Equilibrium Chiral Magnetic Effect: spatial inhomogeneity, finite temperature, interactions

We discuss equilibrium relativistic fermionic systems in lattice regularization, and extend the consideration of chiral magnetic effect to systems with spatial inhomogeneity and finite temperature. Besides, we take into account interactions due to exchange by gauge bosons. We find that the equilibrium chiral magnetic conductivity remains equal to zero.


I. INTRODUCTION
The chiral magnetic effect (CME) [1][2][3][4]6] is one of the non -dissipative transport effects. Unlike most other members of this family it most likely does not appear in true equilibrium. Instead it appears, presumably, in a steady state out of equilibrium in the presence of both external electric field and external magnetic field [5]. Combination of these fields gives rise to chiral imbalance. The latter together with the magnetic field leads to electric current directed along the magnetic field. Experimental evidence of this effect is found in magnetoresistance of Dirac and Weyl semimetals, and its dependence on the angle between magnetic and electric fields [7]. The calculation of the CME conductivity [5,7] demands reference to kinetic theory, and the chiral imbalance appears as a pure kinetic phenomenon. Yet it is not sufficiently clear, is this possible to describe this imbalance using the notion of chiral chemical potential. The resulting expression for the CME current is proportional both to the magnetic field squared and to electric field. In the systems without external electric field the chiral magnetic effect may possibly be observed in the nonequilibrium systems with chiral chemical potential depending on time [33].
As it was mentioned above, at the present moment it is widely believed that the equilibrium version of CME does not exist. This has been proven for the case of homogeneous systems at zero temperature (better to say, those systems that become homogeneous when external magnetic field is removed). In [12][13][14][15] the proof has been given using lattice numerical simulations. In [10] the same conclusion was obtained using analytical methods for the system of finite size with the specfic boundary conditions in the direction of the magnetic field. In [26] the absence of CME was reported for a certain model of Weyl semimetal. The contradiction of equilibrium CME to the no -go Bloch theorem has been noticed in [27]. In [31] it has been shown that at zero temperature the CME current is a topological invariant. Its responce to any parameter (to the chiral chemical potential, as an example) is vanishing, which is an alternative proof.
So far the question about the possibility to observe equilibrium CME current remains open for the systems at finite temperature with explicit dependence of lagrangian on coordinates. (The homogeneous systems at finite temperature have been considered in [32] using zeta regularization.) Moreover, the influence of interactions on the CME conductivity has not been considered in sufficient details. In the present paper we close this gap and demonstrate that the CME conductivity remains vanishing in equilibrium under these circumstances.
Technically we rely on the machinery developed in [28][29][30]. Namely, we use Wigner transformed Green functions in order to express electric current and its response to external fields. Using this technique we will show that the response of total electric current to constant external magnetic field is a topological invariant even in the non -homogeneous systems, and even at finite temperature. Besides, we will demonstrate that interactions due to exchange by gauge bosons do not affect this conclusion at the one and two -loop level. The generalization of this consideration to the higher orders may be given along the lines of [29].
Hystorically Wigner -Weyl calculus has been proposed in order to reformulate quantum mechanics using language of functions in phase space instead of the language of operators [34]. The so -called "deformation quantization" has been developed on the basis of this calculus (see [35,36] and references therein). One of the standard quantites of the Wigner -Weyl calculus is Wigner function W(q, p) that generalizes the notion of distribution in phase space of classical mechanics [37]. It is worth mentioning that Wigner distribution can not be treated as probability distribution [38]. Nevertheless, it is possible to formulate the fluid analog of quantum entropy flux in phase space using Weyl-Wigner calculus [45]. Von Neuman entropy and the other quantites of quantum information theory may be defined using Wigner -Weyl calculus [40,41,45]. We would also like to notice that various applications to quantum mechanics [42][43][44][45] have been followed by the applications of Wigner -Weyl calculus to anomalous transport in quantum field theory [46][47][48][49][50][51].

