Heating and small-size instantons in the $O(3) \:\sigma$ model on the lattice

TL;DR

This paper investigates the influence of small-size instantons on the topological susceptibility in the 2D $O(3) \\sigma$ model, using lattice Monte Carlo methods to achieve high-precision non-perturbative results.

Contribution

It provides a detailed analysis of small instantons' effects on topological susceptibility and offers a high-precision non-perturbative determination of operator mixing.

Findings

Good agreement with perturbative predictions for operator mixing

Distribution of instanton sizes extends to very small scales without ultraviolet cutoff

Mixing with the action density is negligible in the non-perturbative signal

Abstract

We study the role of small-size instantons in the determination of the topological susceptibility of the 2-d $O(3) \: \sigma $ model on the lattice. In particular, we analyze how they affect the non-perturbative determination, by Monte Carlo techniques, of the renormalizations on the lattice. As a result, we obtain a high-precision non-perturbative determination of the mixing with the unity operator, finding good agreement with perturbative computations. We also obtain the size distribution of instantons in the physical vacuum up to very small values of the size in physical units, without observing any ultraviolet cut-off. Moreover, we show by analytical calculation that the mixing of the topological susceptibility with the action density is a negligible part of the whole non-perturbative signal.