Thimble regularization at work: from toy models to chiral random matrix theories

TL;DR

This paper explores the Lefschetz thimble method applied to simple toy models and more realistic chiral random matrix theories, demonstrating its effectiveness in addressing the sign problem and correctly sampling contributions from relevant thimbles.

Contribution

It extends the Lefschetz thimble approach to complex models, analyzing thimble contributions and sampling strategies in both toy and realistic theories.

Findings

Only one thimble contributes in studied parameter regions.

The sampling algorithm accurately reproduces known results.

The method effectively mitigates the sign problem in the models.

Abstract

We apply the Lefschetz thimble formulation of field theories to a couple of different problems. We first address the solution of a complex 0-dimensional phi^4 theory. Although very simple, this toy-model makes us appreciate a few key issues of the method. In particular, we will solve the model by a correct accounting of all the thimbles giving a contribution to the partition function and we will discuss a number of algorithmic solutions to simulate this (simple) model. We will then move to a chiral random matrix (CRM) theory. This is a somehow more realistic setting, giving us once again the chance to tackle the same couple of fundamental questions: how many thimbles contribute to the solution? how can we make sure that we correctly sample configurations on the thimble? Since the exact result is known for the observable we study (a condensate), we can verify that, in the region of parameters we studied, only one thimble contributes and that the algorithmic solution that we set up works well, despite its very crude nature. The deviation of results from phase quenched results highlights that in a certain region of parameter space there is a quite important sign problem. In view of this, the success of our thimble approach is quite a significant one.