Excitonic insulators and Gross-Neveu models

We introduce a generalized Gross-Neveu (GN) model to describe the excitonic instabilities in two different systems: a small overlap semimetal (SM) and a small gap semiconductor (SMC), both in two (2d) and three-dimensions (3d). We identify the excitonic order parameter (EOP) and obtain the effective potential within the large N limit approach where the GN model can be exactly solved. We obtain the excitonic insulator (EI) phase diagrams as a function of temperature, chemical potential, overlap between bands, and gaps of the system. We show that the EI may undergo first- or second-order thermal transitions depending on the regime whereupon this phase is approached. We also investigate the expected thermodynamic signatures for the specific heat above the fine-tuned excitonic quantum critical point (EQCP), in both 2d and 3d, in the SMC regime. We show that the EQCP is a different kind of critical point since although the EOP vanishes at the EQCP, there is always a finite gap in the SMC regime. We find that for high temperatures, the specific heat might exhibit a scaling behavior in the form C-V /T proportional to T(d-z)/z, where d is the dimension of the system and z is the dynamical critical exponent. The very low-temperature behavior has a dominant exponential thermally activated term due to the presence of a gap that does not vanish at the excitonic transition.