Anomalous Hall conductivity of the holographic $\mathbb{Z}_2$ Dirac semimetals

The anomalous spin Hall conductivity in the holographic model of Dirac semimetals with two Dirac nodes protected by the crystal symmetry has been elaborated. Such system besides the chiral anomaly possesses another anomaly which is related to the $\mathbb{Z}_2$ topological charge of the system. The holographic model of the system contains matter action with two $U(1)$-gauge fields as well as the appropriate combination of the Chern-Simons gauge terms. We also allow for the coupling of two gauge fields {\it via} the kinetic mixing parametrised by the coupling $\alpha$. The holographic approach in the probe limit enables us to obtain Hall conductivity. The aim of this work is to describe the phase transitions in the $\mathbb{Z}_2$ Dirac semimetals between the topologically trivial and non-trivial phases. Interestingly the anomalous Hall conductivity plays a role of the order parameter of this phase transition. The holographically found prefactor of the Hall conductivity in the topologically non-trivial phase, depends on the coupling $\alpha$ and the Chern - Simons couplings.


I. INTRODUCTION
Recently it has been argued that near the charge neutrality point in two and three dimensional systems with Dirac massless energy spectrum, a strong interacting plasma (or Dirac fluid) forms. The evidence of such kind of strong interacting charged fluid was observed in experiments indicating the violation of the Wiedemann-Franz (WF) law [1] or the appearance of the viscous flow of charge carriers [2] in the extremely clean graphene near the charge neutrality point. On the theoretical side these facts initiated the emergence of the holographic generalisation of hydrodynamical approach to the aforementioned systems [3,4]. In order to quantitatively explain the experimental data on the thermal conductivity of graphene, the holographic model of strongly coupled plasma with two non-interacting U (1)-gauge currents was proposed [5]. Two fields bounded, respectively, with the electron and hole currents in graphene with Fermi energy coinciding with the Dirac point have been introduced. They allowed for the nearly perfect agreement with experiments.
The model has been further generalised [6] by allowing for the interaction of two U (1)gauge currents via so-called kinetic mixing term with non-zero coupling constant between two fields. The thermoelectric and magneto-transport properties of graphene have been studied. The Hall effect has been found in the standard geometry with the magnetic field perpendicular to the graphene plane, with the electric field and temperature gradients lying in the plane but being perpendicular to each other. The DC-transport coefficients were calculated by the introduction of the axionic field, which provides momentum relaxation mechanism related to the finite mobility of carriers. The auxiliary U (1)-gauge field played an important role and affected the kinetic and transport coefficients via the parameter α, connected with the kinetic mixing term.
The holographic model with two interacting gauge currents predicts [6] that the increase of α-coupling constant leads to the increase of the width of normalised thermal conductivity with the parameter which bounds both U (1) charges. Moreover, the dependence of the WF ratio on the α-coupling constant results in the changes of the width of curves and their heights. This dependence was found to be valid for all charge densities. The dependence of the Seebeck coefficient on the charge concentration, for the different values of mobilities was also studied and a very good agreement with the experimental data was achieved.
After the experimental discovery of the three dimensional analogs of graphene, the so called Weyl and Dirac semimetals [7] it became clear that besides the relativistic massless spectrum the carriers in these systems possess quantum anomalies. In the case of Weyl semimetals it is the chiral anomaly which at the quantum level shows up as a nonconservation of the chiral charge. It has been also revealed that the class of Dirac semimetals with two Dirac nodes at the Fermi level separated in momentum space along the crystallographic rotation axis besides the chiral charge possesses another topological charge and the corresponding anomaly. The underlying symmetry and anomaly, after [8] we call the Z 2 anomaly and the corresponding systems the Z 2 Dirac semimetals. These anomalies are related to chirality and spin degrees of freedom of charge carriers in Dirac cones. The detailed physical discussion of these issues can be found in [8]. While the positive longitudinal charge magnetoconductivity has been predicted as an experimental verification of the chiral anomaly in Weyl semimetals, the positive longitudinal spin magnetoconductivity is the proposed [9] smoking gun of the Z 2 anomaly in the topological Dirac semimetals.
The extension of the aforementioned analysis [5,6] of transport in graphene towards three dimensional Dirac systems was provided in [10]. The hydrodynamical model of the Dirac Z 2 semimetal was studied in the theory with two interacting gauge fields. The implementation of chiral anomaly and Z 2 topological charge [11] was taken into account, where the topological charge was described by the anomaly term in the additional gauge field sector. The existence of Z 2 topological charge modifies equations and leads to the appearance of the new kinetic coefficients connected with vorticity and magnetic field of the auxiliary field.
In [12] the magneto-transport coefficients were found for the five-dimensional Chern-Simons generalisation of the presented holographic model. The model in question enables one to describe the holographic Dirac semimetals with Z 2 symmetry and leads to the positive longitudinal magnetoconductivity, at large B fields, with small region around B=0 characterised by a negative magnetoconductivity, being in agreement with some experimental data. One also should remark that the model with two coupled vector fields, was used in a generalisation of p-wave superconductivity, for the holographic model of ferromagnetic superconductivity [13].
The holographic model of Weyl semimetal which encoded the axial charge dissipation effect was elaborated in [14,15]. It has been revealed that varying the mass parameter the model underwent a sharp crossover at small temperature from a topologically non-trivial state to a trivial one. The holographic renormalisation group flow was a helpful device in the interpretation of the results, leading to the restoration of the time reversal symmetry at the end point of the renormalisation flow in the trivial phase.

