Memory efficient finite volume schemes with twisted boundary conditions

TL;DR

This paper introduces a memory-efficient finite volume scheme with twisted boundary conditions and asymmetric geometry for $SU(N)$ gauge theories, enabling precise strong coupling measurements with reduced computational resources.

Contribution

It proposes a novel finite volume renormalization scheme combining gradient flow, twisted boundary conditions, and asymmetric geometry for improved efficiency and precision.

Findings

Scheme achieves accurate $ mar{MS}$ Lambda parameter consistent with literature.

Reduced memory footprint allows more efficient GPU cluster use.

Asymmetric geometry does not significantly affect scaling violations.

Abstract

In this paper we explore a finite volume renormalization scheme that combines three main ingredients: a coupling based on the gradient flow, the use of twisted boundary conditions and a particular asymmetric geometry, that for $SU(N)$ gauge theories consists on a hypercubic box of size $l^2 \times (Nl)^2$, a choice motivated by the study of volume independence in large $N$ gauge theories. We argue that this scheme has several advantages that make it particularly suited for precision determinations of the strong coupling, among them translational invariance, an analytic expansion in the coupling and a reduced memory footprint with respect to standard simulations on symmetric lattices, allowing for a more efficient use of current GPU clusters. We test this scheme numerically with a determination of the $\Lambda$ parameter in the $SU(3)$ pure gauge theory. We show that the use of an asymmetric geometry has no significant impact in the size of scaling violations, obtaining a value $\Lambda_{\overline{MS}} \sqrt{8 t_0} =0.603(17)$ in good agreement with the existing literature. The role of topology freezing, that is relevant for the determination of the coupling in this particular scheme and for large $N$ applications, is discussed in detail.