INTRODUCTION

The prediction of a family of nondissipative currents [1–7], which appear in an external magnetic field and in a uniform rotation field, stimulated significant scientific interest and led to active studies of nondissipative transport [8, 9]. These effects were commonly called chiral effects. In this work, the chiral magnetic effect [3] and the chiral separation effect [1, 2, 10] are primarily discussed in this work. The former effect is the appearance of the electric current \({{j}^{V}}\) along the external magnetic field in the presence of the chiral chemical potential \({{\mu }_{A}}\), whereas the latter effect is the appearance of the axial current \({{j}^{A}}\) along the external magnetic field in the presence of the conventional chemical potential \({{\mu }_{V}}\). Both effects in the massless case can be written in the form

$${{{\mathbf{j}}}^{{A,V}}} = \frac{1}{{2{{\pi }^{2}}}}{{\mu }_{{V,A}}}{\mathbf{B}}.$$
(1)

Necessity of a correct regularization for the description of the chiral effects was not discussed for a long time. Since currents are calculated in the infrared region and the effects are due primarily to processes on the Fermi surface, it seemed obvious that the effect of ultraviolet physics in this case can be neglected. The invalidity of this conclusion was demonstrated for the first time when calculating the chiral magnetic effect on a lattice and was explained in terms of the Pauli–Villars regularization [11]. Further studies explained the result obtained in [11] in terms of the lattice regularization [12] and developed a technique [13, 14] relating the nondissipative transport to topological invariants in the momentum space [15]. However, the application of this technique to describe the chiral separation effect [14] resulted in ambiguity in the continuous limit, which required regularization. The calculations in [14] were performed by introducing a finite lattice step, which remained the direct derivation of all results in the continuous theory unsolved.

The aim of this work is to describe a regularization that allows one to avoid the above problems and to apply all methods developed in [12, 13] to describe the chiral effects directly in the continuous limit.

NECESSITY OF REGULARIZATION

Let us discuss the problems appearing when describing the chiral separation effect in the continuous theory. When developing a formalism to relate the expression for the axial current to topological invariants in the momentum space [14], the following expression for the axial current in the external gauge field:

$$\begin{gathered} j_{k}^{5} = - \frac{i}{2}T\sum\limits_{n = - \infty }^\infty \int \frac{{{{d}^{3}}p}}{{{{{(2\pi )}}^{3}}}}{\text{Tr}}({{\gamma }^{5}}\mathcal{G}({{\omega }_{n}},{\mathbf{p}}){{\partial }_{{{{p}_{i}}}}}{{\mathcal{G}}^{{ - 1}}}({{\omega }_{n}},{\mathbf{p}}) \\ \times \;{{\partial }_{{{{p}_{j}}}}}\mathcal{G}({{\omega }_{n}},{\mathbf{p}}){{\partial }_{{{{p}_{k}}}}}{{\mathcal{G}}^{{ - 1}}}({{\omega }_{n}},{\mathbf{p}})){{F}_{{ij}}}. \\ \end{gathered} $$
(2)

Here, \(\mathcal{G}({{\omega }_{n}},{\mathbf{p}})\) is the Green’s function in the Euclidean space, which is calculated from the functional integral in the corresponding theory, \({{\mathcal{G}}^{{ - 1}}}({{\omega }_{n}},{\mathbf{p}})\) is the inverse Green’s function, and \({{\gamma }^{\mu }}\) are the Dirac matrices. The propagator of massless noninteracting fermions has the form

$$\begin{gathered} {{\mathcal{G}}^{{ - 1}}}({{\omega }_{n}},{\mathbf{p}}) = - i{{\gamma }^{\mu }}{{p}_{\mu }}, \\ {{p}_{\mu }} = ({\mathbf{p}},{{\omega }_{n}}),\quad {{\omega }_{n}} = 2\pi T\left( {n + \frac{1}{2}} \right). \\ \end{gathered} $$
(3)

Let the magnetic field be directed along the \(z\) axis (i.e., \({{F}_{{12}}} = - {{B}_{z}}\)); then, the substitution of Eq. (3) into Eq. (2) gives the expression for the axial current in the form

$$j_{z}^{5} = - 4Ti\sum\limits_{n = - \infty }^\infty \int \frac{{{{d}^{3}}p}}{{{{{(2\pi )}}^{3}}}}\frac{{{{\omega }_{n}}}}{{{{{(({{\omega }_{n}}{{)}^{2}} + {{p}^{2}})}}^{2}}}}{{B}_{z}};$$
(4)

the integration over momenta reduces Eq. (4) to the form

$$j_{z}^{5} = - \frac{{Ti}}{{2\pi }}\sum\limits_{n = - \infty }^\infty {\text{sgn}}({{\omega }_{n}}){{B}_{z}}.$$
(5)