A. Motivation of the paper
The main motivation standing behind our studies is to envisage how the quantum phase transition from topologically non-trivial to trivial phase looks like, for the Dirac Z 2 semimetals. It will be the key point to visualise the role of the auxiliary U (1)-gauge field and especially the coupling between the two gauge fields in question, in the studied process. Our holographic model possesses the Chern-Simons and kinetic mixing terms, binding various combinations of U (1)-gauge field strengths, with the adequate coupling constants. The topologically non-trivial Z 2 Dirac semimetal phase will be characterised by appearance of spontaneous Hall conductivity. The results can be interpreted in terms of the holographic renormalisation group flow.
In order to obtain the Hall effect in topologically trivial system one needs the magnetic field or breaking the time reversal symmetry by other means. One of the proposals, presented in [14], is to have a closer look at the Weyl semimetal with two Weyl nodes of left and rightchirality, separated in the wave vector space, by a spatial vector b µ (in the present convention b µ b µ > 0). This separation causes the breaking of time reversal symmetry. In the presence of massive fermions M the parameter b separating Weyl nodes along z direction in momentum space gets modified to the effective value b ef f , which for the linear spectrum and for b > M where the axial U (1) symmetry is represented by the gauge field A (5) µ bounded to the field strength F (5) µν . This field is anomalous and it is the source of Chern-Simons part of the above action.
To better understand the above holographic action of the Weyl semimetal and its later generalisation to the Dirac semimetal we shall present here some results on the anomalous Hall effect in a condensed matter lattice of, e.g., simple cubic symmetry (with lattice constant a = 1) described by the Weyl Hamiltonian with two nodes at the points k W = (0, ±b y , 0) with The parameter M denotes the fermion mass [16]. The band structure along the z-axis for k x = k z = 0, γ x = γ y = γ z = −γ m = −1 is shown in the left panel of The non-equilibrium effects in Dirac semimetals with Z 2 topological charge [8] require a suitable generalisation of the above model, which has been done in [11,12]. Two different gauge fields were introduced, one of them coupled to charge of fermions and the other related to spin degrees of freedom. The low energy Hamiltonian describing inter alia two materials argued to be Dirac semimetals and possessing Z 2 charge, namely Cd 3 As 2 and Na 3 Bi, is provided by the relation [8] where a and b(k) are in general k dependent constants, O(k 3 ) denotes terms of higher order in k, τ i are Pauli matrices acting on orbital degrees of freedom, while σ i are Pauli matrices in the spin sector. It is seen that at low energy the z-component of spin is a good quantum As the Hamiltonian in question commutes with spin operator σ z , its eigenvalues can be labelled by the eigenvalues of σ z , i.e., s = ±1. This fact enables us to conclude that the Hamiltonian for each spin projection implies being 2×2 matrix. Using the standard low energy form [9] for b(k) = m 0 −m 1 k 2 z −m 2 (k 2 x +k 2 y ) in the continuous limit one notices that H s for each spin eigenvalue s =↑, ↓ contributes two Weyl nodes at k C 2 W = (0, 0, C 2 m 0 /m 1 ), where C 2 = ±1 denotes the Z 2 charges of the Dirac points. Diagonalising Hamiltonian (5) one notes that the spectrum does nor depend on the spin quantum number s. To get the spectrum shown in Fig. (2) we have put k x = k y = 0 and expanded b(k) near each of the nodes to the linear order in k z . One the expression plotted in the figure for m 0 = m 1 = 1. In equilibrium the nodes are at E(0, 0, k z ) = 0; the additional shifts mimic The spectrum of electrons in Dirac semimetal with two separated Dirac cones shifted in energy by the Z 2 chemical potentials ±µ 2 . Two Dirac nodes harbour different Z 2 charges C 2 as indicated. As visible from Eq. (5) both spin directions lead to the same spectrum, which is thus spin degenerated.
Using the above phenomenological picture we propose the holographic description of the Dirac semimetal. To this end, and in analogy to the chirality in Weyl semimetals [14,15] where the axial current j 5 is related to the axial charge ρ 5 , to treat the corresponding 'axial part of the current between the nodes' we introduce two fields. The first denoted by F µν is standard Maxwell U(1)-gauge field and other, which we call B µν in the following plays the role analogous to F 5 µν present in Weyl semimetals, where the charge and chirality are the only relevant quantum numbers. In principle, we could introduce F 5 µν and and spin tensor field [19] together with additional field playing the role of F µν in spin sector and use all four fields to calculate transport properties of the system. Such procedure, which would allow to calculate, e.g., inverse spin Hall effect is beyond the scope of the present paper. Thus in the analysed Dirac semimetal, the field B µν is the analogue of the field F 5 µν in the Weyl semimetals with axial current. However, in our approach the Z 2 charge related current is induced by electromagnetic gauge field F µν . The calculated below anomalous Hall effect is related to the spin degrees of freedom and is more properly called anomalous spin Hall effect.
The word anomalous is related to the fact that it is intimately related to the vector b and not due to the magnetic field B, which also breaks time reversal symmetry. The magnetic field in Dirac semimetals with separated Dirac nodes is another source of the Hall effect and spin Hall effect, e.g., in the presence of spin dependent scatterings.
In the presence of external electric field E there is a shift of the nodes with a given Z 2 charge in energy and with applied magnetic field the spin current proportional to the B field is expected to appear in the system. The separation of the nodes in wave-vector space in the presence of the electric field contributes to the appearance of standard Drude current along the field and additional spin current perpendicular to the node separation vector b, In what follows we shall also focus on the role of the coupling constant between fields F µν and B µν . The coupling α between the aforementioned gauge fields, in the considered action (6), provides additional degree of freedom. Physically it could be related to the scattering processes. Contrary to the low-energy physics described by the free particle Hamiltonian, the holographic approach takes strong interactions into account.
The paper is organised as follows. In Section II we present the basic assumptions and equations of motion for the model in question. Section III is devoted to the Hall conductivity caused by the elaborated gauge fields. In section IV one derives the equations constituting the description of the longitudinal conductivity. Section V is dedicated to the numerical solutions of the underlying equations of motion, paying special attention to the role of αcoupling constant and the influence of Chern-Simons terms on the physics of the studied phenomena. Section VI concludes our studies. In the appendix we present some comments concerning the boundary currents in the underlying theory.

