The Single Histogram Method and the Quantum Harmonic Oscillator: Accuracy Limits

TL;DR

This paper applies a single histogram Monte Carlo method to quantum harmonic oscillators within the WLQMC framework, identifying a temperature limit due to Trotter-Suzuki expansion accuracy constraints.

Contribution

It demonstrates the application of a single histogram method to quantum systems and quantifies the minimal temperature limit imposed by Trotter-Suzuki expansion accuracy.

Findings

Minimal temperature limit scales as T_min = 1.9(2) N^{-0.80(6)}.

Trotter-Suzuki expansion failure causes the temperature limit.

The method's accuracy is constrained by the Trotter-Suzuki expansion at large epsilon.

Abstract

In a recent work, M. Troyer, F. Alet and S. Wessel \cite{brazilean} proposed a way to extend histogram methods to quantum systems in the World Line Quantum Monte Carlo (WLQMC) formulation. The strategy, also proposed in \cite{josedaniel}, allows to compute quantum averages on a narrow temperature range from a single Monte Carlo run at fixed temperature. This is achieved by fixing $N$, the number of temporal divisions in the Trotter-Suzuki expansion of WLQMC, and by changing $\epsilon$$=$$1/(N \kb T)$. In this work we apply this strategy to construct a single histogram Monte Carlo method for a canonical ensemble of one-dimensional quantum harmonic oscillators and we explore its accuracy limits. We obtain that fixing $N$ imposses a limit of minimal temperature to the properly performance of the method, which is $T_{min}$$=$$1.9(2)N^{-0.80(6)}$ in our example. This limit is a consequence of the fact that the Trotter-Suzuki expansion fails for large $\epsilon$ values, and, therefore, should be taken into account in all applications of this histogram method for quantum systems.