Thus, the axial current in the case of massless fermions in the absence of the medium is formally equal to the infinite sum of terms +1 and –1. If the interaction in the medium results in the transformation \({{\omega }_{n}} \to \) \(f({{\omega }_{n}}),\;{{p}_{i}} \to g({{p}_{i}})\) in \(\mathcal{G}({{\omega }_{n}},{\mathbf{p}})\), according to Eq. (2), the result will depend on sgnf(ωn).

The inclusion of the chemical potential leads to the change \({{\omega }_{n}} \to {{\omega }_{n}} - i\mu \), and the analytical continuation of the sign function on the complex plane is necessary to obtain the expression for the current. this problem can be solved by the lattice regularization [14, 16], but a method described in the next section allows one to obtain all results directly in the continuous theory. It will be shown that the final expression for the current coincides with the expression obtained by the initial summation of Matsubara frequencies and the subsequent integration over momenta in Eq. (2). This means that the formalism developed in [12–14, 16] is directly applicable to describe continuous theories without the introduction of the lattice regularization.

USE OF THE ZETA FUNCTION

In the case of the absence of complex quantities, the sign function in Eq. (5) can be represented in the form

$${\text{sgn}}({{\omega }_{n}}) = \mathop {\lim }\limits_{\alpha \to - 0} A\left( {\sum\limits_{n = 0}^\infty {{{(n + 1{\text{/}}2)}}^{{ - \alpha }}} - \sum\limits_{n = 0}^\infty {{{(n + 1{\text{/}}2)}}^{{ - \alpha }}}} \right);$$
$$A = (2\pi T{{)}^{{ - \alpha }}};$$
(6)
$$\sum\limits_{n = 0}^\infty {{(n + 1{\text{/}}2)}^{{ - \alpha }}} = {{\zeta }_{H}}(\alpha ,1{\text{/}}2),$$
(7)

where ζH(α, 1/2) is the Hurwitz zeta function [17] at the point (α, 1/2). Such a regularization reduces the calculation of the coefficient in the expression for the chiral separation effect to the calculation of the Hurwitz zeta function or its analytical continuation at the point (–0, 1/2). The first prediction in this regularization is zero axial current in the absence of the chemical potentialFootnote 1 with the complex argument. In terms of

$${{\zeta }_{H}}(\alpha ,\eta ) = \sum\limits_{n = 0}^\infty {{\left( {n + \frac{1}{2} - i\frac{\mu }{{2\pi T}}} \right)}^{{ - \alpha }}};$$
(8)
$${{\zeta }_{H}}(\alpha ,\bar {\eta }) = \sum\limits_{n = 0}^\infty {{\left( {n + \frac{1}{2} + i\frac{\mu }{{2\pi T}}} \right)}^{{ - \alpha }}};$$
(9)
$$\eta = \frac{1}{2} - i\frac{\mu }{{2\pi T}},$$
(10)

the axial current can be represented in the form

$$j_{z}^{5} = - \frac{{Ti}}{{2\pi }}\mathop {\lim }\limits_{\alpha \to - 0} {{(2\pi T)}^{{ - \alpha }}}({{\zeta }_{H}}(\alpha ,\eta ) - {{\zeta }_{H}}(\alpha ,\bar {\eta })){{B}_{z}}.$$
(11)

The Hurwitz zeta function at zero is given by the standard formula

$${{\zeta }_{H}}( - 0,\eta ) = \frac{1}{2} - \eta {\kern 1pt} {\kern 1pt} .$$
(12)

The axial current is given by the expression

$$j_{z}^{5} = \frac{\mu }{{2{{\pi }^{2}}}}{{B}_{z}};$$
(13)

or in the vector form

$${{{\mathbf{j}}}^{5}} = \frac{\mu }{{2{{\pi }^{2}}}}{\mathbf{B}}.$$
(14)

This expression coincides with that obtained by the initial summation of Matsubaru frequencies and the subsequent integration over the momentum in Eq. (2). We emphasize that the summations and integrations in well-regularized theories should be permutable.