II. HOLOGRAPHIC MODEL
Topological semimetals being novel quantum states of matter can be classified in main two groups [20]- [22]. The first one, in which the Dirac points fall out at time reversal invariant momenta in the first Brillouin zone and the other one for which the Dirac points happen in pairs, separated in momentum space along a rotation axis. The latter one characterises by a non-trivial Z 2 topological invariant, leading to the emergence of Fermi arc surface states, connecting projections of the node locations on the Brillouin surface. In [8] it was revealed that this kind of Dirac semimetals exhibited, in addition to the chiral the Z 2 quantum anomaly. For the sake of completeness we add that each of the above groups of semimetals can break the Lorentz invariance and be further considered as being of type-I or type-II.
Type-II semimetals are characterised by the over-tilted Dirac cones [23] and the existence of electron and hole pockets and thus finite density of states at the Dirac/Weyl point. In type I systems the cone can also be tilted but the density of states vanishes at the node. Besides the above systems a special class of topologically nontrivial materials, so called nodal line semimetals [24] exists, in which nodes appear along the closed line in the wave vector space.
The quantum model in question was described in terms of AdS/CFT correspondence [12], where the bulk action describing the system with chiral anomaly and Z 2 topological charge was studied in the context of magnetic conductivity. The key point in our model are various combinations of gauge Chern-Simons terms mimicking the quantum properties of the system in question, i.e., relations among the indexes for spin, Z 2 charge and chirality in topological semimetals.
The action is provided by The existence of the kinetic mixing term α has its origin in the cosmology where the two fields are interpreted as visible and dark sectors. The cosmological data related to the abundance of visible and dark matter in the Universe constrain the value of α to be much less than unity (α 1). In the studies of thermal transport properties of graphene the two fields were introduced as a representation of two sorts of carriers existing at finite temperature at the charge neutrality point [5] and shown to lead to quantitatively correct holographic description of the thermal conductivity of the graphene. The mixing between two fields studied in [6] allowed for additional improvements. The condensed matter applications do not lead to such strong constraint on its value as cosmological arguments. However, even though the calculations show that |α| < 2 we generally consider here 0 < α ≤ 1. We adopt this limitation in the present work. Moreover, it turns out that the conservation law of the vector current requires that α 1 = α 3 = 0 (see Appendix).
The transport properties of the model (6) have been studied earlier using the hydrodynamic and holographic approaches [11,12]. We paid special attention to the chiral anomaly and Z 2 topological charge. The Chern-Simons parameters α i have been shown to be directly related to the corresponding chiral anomaly parameters C i and to affect the magnetotransport characteristics of the material by introducing novel kinetic coefficients related, e.g., to the chiral magnetic and chiral vortical effects. The ability of the model (6) to provide the correct description of the magnetotransport of the Dirac system in the hydrodynamic regime [11,12] is the main motivation of its use to describe anomalous Hall conductivity in the holographic approach and the topological to trivial system phase transition with increasing the mass M . Properties of the Dirac and Weyl semimetals have been studied, for the non-interacting systems, by means of standard condensed matter techniques. On that level the Berry phase and the topology of the Fermi surface are responsible for monopole like singularity of the Berry connection which mimics the effect of Chern-Simons terms and chirality [26].