A. Model setup
We start from the fermionic lattice model with partition function written in momentum space [31] By |M| we denote volume of four -dimensional momentum space M. Both p 1 and p 2 are four -momenta. Here matrix elements of lattice Dirac operatorQ are denoted by It is worth mentioning that we use the relativistic units, in which both and c are equal to unity. In addition, elementary charge e is included to the definition of electric and magnetic fields. Wigner transformation of Green function G(p 1 , p 2 ) = p 1 |Ĝ|p 2 , i.e. the Weyl symbol of operatorĜ is defined as In the similar way we define Weyl symbol of operatorQ: If the inhomogeneity is due to the slowly varying external gauge field A(x) (when its variation at the distance of the order of lattice spacing may be neglected) we have ). Under these conditions matrix element Q(p 1 , p 2 ) remains nonzero only when |p 1 − p 2 | is much smaller than the size of compact momentum space. This property remains also for sufficiently weak inhomogeneity caused by any other reasons. Then Wigner transformation of the Green function and Weyl symbol of Dirac operator obey the Groenewold equation [52] Here the Moyal product is introduced Let us notice associativity of the Moyal product, which allows to remove brackets in expressions containing the stars. Besides, we will use below the following property of the star product: This property is valid for any functions A W , B W defined in phase space. We consider external electromagnetic potential that gives rise to magnetic field B(x) = ∇ × A(x) and to varying electric field For definiteness we may chose lattice regularization with Wilson fermions that corresponds to Here γ i are Euclidean gamma -matrices, while g i = sin(π i ) with i = 1, 2, 3, 4; (1 − cos(π i )). µ 5 is chiral chemical potential. For the massless fermions we have m (0) = 0.
We will also consider as a particular example the lattice Dirac operator with the form of where functions g µ depend on π = p − A(x), and in addition depend on x explicitly. However, the results discussed below remain valid also for the lattice regularization with arbitrary form of Q W (p, x) provided that the inhomogeneity is negligible at the distance of the order of lattice spacing.

B. Linear response of Green function to external magnetic field
In the presence of external magnetic field related to Euclidean four -potential Here dependence on magnetic field is encoded in vector potential A(x) while dependence on scalar potential is written as explicit dependence of Q on x. The direct dependence of Q on x may also originate from the other sources like the chiral chemical potential depending on x (see below). The Dirac operator may be written as The Green's function may also be represented as

The Groenewold equation results in
So, we have where [52],

C. Electric current
In the presence of gauge field electric current may be obtained varying the partition function Integral here is over momentum space M. Its volume is equal to M = (2π) D for the model with Wilson fermions. The total integrated current is equal to the integral over coordinates Notice, that here and below we replace in such expressions sum over the lattice points by an integral, which is possible since we are considering the model with the inhomogeneity that is negligible at the distance of the lattice spacing. Using periodic boundary condition, we have Here We denote For the periodic boundary conditions in momentum space j (1) i(k) (x) = 0 since it is an integral of total derivative. The first leading order term with the external gauge field is given by The average current linear in F mn is Here V (4) is the overall volume of Euclidean space-time while V is the three-dimensional volume, T is temperature assumed to be very small. We have the average current [28] In the last expression the star may be inserted and we will arrive at This expression is topological invariant as long as the system is gapped and the integral is convergent. We consider the magnetic field only, which is calculated through the field tensor F i j = ǫ i jk B k . Notice, that we define the field strength in Euclidean space -time, and assume that A k = A k . Therefore, from B = rot A it follows B k = ǫ ki j ∂ i A j . In this case and in the presence of finite chiral chemical potential the system remains gapped in the absence of inhomogeneity. This property is kept also when weak inhomogeneity is added. Then the topological nature of quantity M 4 allows us to come to the conclusion that the CME conductivity vanishes also in the presence of weak inhomogeneity. This is an extension of the result of [31] 1 .

