Harmonic Oscillator Staging Coordinates for Efficient Path Integral Simulations of Quantum Oscillators and Crystals

Abstract

Imaginary-time path integral (PI) is a rigorous tool to compute static properties at finite temperatures. However, the stiff PI internal modes poses a sampling challenge. This is commonly tackled using staging coordinates, in which the free particle (FP) term of the PI action is diagonalized. We introduce novel and simple staging coordinates that diagonalize the entire action of the harmonic oscillator (HO) model, rendering it efficiently applicable to systems with harmonic character, such as quantum oscillators and crystals. The method is not applicable to fluids or systems with imaginary modes. Unlike FP staging, the HO staging provides a unique treatment of the centroid mode. We provide implementation schemes for PIMC and PIMD in NVT ensemble. Sampling efficiency is assessed in terms of precision and accuracy of estimating the energy and heat capacity of a one-dimensional HO and an asymmetric anharmonic oscillator (AO). In PIMC, the HO coordinates propose collective moves that perfectly sample the HO contribution, then (for AO) the anharmonic term is sampled using standard Metropolis method. This results in a high acceptance rate and, hence, high precision, in comparison to the FP staging. In PIMD, the HO coordinates prescribe definitions for the fictitious masses, yielding equal frequencies when applied to HO model. This allows for larger time step sizes relative to standard staging, without affecting accuracy or integrator stability. We also present results using normal mode (NM) coordinates, based on both HO and FP models. While staging and NM coordinates show similar performance (for FP or HO), staging is computationally preferable due to its cheaper scaling with Trotter number. The enhanced sampling of HO coordinates open avenues for efficient estimation of nuclear quantum effects in more complex systems with harmonic character, such as real molecular bonds and quantum crystals.