A. Equations of motion
The main objective of our paper is the action provided by where the scalar field Φ appearing in (6) is charged under B µ gauge field, i.e., where q d is the charge connected with the auxiliary gauge field. The scalar mass is chosen in such way that it fulfils the Breitenlohner-Freedman limit, i.e., the bulk mass satisfies the condition m 2 = −3.
In show that there exist constraints on the allowed values of these parameters.
We also consider the symmetry breaking by the non-zero mass term connected with a nonnormalizable mode of the charged scalar field. It is worth to remark that the aforementioned influence of Chern-Simons term and symmetry breaking by non-zero value of gauged scalar field was elaborated in [25].
In what follows, our convention is trxyz = 1 and Φ = φ(r). The equation of motion for the U (1)-gauge fields with scalar field charged under B µ gauge field can be written as On the other hand, for the auxiliary B µν field strength one arrives at the relation In the next step we can get rid of the terms with α-coupling constant and finally get where the coefficientsα i are given, respectively bỹ As mentioned earlier the conservation law of the vector current requires α 1 = α 3 = 0 (for the detailed discussion of the constraints on coupling constants, from the point of view of gauge-current conservations, see Appendix).
The same procedure of obtaining equations of motion can be applied to the additional gauge field. It reveals the following relation: The scalar field equation charged under B µ field fulfils the following equation of motion: In our consideration, as the background metric we take the line element of AdS-Schwarzschild five-dimensional black brane where f (r) = 1 − r 4 0 r 4 and r 0 is the radius of the black brane event horizon, related to the Hawking temperature by T = r 0 /π. We shall work in the probe limit, neglecting the spacetime metric tensor fluctuations.

III. ANOMALOUS SPIN HALL EFFECT IN THE HOLOGRAPHIC DIRAC SEMIMETAL
To commence with, in this section we consider the case when B µ field will constitute the background field and one will elaborate the A µ field as the fluctuations on the spin U (1)gauge field background. As in [8], we introduce the vector field (in the present notation b µ ) which in the weak coupling description couples to the Dirac fermions. In what follows, without loss of generality we assume and correspondingly, in the holographic model, take the z-component (B z ) of the background field. The background field equations of motion are provided by The boundary conditions are given by the Dirac cones separation parameter b and the fermion mass M parameter, respectively Note, that in this respect both Weyl and Dirac systems are not distinguishable, both the nodes are separated by b.

