# Inhomogeneous charged pion condensation in chiral asymmetric dense quark matter in the framework of NJL$_2$ model

## TL;DR

This study explores the phase structure of a (1+1)-dimensional quark model with four-quark interactions, revealing inhomogeneous charged pion condensation induced by chiral isospin chemical potential, with a duality between chiral symmetry breaking and pion condensation.

## Contribution

It demonstrates the existence of inhomogeneous charged pion condensation in a (1+1)-D quark model with chemical potentials, highlighting a duality and the influence of chiral isospin chemical potential.

## Key findings

- Inhomogeneous charged pion condensate occurs at small chiral isospin chemical potential.
- Duality exists between chiral symmetry breaking and pion condensation phases.
- Charged pion condensation is induced over a large parameter region.

## Abstract

In this paper we investigate the phase structure of a (1+1)-dimensional quark model with four-quark interaction and in the presence of baryon ($\mu_B$), isospin ($\mu_I$) and chiral isospin ($\mu_{I5}$) chemical potentials. Spatially inhomogeneous chiral density wave (for chiral condensate) and single wave (for charged pion condensate) approaches are used. It is established that in the large-$N_c$ limit ($N_c$ is the number of colored quarks) there exists a duality correspondence between the chiral symmetry breaking phase and the charged pion condensation (PC) one. Moreover, it is shown that inhomogeneous charged PC phase with nonzero baryon density is induced in the model by arbitrary small values of the chemical potential $\mu_{I5}$ (for a rather large region of $\mu_B$ and $\mu_I$).