Overlap Dirac Operator at Nonzero Chemical Potential and Random Matrix Theory

We show how to introduce a quark chemical potential in the overlap Dirac operator. The resulting operator satisfies a Ginsparg-Wilson relation and has exact zero modes. It is no longer ${\gamma }_5$ Hermitian, but its nonreal eigenvalues still occur in pairs. We compute the spectral density of the operator on the lattice and show that, for small eigenvalues, the data agree with analytical predictions of non-Hermitian chiral random matrix theory for both trivial and nontrivial topology. We also explain an observed change in the number of zero modes as a function of chemical potential.

Figure 1

Spectrum of $D_{\mathrm{ov}}(\mu )$ for $\mu =0$ and $\mu =0.3$ for a typical configuration. The figure on the right is a magnification of the region near zero.

Figure 2

Density of the small eigenvalues of $D_{\mathrm{ov}}(\mu )$ in the complex plane [after projecting $\lambda \to 2\lambda /(2-\lambda )$ and rescaling by $V\Sigma$] for (from left to right) $\mu =0.1$, 0.2, 0.3, 1.0. The histograms are lattice data for $\nu =0$ and $\nu =1$, and the solid lines are the corresponding RMT prediction of Eq. (8), integrated over the bin size. Top: cut along the imaginary axis, middle: cut along the real axis, bottom: cut parallel to the real axis at $y_c$. Vertical lines indicate the fit interval. For $\mu =1.0$, the eigenvalues are rescaled by $V\Sigma /2\sqrt{\alpha }$, the fit is a one-parameter fit to Eq. (10), and in the bottom plot the data are integrated over the phase.