Optimal Convergence Rate of Lie-Trotter Approximation for Quantum Thermal Averages

TL;DR

This paper rigorously analyzes the convergence rates of the Lie-Trotter approximation for quantum thermal averages, establishing optimal error bounds for systems with unbounded Hamiltonians, thus supporting high-accuracy path integral simulations.

Contribution

It provides the first rigorous error bounds with optimal convergence rates for the Lie-Trotter approximation in quantum systems with unbounded Hamiltonians.

Findings

Optimal convergence rate of O(1/N^2) for periodic potentials.

Nearly optimal rate of O((log N+1)^{3/2}/N^2) for confining potentials.

Mathematical foundation for high-order accuracy in quantum path integral methods.

Abstract

The Lie--Trotter product formula is a foundational approximation for the quantum partition function, yet obtaining rigorous error bounds for the unbounded Hamiltonians common in physics remains a significant challenge. This paper provides a quantitative error analysis for this approximation across two key systems. For a particle in a smooth, periodic potential, we establish an optimal convergence rate of $\mathcal O(1/N^2)$ for both the partition function and thermal averages, where $N$ is the number of imaginary time steps. We then extend this analysis to the more challenging case of a confining potential on $\mathbb R$, proving a nearly optimal rate of $\mathcal O((\log N+1)^{\frac32}/N^2)$. The derived error bounds provide a firm mathematical foundation for the high-order accuracy of path integral simulations in quantum statistical mechanics.