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

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 $d \to 3$.