Accurate and Fast Estimation of the Continuum Limit in Path Integral Simulations of Quantum Oscillators and Crystals

Abstract: Convergence of path integral simulations requires a substantial number of beads when quantum effects are significant. Traditional Trotter scaling approaches estimate the continuum limit through extrapolation, however they are restricted to the asymptotic behavior near this limit. We introduce an efficient extrapolation approach for thermodynamic properties of quantum oscillators and crystals from primitive path integral simulations. The method utilizes a fitting function inspired by the analytic solution of the harmonic oscillator (HO), or the Einstein crystal for solids. The formulation is for first derivative properties, such as energy and pressure; however, extension to second derivative properties, such as elastic constants, is straightforward. We apply the method to a one-dimensional HO and anharmonic oscillator (AO), as well as a three-dimensional Lennard-Jones crystal. Configurations are sampled using path integral molecular dynamics simulations in the canonical ensemble, at a low temperature of $T=0.1$ (simulation units). Compared to Trotter extrapolation approaches, the new method demonstrates substantial accuracy in estimating the continuum limit, using only a few simulations of relatively smaller system sizes. This capability significantly reduces computational cost, providing a powerful tool to facilitate computations for more complex and challenging systems, such as molecules and real crystals.