Extrapolation to infinite Trotter number in path-integral Monte Carlo simulations of solid-state systems

We consider the problem of the extrapolation of path-integral Monte Carlo (PIMC) data to infinite Trotter number P. Finite-P data, being even functions of P, have high-P dependence that is generally well described by a quadratic fit, ${\mathit{a}}_0$+${\mathit{a}}_1$${\mathit{P}}^{\mathrm{-}2}$, where ${\mathit{a}}_0$ is the exact quantum value. However, in order to get convergence it is often necessary to run PIMC codes with rather high P values, which implies long computer times and larger statistical errors of the data. It is well known that also for harmonic systems the finite-P data are not exact; nevertheless, they can be easily calculated by Gaussian quadrature. Starting from this observation, we suggest an easy way to correct PIMC data for anharmonic systems in order to take into account the harmonic part exactly, with strong improvement of the extrapolation to P=∞. Lower Trotter numbers are thus required, with the advantages of computer-time saving and much better accuracy of the extrapolated values, without any change in the PIMC code. In order to demonstrate the effectiveness of the approach, we report finite-P data processing for a single anharmonic particle, whose finite-P data are obtained by the matrix-squaring method, and for a chain of atoms with Morse interaction.