Equilibrium chiral magnetic effect: Spatial inhomogeneity, finite temperature, interactions

Abstract

The chiral magnetic effect (CME) [1–4, 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 non equilibrium systems with chiral chemical potential depending on time [33]. The other non dissipative transport effects have been widely discussed in condensed matter physics and in high energy physics [8–15]. Some of them may also be observed in the recently discovered Dirac and Weyl semimetals [16–22]. The experimental indications of the CME in relativistic heavy ion collisions were discussed, for example, in [3, 23, 24]. As a fluctuation the CME has been reported using lattice simulations [25]. 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 [12–15] 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 [28–30]. 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