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

We investigate the thermodynamics at finite density of a generalized $(2 + 1)$-dimensional 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. [16]. 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, discuss the Levinson theorem and the decomposition of the phase of the exciton correlation into a resonant and scattering part.