DIRAC materials [1–3]. It is also convenient to introduce the SEMIMETAL spatial component of the undeformed vierbein (11):  ν−1/3 0 0  The Dirac semimetal possesses two cones, each of e(0),i ≡ fî = 0 ν−1/3 0 , (12) which hosts one right-handed and one left-handed Weyl a a   0 0 ν2/3 fermion. In the presence of elastic deformations caused by the dislocation the action for a right-handed and left- i î with i, a = 1, 2, 3. The quantity vF ≡ vF fi with fixed i = handed Weyl fermions near a given Fermi point are, re- 1, 2, 3 has a meaning of the anisotropic Fermi velocity in spectively, as follows [39]: i-th direction. The determinant (10) in the undeformed (0) Z case is |e | = vF . 1 4 ¯ µ b ¯ µ b SR= d x|e| Ψieb (x)σ DµΨ − [DµΨ]ieb (x)σ Ψ , (4) The low-energy effective field theory (4), (5) has the 2 (0) Z natural ultraviolet cutoff ΛUV ∼ |K | associated with 1 4 ¯ µ b ¯ µ b SL= d x|e| Ψieb (x)¯σ DµΨ − [DµΨ]ieb (x)¯σ Ψ , (5) the positions of the Dirac cones in the momentum space. 2 In order to determine a natural infrared cutoff we no- where tice that in our field-theoretical approximation the massless quasiparticles do not interact with each other since iDµ = i∇µ + Aµ(x) (6) the effective actions (4) and (5) contain only bilinear fermionic terms while the gauge field Aµ is a classical is the covariant derivative corresponding to the emergent non-propagating field. Therefore, the natural infrared 0 0 a a U(1) gauge field Aµ, σ =σ ¯ = 1, andσ ¯ = −σ with cutoff for our approach is ΛIR = 1/ξ0, where the length a = 1, 2, 3 are the Pauli matrices. The currents of the ξ0 may be identified with the mean free path of the mass- right- and left-handed quasiparticles are, respectively, as less quasiparticles. Indeed at the distances of the order follows: of the mean free path ξ0 we cannot neglect interactions µ µ between the quasiparticles and their scattering on the de- J = Ψ¯ ie (x)σbΨ , (7) R b fects of the atomic lattice which, in general, cannot be µ ¯ µ b JL = Ψieb (x)¯σ Ψ . (8) captured by Eqs. (4) and (5). As an example, we mention that for the Dirac material Throughout this paper the internal SO(3, 1) indices are Cd As the mean free path ξ was estimated in Ref. [8] to denoted by Latin letters a, b, c, ... from the beginning of 3 2 0 be of the order of 200 µm. In the above formulation of the the alphabet while the space-time indices are denoted by low-energy theory, the Dirac point corresponds to zero Greek letters or by Latin letters i, j, k, ... from the middle energy. In real situation the crystals of semimetal may of the alphabet. have nonzero Fermi energy at the level crossing points. The vierbein field eµ = eµ(x) is a 4 × 4 matrix which a a In particular, in [43] the values of Fermi energy of the carries all essential information about anisotropy and the order of 10 meV were reported for Na Bi. In the following elastic deformations (caused, for example, by a disloca- 3 applications we assume, that in the real systems the value tion) of the ion lattice of the Dirac crystal. It is con- of Fermi energy may be neglected, or that the sample is venient to introduce the inverse of the inverse vierbein doped in such a way, that the doping-induced chemical field, ea = ea (x), defined, naturally, as follows: µ µ potential shifts the level crossing to the vanishing energy. µ a µ In the upcoming sections for simplicity we restrict our- ea (x)eν (x) = δν . (9) selves to the case, when the dislocation is an infinite In our paper we always assume that the deformations are straight line directed along the symmetry axis z of the small so that the determinant of the vierbein field crystal, which coincides with the direction of the Dirac (0) |e| ≡ det(ea ) , (10) point K in momentum space. We will return to the µ more general case of an arbitrarily aligned straight dislo- never vanishes. cation in Sect. VI. In the absence of elastic deformations the fields enter- Now let us consider the case when the atomic lattice ing the actions (4) and (5) are simplified. In this case the of a Dirac semimetal is elastically deformed. The defor- i emergent gauge field Aµ vanishes. mation is described by the displacement vector u which In the absence of elastic deformations the vierbein can gives the displacements of the ions with respect to their be chosen in a diagonal form, positions with respect to the unperturbed semimetal. In the approximation of isotropic elasticity for a straight  −1  vF 0 0 0 dislocation directed around the z ≡ x3 axis the displace- −1/3 a (0),µ  0 ν 0 0  ment vector u is given by: ea =  −1/3  , (11)  0 0 ν 0  ba 0 0 0 ν2/3 ua = −θ + ua , (13) 2π cont where the parameter ν 6= 1 reflects the fact that the where θ is the polar angle in the plane orthogonal to the experimentally studied Dirac semimetals are anisotropic dislocation and ba is the Burgers vector. The first term 5 in the right hand side of Eq. (13) is discontinuous vector combination K(0) +A appears as the value of momentum function as it has a jump by ba at θ = 0. The second, P , at which the one-particle Hamiltonian H(x, Pˆ ) van- continuous part of displacement is given by [45] ishes (one substitutes K(0) +A instead of the momentum l γ 3il i k operator Pˆ ): k b 1 − 2σ 3kl |x⊥|e xˆ⊥xˆ⊥ ucont(x⊥)= − log + (14) 4π 1 − σ 2R 1 − 2σ Hx, K(0) + A(x) = 0 (18) where x⊥ = (x1, x2) are the transverse coordinates in As a result we expand the Hamiltonian near the floating i the laboratory reference frame,x ˆ⊥ are respective unit Fermi point K(0) + A(x): angles in the transverse plane and σ is the Poisson ratio which is defined as the negative ratio of transverse to ˆ k a h ˆ (0) i H(x, P ) = |e(x)| ea(x)σ ◦ Pk − K + Ak(x) axial strain of the atomic crystal. Throughout this paper k we shall work in the laboratory reference frame in which +A0(x) , (19) the positions of ions are their real 3d coordinates. where by the symbol ◦ we denote the symmetric product Notice, that for the screw dislocation when the Burgers 1 vector directed along the dislocation axis, b = (0, 0, bz), A ◦ B = (AB + BA). (20) the continuous part of the displacement vector vanishes, 2 k ucont = 0. It is worth mentioning, that while the values of The only possible source of A0(x) is the noncommutativ- k ucont may be large, its derivatives are small for sufficiently ity of momentum Pˆ and coordinates. This means, that small b because after the differentiation the expression in unlike Ak with k = 1, 2, 3 the emergent electric poten- Eq. (14) tends to zero at |x | → ∞. ⊥ tial A0 is proportional to the derivatives of the param- In the presence of elastic deformations, in principle, eters entering H(x, Pˆ ). The field Ak with k = 1, 2, 3 the emergent vielbeins (as well as the emergent gauge is proportional to 1/a times the combination of the di- fields) may differ for the left-handed and the right-handed mensionless parameters while A0 is proportional to their fermions incident at the given Dirac point. derivatives but it does not contain the factor 1/a. For Let us introduce tensor of elastic deformations [40] slow varying elastic deformations this means that A0 may uij = ∂iuj + ∂jui , (15) be neglected. This consideration does not work, however, within the dislocation core, where physics is much where we have neglected the part quadratic in ui by more complicated. The influence of this unknown physics assuming that the deformations are small. In general, on the quasiparticles with small values of momenta (de- the emergent vielbein around the dislocation may be ex- scribed by the action of the form of Eqs. (4), (5)) may pressed, up to the terms linear in displacement vector, as be taken into account through the same emergent fields k follows (see Ref. [39] for the details of the derivation): Aµ, µ = 0, 1, 2, 3 and ea, which become strong within the dislocation core. The component of A0 of emergent i î 1 k nj ˆk i ˆn i jk ea = fa(1 + γknju ) + fa ∂ku − fa γnjku electromagnetic field is not forbidden by any symmetry. 3 Therefore, it appears and gives rise to emergent electric i 1 i jk 0 e0 = − γ0jku , ea = 0 potential (either attractive or repulsive) within the dis- vF location core. 0 1 1 k ij Notice, that the simple model of Weyl semimetal with e0 = (1 + γkiju ) vF 3 cubic symmetry has been considered in [36]. The Dirac 1 semimetal (with cubic symmetry) may, in principle, be |e| = v (1 − ∂ ui − γk uij) F i 3 kij described by the two copies of the model of [36]. a, i, j, k, n = 1, 2, 3 (16)

