# Charged pion condensation and duality in dense and hot chirally and isospin asymmetric quark matter in the framework of NJL$_2$ model

## TL;DR

This paper explores how charged pion condensation occurs in dense, hot, and asymmetric quark matter within a 1+1 dimensional NJL model, revealing duality between chiral symmetry breaking and pion condensation across various conditions.

## Contribution

It demonstrates the emergence of charged pion condensation induced by chiral isospin chemical potential and establishes a duality between chiral symmetry breaking and pion condensation phenomena.

## Key findings

- Charged pion condensation occurs in dense, hot, and asymmetric quark matter.
- Duality exists between chiral symmetry breaking and pion condensation at all temperatures.
- Charged pion condensation can be induced at high temperatures, relevant for astrophysical and collider environments.

## Abstract

In this paper we investigate in the large-$N_c$ limit ($N_c$ is the number of colored quarks) the phase structure of a massless (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 as well as at nonzero temperature. It is established that chiral isospin chemical potential leads to the generation of charged pion condensation (PC) in dense (nonzero baryon density) and chiral asymmetric quark matter for a wide range of isospin densities. It is shown that there exists a duality correspondence between the chiral symmetry breaking and the charged PC phenomena at any values of temperature even for very hot quark gluon plasma. Moreover, it is shown that charged PC phase with nonzero baryon density can be induced in the model at comparatively high temperatures. This opens up new possible physical systems, where it can be of importance, such as heavy ion collisions, just born neutron stars (proto-neutron stars), supernovas as well as neutron star mergers.