One-dimensional QCD in thimble regularization

TL;DR

This paper applies thimble regularization to solve 0+1 dimensional QCD, demonstrating how multiple thimbles contribute to the sign problem and reproducing known analytical results, with insights into thimble dominance at high flavor numbers.

Contribution

It introduces a formalism for SU(N) theories in thimble regularization and analyzes the role of multiple thimbles in addressing the sign problem in a toy QCD model.

Findings

Reproduces analytical results for chiral condensate and Polyakov loop.

Shows single thimble dominance at high N_f.

Demonstrates control over the sign problem across parameter space.

Abstract

QCD in 0+1 dimensions is numerically solved via thimble regularization. In the context of this toy model, a general formalism is presented for SU(N) theories. The sign problem that the theory displays is a genuine one, stemming from a (quark) chemical potential. Three stationary points are present in the original (real) domain of integration, so that contributions from all the thimbles associated to them are to be taken into account: we show how semiclassical computations can provide hints on the regions of parameter space where this is absolutely crucial. Known analytical results for the chiral condensate and the Polyakov loop are correctly reproduced: this is in particular trivial at high values of the number of flavors N_f. In this regime we notice that the single thimble dominance scenario takes place (the dominant thimble is the one associated to the identity). At low values of N_f computations can be more difficult. It is important to stress that this is not at all a consequence of the original sign problem (not even via the residual phase). The latter is always under control, while accidental, delicate cancelations of contributions coming from different thimbles can be in place in (restricted) regions of the parameter space.