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

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.

💡 Research Summary

This paper presents a comprehensive extension of a previously introduced recursive method for constructing short‑time expansions of quantum propagators to the realm of many‑body systems subjected to time‑dependent potentials, and further adapts the formalism from imaginary‑time to real‑time. The authors start by recalling the short‑time representation of the propagator as a Gaussian kernel multiplied by an exponential of an effective potential, (K = (2\pi\varepsilon)^{-Nd/2}\exp{-\frac{(q_f-q_i)^2}{2\varepsilon} - \varepsilon V_{\text{eff}}(q_f,q_i;\varepsilon)}), where (N) is the number of particles and (d) the spatial dimension. By inserting this ansatz into the Schrödinger equation and equating coefficients of powers of the time step (\varepsilon), they derive a hierarchy of recursive relations that generate the coefficients (V^{(n)}) of the effective potential to arbitrary order.

A key innovation is the separation of the velocity‑independent part of the effective potential, (V^{(0)}), which governs diagonal matrix elements and thus the partition function. The authors formulate an independent recursion for (V^{(0)}), enabling analytic access to high‑order corrections of thermodynamic quantities that were previously inaccessible.

The formalism is then generalized to time‑dependent potentials by promoting the effective potential to a function of the midpoint coordinate and the average time, and by systematically incorporating time derivatives. The resulting recursion remains algebraic and can be applied to any Lagrangian that is polynomial in the coordinates and velocities, making the method broadly applicable to non‑linear interactions and external drivings.

To bridge the gap between statistical and dynamical studies, the authors replace the imaginary time step (\varepsilon) with (i t), thereby obtaining a real‑time propagator with the same recursive structure. They demonstrate that the complex coefficients are handled consistently within the recursion, preserving the rapid convergence observed in the imaginary‑time case. This extension opens the door to accurate calculations of real‑time correlation functions, response functions, and quantum dynamics without resorting to costly Trotter‑Suzuki decompositions.

Implementation is carried out in a publicly released Python package named SPEEDUP, built on the SymPy symbolic engine. The code automates differentiation, tensor index contraction, and term collection, delivering the full set of coefficients up to a user‑specified order (typically (\varepsilon^8)) within minutes on a standard workstation.

The theoretical developments are validated on four benchmark models: (i) a one‑dimensional harmonic oscillator, (ii) a many‑body quantum Ising chain with nearest‑neighbor coupling, (iii) a time‑modulated two‑level system exhibiting entanglement dynamics, and (iv) multi‑dimensional particle potentials. In each case, the authors compare the recursive expansion against exact solutions (where available) or high‑precision Monte‑Carlo path‑integral data. The results show that the error scales as (\mathcal{O}(\varepsilon^{p+1})) when the expansion is carried to order (p), with relative errors dropping from (10^{-4}) at (\varepsilon^4) to below (10^{-8}) at (\varepsilon^8). Real‑time simulations reproduce known dynamical features such as coherent oscillations and decoherence rates with comparable accuracy.

The discussion highlights several promising directions. The recursion naturally accommodates higher‑body interactions by extending the tensor structures, suggesting applicability to strongly correlated electron systems and lattice gauge theories. The ability to treat rapidly varying time‑dependent fields makes the method suitable for pump‑probe spectroscopy and Floquet engineering. Moreover, the high‑order analytic expressions could be leveraged to design optimal Trotter‑Suzuki sequences with dramatically reduced discretization errors, a prospect of interest for quantum‑computing algorithms.

In conclusion, the paper delivers a powerful, systematic, and automated framework for generating high‑order short‑time expansions of quantum propagators in many‑body, time‑dependent settings, and successfully bridges imaginary‑time statistical mechanics with real‑time quantum dynamics. The combination of analytic recursion, symbolic automation, and extensive numerical verification establishes a new benchmark for precision path‑integral calculations and paves the way for a wide range of applications in condensed‑matter physics, quantum chemistry, and quantum information science.