Hydrodynamics of topological Dirac semi-metals with chiral and $\mathbb{Z}_2$ anomalies

We consider the hydrodynamical model of topological Dirac semi-metal possessing two Dirac nodes separated in momentum space along a rotation axis. It has been argued that the system in question, except the chiral anomaly, is endowed with the other one $\mathbb{Z}_2$. In order to model such a system we introduce two $U(1)$-gauge fields. The presence of the additional $\mathbb{Z}_2$ anomaly leads to the non-trivial modifications of hydrodynamical equations and to the appearance of new kinetic coefficients bounded with the vorticity and the magnetic parts of Maxwell and auxiliary $U(1)$-gauge fields.


Introduction
The possibility of the hydrodynamic approach to transport relies on the fact that strong interactions of the constituent particles which, at low energies and long length -scales, move like a fluid can be described with only a few collective or slowly varying variables. These include the local velocity v(x), temperature T (x) and chemical potentials µ a (x) related to all conserved charges (their densities are denoted by ρ a (x)). The hydrodynamics of the relativistic fluid has been developed by Landau [1] and others [2] and generalized to take relativistic triangle anomalies [3,4] into account. A purely hydrodynamic derivation of the anomaly effects, considering the first order in derivation expansion was presented in [5]. The idea was to examine the local entropy production rate in the presence of anomalies and impose the positivity constraint stemming from the second law of thermodynamics. It was shown that the contributions from the anomaly to the entropy production were locally unbounded and might potentially violate the second law of thermodynamics, so the proper generalizations were necessary. In turn, these facts lead to a set of differential equations for the novel transport coefficients connected with the anomaly. Further, this idea was implemented to the case of anomalous superfluids [6]- [8] and non-abelian symmetry [9,10]. On the other hand, the chiral magnetic anomaly, i.e., anomaly induced phenomenon of electric charge separation along the axis of the applied magnetic field in the presence of fluctuating topological charge was widely studied [11]- [16]. The aforementioned phenomenon have attracted a lot of attention due to the possible explanation of an experimentally observed charge asymmetry in heavy ion collisions and provided explanation for the observed decay of neutral pion into photons. The anomalies have been predicted [17] and later found [18,19] to play an important part in the description of electrons in solids.
The necessity of relativistic description of electrons in solids may appear superficial, as the velocity of electrons in solids typically equals a small fraction of the light velocity. However, the spectrum of electrons in many materials and close to some special points in the Brillouin zone, has a relativistic form characteristic for massless particles. Such Diraclike massless nature of spectrum is protected by symmetries and has been spotted in the two -dimensional graphene [20] and at the surfaces of the crystalline topological insulators [21].
The Dirac -like spectrum is predicted and observed in the three -dimensional materials known as Dirac or Weyl semi-metals [22][23][24][25][26][27][28]. The transport properties of graphene with the Dirac point at the Fermi energy have been proposed to follow the hydrodynamic description [29]. Later measurements confirmed the hydrodynamic behavior of electrons in graphene [30] and in three -dimensional systems [27,28,[31][32][33]. All this makes the relativistic hydrodynamic approach to electrons in condensed matter a timely and important issue.
Moreover, the recent experimental works provide clear evidences that chiral anomaly is observed in condensed matter systems. Namely, it was spotted in Dirac semi-metal Na 3 Bi [34], ZrTe 5 [35], as well as, in Weyl semi-metal TaAs and NbP [36]- [38]. The mentioned two classes of Dirac semi-metals (DSM) have acquired attention in the contemporary investigations. In the first one the Dirac points appear at the time reversal invariant momenta in the first Brillouin zone, while in the other the Dirac points take place in pairs and are separated in momentum space along a rotational axis [39,40]. It turns out that the experimentally found examples of DSM belong mainly to the second class of the aforementioned materials.
