Abstract
Critical slowing down for the Krylov Dirac solver presents a major obstacle to further advances in lattice field theory as it approaches the continuum solution. We propose a new multi-grid approach for chiral fermions, applicable to both the 5-d domain wall or 4-d Overlap operator. The central idea is to directly coarsen the 4-d Wilson kernel, giving an effective domain wall or overlap operator on each level. We provide here an explicit construction for the Shamir domain wall formulation with numerical tests for the 2-d Schwinger prototype, demonstrating near ideal multi-grid scaling. The framework is designed for a natural extension to 4-d lattice QCD chiral fermions, such as the M\"obius, Zolotarev or Borici domain wall discretizations or directly to a rational expansion of the 4-d Overlap operator. For the Shamir operator, the effective overlap operator is isolated by the use of a Pauli-Villars preconditioner in the spirit of the K\"ahler-Dirac spectral map used in a recent staggered MG algorithm [1].
Highlights
The intersection of the preexascale era and ever-evolving theoretical insights continues to boost the predictive power of computational approaches to quantum field theories, most substantively for lattice quantum chromodynamics [1]
We see that transferring only the boundary component of the fine residual and the coarse error correction is essential for the success of our multigrid algorithm, and present a physical argument for why this “cures” the problem introduced by the spurious small eigenvalues
The problem resembles the spurious small eigenvalues that plagued the direct application of Galerkin projection to the staggered operator prior to the KählerDirac preconditioning. In this instance we posit that the saving grace for the domain wall operator is that the low modes of D−PV1 DDW are bound to the chiral walls, while higher modes bleed more dominantly into the bulk as suggested by Eq (15)
Summary
The intersection of the preexascale era and ever-evolving theoretical insights continues to boost the predictive power of computational approaches to quantum field theories, most substantively for lattice quantum chromodynamics [1]. The successful extension to strong background chromodynamic fields was enabled by the development of fully recursive adaptive geometric MG method for the WilsonDirac fermion discretization [7,8]. While this was and continues to be of great use, it is only one piece of the broader problem: there are two other discretizations, referred to as staggered [9] and domain wall [10] fermions, that are used extensively in high energy applications that more faithfully represent chiral symmetry on the lattice. V we conclude by noting that our core developments apply to the 4D Shamir formulation presented here and to the Möbius [23,24], Borici [25,26], and Zolotarev [27,28] formulations, as well as directly to the overlap operator approximation to the sign function [14,29,30,31]
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