New results on Trotter-like approximations

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.