A. Chiral magnetic conductivity
At finite temperature the partition function receives the form Here ω is the sum over Matsubara frequencies ω = 2πT (n + 1/2), n ∈ Z, 0 ≤ n < N, where N = 1/T is the number of lattice sites in the fourth direction, while temperature T is measured in the lattice units. Integrals are over the Brillouin zone B. Notice that the volume of momentum space at zero temperature is equal to 2π times |B|, where |B| is volume of the three -dimensional Brillouin zone.
In the similar way we obtain that the chiral magnetic current averaged over the sample volume is given bȳ Here and ω is the sum over Matsubara frequencies ω = 2πT (n + 1/2), n ∈ Z, 0 ≤ n < N, where N = 1/T . We are able to add an extra star to the above expression and arrive at Since Q W and G W do not depend explicitly on imaginary time τ, the star operation is reduced to the one without derivatives with respect to ω and τ. One can see that for any given value of ω quantity N 4 (ω) is a topological invariant as long as the expression standing inside the integral does not contain singularity (the proof is similar to that of the similar statement presented in [28]). We know that finite temperature being added to the homogeneous model of massless fermions removes the singularity of the Green function. Weak inhomogeneity cannot change this. We come to conclusion that the response of M 4 to chiral chemical potential at finite temperature (i.e. the CME conductivity) vanishes even for the system with inhomogeneity, at least if the inhomogeneity is sufficiently weak. Below we will check by direct calculation that the CME conductivity indeed vanishes for the particular case of the model with the Dirac operator given by Eq. (6). An example of such a model is given by Wilson fermions, or by more complicated but still standard lattice regularizations.
We will denote r = 2πT 16Vπ 2 and consider the linear response of M 4 to µ 5 : We represent

B. Homogeneous limit
For simplicity first let us consider the limit of homogeneous system without explicit dependence of Q on spatial coordinates. We represent Q = µ g µ γ µ − ig 5 = −iγ 5 a=1,2,3,4,5 Γ a g a = −iγ 5Q whereQ = a Γ a g a . Here Γ 5 = γ 5 and Γ µ = iγ 5 γ µ for µ = 1, 2, 3, 4 and it satisfies the usual anti-commutation relations of Dirac gamma matrices; {Γ a , Γ b } = 2δ ab . So we have G = a Γ a g a ||g|| 2 × (iγ 5 ) =G × (iγ 5 ) whereG = a Γ a f a . We hereafter replace the star product by ordinary product and write the above equation in terms ofQ andG which is given by ForQ = a Γ a g a andG = a Γ a g a ||g|| 2 , we have Using results of Appendix A we come to the following expression of ∂M 4 ∂µ 5 : Let us represent Here ∂ p 4 g σ = f (ω) and ∂ p i ∂ p 4 g σ = 0 and ǫ i jk g µ ||g|| 2 ∂ p i ∂ p j g ν ∂ p k g ρ ||g|| 2 ∂ p 4 g σ = 0 and ǫ i jk g µ ||g|| 2 ∂ p j g ν ∂ p i ∂ p k g ρ ||g|| 2 ∂ p 4 g σ = 0. We come to the expression for the CME conductivity equal to the integral of total derivative = 0.

C. Calculation for the non -homogeneous case
Now let us extend the above calculation to the non -homogeneous case. We still may represent whereQ = a Γ a g a . This is the same expression as for the homogeneous case, except that functions g µ depend both on momenta and on spatial coordinates. Here Γ 5 = γ 5 and Γ µ = iγ 5 γ µ for µ = 1, 2, 3, 4 and it satisfies the usual anti-commutation relations of Dirac gamma matrices; {Γ a , Γ b } = 2δ ab . For the Wigner transformation of Green function we have a more complicated expression In order to determine functions f (p, x), f a (p, x), f ab (p, x) we substitute the above expression to the Groenewold equation. In particular, this allows us to obtain f = 0, and we come to the following expression for ∂M 4 ∂µ 5 : Here we denote As above we obtain This is again the integral of total derivative, and therefore is equal to zero. Let us consider briefly effect of interactions on the CME conductivity. Without interactions the averaged electric current in the presence of external magnetic field is given bȳ