A. Hall conductivity
Our main task will be to find the Hall conductivity, using Kubo formula given by σ xy = lim r→∞ As in [14] the retarded correlation function can be calculated having in mind the fluctuations of the adequate fields above given background. Let us consider Maxwell U (1)-gauge field fluctuations provided by the following: In the next step one defines the quantity binding b x (r) and b y (r) in the form as a ± (r) = a x (r) ± i a y (r).
It leads to the following form of the underlying equations: After a convenient parameterisation we obtain the relations for zero and first order expansions in ω The regularity condition near the black brane event horizon leads to the conclusion that the solution of the equation (25) is given by a The Green function implies Having in mind relation (28), one gets that The first term in equation (29) originates from the Chern-Simons gauge term contribution to the currents (it constitutes the Bardeen-Zumino like polynomial contribution). In order to obtain the correct charge conserving definition of the current and accurate value of the appropriate kinetic coefficient, one ought to subtract it [28]. On this account, at leading order in ω, we achieve the following relation: We remark here that the general value ofα 3 found earlier, reduces toα 3 = − α α 4 3 if the constraints found in Appendix are taken into account. On the other hand, the conductivities in x and y-directions imply respectively

IV. LONGITUDINAL CONDUCTIVITY
This section will be devoted to the longitudinal electric conductivity at zero density. We shall consider the fluctuation in the background, which does not source other modes at zero density. The fluctuation of the charge gauge field is of the form The same consideration as in the previous sections lead to the conclusion that in the ω 0 order ν On the other hand, in ω 1 -order we have Thus the solution is provided by The form of the above relation implies that the component of the conductivity σ zz = r 0 .
Its value does not depend on the considered gauge field, but on the background geometry.
Namely, the radius of the black brane event horizon.

V. NUMERICAL RESULTS
Let us now turn our attention towards the numerical solutions of the equations that had been derived in the former sections. The background spin U (1)-gauge field condensation leads to two coupled ordinary differential equations. We solve them by virtue of a shooting method, with the adequate boundary conditions (20) to be fulfilled on the r → ∞ boundary.
The initial values of the fields φ(r 0 ) and B z (r 0 ) are our shooting parameters, moreover their derivatives are specified by the equations of motion. Therefore our boundary value problem is  On the other hand, the gauge field envisages the monotonic growth tendency, i.e., B z (r) is growing towards its limiting value b as r → ∞. The bigger M/b is taken into account, the smaller value of B z (r) deep in the interior of AdS spacetime one obtains. Next, for all the studied cases of M/b, they attain the same UV value. The UV limit is obtained earlier for smaller M/b ratios. As in [14], the limiting B z (r) value is connected with the Hall conductivity and it authorises the holographic analogue of b ef f .  At this point it will be instructive to examine more carefully the transport coefficient, at lower temperature, as well as, its dependence on the relative ratio of the mass M and temperature T . As we have alluded in the introductory discussion the mass M and the parameter b define a gap in the Dirac semimetal spectrum. It constitutes the relation between the gap in the spectrum and temperature, which affects the observation of the true phase transition. The smooth behaviour of the Hall conductivity at high temperatures has already been observed in Fig. 4.
To commence with, let us investigate the critical behaviour of the system near the quantum phase transition point, driven by the change of b/M scale. As one goes down with the temperature, the smeared tail of Hall conductivity (as it is seen in Fig. 5) vanishes and the phase transition becomes sharp. Naturally in our theoretical set up we cannot achieve the exact zero temperature, as we work in the probe limit and use the gravitational background of a black brane with defined Hawking temperature instead of a gravitational soliton metric. Nevertheless we can lower the temperature to the point that allows us the approximate analysis of the critical behaviour of the order parameter. To obtain quantitative information we fit the anomalous Hall conductivity close to the critical point by a power law in the form Similar analysis for the closely related model has been already done in [29] in the context of holographic disordered Weyl semimetal. However, the presence of α coupling is interesting from the perspective of the phase transition analysis in the Dirac semimetal. The results are presented in Fig. 6. It can be clearly seen that the quantum phase transition point (b/M ) crit is influenced by α. The value of (b/M ) crit clearly grows as the coupling increases.
The value of the critical exponent β ≈ 0.213, calculated at temperature πT /M = 0.1, is in quite good agreement with the previously obtained zero temperature result β ≈ 0.211 [15,29]. It has to be mentioned that the field theory model predicts standard mean field value β = 0.5. We shall also like to add that in the present approach all other components of conductivity tensor are given by r 0 ∝ T .
We also investigated the anomalous Hall conductivity as a function of M/T scale. Interestingly it features two different behaviours. For small values of M/T the function follows the power law, while for the bigger values of the mass (or smaller temperature) it behaves exponentially, see Fig. 7. It has to be noted that similar behaviour has been observed by Ammon and co-workers [29]. This scaling dependence is observed at, or in the vicinity of the critical point. The coupling α mainly changes the position of the critical point, as visible in Fig. 6. This explains our numerical observation in Fig. 7 that the scaling of the Hall conductivity is essentially independent on the coupling constant α. The presence of α-coupling shifts the critical value of the holographic scale parameter.
In the next step, we focus on the properties of Hall conductivity calculated using the equations (18) and (19). To begin with, let us first discuss the behaviour of σ xy , given by equations (29). We take b = 8 and q d = 1, and ignore the constant factor originating from Chern-Simons couplings, considering the normalised conductivity on its own. In this way we can extract the influence of α coupling on the holographic Hall conductivity as a function of the fermionic mass to the gap ratio.
Consequently, let us consider the following dimensionless ratio: In the first one, presented in Fig. 8, we fix the α-coupling constant to α = 0.4 and increase Hawking temperature. It can be observed that for low temperatures the coupling constant causes a strongly peaked shift in the Hall conductivity around a specific mass to gap ratio.
With increasing temperature the peak blurs away due to thermal fluctuations. Apparently, On the other hand, one may fix the temperature and vary the coupling strength parameter, which shows us the similar picture seen from a different perspective. Fig. 9 presents the conductivity shift ratio as a function of M/b. Once again we can see that it is peaked near specific values of M/b with the peak position and its magnitude visibly dependent on α. Namely, at this point one has that the larger value of α-coupling constant we take into account, the bigger δσ one achieves. It is worth to recall that the non-interacting system is expected to have the critical value of M/b = 1. This large renormalisation of the critical value is attributed to strong coupling effects taken into account by holographic approach.
It is also worth to note, that similarly large renormalisation of the critical value of the mass  The result is depicted in Fig. 11. The scaling with power n = 2 is only an approximate one.

