Thermodynamics of a generalized graphene-motivated (2+1) D Gross–Neveu model beyond the mean field within the Beth–Uhlenbeck approach

We investigate the thermodynamics at finite density of a generalized $(2 + 1)$D Gross–Neveu model of $N$ fermion species with various types of four-fermion interactions. The motivation for considering such a generalized schematic model arises from taking the Fierz transformation of an effective Coulomb current–current interaction and certain symmetry-breaking interaction terms, as considered for graphene-type models in Ref. [29]. We then apply path-integral bosonization techniques, based on the large-$N$ limit, to derive the thermodynamic potential. This includes the leading-order mean-field (saddle point) contribution as well as the next-order contribution of Gaussian fluctuations of exciton fields. The main focus of the paper is then the investigation of the thermodynamic properties of the resulting fermion–exciton plasma. In particular, we derive an extended Beth–Uhlenbeck form of the thermodynamic potential, and discuss the Levinson theorem and the decomposition of the phase of the exciton correlation into resonant and scattering parts.