Distributions of individual Dirac eigenvalues for QCD at non-zero chemical potential: RMT predictions and lattice results

TL;DR

This paper derives the distributions of individual Dirac eigenvalues in QCD at non-zero chemical potential using RMT, distinguishes two regimes based on a parameter, and confirms predictions with lattice data.

Contribution

It introduces a method to order individual eigenvalues in the complex plane and derives their distributions from RMT for different parameter regimes, validated by lattice results.

Findings

Excellent agreement between RMT predictions and lattice data in multiple topological sectors.

Effective approximation of eigenvalue distributions using a few terms in the Fredholm determinant expansion for small lpha.

Exact spectral correlation results obtained for large lpha.

Abstract

For QCD at non-zero chemical potential $\mu$, the Dirac eigenvalues are scattered in the complex plane. We define a notion of ordering for individual eigenvalues in this case and derive the distributions of individual eigenvalues from random matrix theory (RMT). We distinguish two cases depending on the parameter $\alpha=\mu^2 F^2 V$, where $V$ is the volume and $F$ is the familiar low-energy constant of chiral perturbation theory. For small $\alpha$, we use a Fredholm determinant expansion and observe that already the first few terms give an excellent approximation. For large $\alpha$, all spectral correlations are rotationally invariant, and exact results can be derived. We compare the RMT predictions to lattice data and in both cases find excellent agreement in the topological sectors $\nu=0,1,2$.