The emergent gauge field is given by III. EMERGENT MAGNETIC FLUX CARRIED BY THE DISLOCATION 1 A ≈ −∇ (u · K(0))) + β ujk, (17) i i a ijk 1 In order to calculate the emergent magnetic field we A = β ujk, i, j, k = 1, 2, 3 should use integral equation 0 a 0jk Z Z The tensors β and γ, which are the analogues to the 1 i j k k ijk H dx ∧ dx ≡ Akdx , (21) Gruneisen parameters in graphene, may, in principle 2 S ∂S be different for the right-handed and the left-handed where the integration goes over a surface in the transverse fermions. The analogy to graphene prompts that their plane which includes the position of the dislocation. For values could be of the order of unity. Notice, that in the considered solution of elasticity equations (17) we graphene the emergent electric potential A does not 0 represent the right hand side of this expression as follows arise outside of the dislocation core [33]. In the same Z Z way we assume, that in the semimetal the parameters k i (0) 1 jk i Akdx = b Ki + βijk u dx (22) β0jk may be neglected. The reason for this is that the ∂S a ∂S 6

The first term in this expression gives the following sin- where Ni ∈ Z. The vectors mi gular contribution to magnetic field: 3a m0 = zˆ , Z 2ζ k i (0) k (3) √ Hsing(x) ≈ b Ki dy (s)δ (x − y(s)), (23) 3a 3a l0 m1 = −l1 + l2 ≡ xˆ + yˆ , √2 2 where the integration over y goes along the dislocation m2 = l3 − l2 ≡ − 3ayˆ , (29) axis l0. √ 3a 3a One can check that the solutions of elasticity equations m3 = l1 − l3 ≡ − xˆ + yˆ , give ujk ∼ 1/r at r → ∞. Therefore, the integral along 2 2 the circle Cr ≡ ∂S at r → ∞ (with the dislocation at are constructed from the unit vectors l1, l2 and l3 which its center) in the second term of Eq. (22) gives finite correspond to the nearest-neighbor Na-Bi bonds of the contribution to the normalized total flux of the singular honeycomb lattice in the transverse planes of Na3Bi: gauge field H : sing l = −a xˆ, 1 √ Z Φ(r) 1 k 1 3 Φ(ˆ r) = = A dx , (24) l2 = a xˆ + yˆ , (30) Φ 2π k 2 2 0 Cr √ 1 3 l = a xˆ − yˆ . where 3 2 2 For a screw dislocation perpendicular to the layers of Φ0 = 2π (25) Na3Bi the displacement vector is given by Eq. (13). The is the elementary flux (in out units the electric charge is only nonzero components of the corresponding deformation tensor (15) are unity e = 1). At the same time the function Φ(ˆ ∞)−Φ(ˆ r) takes its maximum at r = 0 and decreases fast out of the b 3abxb u3a(x ) ≡ ua3(x ) = 3 ⊥ , (31) core of the dislocation. ⊥ ⊥ 4πx2 In the considered crystals there exist several excep- ⊥ tional vectors Gi (i = 0, 1, 2, ...), which generate the and the emergent electromagnetic field (17) is given by symmetry of Brillouin zone, i.e. momenta k and k + Gi the following expression: are equivalent. The unperturbed Fermi point is directed 0 β 3i β 3j along G0 and is also defined up to the transformations Ai = −∇i(uK) + u + 3iju ,A0 = 0 , (32) (0) (0) a a K → K + Gi. This corresponds to the change of 0 the magnetic flux by with some material-dependent constants β and β . No- tice that our expression (32) differs from that of Ref. [36]. ˆ Equation (32) leads to the following expression for the ∆Φ = b · Gi = 2πN , N ∈ Z . (26) (normalized) magnetic flux (24) of the emergent mag- Such a change of the magnetic flux is unobservable for netic field H: Weyl fermions and Eq. (26) is thus posing certain restric- K(0)b β tions on the choice of the Burgers vectors. For example, Φ(ˆ ∞) = + b3 (33) for the layered hexagonal structure of Na and Bi atoms 2π 4πa in the compound Na3Bi we have (0) π For example, in Na3Bi the value of K ≈ 0.26 zˆ, az √ where a is the lattice spacing in z direction [1, 47]. The 4π 4π 1 3 z G1 = xˆ, G2 = xˆ + yˆ , value of b3 = Naz is proportional to az. Therefore, the 3a √ 3a 2 2 topological contribution to magnetic flux of the disloca- 4π 1 3 4π K(0)b 0.26πN G3 = xˆ − yˆ , G0 = ζ zˆ. (27) tion is 2π ≈ 2π ≈ 0.13 N. Following an analogy 3a 2 2 3a to graphene, where Gruneisen parameter β ∼ 2 we may Here a is the interatom distance within each layer in the roughly estimate the second term in Eq. (33) as ∼ 0.2 N. Then the emergent magnetic flux incident at the disloca- plane orthogonal to G k K(0) and the material parame- 0 tion, presumably, reaches the value of 2π at N ∼ 30. ter ζ determines the interlayer distance a = 3a . Due to z 2ζ As it was mentioned above, we may neglect the zero the hexagonal (honeycomb) structure of the Na3Bi lay- component of the emergent electromagnetic field A0 at ers in xy plane, we may construct the Burgers vectors large distances r a, where the elasticity theory works. similarly to the case of graphene [35] which has also the However, such a potential may be present within the dis- hexagonal structure. Condition (26) gives us the follow- location core because of the essential change in the mi- ing general expression for the Burgers vectors: crophysics at the interatomic scales. Thus we assume the X existence of either repulsive or attractive potential b = N m (28) i i −1/3 i A0(x⊥) = vF ν φ(x⊥) , (34) 7 localized at the dislocation core. We neglect possible ap- x⊥ = (x1, x2) using pearance of such potential far away from the dislocation axis. x1 = r cos θ, x2 = r sin θ, (39) i pˆ = −i∂ , pˆ = − ∂ , (40) r r θ r θ IV. FERMION ZERO MODES PROPAGATING and the radial sigma matrices: ALONG THE DISLOCATION −iθ −iθ r 0 e θ 0 −ie σ = iθ , σ = iθ (41) Momentum of quasiparticles that are described by the e 0 ie 0 action of Eq. (4) should be much smaller than 1/a. At (R) the same time the emergent gauge field A within the Then equation for the function ψe is He⊥ ψe = 0, where dislocation core may be as large as ∼ 1/a. Therefore, (R) r θ the contribution of emergent gravity to Eq. (4) is always He⊥ ≈ φ(r, θ) + σ pˆr + σ pˆθ , (42) small compared to the contribution of the emergent gauge or field. Nevertheless, for the completeness we consider this ! contribution in Appendix A. iθ (R) † (R) 1 e H+ φ(r, θ) (R) h (R)i In this section we show that the dislocations in a He⊥ ≈ σ (R) , H− = H+ φ(r, θ) e−iθH Dirac semimetal host a topologically protected massless − (quasi)fermion mode, which propagates along the dis- (43) location with the Fermi velocity. This fermionic mode with corresponds to the zero mode of the transverse Hamil- (R) H± ≈ pˆr ± ipˆθ (44) tonian (103). The appearance of the propagating mode localized in the vicinity of the dislocation is also known In the absence of the electric potential φ(r, θ) the zero to emerge in topological insulators with lattice disloca- modes (if they exist) have a definite value of the spin tions [46]. projection s = ±1/2 on the z axis. At large r the corre- Let us neglect the emergent gravity due to its weakness sponding coordinate parts of their wave functions satisfy and concentrate first on the case of the screw dislocation, the relations when A3 = 0. We may apply the gauge transformation, (m) which brings the gauge field to the form (ˆpr ± ipˆθ)ψe± = 0 . (45) Next, we chose Ai = 3ij∂jf(x⊥) , (35) Z r ˆ dr where f is a function of transverse coordinates. Then the f(r, θ) = Φ(r, θ) , (46) 0 r Hamiltonian (98) receives the form: so that the angular θ-component of the gauge poten- (R) 2/3 3 −1/3 (R) tial (35) gets the following form: H = vF ν σ pˆ3 + vF ν H⊥ (36) Φ(ˆ r, θ) with A = . (47) θ r (R) X a H⊥ ≈ φ(x⊥) + σ pâ + Aa(x) . (37) The axial symmetry of the problem implies that at large a=1,2 distances r the function Φ(ˆ r, θ) is independent of the polar angle θ. Therefore, at large r the function Φ(ˆ r) is the The zero modes of the transverse Hamiltonian (37) are magnetic flux within the circle Sr of radius r: defined as the solutions of equation 1 Z 1 Φ(ˆ r) = dxj ∧ dxkHi(x, y) , (48) (R) 2π 2 ijk H⊥ ψ = 0 . (38) Sr

