The M\"obius Domain Wall Fermion Algorithm

TL;DR

This paper reviews the M"obius domain wall fermion algorithm, highlighting its efficiency, reduced chiral violations, and improved scaling behavior compared to previous methods, by tuning a new parameter and optimizing computational techniques.

Contribution

It introduces a M"obius transformation-based domain wall fermion algorithm with a new scaling parameter that reduces chiral violations and improves the scaling of residual mass.

Findings

Chiral violations are reduced by an order of magnitude at fixed fifth dimension.

The residual mass scales as 1/L_s^2 with the M"obius algorithm.

The M"obius algorithm interpolates between existing domain wall formulations without extra Dirac applications.

Abstract

We present a review of the properties of generalized domain wall Fermions, based on a (real) M\"obius transformation on the Wilson overlap kernel, discussing their algorithmic efficiency, the degree of explicit chiral violations measured by the residual mass ($m_{res}$) and the Ward-Takahashi identities. The M\"obius class interpolates between Shamir's domain wall operator and Bori\c{c}i's domain wall implementation of Neuberger's overlap operator without increasing the number of Dirac applications per conjugate gradient iteration. A new scaling parameter ($\alpha$) reduces chiral violations at finite fifth dimension ($L_s$) but yields exactly the same overlap action in the limit $L_s \rightarrow \infty$. Through the use of 4d Red/Black preconditioning and optimal tuning for the scaling $\alpha(L_s)$, we show that chiral symmetry violations are typically reduced by an order of magnitude at fixed $L_s$. At large $L_s$ we argue that the observed scaling for $m_{res} = O(1/L_s)$ for Shamir is replaced by $m_{res} = O(1/L_s^2)$ for the properly tuned M\"obius algorithm with $\alpha = O(L_s)$