Complex spectrum of finite-density lattice QCD with static quarks at strong coupling

TL;DR

This paper analyzes the eigenvalue spectrum of transfer matrices in finite-density lattice QCD with static quarks, revealing complex eigenvalues linked to the sign problem and particle-hole symmetry, with implications for lattice simulations.

Contribution

It provides a detailed calculation of the transfer matrix spectrum in finite-density lattice QCD, highlighting the conditions for real and complex eigenvalues and their physical implications, especially at heavy quark masses.

Findings

Complex eigenvalues indicate the sign problem in finite-density QCD.

Eigenvalues are real or form complex pairs depending on chemical potential and particle-hole symmetry.

Spectrum features can be tested in lattice simulations and confirm previous PNJL model results.

Abstract

We calculate the spectrum of transfer matrix eigenvalues associated with Polyakov loops in finite-density lattice QCD with static quarks. These eigenvalues determine the spatial behavior of Polyakov loop correlations functions. Our results are valid for all values of the gauge coupling in $1+1$ dimensions, and valid in the strong-coupling region for any number of dimensions. When the quark chemical potential $\mu$ is nonzero, the spatial transfer matrix $T_s$ is non-Hermitian. The appearance of complex eigenvalues in $T_s$ is a manifestation of the sign problem in finite-density QCD. The invariance of finite-density QCD under the combined action of charge conjugation $\mathcal{C}$ and complex conjugation $\mathcal{K}$ implies that the eigenvalues of $T_s$ are either real or part of a complex pair. Calculation of the spectrum confirms the existence of complex pairs in much of the temperature-chemical potential plane. Many features of the spectrum for static quarks are determined by a particle-hole symmetry. For $\mu$ small compared to the quark mass $M$, we typically find real eigenvalues for the lowest lying states. At somewhat larger values of $\mu,$ pairs of eigenvalues may form complex-conjugate pairs, leading to damped oscillatory behavior in Polyakov loop correlation functions. However, near $\mu=M$, the low-lying spectrum becomes real again. This is a direct consequence of the approximate particle-hole symmetry at $\mu=M$ for heavy quarks. This behavior of the eigenvalues should be observable in lattice simulations and can be used as a test of lattice algorithms. Our results provide independent confirmation of results we have previously obtained in PNJL models using complex saddle points.