# Random matrix approach to three-dimensional QCD with a Chern-Simons term

## TL;DR

This paper develops a random matrix model for three-dimensional QCD with a Chern-Simons term, deriving symmetry breaking patterns and spectral properties, providing universal predictions for the Dirac operator's spectral density in this context.

## Contribution

It introduces a novel random matrix model incorporating a Chern-Simons term, revealing symmetry breaking and spectral characteristics specific to this setting.

## Key findings

- Derived the symmetry breaking pattern U(2N_f) to U(N_f+k)×U(N_f−k).
- Obtained explicit spectral density and correlation functions for the Dirac operator.
- Predicted universal spectral oscillations that cancel phase contributions in the model.

## Abstract

We propose a random matrix theory for QCD in three dimensions with a Chern-Simons term at level $k$ which spontaneously breaks the flavor symmetry according to U($2N_{\rm f}$) $\to $ U($N_{\rm f}+k$)$\times$U($N_{\rm f}-k$). This random matrix model is obtained by adding a complex part to the action for the $k=0$ random matrix model. We derive the pattern of spontaneous symmetry breaking from the analytical solution of the model. Additionally, we obtain explicit analytical results for the spectral density and the spectral correlation functions for the Dirac operator at finite matrix dimension, that become complex. In the microscopic domain where the matrix size tends to infinity, they are expected to be universal, and give an exact analytical prediction to the spectral properties of the Dirac operator in the presence of a Chern-Simons term. Here, we calculate the microscopic spectral density. It shows exponentially large (complex) oscillations which cancel the phase of the $k=0$ theory.