The validity of the proposed regularization and Eq. (11) can also be confirmed as follows. The authors of [16] proposed a generalization of the technique to seek topological invariants in the momentum space in the lattice regularization to inhomogeneous media [16] and obtained a formula for the axial current density within the linear response theory. In the case of generalization to the continuous theory,Footnote 2 this formula can be represented as

$$j_{k}^{5}(x) = - \frac{i}{2}\sum\limits_n T\int \frac{{{{d}^{3}}p}}{{{{{(2\pi )}}^{3}}}}tr({{\gamma }^{5}}I{{F}_{{ij}}}),$$
(15)
$$I = (G_{W}^{0} \star ({{\partial }_{{{{p}_{i}}}}}Q_{W}^{0}) \star G_{W}^{0} \star ({{\partial }_{{{{p}_{j}}}}}Q_{W}^{0}) \star G_{W}^{0}){{\partial }_{{{{p}_{k}}}}}Q_{W}^{0}.$$
(16)

Here, \(G_{W}^{0}\) is the Weyl symbol of the Green’s function, which can describe inhomogeneous systems, the superscript 0 indicates that it is calculated in a theory unperturbed in external field [16], \( \star \) means the operator \(\exp \left( {\frac{i}{2}({{{\overleftarrow \partial }}_{x}}{{{\overrightarrow \partial }}_{p}} - {{{\overleftarrow \partial }}_{p}}{{{\overrightarrow \partial }}_{x}})} \right.\), where arrows specify the direction of action of the corresponding derivatives, and \({{Q}_{W}} \star {{G}_{W}} = 1\). Now, the divergence of the axial current along the z axis is calculated under the assumption of a weak inhomogeneity of the chemical potential (\({{l}^{2}}{{\partial }_{i}}\mu ({\mathbf{x}}) \ll 1\), where l is the characteristic scale of the system). In this case, the operator \( \star \) in Eq. (16) can be omitted in the first expression; then, the axial current density in the continuous theory can be written in the form

$$j_{z}^{5}({\mathbf{x}}) = - 4iT\sum\limits_{n = - \infty }^\infty \int \frac{{{{d}^{3}}p}}{{{{{(2\pi )}}^{3}}}}\frac{{{{\omega }_{n}} - i\mu ({\mathbf{x}})}}{{{{{(({{\omega }_{n}} - i\mu ({\mathbf{x}}{{{))}}^{2}} + {{p}^{2}})}}^{2}}}}{{B}_{z}}.$$

The integration over the momenta yields

$${{j}^{{5z}}}({\mathbf{x}}) = - \frac{{Ti}}{{2\pi }}\mathop {\lim }\limits_{\alpha \to - 0} A({{\zeta }_{H}}(\alpha ,\eta ({\mathbf{x}})) - {{\zeta }_{H}}(\alpha ,\bar {\eta }({\mathbf{x}}))){{B}^{z}}.$$

In this case, the chemical potential in Eq. (10) becomes dependent on the spatial coordinates and the coefficient A is specified by Eq. (6). Then, the expression for the divergence takes the form

$${{\partial }_{z}}{{j}^{{5z}}} = \frac{{Ti}}{{2\pi }}\mathop {\lim }\limits_{\alpha \to - 0} \alpha A{{W}_{z}}{{B}^{z}}.$$
(17)

Here,

$${{W}_{z}} = {{\zeta }_{H}}(\alpha + 1,\eta ({\mathbf{x}})){{\partial }_{z}}\eta ({\mathbf{x}}) - {{\zeta }_{H}}(\alpha + 1,\bar {\eta }({\mathbf{x}})){{\partial }_{z}}\bar {\eta }({\mathbf{x}}),$$

where

$${{\partial }_{z}}\eta = - \frac{i}{{2\pi T}}{{\partial }_{z}}\mu ({\mathbf{x}}) \equiv \frac{i}{{2\pi T}}{{E}_{z}}({\mathbf{x}}).$$
(18)

