Diagonal Kenney-Laub Rational Approximation to the Overlap Operator using Wilson and Brillouin Kernel

Technical Summary: Diagonal Kenney-Laub Rational Approximation to the Overlap Operator

Problem Statement
The overlap Dirac operator is a lattice discretization of the QCD Dirac operator that preserves chiral symmetry properties of the continuum theory, adhering closely to the Ginsparg-Wilson (GW) relation. However, its practical implementation faces significant computational challenges, primarily due to the need to approximate the matrix sign function, $\text{sgn}[X]$, where $X$ is a Hermitian kernel operator. Traditional approaches, such as Chebyshev polynomial approximations or single-fraction rational approximations, often suffer from high computational costs, require prior spectral information (eigenvalue estimates) for tuning, or exhibit oscillatory convergence behavior that complicates the selection of approximation orders. Furthermore, the choice of the kernel operator (e.g., Wilson vs. improved kernels) significantly impacts the condition number of $X$ and, consequently, the efficiency of the sign function approximation.

Methodology
The authors propose a formulation of the overlap operator that integrates three specific components:

1. Diagonal Kenney-Laub (KL) Iterates: Instead of standard polynomial approximations, the authors employ diagonal $(n, n)$ KL iterates, a globally convergent rational iterative scheme for approximating the matrix sign function. A key feature of this approach is that it does not require prior estimates of the extreme eigenvalues of the kernel operator.
2. Partial Fraction (PF) Decomposition: The diagonal KL iterates are expressed via their partial fraction decomposition. This transforms the approximation into a sum of terms of the form $(X^2 + \sigma_i)^{-1}$. This form allows for the simultaneous inversion of all terms using the Multi-Shift Conjugate Gradient (MSCG) algorithm, significantly reducing computational overhead compared to inverting a single high-degree polynomial fraction.
3. Kernel Selection: The study evaluates two kernel operators: the standard Wilson-Dirac operator and the Brillouin operator. The Brillouin operator is a parameter-free, improved Wilson-like operator known to exhibit a spectrum closer to ideal Ginsparg-Wilson fermions, potentially offering better chiral properties.

The authors implement this formulation in a quenched lattice QCD setting ($\beta=6.2$, $48 \times 24^3$ lattice) and benchmark it against the well-established Chebyshev polynomial approximation. They analyze the violation of the GW relation, the convergence of the sign function, and the calculation of physical observables, specifically the PCAC quark mass and the critical bare quark mass ($m_{bare}^{critical}$).

Key Contributions

• Algorithmic Implementation: The paper demonstrates a practical implementation of diagonal KL iterates using partial fraction decomposition, enabling the use of robust MSCG solvers without the need for spectral estimation.
• Systematic Improvement: The authors show that increasing the diagonal KL order $n$ leads to a systematic, monotonic improvement in the approximation of the sign function and the satisfaction of the GW relation.
• Kernel Comparison: The study provides a detailed comparison between Wilson and Brillouin kernels within the KL framework, highlighting the Brillouin kernel's superior condition numbers and its impact on convergence rates.
• Benchmarking: A comprehensive comparison is made between the KL-PF approach and the Chebyshev polynomial method, evaluating both sign function convergence and the computational cost required to reach specific physical precision targets.

Results

• Efficiency of Partial Fraction vs. Single Fraction: The PF decomposition proves vastly superior to the single-fraction (SF) form. For higher orders (e.g., $n=5$), the SF approach becomes prohibitively expensive (costs exceeding 384 core-hours for Wilson), whereas the PF approach remains efficient (approx. 0.22 core-hours).
• Sign Function Convergence: Increasing the KL order $n$ yields monotonic reduction in the "sign-squared violation" ($\delta_{sgn^2}$). The Brillouin kernel generally achieves a given violation threshold with lower computational cost than the Wilson kernel, particularly for well-conditioned configurations, due to its smaller condition number.
• Physical Observables: The PCAC mass and pion effective mass stabilize rapidly as $n$ increases, with convergence effectively reached by $n \geq 3$. The critical bare mass ($m_{bare}^{critical}$) approaches zero monotonically with increasing $n$.
• KL vs. Chebyshev:
  • Convergence Behavior: The KL method exhibits monotonic convergence toward the target precision, whereas the Chebyshev method displays oscillatory behavior, making the selection of the optimal order more difficult.
  • Computational Cost: To reach a reference precision of $|m_{bare}^{critical}| \approx 4 \times 10^{-4}$, the KL-Brillouin combination requires approximately 20% less computational cost than the Chebyshev-Wilson combination. The KL-Brillouin combination is the most efficient among all tested method-kernel pairs.

Significance and Claims
The paper claims that the combination of diagonal KL rational approximations with the Brillouin kernel offers a practical and effective implementation of the overlap operator. The primary significance lies in the computational efficiency and predictability of the method. By utilizing partial fraction decomposition, the authors overcome the scaling issues of higher-order rational approximations, making high-order KL iterates feasible. The monotonic convergence of the KL method reduces the tuning effort required to achieve a target precision compared to the oscillatory Chebyshev approach. Furthermore, the results suggest that the Brillouin kernel, despite its higher per-application cost, provides a net computational advantage when paired with the KL approximation due to its superior spectral properties. The authors conclude that this combination warrants implementation in lattice QCD computations for vector-like gauge theories.