IV. EFFECT OF INTERACTIONS
Here G W is Wigner transformation of fermion propagator in the presence of magnetic field, while Q W is Weyl symbol of its inverse operator. For definiteness we consider interaction caused by gauge bosons. Then in one loop approximation (Fig. 1 a)) we obtain the following correction to electric current: Here D ab µν (k) is propagator of gauge boson whilet a is generator of the gauge group. We assume that bare action of the gauge boson is unform, i.e. bare Wigner transformed propagator depends on momentum only and does not depend on coordinates. Using integration by parts we represent We can always chose the gauge, in which the gauge boson propagator obeys As a result we obtainJ (1) 2. An example of the two loop contribution to self energy, which gives rise to a three loop contribution to electric current J (2,cross) k . The cross marks the position of derivative ∂ p k Q W .
For Abelian theory this proof may be extended to all orders of perturbation theory along the lines of [29], which will give that the total integrated electric current is not renormalized by interactions. The same may be true also for the case of the non -Abelian gauge theory, but for the non -Abelian theory we restrict ourselves by the one -loop order. Below we will consider the sketch of the proof that the higer order diagrams do not renormalize electric current for the case of Abelian gauge theory without self -interactions of gauge field quanta.
Let us consider first the case of the diagram of Fig. 1 b). The corresponding contribution to electric current is given by Eq. (24) with gauge boson propagator substituted by its one -loop correction . 3. a) The progenitor for a two -loop rainbow contribution to self energy/electric current. b) The progenitor for the two -loop crosscontribution to electric current.
Here Π ρσ W,cd (x, k) is Wigner transformation of polarization operator Π ρσ cd (x, y) = Trγ ρtc G(x, y)γ σtd G(y, x) One can check that This will give us vanishing contribution to electric current. The next example is the two -loop diagram of Fig. 1 e). On Fig. 2 we represent the corresponding diagram of self energy. It gives the contribution to electric current We use here notations of Feynmann diagram technique designed for the Wigner transformed propagators (it has been proposed in [30]). Here the Moyal product * = e i ← − In [30] also another type of the star product has been introduced Here derivatives with the right arrow act on Wigner transformed gauge boson propagator D W(i) (R, k) while the derivatives with the left arrow act to all propagators of fermions that stand left to this symbol. In a similar way in i • the derivatives with the right arrow act on one function that stands right to the symbol, and the derivatives with the left arrow act on gauge boson propagator D W(i) (R, k). However, at the given order the Wigner transformed boson propagator does not depend on R, it depends on momentum (k or q) only, and is given by an ordinary Fourier transform. Therefore, the circle products may be omitted completely at this orider. However, they should be present in the next orders of perturbation expansion because the (Wigner transformed) gauge boson propagator becomes depending both on R and k due to the non -homogeneity in G.
The last expression corresponds to the so -called progenitor diagram of Fig. 3 b) with the derivatives inserted inside the integral over p. The integral of a total derivative is zero, and we come to vanishing contribution of the two -loop crossdiagram. Correspondingly, the diagram of Fig. 3 a) is a progenitor for the rainbow contributions to electric current of Fig. 1  c) and d). The corresponence between the two is represented on Fig. 4. In the similar way it may be shown that the latter contributions vanish as well.
Moreover, we know that Eq. (22) is topological invariant. Therefore, in this expression we may replace bare G W (without interaction corrections) by Wigner transformation of complete interacting Green function and Q W by Weyl symbol of its inverse. As a result the CME conductivity for the system in the presence of interactions may be written as Eq. (15) with complete interacting Q and G.

V. CONCLUSIONS
We extended in the present paper the methodology of [31]. The Wigner -Weyl calculus is used to express response of electric current to external magnetic field through the topological invariant composed of the Green functions. In [31] the lattice models of relativistic quantum field theory were discussed. The theory was in thermal equilibrium at zero temperature and was assumed homogeneous. Besides, the interactions were not taken into account. Here we take into account dependence of lattice Dirac operator on spatial coordinates. The dependence may be almost arbitrary, but it should be weak in the sense that the variations on the distance of the order of lattice spacing are negligible. In addition we do not restrict ourselves by zero temperature, and consider the theory at arbitrary finite temperature. Finally, we take into account inter -fermion interactions due to exchange by gauge bosons. Explicit calculations are presented in one and two loop approximations, but they may be extended easily to arbitrarily high orders (at least for Abelian gauge group) in direct analogy to the consideration of [29].