It is now necessary to use the formula

$$\mathop {\lim }\limits_{\alpha \to 0} \left( {{{\zeta }_{H}}(\alpha + 1,\eta ) - \frac{1}{\alpha }} \right) = - \psi (\eta ),$$
(19)

where \(\psi (\eta )\) is the digamma function, which is regular at this set of parameters, leading to

$${{\partial }_{\mu }}{{j}^{{5\mu }}} \equiv {{\partial }_{z}}{{j}^{{5z}}} = \frac{{iT}}{{2\pi }}\mathop {\lim }\limits_{\alpha \to - 0} \frac{{iA}}{{\pi T}}{{E}_{z}}{{B}^{z}}\alpha \left( {\frac{1}{\alpha } + ...} \right),$$
(20)

where ellipsis stands the regular part. Thus, from the chiral separation effect, an expression for the axial anomaly was obtained, which can be represented in the vector form

$${{\partial }_{\mu }}{{j}^{{5\mu }}} = - \frac{1}{{16{{\pi }^{2}}}}{{\epsilon }^{{\mu \nu \lambda \rho }}}{{F}_{{\mu \nu }}}{{F}_{{\lambda \rho }}}.$$
(21)

DZETA REGULARIZATION AND THERMODYNAMICS

The use of the Hurwitz zeta function is not novel in the description of the thermodynamics of fermion systems [20], but it is of interest in this work to illustrate a relation between the partition function and the zeta function of the determinant of the operator in the sense considered by Hawking [18].

One would think that the regularization method in Eq. (11) is successfully guessed, but it is shown below that this method is dictated by the thermodynamic description of the system [18]. The calculation of the partition function implies the determination of the zeta function of the determinant of the operator appearing in it. For clarity, all calculations are performed for massless fermions:

$$Z = \tilde {C}\det (D),D = \mathop \Pi \limits_{\{ P\} } {{(\det ({{P}^{2}} \cdot {{{\mathbf{1}}}_{{4 \times 4}}}))}^{{1/2}}}.$$
(22)

Here, P are the quantum numbers of all possible states of fermions. This determinant can be represented in the form

$$Z = \tilde {C}\mathop \Pi \limits_{\{ P\} } {{({{P}^{2}})}^{2}}.$$
(23)

The most important fact that Eq. (22) dictates the method of introducing the zeta regularization.

Let \({{\lambda }_{n}}\) be the eigenvalue of the operator D, and the zeta function be defined as

$$\zeta (s) = \sum\limits_{n = 0}^\infty {{{{({{\lambda }_{n}})}}^{{ - s}}}} .$$
(24)

In our case,

$${{\lambda }_{n}} = ((2\pi T(n + 1{\text{/}}2) + i\mu {{)}^{2}} + {{k}^{2}}).$$
(25)

Expression (24) is written in a symbolic form because it is necessary to take into account the eigenvalue number density, which is specified in the continuous theory in the form

$$\frac{V}{{{{{(2\pi )}}^{3}}}}\int {{d}^{3}}k.$$
(26)

Thus,

$$\det D = \exp \left( { + 2{{{\left. {\frac{{d\zeta }}{{ds}}} \right|}}_{{s = 0}}}} \right).$$
(27)

The inclusion of the constant \(\tilde {C}\) in Eq. (23) leads to the expression

$${\text{ln}}Z = 2{{\left. {\frac{{d\zeta }}{{ds}}} \right|}_{{s = 0}}} + \tilde {C}\zeta (0).$$
(28)

Now, it is possible to calculate the zeta function for the operator beginning with the case of zero chemical potential:

$$\zeta (s) = \frac{{8\pi V}}{{{{{(2\pi )}}^{3}}}}\int {{k}^{2}}dk{{\left( {\sum\limits_{n = 0}^\infty ((2\pi (n + 1{\text{/}}{{{2))}}^{2}} + {{k}^{2}}} \right)}^{{ - s}}}.$$
(29)

The integration by parts gives

$$\begin{gathered} \zeta (s) = - \frac{{8\pi V}}{{{{{(2\pi )}}^{3}}}}{{(2\pi T)}^{{3 - 2s}}}{{\zeta }_{H}}(2s - 3,1{\text{/}}2) \\ \times \,{{(4 - 4s)}^{{ - 1}}}\frac{{\Gamma (1{\text{/}}2)\Gamma (s - 3{\text{/}}2)}}{{\Gamma (s - 1)}}. \\ \end{gathered} $$
(30)

In the limit s → 0, the Euler gamma function has a pole; therefore, ζ(0) = 0 according to Eq. (30), but

$$\zeta '(0) = - {{T}^{3}}\frac{{8{{\pi }^{2}}}}{3}{{\zeta }_{H}}( - 3,1{\text{/}}2).$$
(31)