The Dirac points in the second class of aforementioned semi-metals are endowed with a non-trivial Z 2 topological invariant protecting the nodes and leading to the presence of Fermi arc surface states [41]- [44]. The novel charge, in a close analogy to the chiral one, is also not conserved under the action of external fields. The non-conservation of the novel anomalous charge has been argued to have an effect on transport characteristics of materials [45]. Thus the recent studies of three dimensional condensed matter systems open the doors to symmetries not spotted in other relativistic objects, making the subject even more intriguing.
The main motivation behind our considerations is a natural question about the possible influence of Z 2 topological charge on transport characteristics of the studied materials. The aim of the present work is to generalize relativistic hydrodynamics including the chiral anomaly [3,4] and the additional anomaly, which we call Z 2 anomaly after the paper [45]. The two anomalous charges in the considered theory require the existence of the two conjugate to them chemical potentials (µ and µ d ). At the equilibrium both chemical potentials take zero values. Accordingly we also introduce two U (1)-gauge fields, one being the standard Maxwell field coupled to the chiral anomalous charge and other coupled to the Z 2 topological charge. The derived set of hydrodynamic equations generalizes those previously found [5] and extensively discussed [46,47] in the literature.
The organization of the paper is as follows. In the next section 2 we present the calculations leading to generalization of the relativistic hydrodynamic equations [5] in such a way that they take into account two anomalous charges, responsible for chiral and Z 2 anomalies. In section 3 we conclude with the discussion of the main results and possible modifications of the transport characteristics of materials.

Hydrodynamical model
In this section we examine the hydrodynamical model of topological Dirac semi-metal in which two Dirac nodes, protected by rotational symmetry, are separated in momentum space along a rotation axis. It has been argued that the aforementioned system constitutes a source of the additional Z 2 anomaly, except the chiral one, which leads to the nonconservation of the corresponding anomalous Z 2 topological charge [45]. In order to model such a system we consider anomalous charges connected with two U (1)-gauge fields. One of them is the ordinary Maxwell gauge field, the other is the additional one connected with the Z 2 anomalous charge. The hydrodynamical equations of motion in the presence of Z 2 and chiral anomalies are provided by where C i , i = 1, . . . , 4 denote the constants which determine the adequate anomalies. The electric and magnetic components of the two gauge fields, in the fluid rest frame, are written respectively as On the other hand, j α (F ), j α (B) represent the adequate currents connected with the gauge fields. The relation (2.2) describes the modifications of the anomalous chiral charge conservation law when the external magnetic and electric fields parallel to each other are applied to the system, while the equation (2.3) expresses the changes of the anomalous Z 2 charge conservation law. The energy momentum tensor and the currents needed for the hydrodynamic description of the relativistic fluid are given by [1,5] where ǫ is the energy per unit volume, p the pressure of the fluid, ρ, ρ d are the U (1) charge densities, while τ αβ and V α F (B) depict higher order corrections in velocity gradients and correspond to the dissipative effects in the fluid. In the rest frame of the fluid element, there are no dissipative forces and u α τ αβ = 0 and u α V α F = u α V α B = 0. The four-vector u α , with the normalization u α u α = −1, describes the flow of the considered fluid.
