Supersymmetric Nonlinear O(3) Sigma Model on the Lattice

TL;DR

This paper investigates the lattice formulation of a supersymmetric nonlinear O(3) sigma model, demonstrating the challenges in preserving symmetries exactly and proposing an alternative approach with partial symmetry restoration in the continuum limit.

Contribution

It introduces an alternative lattice action for the supersymmetric O(3) sigma model that preserves O(3) symmetry and compares different derivative operators, analyzing symmetry restoration.

Findings

Exact O(3) symmetry achieved with the new lattice action.

Step scaling function closely matches the continuum value.

Supersymmetry and chiral symmetry are restored in the continuum limit via fine tuning.

Abstract

A supersymmetric extension of the nonlinear O(3) sigma model in two spacetime dimensions is investigated by means of Monte Carlo simulations. We argue that it is impossible to construct a lattice action that implements both the O(3) symmetry as well as at least one supersymmetry exactly at finite lattice spacing. It is shown by explicit calculations that previously proposed discretizations fail to reproduce the exact symmetries of the target manifold in the continuum limit. We provide an alternative lattice action with exact O(3) symmetry and compare two approaches based on different derivative operators. Using the nonlocal SLAC derivative for the quenched model on moderately sized lattices we extract the value {\sigma}(2, u_0) = 1.2604(13) for the step scaling function at u_0 = 1.0595, to be compared with the exact value 1.261210. For the supersymmetric model with SLAC derivative the discrete chiral symmetry is maintained but we encounter strong sign fluctuations, rendering large lattice simulations ineffective. By applying the Wilson prescription, supersymmetry and chiral symmetry are broken explicitly at finite lattice spacing, though there is clear evidence that both are restored in the continuum limit by fine tuning of a single mass parameter.