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

Technical Summary: Sign of Wilson-Fermion Determinant using Contour-Integral Spectral Projection

Problem Statement
In lattice Quantum Chromodynamics (QCD) utilizing Wilson fermions, the sign of the fermion determinant, $\text{sign}(\det D)$, is a critical quantity. While $\gamma_5$-Hermiticity ensures the determinant is real, it is not guaranteed to be positive for single-flavor or mass-nondegenerate Wilson fermions. The sign is exclusively determined by the parity of $N_-$, the number of eigenvalues of the non-Hermitian Wilson-Dirac operator ($D$) lying on the negative real axis. Determining this signature is computationally challenging, particularly in parameter regimes featuring near-zero modes or "exceptional configurations" where the operator is poorly conditioned. Existing methods, such as the spectral flow method (tracking eigenvalues of the Hermitian operator $Q=\gamma_5 D$ as a function of mass) or direct iterative eigensolvers on the non-Hermitian matrix, often require resolving broad low-lying spectral regions or rely on polynomial/rational approximations that necessitate reweighting factors ($W_{\text{sign}}$) to correct accumulated errors.

Methodology
The authors propose a targeted spectral projection method based on contour integrals to count negative-real eigenvalues directly, bypassing the need to resolve the entire low-lying spectrum. The core components of the methodology are:

1. Contour-Integral Spectral Projection:
   The method utilizes a contour $\Gamma$ in the complex Argand plane, specifically an ellipse symmetric about the real axis, to isolate the spectral region of interest. The spectral projector is defined as:
   $$P_\Gamma = \frac{1}{2\pi i} \oint_\Gamma (zI - A)^{-1} dz$$
   where $A$ is the non-Hermitian Wilson-Dirac operator. This integral is approximated numerically using a finite quadrature rule with $N_q$ nodes ($z_j$) and weights ($\omega_j$). The resulting approximate projector $\hat{P}_\Gamma$ maps the operator onto a reduced subspace.

2. Exploitation of $\gamma_5$-Hermiticity:
   Due to the $\gamma_5$-Hermiticity of the Wilson-Dirac operator, the left and right eigensystems are related ($r_k^\dagger \gamma_5$ acts as the left eigenvector for $\lambda_k^*$). Consequently, the method only requires solving for the right projected subspace, avoiding the independent determination of the left eigensystem.

3. Shifted Linear Systems and Solver Adaptation:
   The projection involves solving shifted linear systems $(z_j I - \hat{D})q_j = Y$ at each quadrature node, where $\hat{D}$ is the even Schur complement of $D$. The authors adapt the DFL-SAP-GCR solver (originally designed for the full operator $D$) to act exclusively on the shifted even Schur complement $\hat{D} - zI$. This adaptation is crucial because the Schur complement of the shifted full operator does not equal the shifted Schur complement of the operator.

4. Singular-Value Exclusion for Spectral Bounds:
   A critical challenge is ensuring the contour's left boundary ($X_L$) encompasses all negative real eigenvalues without being excessively large (which would increase computational cost). The authors introduce a pragmatic strategy using singular-value estimates. By evaluating the smallest singular value $\sigma_{\min}$ of shifted systems $\hat{D} - s_i I$ at a sequence of points $s_i$, they identify exclusion intervals free of eigenvalues. This allows for a conservative yet efficient determination of $X_L$, avoiding the over-conservative bounds derived solely from the Rayleigh quotient of the Hermitian part.

5. Subspace Iteration:
   The FEAST package is employed to realize the trial subspace vectors. The dimension of the trial subspace ($m$) is chosen to exceed the expected number of enclosed eigenvalues. The method demonstrates rapid convergence, typically resolving all negative-real modes within two FEAST iterations.

Key Results
The method was tested on a diverse set of configurations from lattice QCD ensembles generated by the CLS consortium. Key findings include:

• Robustness and Accuracy: The procedure successfully reproduces all negative real solutions of $\hat{D}$ up to machine precision across various configurations, including those with no negative solutions, shallow negative solutions, deep negative solutions, and cases with multiple solutions differing by orders of magnitude.
• Parameter Efficiency: For most configurations, a minimal quadrature node count of $N_q = 8$ and a shifted-solver tolerance of $10^{-3}$ were sufficient. All negative-real eigenvalues were resolved after the second FEAST iteration.
• Handling Extreme Cases: In configurations with eigenvalues spanning two orders of magnitude (e.g., configuration C102r005n56), increasing $N_q$ to 12 and tightening the solver tolerance to $10^{-5}$ allowed for faithful capture of all modes without significant computational overhead.
• Computational Cost: The dominant cost arises from solving the shifted linear systems. The overhead associated with selecting the contour and quadrature nodes is less than a few percent of the total evaluation. The singular-value exclusion strategy effectively minimizes the integration domain, reducing the number of required quadrature nodes compared to using a bound derived from the Hermitian part alone.

Significance and Claims
The paper claims that this contour-integral-based spectral projection offers a systematic and efficient means to determine the fermion determinant sign without the computational overhead of resolving unwanted physical modes or relying on broad spectral flow analysis. By restricting the eigensolver to the exact domain governing the determinant's signature, the method provides a robust alternative to traditional iterative eigensolvers supplemented by polynomial transformations.

Beyond the specific application of evaluating $W_{\text{sign}}$ in lattice QCD, the authors suggest the approach provides a general strategy for isolating physically relevant eigenvalues embedded in dense complex spectra. This has potential applications in non-Hermitian quantum systems, stability analyses, and large-scale eigenvalue problems, offering a targeted filter for specific subsets of a spectrum rather than a broad search. The work emphasizes that reliable determination of the determinant sign is essential for correcting rational-approximation errors in simulations (via reweighting) or for validating the positivity of the measure in single-flavor scenarios.