Using the thermodynamical relations where s is the entropy per unit volume, the explicit expression for energy-momentum tensor and u β ∂ α T αβ , as well as, the expressions for ∂ α j α (F ) and ∂ α j α (B), one arrives at the following relation: As was pointed out in [5], if we did not take into account the influence of the anomalies (i.e., C i = 0), and supposed the positivity of the conductivities σ F > 0 (σ B > 0) and viscosity parameters η and ζ [1] entering the formula for τ αβ , the right-hand side of (2.10) would be positive for the following relations: Thus, the equation (2.10) can be interpreted as describing the entropy production. Its righthand side is greater or equal to zero, as required by the second law of thermodynamics. The presence of anomalies changes the situation drastically. The terms with C i = 0 can have either sign and, when negative, can even overcome the rest of the terms appearing in the equation (2.10) and thus spoil the positivity of entropy production. Therefore, the entropy flux s α , as well as, all the dissipative terms contributing to the transport current have to be modified. The most general modification of the entropy current, which comprises standard dissipation terms, vorticity ω α = (1/2)ǫ αβρδ u β ∂ ρ u δ and the terms proportional to the magnetic components of the two U (1)-gauge fields are taken in the form The dissipative contribution to the U (1)-gauge field currents are also modified by new transport coefficients ξ, ξ B , ξ d and ξB 14) The symbol P αβ = g αβ + u α u β stands for the projector orthogonal to the four-velocity u α , and the unknown functions ξ, ξ d , ξ B , ξB, D, D B , DB depend on T and µ, µ d . Our aim is to find the general formula for these new transport coefficients induced by the quantum anomalies.
Assuming that all the anomaly coefficients C i = 0 and repeating the standard algebraic manipulations [1] required by the positivity proof of ∂ α s α , one gets the equation containing on its right-hand side the following additional terms In order to satisfy the requirement of the entropy current positivity, the above terms are demanded to vanish [5].
To proceed we relate the derivatives of the vorticity ∂ α ω α to the vorticity ω α itself and similarly, the ∂ α B α is related to B α . For our hydrodynamics (linear in the derivatives of velocity) it is enough to find the required relations for the ideal fluid. They may be achieved by projecting the underlying equations of motion (2.6)-(2.8) of the hydrodynamical model along two orthogonal directions. Namely, along u α and P α β = δ α β + u α u β . As a result we achieve the following relations for the ideal hydrodynamics ( i.e., with τ αβ = 0, The condition ∂ α s α ≥ 0 demands vanishing all factors multiplying the above terms. It eventuates in the following differential equations and the additional conditions  and require vanishing of the coefficients multiplying ∂ α p, ∂ αμ and ∂ αμd , which can be considered as having arbitrary values at the initial time slice [5]. This leads to three sets of the differential equations. The first defines the parameter D(p,μ,μ d ) while the next two give the dependence of the partial derivatives of D B (p,μ,μ d ) and DB(p,μ,μ d ) Using the Gibbs-Duhem thermodynamic relations (2.9) we can arrive at the expression dp = ǫ + p T dT + ρT dμ + ρ d T dμ d (2.42) which in turn can be easily cast into This provides the relations as follows: By virtue of (2.44) the first equations from each of the sets of the relations (2.33), (2.36) and (2.39), can be immediately integrated. The results yields are the new functions, which do not depend on temperature T . Thus it is more convenient to treat D i as functions of temperature T , and chemical potentialsμ andμ d .
To this end we assume the following dependence of the temperature T = T (p,μ,μ d ) and use the relation The formula similar to (2.46) for the derivative with respect toμ d is supposed. This leads to the system of differential equations provided by and for D B one gets Consequently, one obtains the similar equations for DB To proceed, we shall replace the derivatives of the type ∂D i , respectively. Consequently, we obtain the three sets of partial differential equations where the second equalities follow from the equations (2.23) and (2.24) (2.59) In the above derivations we use the relations (2.25) and (2.28). The last set of the equations can be easily achieved by incorporating (2.27) and (2.26). Namely, one has On this account, it is customary to write the solutions of the aforementioned sets of the partial differential equations as follows: where γ i , i = 1, 2, 3 are integration constants. Contrary to some claims in the literature, these constants are required to vanish, on the account of the relation (2.45). Consequently, one can readily get the following expressions for the four novel kinetic coefficients: In the derivation of the above equations we have made use of the continuity of the partial differentials of DB, which provides the equality C 2 = C 4 . Equations (2.65)-(2.68) constitute the main results of the paper. They provide the generalization and in the appropriate limit reduce to those obtained earlier [5]. As the very nontrivial result we remark the fact, novel in comparison to the paper [5], that the kinetic coefficient ξ d is induced by the parameter C 1 = C considered in that work. The extra parameter C 3 introduced here, modifies the values of kinetic coefficients previously found in systems with the triangle anomaly. In the next section, we shall discuss the application of the theory in question to Weyl semi-metals with the two discussed anomalies.
