Detecting inhomogeneous chiral condensation from the bosonic two-point function in the $(1 + 1)$-dimensional Gross-Neveu model in the mean-field approximation

TL;DR

This paper reanalyzes the phase diagram of the (1+1)-dimensional Gross-Neveu model, detecting inhomogeneous chiral condensation via bosonic two-point functions and stability analysis, confirming the existence of spatially varying condensates.

Contribution

It demonstrates how to accurately locate second-order phase transition lines to inhomogeneous phases using stability analysis of the homogeneous phase within the mean-field approximation.

Findings

Detection of inhomogeneous phase via bosonic two-point function

Confirmation of second-order phase transition lines without condensation

Identification of a moat regime with negative wave function renormalization

Abstract

The phase diagram of the $(1 + 1)$-dimensional Gross-Neveu model is reanalyzed for (non-)zero chemical potential and (non-)zero temperature within the mean-field approximation. By investigating the momentum dependence of the bosonic two-point function, the well-known second-order phase transition from the $\mathbb{Z}_2$ symmetric phase to the so-called inhomogeneous phase is detected. In the latter phase the chiral condensate is periodically varying in space and translational invariance is broken. This work is a proof of concept study that confirms that it is possible to correctly localize second-order phase transition lines between phases without condensation and phases of spatially inhomogeneous condensation via a stability analysis of the homogeneous phase. To complement other works relying on this technique, the stability analysis is explained in detail and its limitations and successes are discussed in context of the Gross-Neveu model. Additionally, we present explicit results for the bosonic wave-function renormalization in the mean-field approximation, which is extracted analytically from the bosonic two-point function. We find regions -- a so-called moat regime -- where the wave function renormalization is negative accompanying the inhomogeneous phase as expected.