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

TL;DR

This paper investigates the phase structure of the (2+1)-dimensional Thirring model, revealing spontaneous non-Hermiticity with real quasiparticle spectra and identifying both Hermitian and non-Hermitian phases, including in the massive case.

Contribution

It demonstrates the spontaneous emergence of non-Hermitian mass terms in the Thirring model, extending previous findings to the massive case and analyzing their spectral properties.

Findings

Existence of non-Hermitian solutions with real spectra.

Identification of both PT-symmetric and non-symmetric phases.

Spontaneous non-Hermiticity observed even in the massive model.

Abstract

Using the Cornwall-Jackiw-Tomboulis effective action $\Gamma(S)$ 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 $\Gamma(S)$ 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 exist a well-defined dependence of $G\equiv G(\Lambda)$ on the cutoff parameter $\Lambda$ under which the HF equation is renormalized. In general, it has two sets, (i) and (ii), of solutions for 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-Hermitiam 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 $\cP\cT$ symmetrical and non-symmetrical phases. Moreover, in contrast with previous investigations of this effect in other models, we have observed the spontaneous non-Hermiticity phenomenon also in the {\it massive} (2+1)-dimensional Thirring model.