One-Dimensional World as a Tool to Study Chiral Effects

An analog of the chiral separation effect and the chiral magnetic effect has been studied for fermions in one spatial dimension. The relation of these effects to axial anomaly in the epsilon substitution method has been demonstrated. The grand canonical potential has been calculated for chiral fermions in one spatial dimension in the presence of the chiral chemical potential.


INTRODUCTION
Chiral effects are associated with the appearance of nondissipative currents in media in thermodynamic equilibrium. The most famous examples in four dimensions are the chiral separation effect [1,2], chiral magnetic effect [3], and chiral vortical effect [4]. Great attention is paid to the relation between these effects and anomalies in quantum field theory. It was assumed for a long time that all chiral effects occur in the infrared region and can thereby be calculated within quantum field theory without regularization. However, the authors of [5,6] showed that this is not the case and that the chiral magnetic effect is absent. The authors of [7] earlier explained the universality of the experimental relation between the current in the effectively one-dimensional thin wires and the applied voltage. In the same way, an expression was obtained for the chiral magnetic effect [7], which makes the mentioned universality doubtful [5,6].
In this work, a simple approach is proposed to study the main characteristics of chiral effects in thermodynamic equilibrium with the focus on two spacetime dimensions. All results are obtained within a well-regularized theory, which makes it possible to examine all features and all significant differences between the chiral separation effect and the chiral effect, which is an analog of the chiral magnetic effect, and the relation of these effects to an anomaly.  (2) where is an analog of the chiral magnetic effect and is an analog of the chiral separation effect. In classical theory, it is simply necessary to have a nonzero charge density. In the quantum case, all results should be obtained in a well-regularized theory.

RELATION BETWEEN THE CHIRAL SEPARATION EFFECT AND THE ANOMALY
The axial anomaly in two spacetime dimensions implies the exact relation between the divergence of the axial current and the gauge field strength: This relation can be treated as an operator relation. To understand the appearance of the axial anomaly, it is necessary to use regularization, which keeps the gauge invariance and relates the anomaly to the physics at small distances [8]: Here,

OF THEORETICAL PHYSICS
In the real time formalism, the propagator of a massless fermion has the form [9] (5) The substitution of the propagator into the expression for the current gives two contributions where and The infrared part is responsible for the chiral separation effect and is free of divergence. The divergence of the axial current in the nonzero-mass case has the form (6) The Wick rotation reduces the integral to an integral over an infinite one-dimensional sphere: Here, the last integral is calculated over the total solid angle equal to 2π in one dimension. Taking into account the expression for the propagator, we obtain (13) It is noteworthy that the anomaly within the epsilon substitution method is a significantly ultraviolet effect and is reduced to the flux through an infinitely remote surface, whereas the chiral effect is associated with the infrared behavior in the theory. 1 The Fermi distribution obviously does not affect the coefficient in the anomaly because its integral is zero. The above consideration shows that the interaction at these scales can provide different effects. In particular, the anomaly in the case of fermions with a nonzero mass does not change, whereas the chiral separation effect changes as . This consideration can give examples of renormalizable theories where the interaction provides a correction to the chiral separation effect.