Equation (38) is well known in particle physics as it where the surface Sr belongs to the plane which is or- determines zero eigenmodes of a fermion field in a back- thogonal to the dislocation. We come to the following ground of an abelian vortex [48]. The magnetic flux solutions of Eq. (38) for the right-handed zero modes [44]: of the abelian vortex is equal to the quantized vortic- (m) R r ˆ dr m ±imθ∓ Φ(r,θ) r ity number n. There are exactly |n| isolated, linearly- ψ± (r, θ) ∼ r e 0 , (49) independent, zero-energy bound states. These bound where the integer m is the angular quantum number. states are topologically protected by index theorems. For The solutions (49) are localized in a small vicinity of the sake of convenience here we repeat below the deriva- the dislocation core provided that the angular quantum tion of Ref. [48]. 3 number satisfies the following condition We represent ψ = e−σ f ψe and rewrite the Hamiltonian in the polar coordinates r, θ in the transverse plane of m − 2sΦˆ < −1, (50) 8 where Then far from the dislocation core one gets Φ ˆ ˆ iθ (R) ! Φ ≡ = lim Φ(r, θ) , (51) (R) e H νA (r, θ) Φ r→∞ H = σ1 + 3 (57) 0 e⊥ −iθ (R) −νA3(r, θ) e H is the total flux of the intrinsic magnetic field H normal- − ized by the elementary magnetic flux (25). If Eq. (50) is Perturbative corrections to the eigenenergy due to the satisfied then the corresponding probability distribution presence of A3 may be nonzero, in principle, but for the is convergent at large r: mode with m = [2sΦ(ˆ ∞)] those corrections may be ne- Z ∞ glected because all integrals are dominated by the regions 2 rdrdθ|ψ| = 1 . (52) with r → ∞ while A3 ∼ 1/r at large distances. ξ Thus we come to the conclusion, that the only zero Notice that the ultraviolet cutoff ξ is of the order of the mode existing around the dislocation is the one with lattice constant a. 1 In addition, there exist two solutions of Eq. (45), which m = [2sΦ(ˆ ∞)] , s = sign Φ(ˆ ∞) . (58) may not be normalized and which have their maxima at 2 the dislocation core provided that The zero mode (49), (58) of the transverse Hamiltonian (R) m = [2sΦ(ˆ ∞)], (53) H⊥ corresponds to the zero mode of the full Hamilto- nian H(R) provided the longitudinal momentum is zero ˆ ˆ where [2sΦ] is the integer part of 2sΦ, which is the max- p3 = 0. At the same time it corresponds to a linear imal integer number that is not larger than 2sΦ.ˆ branch of spectrum of the full Hamiltonian H(R) with The probability distributions of the considered solu- the corresponding dispersion law: tions are convergent at small r for m ≥ 0. Therefore, (R) 2/3 ˆ in the absence of both the nontrivial vielbein and the E ≈ vF ν sign(Φ) p3 . (59) electric potential, the zero modes that are not singular This branch crosses zero energy level at p = 0. at r → 0 and are not localized on the boundaries of the 3 ˆ Similar considerations can also be applied to the left- system, should satisfy 0 ≤ m ≤ 2sΦ(∞). Such modes handed Hamiltonian H(L), where the only physical zero exist for sΦ(ˆ ∞) > 0 and are enumerated by the values of (L) mode of the corresponding transverse part H is orbital momentum ⊥ (m) R r ˆ dr m i2smθ−2s 0 Φ(r) r m = 0, ..., [2sΦ]ˆ , (54) ψ2s (r, θ) ∼ r e , (60) We neglected in this derivation the potential φ. How- with the quantum numbers ever, it is localized at the dislocation. Therefore, the 1 zero modes in the presence of electric potential (if they m = [2sΦ]ˆ , s = sign Φˆ . (61) exist) have the form of Eq. (49) at r a. Recall, that 2 Eq. (4) works for the momenta of quasiparticles much This mode corresponds to the branch of spectrum with smaller than 1/a. Therefore, the solutions of Eq. (38) the dispersion localized at the dislocations, presumably, do not repre- (L) 2/3 sent physical zero modes. The only solution that remains E ≈ −vF ν sign(Φ)ˆ p3 . (62) is the one with The right-handed and left-handed fermionic modes 1 m = [2sΦ]ˆ , s = sign Φˆ (55) propagate along the dislocation with the velocity 2 v = −v = v ν2/3sign(Φ)ˆ , (63) Fortunately, the field φ cannot affect the energy of this R L F solution because the probability density corresponding to which is nothing but the corresponding component of this solution of Eq. (45) is dominated by the distances the anisotropic Fermi velocity. Thus, the right-handed far from the dislocation core, so that we can neglect com- massless quasiparticle propagates up or down along the pletely the region of the dislocation core. The vielbein dislocation depending on the sign of the flux Φ. The left- for