Spontaneous non-Hermiticity in the (2+1)-dimensional Gross-Neveu model

Using a nonperturbative approach based on the Cornwall-Jackiw-Tomboulis effective action $\Gamma(S)$ for composite operators ($S$ is the full fermion propagator), the phase structure of the simplest massless (2 + 1)-dimensional Gross-Neveu model is investigated. We have calculated $\Gamma(S)$ and its stationary (or Dyson-Schwinger) equation in the first order of the bare coupling constant $G$ and have shown that there exist a well-defined dependence of $G\equiv G(\Lambda)$ on the cutoff parameter $\Lambda$, such that the Dyson-Schwinger equation is renormalized. It has three different solutions for fermion propagator $S$ corresponding to possible dynamical appearance of three different mass terms in the model. One is a Hermitian, but two others are non-Hermitian and $\cP\cT$ even or odd. It means that two phases with spontaneous non-Hermiticity can be emerged in the system. Moreover, mass spectrum of quasiparticles is real in these non-Hermitian and $\cP\cT$ even/odd phases.