Fast Converging Path Integrals for Time-Dependent Potentials II: Generalization to Many-body Systems and Real-Time Formalism

TL;DR

This paper extends a recursive path integral formalism to many-body, time-dependent quantum systems, enabling high-order short-time expansions and real-time analysis with verified numerical accuracy.

Contribution

It introduces a comprehensive recursive method for many-body systems with time-dependent potentials and extends the formalism to real-time dynamics, enhancing analytical and computational capabilities.

Findings

High-order short-time expansions derived analytically.

Extension to real-time formalism demonstrated.

Numerical verification confirms accuracy across models.

Abstract

Based on a previously developed recursive approach for calculating the short-time expansion of the propagator for systems with time-independent potentials and its time-dependent generalization for simple single-particle systems, in this paper we present a full extension of this formalism to a general quantum system with many degrees of freedom in a time-dependent potential. Furthermore, we also present a recursive approach for the velocity-independent part of the effective potential, which is necessary for calculating diagonal amplitudes and partition functions, as well as an extension from the imaginary-time formalism to the real-time one, which enables to study the dynamical properties of quantum systems. The recursive approach developed here allows an analytic derivation of the short-time expansion to orders that have not been accessible before, using the implemented SPEEDUP symbolic calculation code. The analytically derived results are extensively numerically verified by treating several models in both imaginary and real time.