VI. CONCLUSIONS
Our theory describes the holographic scenario of the quantum phase transition in Z 2 Dirac semimetals, from the topologically non-trivial to the trivial phase. This system has been modelled by the action with Chern-Simons terms binding various combinations of U (1)gauge field strengths with adequate coupling constants. The important ingredient, which we paid special attention to, is the direct coupling between both fields. The coupling constant In the case of the longitudinal conductivity we have found that it is independent on parameters M and b. Namely it has the constant value σ xx = σ yy = σ zz = r 0 = πT . Such linear dependence on temperature is characteristic for the gap-less topological phase.

ACKNOWLEDGMENTS
The work of KIW has been supported by the National Science Centre (Poland) through the grant no. DEC-2017/27/B/ST3/01911.
We thank the unknown Referee for suggesting to analyze the conservation of the currents. does not lead to the collapse of data onto the single curve. A slightly different power 2.03 < n < 2.04 leads to much better convergence of the curves, nevertheless it is still not ideal.
Appendix A: Constraints from the current conservation Now we pay some attention to the analysis of the boundary currents bounded with the Chern-Simons terms in the underlying theory and the conservation of currents. In order to obtain some constraints on the coupling coefficients we shall study currents in our theory, i.e., one expands the holographic action about fixed background gauge fields of the form A µ → A µ + δA µ , B µ → B µ + δB µ , to the second order in fluctuations [30]. After tedious calculations one finds that the variation of the action can be grouped into parts, one connected with the bulk action which is responsible for the equations of motion and the boundary part from which we get expressions for the searched currents. They yield J α (F ) = δS(gauge) δA α | r→∞ = √ −g F αr + α 2 B αr | r→∞ (A1) The important point is that in both currents we have mixture of gauge fields. The current J α (F ) above corresponds to the charge current in the system and thus should be conserved.
This immediately leads to α 1 = α 3 = 0. On the other hand, the current J α (B), which we interpret as the spin current does not vanish and is related to the Z 2 anomaly of the studied model.
There is no obvious constraints on the couplings α 2 and α 4 . In this context we add the following comment. It may be recalled that the appearance of the Chern-Simons terms, similar to those encountered in the action (1), has been discussed for the (2 + 1)-dimensional condensed matter system [31,32]. The couplings have been found to depend on the lifetime τ of the fermions in the system. On the other hand, the lifetime changes with the strength of disorder, electron-phonon interactions, etc. If a similar physics is realised in the considered Z 2 Dirac systems the mentioned couplings may be system dependent.