# QCD in One Dimension at Nonzero Chemical Potential

## TL;DR

This paper calculates the phase factor expectation value in 1D lattice QCD with nonzero chemical potential, revealing connections to chiral symmetry, random matrix theory, and a novel understanding of the chiral condensate discontinuity.

## Contribution

It applies a recent integration formula to analyze 1D lattice QCD at nonzero chemical potential, linking it to three-dimensional QCD and random matrix theory, and introduces an alternative to the Banks-Casher formula.

## Key findings

- Chemical potential absorbed into quark masses
- Eigenvalues follow 3D QCD random matrix theory
- Discontinuity explained by new condensate formula

## Abstract

Using an integration formula recently derived by Conrey, Farmer and Zirnbauer, we calculate the expectation value of the phase factor of the fermion determinant for the staggered lattice QCD action in one dimension. We show that the chemical potential can be absorbed into the quark masses; the theory is in the same chiral symmetry class as QCD in three dimensions at zero chemical potential. In the limit of a large number of colors and fixed number of lattice points, chiral symmetry is broken spontaneously, and our results are in agreement with expressions based on a chiral Lagrangian. In this limit, the eigenvalues of the Dirac operator are correlated according to random matrix theory for QCD in three dimensions. The discontinuity of the chiral condensate is due to an alternative to the Banks-Casher formula recently discovered for QCD in four dimensions at nonzero chemical potential. The effect of temperature on the average phase factor is discussed in a schematic random matrix model.