this solution also gives small corrections compared to handed mode always propagates in the opposite direction the contribution of emergent magnetic field. Therefore, compared to the right-handed mode. the strong gravity and the potential φ at r ∼ ξ cannot Notice that Eqs. (59) and (62) were derived in the as- affect the main properties of this solution: it certainly sumption that the magnetic fluxes of the emergent mag- survives as the zero mode and still has the definite value netic field for the right-handed ΦR and the left-handed of the projection of spin to the z axis. ΦL quasiparticles are the same, ΦR = ΦL ≡ Φ. However, In the case of edge or mixed dislocation we should take if in a Dirac semimetal the constants βijk differ for the into account the appearance of a nonzero third compo- left-handed and the right-handed fermions, then the cor- nent of the emergent gauge field: responding gauge fields (17) are also different, and in this 1 case the magnetic flux entering Eq. (59) will be different A (x ) ≈ β b x + β bi xj . (56) 3 ⊥ r2 1 ⊥ ⊥ 2 3ij ⊥ ⊥ from the flux in Eq. (62). 9

V. CHIRAL ANOMALY IN DIRAC where the function f0 can be read from Eqs. (49), (60): SEMIMETALS ALONG THE DISLOCATION −2(|Φˆ|−[|Φˆ|]) x⊥ exp (−|x⊥|/ξ0) ξ0 In the presence of external electric field E the states f0(x⊥) = . (71) 2 ˆ ˆ that correspond to the described above zero modes flow 2πξ0 Γ(−2|Φ| + 2[|Φ|] + 2) in the correspondence with the following equation: This function is normalized in such a way, that Z ∞ hp˙3i = E3 . (64) 2π rdrf0(r) = 1 . (72) 0 Now let us take into account, that the studied model The quantity 1/ξ0 has the meaning of the infrared cutoff has the infrared cutoff 1/ξ , where ξ a. Then the zero 0 0 of the theory and the factor exp (−|x⊥|/ξ0) appears as (R) (L) modes of H⊥ and H⊥ , and the corresponding branches the infrared regulator. We imply that the size of the of spectrum of the propagating modes of the full Hamil- semimetal sample is much larger than the infrared cutoff (R) (L) tonians H and H obey the following properties: ξ0, while ξ0 is much larger than the size of the dislocation core ξ ∼ a, ξ0 ξ. Thus the chiral anomaly due to the 1. The propagating fermion modes are not localized zero mode with m = [|Φˆ|] is localized within the tube of at the dislocation core. Instead, the region of space size ξ0 centered at the dislocation. around the dislocation of size ξ0 dominates, where It is worth mentioning, that for a single dislocation, if 1/ξ0 is the infrared cutoff of the theory. for some reasons the contributions to the emergent magnetic flux of the dislocation due to βijk in Eq. (17) may 2. The propagating fermion modes have the definite be neglected, the typical values of the Burgers vector are value of the spin projection on the dislocation axis: such that |Φˆ| < 1. Therefore, according to Eqs. (58) and 1 ˆ s = 2 sign Φ. The corresponding branch of spec- (61), for a single dislocation the zero mode corresponds trum for the right- and the left-handed fermions is to the angular momentum m = 0. given, respectively, by the following dispersion re- We may rewrite the expression for a chiral anomaly lations: caused by a single dislocation in a Dirac semimetal as follows 2/3 ER/L(p3) = ±2svF ν p3 . (65) EB h∂ jµ(x)i = , (73) µ 5 2π2 3. The propagating mode appears for any dislocations where the effective magnetic field B responsible for the ˆ including those ones, in which the magnetic flux Φ chiral anomaly is given by is smaller than unity. B(x⊥) = 2πn f0(x⊥) sign Φˆ , (74) The total production of the right-handed quasiparticles where the function f0 is given by Eq. (71). per unit length of the dislocation is given by: It is worth mentioning, that the above consideration En refers only to those branches of spectrum, which are de- q˙R = sign Φˆ, (66) scribed by the low energy effective field theory. In the 2π presence of electric field the pumping of the quasipar- where the unit vector n is directed along the disloca- ticles from vacuum may also occur at another branches tion. In the following we assume for simplicity, that of spectrum. Ideally, this pumping process should be the signs of the emergent fluxes Φˆ experienced by the treated with the help of a microscopic theory and is out right-handed and the left-handed fermions coincide in of the scope of the present paper. the Dirac semimetal. Therefore, the production of the left-handed quasiparticles in Dirac semimetal is given by VI. THE CASE OF DISLOCATION DIRECTED q˙L = −q˙R . (67) ARBITRARILY

