New results on Trotter-like approximations

We explore the errors in the free energy and in operator expectation values when the quantum operator $e^{\mathrm{-}\beta H}$=${\prod }_{l=1\mathrm{}}^L$ $e^{\mathrm{-}\left(\Delta \tau \right)H}$ is approximated by ${\prod }_{l=1\mathrm{}}^L$f, where Δτ=β/L and f is an approximant to $e^{\mathrm{-}\left(\Delta \tau \right)H}$. We determine analytically the dependence of the resulting errors on Δτ for Δτ small, on β for β 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 $e^{\mathrm{-}\left(\Delta \tau \right)H}$ in powers of (Δτ)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.