Inhomogeneous condensation in the Gross-Neveu model in noninteger spatial dimensions $1 \leq d < 3$. II. Nonzero temperature and chemical potential

TL;DR

This paper investigates the phase structure of the Gross-Neveu model in noninteger spatial dimensions between 1 and 3, focusing on inhomogeneous phases at nonzero temperature and chemical potential, and analyzes the effects of renormalization.

Contribution

It extends the analysis of inhomogeneous phases in the Gross-Neveu model to noninteger dimensions with nonzero temperature, clarifying the conditions for their existence and the impact of renormalization.

Findings

Inhomogeneous phases exist for 1 ≤ d < 2 but vanish for 2 ≤ d < 3 after proper renormalization.

The stability analysis reveals the presence or absence of inhomogeneous phases depending on the dimension and regulator.

The study discusses the implications of spatial dimension and regulator effects on the phase diagram and the limit as d approaches 3.

Abstract

We continue previous investigations of the (inhomogeneous) phase structure of the Gross-Neveu model in a noninteger number of spatial dimensions ($1 \leq d < 3$) in the limit of an infinite number of fermion species ($N \to \infty$) at (non)zero chemical potential $\mu$. In this work, we extend the analysis from zero to nonzero temperature $T$. The phase diagram of the Gross-Neveu model in $1 \leq d < 3$ spatial dimensions is well known under the assumption of spatially homogeneous condensation with both a symmetry broken and a symmetric phase present for all spatial dimensions. In $d = 1$ one additionally finds an inhomogeneous phase, where the order parameter, the condensate, is varying in space. Similarly, phases of spatially varying condensates are also found in the Gross-Neveu model in $d = 2$ and $d = 3$, as long as the theory is not fully renormalized, i.e., in the presence of a regulator. For $d = 2$, one observes that the inhomogeneous phase vanishes, when the regulator is properly removed (which is not possible for $d = 3$ without introducing additional parameters). In the present work, we use the stability analysis of the symmetric phase to study the presence (for $1 \leq d < 2$) and absence (for $2 \leq d < 3$) of these inhomogeneous phases and the related moat regimes in the fully renormalized Gross-Neveu model in the $\mu, T$-plane. We also discuss the relation between "the number of spatial dimensions" and "studying the model with a finite regulator" as well as the possible consequences for the limit $d \to 3$.