The production of the quasiparticles may be written In this section we consider the dislocation directed ar- as the anomaly in their currents bitrarily. Without loss of generality we consider the dis- jµ = |e(x)|J µ(x) , jµ = |e(x)|J µ(x) , (68) location directed along an axis, which belongs to the (yz) L L R R plane. The angle between the dislocation and the z axis j = jR + jL , j5 = jR − jL , (69) is denoted by ϕ. Let us rotate the reference frame in such where the covariant currents are deﬁned according to a way, that the z axis is directed along the dislocation. ˆ Eqs. (7) and (8). In a local form the anomaly may be In the new reference frame the tensor f has the form: expressed as follows:  ν−1/3 0 0  fˆ = 0 ν−1/3 cos ϕ ν−1/3 sin ϕ (75) µ En   h∂ j (x)i = f (x ) sign Φˆ , (70) 2/3 2/3 µ 5 π 0 ⊥ 0 −ν sin ϕ ν cos ϕ 10

i α σ3 Let us apply the transformation of spinors ψ → e 2 fˆ is modiﬁed: with tg α = ν tg ϕ. In the transformed frame the tensor

 ν−1/3 0 0  p (1−ν2) sin 2ϕ ˆ  0 ν1/3 ν−4/3 cos2 ϕ + ν2/3 sin2 ϕ √  f =  −4/3 2 2/3 2  (76)  2ν ν cos ϕ+ν sin ϕ  0 0 √ 1 ν−4/3 cos2 ϕ+ν2/3 sin2 ϕ

The one-particle Hamiltonian for the right-handed At the distances larger than the size of the disloca- fermions becomes as follows tion core r ξ ∼ a we may neglect the presence of the emergent magnetic field H of the dislocation. Therefore, (R) ˆ3 3 ˆ2 2 ˆ1 (R) H = vF f σ pˆ3 + vF f σ pˆ3 + vF f H (77) 3 3 1 ⊥ in order to relate the chiral chemical potential µ5 with with the chiral density ρ5 at finite temperature T we use an approximation, in which the relevant modes of the quasi- ˆ2 (R) 1 f2 2 particles are the plane waves of the continuous spectrum. H ≈ σ (ˆp1 − A1) + σ (ˆp2 − A2) ⊥ ˆ1 Thus, we neglect gauge field completely and calculate the f1 thermodynamical potential: fˆ3 − 3 σ A (x ) + φ(x ) (78) 3 3 ⊥ ⊥ Z 3 ω +cµ ˆ1 X X d p − p,s 5 f1 Ω = T log 1 + e T , (80) (2π)3 Now we perform the coordinate transformation s=±1 c=±1

ˆ2 ˆ1 where the quantity c = ±1 labels right- and left-handed f2 f1 y → y , Ay → Ay , (79) chiralities, s = ±1 is the projection of spin (multiplied fˆ1 fˆ2 î 1 2 by two) to the auxiliary vector ka(p) = fapi, while pi and notice that the equation for the zero mode of the is momentum of the quasiparticle. The chiral chemical potential µ is the difference between the chemical po- Hamiltonian H⊥ becomes the same as the one discussed 5 in Section IV. Thus we arrive at the expression for the tentials associated with the fermions of right-handed and anomaly in quasiparticle current of Eq. (66). The re- left-handed chiralities: sulting expression for the chiral anomaly in the Dirac 1 µ = (µ − µ ) . (81) semimetal is again given by Eq. (73). 5 2 R L In Eq. (80) the dispersion of the quasiparticles in terms of the vectors p and k is as follows: VII. CHIRAL DENSITY AND CHIRAL CHEMICAL POTENTIAL AROUND THE q î ˆj DISLOCATION IN THE PRESENCE OF ωp,s = c s vF fafa pipj = c s vF |k (p)| , (82) ELECTRIC FIELD where vF is the Fermi velocity (63) which enters the dispersion relation for the chiral fermions (65). The matrix The Dirac semimetal possesses two cones, each of fˆ is given in Eq. (12). For the momentum parallel to which hosts one right-handed and one left-handed Weyl the z axis, p = (0, 0, p ), Eq. (82) leads to the disper- fermion. Since the processes operating in these two cones 3 sion (65). In our calculation we work (following, e.g., are equivalent, we concentrate on one cone hereafter tak- Ref. [4]) in the adiabatic approximation assuming that ing into account the fact of the double degeneracy later. the chiral chemical potential is a slowly varying function The evolution of the local chiral density around the of space and time. dislocation is governed by (i) the generation of the local Since the determinant of the matrix fˆ is equal to unity, chiral charge due to quantum anomaly at the dislocation we get for the thermodynamical potential (80): given by Eqs. (73) and (74), (ii) the spatial diffusion of Z 3 the chiral charge and (iii) the dissipation of the chiral X X d k − c s vF |k|+cµ5 Ω = T log 1 + e T . (83) charge density (85). In the other words, in the presence (2π)3 of the external electric field, the zero modes, distributed s=±1 c=±1 around the dislocation and propagating along the dislo- The chiral density is given by the derivative of the ther- cation, accumulate the chiral charge. The accumulated modynamical potential Ω with respect to chiral chemical chiral charge diffuses (due to, basically, thermal diffusion potential µ5: and scattering) and also dissipates (due to the chirality- ∂Ω 1 X X Z c k2dk changing processes) around the dislocation. We estimate ρ5 = = − (84) ∂µ 2π2 c s vF k+cµ5 these effects below. 5 s=±1 c=±1 1 + e T 11

