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

T G Khunjua,K G Klimenko,R N Zhokhov

doi:10.1103/physrevd.105.025014

Abstract

Using a nonperturbative approach based on the Cornwall-Jackiw-Tomboulis (CJT) effective action $\mathrm{\ensuremath{\Gamma}}(S)$ 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 $\mathrm{\ensuremath{\Gamma}}(S)$ 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 $G\ensuremath{\equiv}G(\mathrm{\ensuremath{\Lambda}})$ on the cutoff parameter $\mathrm{\ensuremath{\Lambda}}$, 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 $\mathcal{P}\mathcal{T}$ 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 $\mathcal{P}\mathcal{T}$ even/odd phases.

Highlights

For a long time, it was believed that to describe quantum systems it is necessary to use theories with Hermitian Hamiltonians, since in this case the energy spectrum is real
In recent decades, it has been understood that there are situations, especially in open physical systems interacting with the environment, which can be effectively considered in the framework of nonHermitian Hamiltonians
One of them is characterized by dynamical appearance of the Hermitian mass term MH in the Lagrangian, i.e., the ground state of the system remains Hermitian

Summary

INTRODUCTION

It was believed that to describe quantum systems it is necessary to use theories with Hermitian Hamiltonians (or Lagrangians), since in this case the energy spectrum is real. Nonperturbative method has emerged for calculating various multifermion Green’s functions based on functional equations of the Dyson-Schwinger type In this CJT effective action approach it is possible to investigate the possibility of dynamical generation of the fermion mass and chiral symmetry breaking, etc., as it was demonstrated, e.g., in the framework of the (1 þ 1)-D GN model in Ref. In the present paper, we show that non-Hermiticity can arise spontaneously in the (2 þ 1)-D GN model under consideration just in the framework of the CJT composite operator approach It means that for a certain well-defined behavior of the bare coupling constant, the ground state of the system can be characterized by a dynamically arising non-Hermitian mass term of the Lagrangian, which can be both PT - and anti-PT symmetric. The mass term MNH1 is PT even (symmetric), but the non-Hermitian mass term MNH2 is PT odd, i.e., it changes the sign under this transformation

CJT effective action of the model

Dynamical generation of the non-Hermitian mass terms MNH1 and MNH2

Conclusions

THE CASE OF NONZERO HERMITIAN BARE m5 MASS

SUMMARY AND CONCLUSIONS