# Affinity of NJL$_2$ and NJL$_{4}$ model results on duality and pion condensation in chiral asymmetric dense quark matter

## TL;DR

This paper explores how chemical potentials influence phase transitions in (1+1) and (3+1)-dimensional quark models, revealing a duality between chiral symmetry breaking and pion condensation, with consistent results across models.

## Contribution

It demonstrates the duality between chiral symmetry breaking and pion condensation in both models, confirming the robustness of these phenomena in dense quark matter.

## Key findings

- Chemical potential $$ promotes charged pion condensation with nonzero baryon density.
- Qualitative similarity of results in both models supports the validity of predictions.
- Existence of a duality correspondence in the large-$N_c$ limit between phases.

## Abstract

In this paper we investigate the phase structure of a (1+1) and (3+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. It is shown that the chemical potential $\mu_{I5}$ promotes the appearance of the charged PC phase with nonzero baryon density. Results of both models are qualitatively the same, this fact enhances one's confidence in %the faith in the obtained predictions. 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 one.