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

We continue previous investigations of the (inhomogeneous) phase structure of the Gross-Neveu model in a noninteger number of spatial dimensions ($1\le d<3$) in the limit of an infinite number of fermion species ($N\to \infty$) at (non)zero chemical potential $\mu$ [L. Pannullo, Inhomogeneous condensation in the Gross-Neveu model in noninteger spatial dimensions $1\le d<3$, Phys. Rev. D 108, 036022 (2023)]. In this work, we extend the analysis from zero to nonzero temperature $T$. The phase diagram of the Gross-Neveu model in $1\le d<3$ 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 $1\le d<2$) and absence (for $2\le d<3$) 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 $d\to 3$.