Solution of the Thirring model in thimble regularization

F Di Renzo,K Zambello

doi:10.1103/physrevd.105.054501

Abstract

Thimble regularisation of lattice field theories has been proposed as a solution to the infamous sign problem. It is conceptually very clean and powerful, but it is in practice limited by a potentially very serious issue: in general many thimbles can contribute to the computation of the functional integrals. Semiclassical arguments would suggest that the fundamental thimble could be sufficient to get the correct answer, but this hypothesis has been proven not to hold true in general. A first example of this failure has been put forward in the context of the Thirring model: the dominant thimble approximation is valid only in given regions of the parameter space of the theory. Since then a complete solution of this (simple) model in thimble regularisation has been missing. In this paper we show that a full solution (taking the continuum limit) is indeed possible. It is possible thanks to a method we recently proposed which de facto evades the need to simulate on many thimbles.

Highlights

On the lattice physical quantities are defined as high-dimensional integrals which can be computed via importance sampling by interpreting the factor e−S that appears in the integrands as a probability distribution
The Thirring model has been extensively studied as a playground to test our ability to tackle finite-density theories plagued by a sign problem
This work aimed at settling an old story: is the thimble approach really failing for the Thirring model? Thimble regularization of lattice field theories is a conceptually nice solution to the sign problem, but it can be strongly limited by the need for multithimble simulations

Summary

INTRODUCTION

The lattice regularization of a quantum field theory enables access to nonperturbative aspects of the theory by numerical simulations. Semiclassical arguments would suggest that the thimble attached to the critical point having the minimum real part of the action ( known as the fundamental thimble) gives the dominant contribution This contribution is expected to be further enhanced in the continuum limit. [6] we proposed a twostep process in which one first computes the weights in the semiclassical approximation and computes the relevant corrections This worked fairly well for the simple version of heavy-dense QCD we had considered. [19] in which observables are reconstructed by merging via Padeapproximants different Taylor series carried out around points where the single-thimble approximation is a good one This last method proved to be more effective and it allowed to repeat the simulations towards the continuum limit. This strategy allowed us to carry out the study of the continuum limit

MONTE CARLO INTEGRATION ON THIMBLES

ONE-THIMBLE SIMULATIONS

MULTITHIMBLE SIMULATIONS

TAYLOR EXPANSIONS ON THE FUNDAMENTAL THIMBLE

Ta and β

CONCLUSIONS