Optimal Convergence Rate of Lie-Trotter Approximation for Quantum Thermal Averages
For researchers in quantum statistical mechanics, this work provides rigorous mathematical foundations for the second-order accuracy of path integral simulations, addressing a long-standing gap in error analysis for unbounded Hamiltonians.
This paper establishes optimal and nearly optimal convergence rates for the Lie-Trotter approximation of quantum thermal averages in two systems: a particle in a smooth periodic potential (O(1/N^2)) and a confining potential on R (O((log N+1)^{1.5}/N^2)), providing rigorous error bounds for path integral simulations.
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 challenge. This paper provides a quantitative error analysis for this approximation across two 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 $\mathcal O((\log N+1)^{1.5}/N^2)$. The derived error bounds provide a firm mathematical foundation for the second-order accuracy of path integral simulations in quantum statistical mechanics.