Chiral Modulations in Curved Space I: Formalism

TL;DR

This paper develops a formalism for analyzing four-fermion theories at finite temperature and density in curved space, enabling the study of inhomogeneous phases beyond traditional approximations.

Contribution

It introduces a non-perturbative heat-kernel ansatz and a series representation for the effective action, extending analysis capabilities to curved spacetimes with arbitrary chemical potentials.

Findings

Formalism supports inhomogeneous and anisotropic phases.

Series representation valid under certain chemical potential constraints.

Illustration with static Einstein spaces at zero chemical potential.

Abstract

The goal of this paper is to present a formalism that allows to handle four-fermion effective theories at finite temperature and density in curved space. The formalism is based on the use of the effective action and zeta function regularization, supports the inclusion of inhomogeneous and anisotropic phases. One of the key points of the method is the use of a non-perturbative ansatz for the heat-kernel that returns the effective action in partially resummed form, providing a way to go beyond the approximations based on the Ginzburg-Landau expansion for the partition function. The effective action for the case of ultra-static Riemannian spacetimes with compact spatial section is discussed in general and a series representation, valid when the chemical potential satisfies a certain constraint, is derived. To see the formalism at work, we consider the case of static Einstein spaces at zero chemical potential. Although in this case we expect inhomogeneous phases to occur only as meta-stable states, the problem is complex enough and allows to illustrate how to implement numerical studies of inhomogeneous phases in curved space. Finally, we extend the formalism to include arbitrary chemical potentials and obtain the analytical continuation of the effective action in curved space.