Obviously, for µ5 = 0 the chiral density vanishes. For times t τV . The equilibrium chiral charge density is |µ5| T we may evaluate the term in ρ5 linear in µ5 given by a solution of Eq. (88) with vanishing left hand diﬀerentiating the last expression with respect to µ5: side:

c s v k 1 Z F 2 3 (3) µ Z e T k dk ρ5(x) = d y G (x − y; λ) B(y) · E(y) , (89) 5 X 2π2D ρ5 ≈ 2 2 5 2π T c s vF k s=±1,c=±1 1 + e T where Z 2 2 2µ5 k dk µ5 T −2 (3) = = . (85) −∆ + LV G (x − y; λ) = δ(x − y) , (90) π2 T cosh2 vF k 3 v3 2T F is the three-dimensional Green’s function and Equation (85) is valid provided that certain conditions p LV = D5τV , (91) are satisfied. First, the size of the dislocation core ξ ∼ a (where the low-energy physics, and, consequently, the is a characteristic length which controls spatial diffusion thermodynamic relation (85) both become inapplicable), of the chiral charge. should be small compared to the wavelength of the typ- Working in linear approximation we consider weak ical thermal momentum λT ∼ 1/pT ∼ vF /T that con- electric field E, so that the chiral imbalance can always tributes to relation (85). According to our estimates (see be treated as a small quantity, µ5 T , so that the lin- below, Section VIII) this condition is satisfied even for ear approximation in Eq. (85) is justified. In the absence the room temperature T ∼ 300 K (with corresponding of the usual chemical potential µ, one gets from Eq. (85) −8 −9 λT ∼ 4 × 10 m) because ξ ∼ 10 m λT according the following relation between the chemical potential and to Eq. (114). Second, the magnetic field of the dislocation the chiral charge density (89): should not affect considerably the plane waves contribut- 3 v3 ing to Eq. (85). To this end one can consider a wavefunc- F µ5(x) = 2 ρ5(x) . (92) tion of a particle that circumferences the dislocation, and T compare the contributions to its phase coming from the Thus we see, that the dislocation produces the chiral magnetic field and from the usual kinetic factor eipx. The charge which spreads around the dislocation, effectively former reaches its maximum at r = ξ0, being equal to the creating an excess of the chiral chemical potential at the total flux δφΦ = Φ ∼ 2π while the later can be estimated characteristic distance LV from the dislocation axis. No- as δφT = 2πpT ξ0. Thus, the second condition requires tice, that this chiral chemical potential corresponds to a δφΦ δφT or λT ξ0 which is also satisfied according single Dirac point with a pair of Weyl fermions. to Eq. (113). In the above derivation we neglect the gradient of tem- The nonconservation of the axial charge can conve- perature. This may be done for sufficiently small exter- niently be written in the following form: nal electric field, when the temperature remains almost constant at the characteristic length of the problem that dρ5 ρ5 1 is ξ0. Very roughly, in equilibrium the heat generated + ∇j5 = − + 2 BE , (86) dt τV 2π by electric field ∼ σE2 (where σ is the total conductivity where the first term in the right hand side corresponds that includes Ohmic contribution) should be equal to the to the dissipation of the chiral charge with the rate given divergence of the heat flow κ∇T (where κ is the thermal conductivity). For our estimate we use the Wiedemann- by the chirality-changing scattering time τV while the second term describes the generation of the chiral charge Franz law κ ∼ σT . This gives for the characteristic due to the quantum anomaly around the dislocation. The length ξT (at which the temperature is changed consid- chiral current, erably): 1 ∆T E2 j5 = −D5∇ρ5 , (87) 2 ∼ ∼ 2 (93) ξT T T is given by the diffusion of the chiral charge ρ5 with the corresponding diffusion constant D5. We assume that According to our estimates (see below Sect. VIII) at the Dirac semimetal has zero usual chemical potential room temperatures the condition ξT ξ0 leads to for the Dirac quasiparticles, µ = 0. Moreover, we con- |E| 1 V/cm. This condition provides, that tempera- sider a linear approximation so that the transport effects, ture remains constant within the region of size ξ0 around which are discussed here, do not generate a nonzero µ. the dislocation. However, temperature may vary within Substituting Eq. (87) into Eq. (86) one gets the following the whole semimetal sample if its size is much larger than equation for the chiral charge density: ξ0. Now let is consider practically interesting case when dρ5 ρ5 1 the dislocation is a strait line centred at the origin, x1 = = − + D5∆ρ5 + 2 BE . (88) dt τV 2π x2 = 0 and directed along the x3 axis. The dislocation induces the intrinsic magnetic field In the constant electric field, dE/dt = 0, the chiral charge ρ5 relaxes towards equilibrium dρ5/dt = 0 at late B(x⊥) = Bz(x⊥)n , (94) 12 which is also directed along the x3 ≡ z axis (here n is VIII. CONCLUSIONS the unit vector in z direction). The intrinsic magnetic field is a function of the transverse coordinates x⊥ = In this work we discussed certain effects of anomalies (x , x ) which takes nonzero values in a (small) core of 1 2 in the Dirac semimetals Na3Bi and Cd3As2 caused by the the dislocation. In our model approach we consider the dislocations in their atomic lattices. This chiral anomaly field given by Eqs. (71) and (74) is operational without any external magnetic field unlike the conventional chiral anomaly that was discussed for −2(|Φˆ|−[|Φˆ|]) x⊥ Dirac semimetals, for example, in [4]. The dislocation exp (−|x⊥|/ξ0) ξ B (x ) = sign Φ 0 , (95) appears as a source of the emergent magnetic field as it z ⊥ 2 ˆ ˆ carries the emergent magnetic flux. The emergent flux ξ0 Γ(−2|Φ| + 2[|Φ|] + 2) gives rise to the zero mode of the one-particle Hamilto- where nian for the fermionic quasiparticles. The fermionic mode is a gapless excitation which propagates along the dislo- α = |Φˆ| − [|Φˆ|] , (96) cation being localized in the area of the size ξ0 around the dislocation (here 1/ξ0 is the infrared cutoff of the ˆ field theoretical approximation used in our approach). is the fractional charge of the normalized flux |Φ|. The The length ξ may also be identified with the mean free effective magnetic field (95) is distributed around the dis- 0 path of the quasiparticles. For example, for Cd3As2 it location with the characteristic length ξ0 that is much is of the order of 200 µm. This propagating zero mode larger than the size ξ of the dislocation core (the latter corresponds to the branch of spectrum of the quasiparti- is of the order of a few lattice spacings a). In Fig. (3) we cles with the spin directed along the magnetic flux of the show the field (95) for a few values of the fractional part dislocation. of the flux α. In the presence of an external electric field the spectral flow along the zero-mode branch of the spectrum leads to the pumping of the quasiparticles from the vacuum.   Since the right-handed and left-handed quasiparticles are α=|Φ|-[|Φ|] 1.5 produced with opposite rates, the pumping process corre-

