Convergence Characteristics of the Cumulant Expansion for Fourier Path Integrals

TL;DR

This paper analyzes the convergence properties of the cumulant expansion in Fourier path integrals, demonstrating that truncating at order p yields an asymptotic convergence rate of N^{-(2p+1)} with manageable numerical costs.

Contribution

It introduces a diagrammatic approach and linked-cluster theorem to simplify the cumulant expansion, enabling efficient and accurate Fourier path integral calculations.

Findings

Convergence rate behaves like N^{-(2p+1)} when truncated at order p.

The diagrammatic representation simplifies complex algebraic proofs.

Numerical cost is approximately linear in potential energy evaluations.

Abstract

The cumulant representation of the Fourier path integral method is examined to determine the asymptotic convergence characteristics of the imaginary-time density matrix with respect to the number of path variables $N$ included. It is proved that when the cumulant expansion is truncated at order $p$, the asymptotic convergence rate of the density matrix behaves like $N^{-(2p+1)}$. The complex algebra associated with the proof is simplified by introducing a diagrammatic representation of the contributing terms along with an associated linked-cluster theorem. The cumulant terms at each order are expanded in a series such that the the asymptotic convergence rate is maintained without the need to calculate the full cumulant at order $p$. Using this truncated expansion of each cumulant at order $p$, the numerical cost in developing Fourier path integral expressions having convergence order $N^{-(2p+1)}$ is shown to be approximately linear in the number of required potential energy evaluations making the method promising for actual numerical implementation.