Chiral magnetic effect at finite temperature in a field-theoretic approach

We investigate the existence (or lack thereof) of the chiral magnetic effect (CME) in the framework of finite temperature field theory, studied through the path integral approach and regularized via the zeta function technique. We show that, independently of the temperature, gauge invariance implies the absence of the effect, a fact proved, at zero temperature and in a Hamiltonian approach, by Yamamoto. Indeed, the effect only appears when the manifold is finite in the direction of the magnetic field and gauge-invariance breaking boundary conditions are imposed. We present an explicit calculation for antiperiodic and periodic boundary conditions, which do allow for a CME, since only 'large' gauge transformations are, then, an invariance of the theory. In both cases, the associated current does depend on the temperature, a well as on the size of the sample in the direction of the magnetic field, even for a temperature-independent chiral chemical potential. In particular, for antiperiodic boundary conditions, the value of this current only agrees with the result usually quoted in the literature on the subject in the zero-temperature limit, while it decreases with the temperature in a well-determined way.