) sponds to the chiral anomaly (73). The production rate 0

/ξ of the chiral density is controlled by a scalar product ⊥ 1.0 α=0.5 of the usual (external) electric field E and the effective (internal) magnetic field B carried by the dislocation. B(x 2 0

ξ 0.5 One should stress the following subtle fact: the disloca- α=0.9 α=0.1 tion carries also the emergent magnetic field H [given in Eq. (33) for the example case of Na3Bi] which gives 0.0 rise to the existence of the mentioned propagating zero 0.0 0.5 1.0 1.5 2.0 2.5 3.0 mode. However, the emergent field H does not directly x /ξ contribute to the chiral anomaly (73): it is the effective ⊥ 0 magnetic field B – that is expressed via the density of the zero mode (71) – which enters the anomaly relation (73). FIG. 3. The effective intrinsic magnetic field B (x ) in z ⊥ In the other words, the emergent magnetic field H with Eq. (95) vs. the distance from the dislocation core x⊥ plotted for a few values of the fractional part of the absolute value of the flux (33) leads to appearance of the right-handed (49) the normalized flux (51) α = 0.1, 0.5, 0.9. and the left-handed modes (60) localized at the dislocation and propagating along its axis. The external electric field E, parallel to the dislocation axis, produces This effective magnetic field carries the unit of the el- the chiral charge by pumping these modes form the vac- R 2 ementary flux (25): d x⊥B(x⊥) = 2π sign Φ. As we uum at unequal (in fact, opposite) rates proportional to have discussed earlier, this effective magnetic field is as- the scalar product EB. The process can be formulated sociated with the propagating zero modes bounded at via the chiral anomaly equation (73), in which, however, the dislocation. In Eq. (95) the total flux Φ of the in- the magnetic field B is expressed through the density of trinsic magnetic field H is of a geometrical origin. The the wave functions of the mentioned zero modes (with flux is a quantity of the order of unity (51), in terms of B 6= H). the elementary magnetic flux (25). For the straight dislo- The chiral anomaly gives rise to a nonzero charge den- cation (95) the axial anomaly generates the axial charge sity localized around the dislocation axis with the char- which spreads in the semimetal in the transverse direc- acteristic localization length LV ∼ 2 µm for Cd3As2. tions according to the equilibrium formula (89). The slowly varying chiral density can be expressed via For the completeness we represent the numerical esti- a (space-dependent) chemical potential. mates for the encountered above constants that charac- In principle, there are various ways to create a Dirac terize the semimetal in Appendix B. semimetal with dislocations. In general, the growth of an 13 atomic crystal may be organized in such a way, that the APPENDIX A. EMERGENT GRAVITY AROUND dislocations appear along the chosen direction with the THE DISLOCATION chosen values of the Burgers vector. In addition, dislocations may also appear as a result of plastic deformations of the crystals [41]. This opens a possibility to observe In this section we briefly consider emergent gravity the effects of the dislocation-induced anomaly experimen- around the dislocation. Let us represent the action for tally. The chiral density that is formed around the dis- the right-handed fermion in the following way: location in the presence of external electric field should Z 3 ¯ h 0 affect transport properties of the semimetal. In order to SR = d x dt Ψ(x, t) |e(x)|e0(x)i∂t calculate the corresponding observable quantities, this is k a ˆ i necessary to use kinetic theory modified accordingly in −|e(x)|ea(x)σ ◦ (Pk − Ak) Ψ(x, t) order to take into account the appearance of the chiral Z density around the dislocations driven by chiral anomaly. 3 ¯ h (R)i = d x dt Ψ(e x, t) i∂t − H Ψ(e x, t), (97) However, the solution of this problem is out of the scope of the present paper. ˆ p 0 where Pk = −i∇k, Ψe = |e(x)|e0(x) Ψ, a = 0, 1, 2, 3 and k = 1, 2, 3. The one-particle Hamiltonian is given by ACKNOWLEDGMENTS (R) k a ˆ H = fa (x)σ ◦ (Pk − Ak), (98)

The part of the work of M.A.Z. performed in Russia where was supported by Russian Science Foundation Grant No ek(x) f k(x) = a . (99) 16-12-10059 (Sections II, III, IV) while the part of the a e0(x) work made in France (Sections V, VI, VII) was supported 0 by Le Studium Institute of Advanced Studies. We used here the following chain of relations:

( ¯ " ¯ # ) Z Ψ(e x, t) Ψ(e x, t) Ψ(e x, t) Ψ(e x, t) d3x dt |e(x)|ek(x)σa∂ − ∂ |e(x)|ek(x)σa p 0 a i p 0 i p 0 a p 0 |e(x)|e0(x) |e(x)|e0(x) |e(x)|e0(x) |e(x)|e0(x) Z 3 ¯ 1 k a h ¯ i k a 1 = d x dt Ψ(e x, t) 0 |e(x)|ea(x)σ ∂iΨ(e x, t) − ∂iΨ(e x, t) |e(x)|ea(x)σ 0 Ψ(e x, t) (100) |e(x)|e0(x) |e(x)|e0(x) Z Z 3 n ¯ k a h ¯ i a k o 3 ¯ k a = d x dt Ψ(e x, t)fa (x) σ ∂iΨ(e x, t) − ∂iΨ(e x, t) σ fa (x)Ψ(e x, t) ≡ 2 d x dt Ψ(e x, t) fa (x) σ ◦ ∂i Ψ(e x, t).