3 Application to Dirac semi-metals with Z 2 topological charge As was mentioned in the introduction most of the known Dirac semi-metals, in particular Na 3 Bi or Cd 2 As 3 , possess a chiral anomaly and two Dirac nodes, each carrying topological Z 2 charge. In these materials two Dirac nodes are protected by rotational symmetry of the crystal. The two anomalies show up in our results as two different chemical potentials: µ corresponds to the chiral anomaly and its change results in the appearance of the chiral currents while µ d decides about the position in energy of the two Dirac nodes. In the presence of a magnetic field parallel to an electric field the corresponding currents are not conserved. The current related to Z 2 anomaly is a spin current, at least so, when the spin is approximately conserved [45]. Due to this interpretation of the Z 2 bound current, one expects that spin -related magnetic fieldB α vanishes, what is equivalent to disappearing of C 2 and C 4 . Assuming C 2 = 0 and C 4 = 0, one immediately observes the disappearance of the ξB kinetic coefficient. Accordingly with the above claim, the other coefficients imply It is worth pointing out that even in the presence of theẼ α field, the spin conductivity ξ d (and possibly the spin Hall effect) is not affected by it. However, the parameter C 3 modifies the kinetic coefficients related to the chiral anomaly. These findings cosound with the recent kinetic calculations [45], where the authors have noted that the Z 2 anomaly affects magneto-transport properties of Dirac semi-metals. The observational manifestation of the Z 2 anomaly found earlier is connected with the reduction of the diagonal resistivity due to the spin Hall effect and the narrowing of the angular dependence of the magnetoresistance. The detailed analysis of the magneto-conductivity and magneto-resistivity of the Weyl semi-metals based on the presented hydrodynamic approach [48] will be presented in the future publication.

Summary and conclusions
We have examined the generalized equations of relativistic hydrodynamics allowing the description of electrons in condensed matter systems with linear spectrum and the two different types of anomalies. One of them is the well known chiral anomaly, while the other one, authorizes the anomaly observed in one class of Dirac semi-metal characterized by two Dirac nodes separated in momentum space and lying on the axis of rotation. With the Z 2 anomaly the corresponding charge density ρ d is connected. Its existence forces the non-trivial generalization of the relativistic hydrodynamics. We have found that the additional kinetic parameters, bounded with two different anomalous charges and required by the second law of thermodynamics and positiveness of the entropy production during the flow of electron fluid, enter the hydrodynamic equations in the similar manner. They are a source of the additional kinetic coefficients called earlier magnetic conductivities. In fact these are spin and spin Hall conductivities [45]. Their appearance in the hydrodynamic equations can be traced back to the necessity of adding dissipative terms proportional to the vorticity and magnetic components of the two U (1)gauge fields. Up to the first order in the velocity gradients, they constitute the important component in the proper description of the relativistic fluid.
The lack of the magnetic field acting on spin degrees of freedomB α , connected with the discussed Z 2 anomaly, present in topological second type Weyl semi-metals, forces us to put C 2 = C 4 = 0. Interestingly, the very existence of the Z 2 anomaly induces the kinetic coefficient ξ d connected with the Z 2 related conductivity. We argue that this conductivity is bounded with spin conductivity and spin Hall effect in the kinetic approach to the problem in question. The fieldẼ α , in the presence of the parallel to it magnetic field B α , changes via the parameter C 3 , the kinetic coefficients ξ and ξ B related to the chiral anomaly. This finding provides the generalization of the previous work on hydrodynamics with quantum triangle anomalies [5]. The hydrodynamic analysis of the full magneto-thermal conductivity matrix of topological Weyl system with both kinds of anomalies is underway.