On the efficient numerical solution of lattice systems with low-order couplings

TL;DR

This paper compares Quasi Monte Carlo and recursive numerical integration methods for solving lattice quantum systems, demonstrating superior performance over traditional methods in certain models, with some limitations.

Contribution

It introduces and evaluates recursive numerical integration as an effective approach for lattice systems with low-order couplings, outperforming standard Monte Carlo methods in specific cases.

Findings

QMC outperforms Markov Chain Monte Carlo for the anharmonic oscillator.

Recursive numerical integration achieves exponential error scaling for the rotor model.

QMC was unsuccessful for the rotor model, but recursive integration succeeded.

Abstract

We apply the Quasi Monte Carlo (QMC) and recursive numerical integration methods to evaluate the Euclidean, discretized time path-integral for the quantum mechanical anharmonic oscillator and a topological quantum mechanical rotor model. For the anharmonic oscillator both methods outperform standard Markov Chain Monte Carlo methods and show a significantly improved error scaling. For the quantum mechanical rotor we could, however, not find a successful way employing QMC. On the other hand, the recursive numerical integration method works extremely well for this model and shows an at least exponentially fast error scaling.