Quantum Statistical Calculations and Symplectic Corrector Algorithms

TL;DR

This paper explores the limitations of symplectic corrector algorithms in quantum statistical calculations, proving that positive time step propagators cannot be corrected beyond second order without additional operators, and provides explicit correctors.

Contribution

It generalizes the Sheng-Suzuki theorem and derives conditions for correctable second order propagators in quantum Monte Carlo methods.

Findings

Positive time step propagators cannot be corrected beyond second order.

Fourth order trace correction requires an additional commutator.

Explicit derivations of four correctable second order propagators.

Abstract

The quantum partition function at finite temperature requires computing the trace of the imaginary time propagator. For numerical and Monte Carlo calculations, the propagator is usually split into its kinetic and potential parts. A higher order splitting will result in a higher order convergent algorithm. At imaginary time, the kinetic energy propagator is usually the diffusion Greens function. Since diffusion cannot be simulated backward in time, the splitting must maintain the positivity of all intermediate time steps. However, since the trace is invariant under similarity transformations of the propagator, one can use this freedom to "correct" the split propagator to higher order. This use of similarity transforms classically give rises to symplectic corrector algorithms. The split propagator is the symplectic kernel and the similarity transformation is the corrector. This work proves a generalization of the Sheng-Suzuki theorem: no positive time step propagators with only kinetic and potential operators can be corrected beyond second order. Second order forward propagators can have fourth order traces only with the inclusion of an additional commutator. We give detailed derivations of four forward correctable second order propagators and their minimal correctors.