# Nambu-Jona Lasinio and Nonlinear Sigma Models in Condensed Matter Systems

## TL;DR

This review explores the connections between Nambu-Jona Lasinio and nonlinear sigma models in condensed matter, highlighting dualities, inhomogeneous condensates, and recent progress in finite systems.

## Contribution

It introduces a duality framework between the Nambu-Jona Lasinio and nonlinear sigma models, revealing new inhomogeneous solutions and phase structures in condensed matter systems.

## Key findings

- Identification of duality between models
- Discovery of new inhomogeneous solutions
- Analysis of inhomogeneous phases in finite systems

## Abstract

We review various connections between condensed matter systems with the Nambu-Jona Lasinio model and nonlinear sigma models. The field theoretical description of interacting systems offers a systematic framework to describe the dynamical generation of condensates. Resent findings of a duality between the Nambu-Jona Lasinio model and the nonlinear sigma model enables us to investigate various properties underlying both theories. In this review we mainly focus on inhomogeneous condensations in static situations. The various methods developed in the Nambu-Jona Lasinio model reveal the inhomogeneous phase structures and also yield new inhomogeneous solutions in the nonlinear sigma model owing to the duality. The recent progress on interacting systems in finite systems is also reviewed.