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We investigate the thermodynamics at finite density of a generalized $(2 +1)$-dimensional Gross-Neveu model of $N$ fermion species with various types offour-fermion interactions. The motivation for considering such a generalizedschematic model arises from taking the Fierz-transformation of an effectiveCoulomb current-current interaction and certain symmetry breaking interactionterms, as considered for graphene-type models in Ref. [16]. We then applypath-integral bosonization techniques, based on the large $N$ limit, to derivethe thermodynamic potential. This includes the leading order mean-field (saddlepoint) contribution as well as the next-order contribution of Gaussianfluctuations of exciton fields. The main focus of the paper is then theinvestigation of the thermodynamic properties of the resulting fermion-excitonplasma. In particular, we derive an extended Beth-Uhlenbeck form of thethermodynamic potential, discuss the Levinson theorem and the decomposition ofthe phase of the exciton correlation into a resonant and scattering part.

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