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Physical review. B, Condensed matter

New results on Trotter-like approximations

Richard M. Fye

Published May 1, 1986• 149 citations
Publisher

Abstract

We explore the errors in the free energy and in operator expectation values when the quantum operator ${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$=${\ensuremath{\prod}}_{l=1}^{L}$${e}^{\mathrm{\ensuremath{-}}(\ensuremath{\Delta}\ensuremath{\tau})H}$ is approximated by ${\ensuremath{\prod}}_{l=1}^{L}$f, where \ensuremath{\Delta}\ensuremath{\tau}=\ensuremath{\beta}/L and f is an approximant to ${e}^{\mathrm{\ensuremath{-}}(\ensuremath{\Delta}\ensuremath{\tau})H}$. We determine analytically the dependence of the resulting errors on \ensuremath{\Delta}\ensuremath{\tau} for \ensuremath{\Delta}\ensuremath{\tau} small, on \ensuremath{\beta} for \ensuremath{\beta} large, and on the size of the system in the limit of a large system. We focus on Trotter approximations, as well as on the expansion of ${e}^{\mathrm{\ensuremath{-}}(\ensuremath{\Delta}\ensuremath{\tau})H}$ in powers of (\ensuremath{\Delta}\ensuremath{\tau})H. Our results are particularly relevant to Monte Carlo studies of quantum systems, and can be used effectively to eliminate the error due to Trotter-like approximations and to provide a guide for choosing a particular type of approximation.

DOI
10.1103/physrevb.33.6271

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The Trotter approximation has been used extensively in quantum Monte Carlo simulations. We consider here different ways of calculating the general zero-field susceptibility \ensuremath{\chi} and the specific heat C when a Trotter approximation is used, deriving analytically the error dependence on the imaginary-time increment \ensuremath{\Delta}\ensuremath{\tau} as well as the low-temperature behavior. We find that certain definitions of \ensuremath{\chi} and C exhibit spurious divergences at low temperatures, and suggest the most appropriate ways to extract these quantities. We test our general predictions on two models, and discuss the implications of our results for numerical simulations.
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