Topological Lattice Actions

TL;DR

This paper investigates lattice field theories with topological actions that are invariant under small deformations, demonstrating they can reach the correct quantum continuum limit despite not having the proper classical limit or perturbative treatment.

Contribution

It provides analytic and numerical evidence that topological lattice actions, even with infinite barriers and classical limitations, yield the correct quantum continuum limit in 1-d and 2-d O(n) models.

Findings

Topological actions can produce the correct quantum continuum limit.

Some topological actions violate the lattice Schwarz inequality.

Topological susceptibility diverges logarithmically, but local correlators have finite limits.

Abstract

We consider lattice field theories with topological actions, which are invariant against small deformations of the fields. Some of these actions have infinite barriers separating different topological sectors. Topological actions do not have the correct classical continuum limit and they cannot be treated using perturbation theory, but they still yield the correct quantum continuum limit. To show this, we present analytic studies of the 1-d O(2) and O(3) model, as well as Monte Carlo simulations of the 2-d O(3) model using topological lattice actions. Some topological actions obey and others violate a lattice Schwarz inequality between the action and the topological charge Q. Irrespective of this, in the 2-d O(3) model the topological susceptibility \chi_t = \l< Q^2 >/V is logarithmically divergent in the continuum limit. Still, at non-zero distance the correlator of the topological charge density has a finite continuum limit which is consistent with analytic predictions. Our study shows explicitly that some classically important features of an action are irrelevant for reaching the correct quantum continuum limit.