Consistent Definitions of Flux and Electric and Magnetic Current in Abelian Projected SU(2) Lattice Gauge Theory

TL;DR

This paper proposes a consistent, lattice-spacing independent definition of flux and currents in Abelian projected SU(2) gauge theory, improving the accuracy of field distribution analyses and potentially influencing gauge choice interpretations.

Contribution

It introduces a unified flux definition based on lattice Ward-Takahashi identities, reducing errors and clarifying monopole and vortex analyses in lattice gauge theories.

Findings

The new flux definition aligns with Maxwell's equations at any lattice spacing.

Order a^2 errors from previous monopole definitions are minimized, leading to more accurate vortex profiles.

In the maximal Abelian gauge, the impact on dual Abrikosov vortex analysis is minimal, but other gauges may show significant effects.

Abstract

Through the use of a lattice U(1) Ward-Takahashi identity, one can find a precise definition of flux and electric four-current that does not rely on the continuum limit. The magnetic four-current defined for example by the DeGrand-Toussaint construction introduces order a^2 errors in the field distributions. We advocate using a single definition of flux in order to be consistent with both the electric and magnetic Maxwell's equations at any lattice spacing. In a U(1) theory the monopoles are slightly smeared by this choice, i.e. are no longer associated with a single lattice cube. In Abelian projected SU(2) the consistent definition suggests further modifications. For simulations in the scaling window, we do not foresee large changes in the standard analysis of the dual Abrikosov vortex in the maximal Abelian gauge because the order a^2 corrections have small fluctuations and tend to cancel out. However in other gauges, the consequences of our definitions could lead to large effects which may help in understanding the choice of gauge. We also examine the effect of truncating all monopoles except for the dominant cluster on the profile of the dual Abrikosov vortex.