Lefschetz Thimble Quantum Monte Carlo for Spin Systems

TL;DR

This paper extends Lefschetz thimble Monte Carlo methods to spin systems using spin coherent states, demonstrating potential to mitigate sign problems but also revealing limitations and systematic errors in the approach.

Contribution

It introduces a novel application of Lefschetz thimbles to spin systems via coherent state path integrals, highlighting both its potential and limitations.

Findings

Lefschetz thimbles reduce sign problems in toy spin systems.

Significant systematic errors observed in large spin regimes.

Breakdown of spin coherent path integral demonstrated.

Abstract

Monte Carlo simulations are useful tools for modeling quantum systems, but in some cases they suffer from a sign problem, leading to an exponential slow down in their convergence to a value. While solving the sign problem is generically NP-hard, many techniques exist for mitigating the sign problem in specific cases; in particular, the technique of deforming the Monte Carlo simulation's plane of integration onto Lefschetz thimbles (complex hypersurfaces of stationary phase) has seen significant success in the context of quantum field theories. We extend this methodology to spin systems by utilizing spin coherent state path integrals to re-express the spin system's partition function in terms of continuous variables. Using some toy systems, we demonstrate its effectiveness at lessening the sign problem in this setting, despite the fact that the initial mapping to spin coherent states introduces its own sign problem. The standard formulation of the spin coherent path integral is known to make use of uncontrolled approximations; despite this, for large spins they are typically considered to yield accurate results, so it is somewhat surprising that our results show significant systematic errors. Therefore, possibly of independent interest, our use of Lefschetz thimbles to overcome the intrinsic sign problem in spin coherent state path integral Monte Carlo enables a novel numerical demonstration of a breakdown in the spin coherent path integral.