Diagonal Kenney-Laub Rational Approximation to the Overlap Dirac Operator

TL;DR

This paper addresses efficient, accurate implementation of the overlap Dirac operator by combining the diagonal Kenney–Laub rational approximants for the matrix sign function with the Brillouin kernel. The approach yields global convergence without eigenvalue estimation and leverages a partial fraction decomposition amenable to multi-shift conjugate gradient solves, while the Brillouin kernel improves spectral properties and conditioning. Numerical tests on quenched lattices show monotonic reduction of Ginsparg–Wilson violation with KL order and rapid convergence of PCAC and pion observables, with the Brillouin kernel outperforming the Wilson kernel in key metrics. The KL–Brillouin combination attains performance comparable to, and in some cases favorable to, Chebyshev-based approaches, offering a practical, tunable path toward efficient overlap fermion simulations and motivating further studies in dynamical settings and alternative sign-function approximations.

Abstract

We propose a practical formulation of the overlap Dirac operator in lattice QCD that employs the diagonal Kenney-Laub rational iterates - expressed via their partial fraction decomposition - to approximate the matrix sign function. We investigate this approximation using the Brillouin operator as kernel, in addition to the standard Wilson Dirac operator. Numerical results show improved chiral symmetry preservation and computational efficiency compared to the Chebyshev polynomial approach.

Figures (8)

• Figure 1: Scalar sign function (solid blue line) compared to approximations using diagonal KL iterates, defined on all of $\mathbb{R}$ (left), and Chebyshev polynomials, defined on $|\lambda| \in [0.01, 7.0]$ (right). The latter interval approximately spans the eigenvalue range for a Wilson kernel in 4D with $\rho=1$.
• Figure 2: Eigenvalue spectra for the Wilson (left) and Brillouin (right) operators on thermalized configurations. The Brillouin eigenvalue spectrum is compared to the ideal GW form (black circle) for increasing number of APE smearing steps as indicated in the legend. Figures adapted from Ref. 10.1103/physrevd.83.114512 (left) and Ref. 10.48550/arxiv.1701.00726 (right).
• Figure 3: GW violation vs. diagonal KL order for the Wilson (left) and Brillouin (right) kernels. Same color and symbol combination refers to the same configuration, whose condition number $\kappa_{\mathbb{X}^2}$ for that kernel is given in the respective legend.
• Figure 4: PCAC mass (left) and pion effective mass (right) for the Brillouin kernel at $\rho=1$ and $am = 0.02$ with diagonal KL orders $n = 1, \ldots, 7$. The number of configurations is $O(20)$.
• Figure 5: Plateau PCAC mass vs. diagonal KL order for $am = 0.02$ (left) and $am = 0.07$ (right), comparing Wilson (filled) and Brillouin (empty) kernels. The statistics are the same as in Fig. \ref{['fig:PCAC_pion']}.