Lefschetz-thimble techniques for path integral of zero-dimensional $O(n)$ sigma models

TL;DR

This paper explores Lefschetz-thimble techniques for zero-dimensional $O(n)$ sigma models, revealing how symmetry-breaking influences the structure of integration cycles and proposing methods to efficiently compute these cycles.

Contribution

It introduces an efficient method to identify slow motions in Lefschetz thimbles and analyzes the effects of symmetry restoration on the convergence of integration cycles.

Findings

Downward flows follow pseudo classical points and branch into middle-dimensional cycles.

Only specific combinations of Lefschetz thimbles remain convergent as symmetry is restored.

Properties hold true for both bosonic and fermionic systems.

Abstract

Zero-dimensional $O(n)$-symmetric sigma models are studied by using Picard--Lefschetz integration method in the presence of small symmetry-breaking perturbations. Due to approximate symmetry, downward flows turn out to show significant structures: They slowly travel along the set of pseudo classical points, and branch into other directions so as to span middle-dimensional integration cycles. We propose an efficient way to find such slow motions for computing Lefschetz thimbles. In the limit of symmetry restoration, we figure out that only special combinations of Lefschetz thimbles can survive as convergent integration cycles: Other integrations become divergent due to non-compactness of the complexified group of symmetry. We also compute downward flows of $O(2)$-symmetric fermionic systems, and confirm that all of these properties are true also with fermions.