Homotopy data as part of the lattice field: A first study

TL;DR

This paper introduces an extended lattice field formulation incorporating local topological information, enabling accurate representation of continuum topological properties and charges through analytical and simulation methods.

Contribution

It proposes a novel lattice approach that includes homotopy data to faithfully reproduce continuum topological features.

Findings

Extended lattice fields successfully encode topological charges.

Monte Carlo simulations confirm the method's effectiveness.

Analytical results support the lattice formulation's accuracy.

Abstract

Fields exhibit a variety of topological properties, like different topological charges, when field space in the continuum is composed by more than one topological sector. Lattice treatments usually encounter difficulties describing those properties. In this work, we show that by augmenting the usual lattice fields to include extra variables describing local topological information (more precisely, regarding homotopy), the topology of the space of fields in the continuum is faithfully reproduced in the lattice. We apply this extended lattice formulation to some simple models with non-trivial topological charges, and we study their properties both analytically and via Monte Carlo simulations.