k We represent fa (x) as follows where the transverse part of the Hamiltonian is

h i (R) X h a X a k i k ˆk ˆk b H ≈ σ pâ − Aa(x⊥) − σ δe (x⊥) ◦ pˆk fa (x) ≈ vF fa − fb δea(x) ⊥ a a=1,2 k=1,2 f k(x) ≈ −v fˆkδeb (x), a, b, k = 1, 2, 3 , (101) 3 0 F b 0 +φ(x, y) − νσ A3(x⊥) (103) X h 3 k k i − σ δe3 (x⊥) + δe0 (x⊥) ◦ pˆk . k=1,2 where the expressions for the small variations of the vier- µ bein field δea can be read off from Eq. (16). The one-particle Hamiltonian for the right-handed In a general form, the dislocation-induced deforma- fermions in the presence of a dislocation along the z axis µ tions of the vielbein field δea in the Hamiltonian (102), is given by i (103) can be expressed via components tensor γjkl of Eq. (16) and the relations given in Eqs. (99) and (101). However, in certain symmetric cases the form of the 3 Hamiltonian may be simplified. Consider, for example, (R) 2/3 3 2/3 X a 3 H = vF ν σ pˆ3 − vF ν σ δeapˆ3 the case, when the screw dislocation is directed along the a=0 z axis of the Na3Bi atomic lattice (or, equivalently, along −1/3 (R) (0) +vF ν H⊥ , (102) the vector K ). Then, one can write the following ex- 14 pression for the deformations of the veilbein: APPENDIX B. CERTAIN NUMERICAL ESTIMATES k akj 3j akj 3j δea = γ1K u + γ2Ke u , a, k = 1, 2, (104) k 3j 3k We take for a reference the Dirac semimetal Cd3As2. δe3 = γ33kju + γ4u , k = 1, 2, (105) 3 3k 3j 3k The diffusion length of the axial charge for this semimetal δe = γ5u + γ63kju + u , k = 1, 2, (106) k was experimentally estimated in Ref. [7] as LV ≈ 3 −6 δe3 = 0, (107) 2 × 10 m. This quantity turns out to be almost k temperature-independent in a wide range of tempera- δe0 = 0, k = 1, 2. (108) tures T = (50 ∼ 300) K. A rough estimate of Ref. [8] −10 Here we have used the fact that the only nonzero com- gives for the relaxation time τV ∼ τtr ≈ 2 × 10 s. ponents of the tensor of elastic deformations (15) are Then from Eq. (91) one finds u3i = ui3 with i = 1, 2 given in Eq. (31). Moreover, we D = L2 /τ ≈ 2 × 10−2m2/s . (112) took into account that the dislocation is directed along 5 V V the z axis which is perpendicular to layers of honeycomb Correspondingly the inverse infrared cutoff ξ0 may be lattices formed by Na and Bi atoms in the transverse estimated as follows: (x, y) plane. The requirement to respect the C3 rota- 1 m tional symmetry of the honeycomb lattice in the (x, y) ξ ∼ v τ ∼ 300 · 106 · 2 × 10−10s = 3 · 10−4 m. plane allows us to define two tensors from the nearest- 0 F V 200 s neighbor vectors (30): (113) In this estimate we use the value of vF for Cd3As2 that 4 X is around 1/200 speed of light. Kijk = − li ljlk (109) 3a3 b b b The value of ξ0 should be compared to the size of the b=1,2,3 dislocation core ijk 4 X i j m Ke = − l l l 3mk (110) −9 3a3 b b b ξ ∼ 10 m , (114) b=1,2,3 and to the value of the diffusion length which enter Eq. (104) with the material-dependent pref- −6 actors γ1 and γ2, respectively. The only nonzero elements LV ∼ 2 · 10 m . (115) of these tensors are: Thus we see that in practice the suggested limiting case −K111 = K122 = K212 = K221 = 1 , is indeed realized: 112 121 211 222 Ke = Ke = Ke = −Ke = 1 . (111) LV ξ0 (116)

The tensor (109) was first introduced in Refs. [29, 30]. and the typical value of parameter x = ξ0/LV is x ∼ 100. The appearance of the second tensor structure (110) in Notice, that we used in the present paper the relativis- Eq. (104) is a nontrivial fact because the tensor Ke ijk tic system of units, in which the only dimensional unit is not invariant under P -parity transformation of the 3d is the electron-volt (eV). For example, our distances are −1 space. The P -parity odd part is justified, however, by the measured in eV . We give the estimate in relativistic chiral property of the screw dislocation, because the left- units, where it is expressed through eV or 1/m, where the −1 handed screws and right-handed screws are not equiva- unit of distance (m) is related to eV according to the −1 −15 lent as they cannot be superimposed on each other with standard relation [200 MeV] ≈ 1 fm = 10 m. Then the help of rotations only. Therefore, P -parity odd terms the quantities under consideration may be expressed in may appear in the Hamiltonian. the SI system using the definition of its unit of electric Similar arguments lead to the appearance of the other current (A) as Coulomb/s. The SI current equal to one four material-dependent terms in Eqs. (105) and (106) Ampere corresponds to the relativistic current equal to 1/(ec) in the units of 1/m, where e is the charge of elec- with parameters γ3, . . . , γ6. Equation (108) originates from the supposition that the dislocation does not break tron (in Coulombs) while c is the speed of light (in m/s). T invariance so that all components of the vielbein in- Let us notice, that the room temperature corresponds to T ∼ 300 K ≈ 0.025 eV. At the same time D5/c ≈ volving one temporal and one spatial components must −11 4 −4 −1 0 6.7 · 10 m = 6.7 · 10 fm ≈ 3 · 10 eV . One should be zero. Notice that the deformation of the e0 does not 0 also take into account that the typical value of vF in Dirac enter the Hamiltonian (98) because f0 ≡ 1 according to Eq. (99). semimetals is of the order of ∼ 1/200 of the speed of light. We denote by ν the degree of anisotropy of the FErmi One can see, that even in this relatively simple case, ˆ ˆ the expressions in Eqs. (104)-(108) contain six phe- velocity. In practise in Cd3As2 [2] vF f1 ∼ vF f2 ∼ c/200 while fˆ ∼ 0.1fˆ . In Na Bi [1] v fˆ ≈ 4.17 × 105m/s, nomenological parameters γi, and the resulting Hamil- 3 1 3 F 1 (R) ˆ 5 ˆ tonian H , given in Eqs. (103) and (102), is rather vF f2 ≈ 3.63 × 10 m/s ∼ c/800, while vF f3 ≈ 1.1 × 5 ˆ ˆ complicated. 10 m/s. Thus here f3 ≈ 0.27f1. 15

Notice, that for small values of the Burgers vector the always be made close to unity. Besides, the contribution value of Φˆ may be much smaller than unity. Say, in Na3Bi of the second term to the magnetic ﬂux in Eq. (33) also the minimal topological contribution to Φˆ is of the order increases the total value of Φ.ˆ The chiral density should of 0.1 (see Section III). However, for larger values of the also be enhanced in a “forest” of dislocations, which are component b3 of the Burgers vector, the value of Φˆ may parallel to each other.

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