The QCD vacuum as a disordered medium: A simplified model for the QCD Dirac operator

TL;DR

This paper models the QCD Dirac operator using a power-law random banded matrix to replicate spectral correlations, showing good agreement with instanton liquid models and chiral perturbation theory, especially regarding the Thouless energy scaling.

Contribution

It introduces a simplified RBM model for the QCD Dirac operator that captures spectral correlations and scaling behaviors observed in more complex models and theories.

Findings

Spectral correlations of the RBM match those of the ILM beyond the Thouless energy.

The Thouless energy scales with the square root of system size in the bulk, consistent with ILM and chiral perturbation theory.

Near the origin, the Thouless energy scaling agrees with chiral perturbation theory but not with ILM.

Abstract

We model the QCD Dirac operator as a power-law random banded matrix (RBM) with the appropriate chiral symmetry. Our motivation is the form of the Dirac operator in a basis of instantonic zero modes with a corresponding gauge background of instantons. We compare the spectral correlations of this model to those of an instanton liquid model (ILM) and find agreement well beyond the Thouless energy. In the bulk of the spectrum the (dimensionless) Thouless energy of the RBM scales with the square root of system size in agreement with the ILM and chiral perturbation theory. Near the origin the scaling of the (dimensionless) Thouless energy in the RBM remains the same as in the bulk which agrees with chiral perturbation theory but not with the ILM. Finally we discuss how this RBM should be modified in order to describe the spectral correlations of the QCD Dirac operator at the finite temperature chiral restoration transition.