CHIRAL EFFECT IN THE THEORY WITH REGULARIZATION
The problem of existence of the chiral effect is more difficult in view of the problems with the regularization of the theory (see below). There are two interesting questions: (i) can the expression for the current be derived from the expression for the axial anomaly, and (ii) can this current exist in equilibrium in a welldefined theory? To answer the first question, we consider the action [11] Under the transformation (14) (15) where , and is the differential parameterizing the infinite set of such infinitely small gauge chiral transformations, and after a change of variables, the functional integral takes the form (16)   (17) As a result, (18) and the action is obtained in the form This expression can be transformed to (20) 1 Without the epsilon substitution, we would have divergence described in [10]. This result can be formally interpreted as the vector current induced by the axial charge density, but the above mechanism does not imply this interpretation, although serves at first glance as the chemical potential. We now consider thermodynamic equilibrium and introduce the chemical potential through the standard rules The chemical potential is conventionally introduced with respect to conserving charges, which implies the condition m = 0, but the case m ≠ 0 will also be discussed. First, let m = 0; then, the charges of the leftand right-hand fermions are conserved separately: Calculations for left-handed fermions give the potential (26) Since the spectrum is symmetric with respect to zero, this potential can be written in the form In this case, the vacuum contribution cannot be separated and the thermodynamic description becomes problematic; in particular, the integral in Eq. (28) is This also leads to problems with some regularizations, e.g., the Pauli-Villars regularization because the ghost degrees of freedom have a nonzero mass and, therefore, the standard reason that their contribution is suppressed according to the distribution function , , is inapplicable.
We obtain the expression for the chiral effect in the case m = 0 using the epsilon substitution method. Let the system have a finite size L (in the final expression, ) and periodic boundary conditions in order to understand the origin of divergences. The spectrum of the Hamiltonian is known, and it is known that [Q L/R , H] = 0. At finite temperatures and a nonzero chemical potential, it is necessary to calculate the sums Since n f (-E -μ) = 1 -n f (E + μ), the sum in Eq. (29) is separated into a well-defined converging part and the vacuum contribution: The infinite sum of unit terms can be calculated by introducing a regularizer holding the gauge invariance [12]: (32) In the limit , we obtain the expression (33) The right-handed charge is given by a similar expression The expression for the anomaly (Q = Q L + Q R , ) can be obtained from the vacuum part. We now take into account the effect of the gauge field on the Fermi distribution. We introduce an infinitely small addition to the momentum and calculate the change in the left-handed charge. In this case, the charge in the thermal part also changes. However, this example shows that the same contribution is taken into account twice. The reason for this error is very simple: thermodynamic potentials are constructed with respect to the ground state; therefore, there arises the question: from which ground state are excited levels, charges, etc., mea- sured? If the external field is considered adiabatically and the transition is treated as a set of equilibrium states, all parameters of the system are measured with respect to the initial vacuum; consequently, this process can be considered as a process affecting only the Fermi distribution because the vacuum contribution does not change. The second method involves a transition to a new vacuum state and then the thermodynamic description (such a description affects, e.g., the temperature measurement onset). Both methods give the same results in the absence of interaction and zero mass.
The current density can be written in the form Together with the expression for the grand canonical potential, the current density in the chiral effect is given by the expression (36) 5. PHYSICAL REASONS It has been proven that the chiral effect occurs in equilibrium but disregarding the effect of the current itself. It has been demonstrated that the result can depend on regularization and, therefore, physical reasons are necessary to answer the question of the existence of the equilibrium chiral effect. Considering equations for the electric field taking into account the expression for the theta term, we obtain (37) This means that the system cannot be stationary because an unlimited increase in the electric field appears already in the equations of motion including quantum corrections and, consequently, the chiral effect in the case of two spacetime dimensions is only a nonequilibrium effect. Such reasons are inapplicable for the chiral separation effect; therefore, this effect can occur in equilibrium. If the system is effectively spatially one-dimensional, these reasons are inapplicable because the electric current induces only the magnetic field, but other problems can arise in this case [5,6,13]. 6. CONCLUSIONS It has been shown that the world with one spatial dimension is an excellent tool to study chiral effects. The chiral separation effect and the chiral effect have been calculated using the epsilon substitution method within theories with regularization. The relation of both effects to the axial anomaly has been demonstrated. It has been shown that the nonzero mass changes the expression for the current in the chiral separation effect, but it does not affect the expression for the anomaly. The grand thermodynamic potential , . x x x F j has been obtained for the theory with chiral fermions and the chiral potential. The difficulty with the calculation in the case of a nonzero mass resulting in the problem with the regularization of the theory with the nonzero mass has been demonstrated. Some regularization procedures to reduce the vacuum contribution have been discussed. This problem is absent in the case of the conventional vector potential. In particular, the ghost degrees of freedom in the Pauli-Villars regularization do not contribute in the thermodynamic region because of suppression in thermal mass distributions. All these effects are significant in the presence of the interaction because the behaviors of systems with the vector and axial potentials are different.
The effect of the interaction on the problem under consideration is very interesting. The conductivity in an experiment with thin wires slightly depends on the interaction, and it would be expected that it is expressed in terms of a topologically protected parameter in the momentum space [14][15][16][17]. It can possibly be described by analogy with the Hall conductivity [18]. FUNDING This work was supported by the Russian Science Foundation (project no. 21-12-00237).