Competition and duality correspondence between inhomogeneous fermion-antifermion and fermion-fermion condensations in the NJL$_2$ model

TL;DR

This paper explores the phase structure of a (1+1)-dimensional NJL model with chiral and superconducting condensates, revealing a duality correspondence between fermion-antifermion and fermion-fermion pairings at various chemical potentials and temperatures.

Contribution

It demonstrates a duality between chiral symmetry breaking and superconducting phases in the NJL$_2$ model, including inhomogeneous condensate configurations, at nonzero temperature and chemical potentials.

Findings

Phase structure exhibits duality between CSB and SC phases.

Inhomogeneous condensates enable degeneracy between phases.

Self-duality occurs when coupling constants are equal.

Abstract

We investigate the possibility of spatially homogeneous and inhomogeneous chiral fermion-antifermion condensation and superconducting fermion-fermion pairing in the (1+1)-dimensional model by Chodos {\it et al.} [ Phys. Rev. D 61, 045011 (2000)] generalized to continuous chiral invariance. The consideration is performed at nonzero values of temperature $T$, electric charge chemical potential $\mu$ and chiral charge chemical potential $\mu_5$. It is shown that at $G_1<G_2$, where $G_1$ and $G_2$ are the coupling constants in the fermion-antifermion and fermion-fermion channels, the $(\mu,\mu_5)$-phase structure of the model is in a one-to-one correspondence with the phase structure at $G_1>G_2$ (called duality correspondence). Under the duality transformation the (inhomogeneous) chiral symmetry breaking (CSB) phase is mapped into the (inhomogeneous) superconducting (SC) phase and vice versa. If $G_1=G_2$, then the phase structure of the model is self-dual. Nevertheless, the degeneracy between the CSB and SC phases is possible in this case only when there is a spatial inhomogeneity of condensates.