The Hurwitz zeta function can be expressed in terms of the fourth-degree Bernoulli polynomial as

$${{\zeta }_{H}}( - 3,1{\text{/}}2) = - \frac{{{{B}_{4}}(1{\text{/}}2)}}{4} = - \frac{7}{{960}},$$
(32)

where

$${{B}_{4}}(x) = {{x}^{4}} - 2{{x}^{3}} + {{x}^{2}} - \frac{1}{{30}}.$$
(33)

Finally,

$${\text{ln}}Z{\text{/}}V = \frac{{7{{\pi }^{2}}}}{{180}}{{T}^{3}}.$$
(34)

To obtain the result in the case of a nonzero chemical potential, it is necessary to carry out the same calculations but taking into account that

$$2\sum\limits_{n = 0}^\infty {{((2\pi T(n + 1{\text{/}}{{2))}^{2}} + {{k}^{2}})}^{{ - s}}} \to {{I}_{1}} + {{\bar {I}}_{1}};$$
(35)
$${{I}_{1}} = \sum\limits_{n = 0}^\infty {{((2\pi T(n + 1{\text{/}}2) - i\mu {{)}^{2}} + {{k}^{2}})}^{{ - s}}};$$
(36)
$${{\bar {I}}_{1}} = \sum\limits_{n = 0}^\infty {{((2\pi T(n + 1{\text{/}}2) + i\mu {{)}^{2}} + {{k}^{2}})}^{{ - s}}}.$$
(37)

As a result,

$$\zeta '(0) = - {{T}^{3}}\frac{{8{{\pi }^{2}}}}{6}({{\zeta }_{H}}( - 3,\eta ) + {{\zeta }_{H}}( - 3,\bar {\eta })),$$
(38)

and, finally,

$$\ln \det D = \ln (Z){\text{/}}V = \frac{{7{{\pi }^{2}}}}{{180}}{{T}^{3}} + \frac{{{{\mu }^{4}}}}{{12T{{\pi }^{2}}}} + \frac{{{{\mu }^{2}}T}}{6}.$$
(39)

Just the calculation of the zeta function for the determinant of the operator D allows us to introduce a correct regularization for Eq. (11). Furthermore, according to this consideration, chiral effects can be calculated in effective theories without regulators if the partition function can be calculated or, equivalently, the thermodynamic description of the system can be developed.

CONCLUSIONS

To summarize, the possibility of using the zeta regularization to calculate the axial current in the chiral separation effect has been analyzed. It has been shown that divergence in the expression for the axial current can be regularized using the Hurwitz zeta function. The validity of this regularization and Eq. (11) can be verified by calculating the expression for the axial anomaly, which is related to residue at the pole of the zeta function. It has been shown that such a regularization makes it also possible to obtain the complete thermodynamic description of the system and does not require any additional assumptions. These statements in the massless case in principle reduce the problem of the existence of chiral effects to the problem of the correct determination of the thermodynamic equilibrium of the system because allow the correct calculation of the expression for the determination of the operator entering the partition function. If this expression is determined correctly, any regularization in the expression for currents is not necessary as the ultraviolet regularization is not necessary at the calculation within the terminal field theory.

According to this work, the formalism considered in [12–14] can be applied without restrictions in the continuous theory. In particular, the results obtained with the lattice regularization can be easily generalized and the absence of the equilibrium chiral magnetic effect both in the continuous four-dimensional theory and in its two-dimensional analog can be demonstrated.Footnote 4 This result is particularly interesting in the massless case because the results obtained in [6] for a system with the exact chiral symmetry that describes experimental data for thin wire contradict the results obtained in [11] by means of the lattice simulation, which leads either to the absence of the exact chiral symmetry or to the problem of duplicates [21]. The considered relation between the regularization of the expression for the chiral effect and the calculation of the determinant of the operator entering the partition function allow the assumption that the first significant signature of the absence of the effect in the system with the chiral chemical potential is the impossibility of the development of conventional thermodynamics due to divergences. In the case of an analog of the chiral magnetic effect in two dimensions, such a consideration indeed results in problems with the development of thermodynamics in the Pauli–Villars regularization [22].

One of the significant consequences of the considered method is the possibility of studying the relation between chiral effects and anomalies because anomalies are very simply calculated in theories with the zeta regularization.