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

Abstract

Using the Cornwall-Jackiw-Tomboulis effective action $\mathrm{\ensuremath{\Gamma}}(S)$ for composite operators ($S$ is the full fermion propagator), the phase structure of the massless ($2+1$)-dimensional Thirring model with four-component spinors is investigated in the Hartree-Fock (HF) approximation. In this case both $\mathrm{\ensuremath{\Gamma}}(S)$ and its stationary (or HF) equation for the full fermion propagator $S$ are calculated in the first order of the bare coupling constant $G$. We have shown that there exists a well-defined dependence of $G\ensuremath{\equiv}G(\mathrm{\ensuremath{\Lambda}})$ on the cutoff parameter $\mathrm{\ensuremath{\Lambda}}$ under which the HF equation is renormalized. In general, it has two sets, (i) and (ii), of solutions for the fermion propagator corresponding to dynamical appearance of different mass terms in the model. In the case of set (i) the mass terms are Hermitian, but the solutions from the set (ii) correspond to a dynamical generation of the non-Hermitian mass terms, i.e., to a spontaneous non-Hermiticity of the Thirring model. Despite this, the mass spectrum of the quasiparticle excitations of all non-Hermitian ground states is real. In addition, among these non-Hermitian phases there are both $\mathcal{P}\mathcal{T}$ symmetrical and nonsymmetrical phases. Moreover, in contrast with previous investigations of this effect in other models, we have observed the spontaneous non-Hermiticity phenomenon also in the massive ($2+1$)-dimensional Thirring model.