Sign of Wilson-Fermion Determinant using Contour-Integral Spectral Projection

Abstract

Key properties of a large sparse operator can often be determined by only a small, spectrally localized subset of its eigenmodes. In lattice QCD with Wilson fermions, the sign of the fermion determinant is determined by the parity of the number of eigenvalues lying on the negative real axis of the non-Hermitian Wilson-Dirac operator. Determination of this signature can be challenging, particularly in parameter regimes where near-zero modes can lead to exceptional configurations. We present a contour-integral-based spectral projection method to count the negative-real eigenvalues of the non-Hermitian Wilson-Dirac operator $D$ that determine the sign of its determinant. The spectral projection is supplemented by an adapted deflation-accelerated solver for shifted linear systems and singular-value exclusions to reliably set spectral bounds and contour boundaries. We demonstrate the robustness of the method through systematic convergence tests, including variations of the contour size, and show its ability to resolve eigenvalues close to the negative real axis. Beyond fermion determinant sign evaluation, the approach provides a systematic means of isolating physically relevant eigenvalues embedded in dense complex spectra, with potential applications to non-Hermitian quantum systems, stability analyses, and large-scale eigenvalue problems.