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. © 2019 The Author(s) 2019. Published by Oxford University Press on behalf of the Physical Society of Japan.