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The Journal of Chemical Physics

Small matrix path integral in imaginary time

Rapti Pal, Nancy Makri

Published Sep 28, 2025• 4 citations
Publisher

Abstract

Thermal equilibrium properties are usually obtained from the imaginary-time path integral representation of the Boltzmann operator in combination with Monte Carlo integration methods. In some situations (identical fermions or frustrated Hamiltonians), the Boltzmann matrix leads to terms of alternating sign, which leads to a sign problem that severely impacts convergence. In this paper, we develop a robust and efficient quadrature-based method suitable for computing the Boltzmann matrix for discrete systems coupled to common or local harmonic baths. By expressing the discretized path integral with the influence functional in terms of a sum of matrix products, we develop a small matrix path integral (SMatPI) decomposition that allows iterative propagation in imaginary time while circumventing the storage of tensors employed in earlier work. The method is illustrated with several examples that involve two- and three-level systems coupled to common or local baths. We show that cyclic tight-binding Hamiltonians with positive coupling parameters give rise to Boltzmann matrix elements with alternating signs, presenting a severe sign problem to Monte Carlo approaches, while the SMatPI algorithm is stable and efficient.

DOI
10.1063/5.0285317

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Thermal equilibrium properties are usually obtained from the imaginary-time path integral representation of the Boltzmann operator in combination with Monte Carlo integration methods. In some situations (identical fermions or frustrated Hamiltonians), the Boltzmann matrix leads to terms of alternating sign, which leads to a sign problem that severely impacts convergence. In this paper, we develop a robust and efficient quadrature-based method suitable for computing the Boltzmann matrix for discrete systems coupled to common or local harmonic baths. By expressing the discretized path integral with the influence functional in terms of a sum of matrix products, we develop a small matrix path integral (SMatPI) decomposition that allows iterative propagation in imaginary time while circumventing the storage of tensors employed in earlier work. The method is illustrated with several examples that involve two- and three-level systems coupled to common or local baths. We show that cyclic tight-binding Hamiltonians with positive coupling parameters give rise to Boltzmann matrix elements with alternating signs, presenting a severe sign problem to Monte Carlo approaches, while the SMatPI algorithm is stable and efficient.
The small matrix decomposition of the quasi-adiabatic propagator path integral (SMatPI) for a system coupled to a harmonic bath, which accounts for multitime memory correlations in the influence functional without the use of tensors, is extended to include a time-dependent term that drives the system. In the case of a periodic field, the algorithm requires the construction of SMatPI matrices initialized over a short time interval. The SMatPI algorithm circumvents the large array storage of tensor-based iterative path integral decompositions and, in the case of a periodic field, also eliminates the demanding tensor multiplication at each time step, leading to dramatic savings which allow the fully quantum mechanical treatment of multistate systems and long-memory environments.
The discretized path integral expression for the reduced density matrix (RDM) of a system interacting with a dissipative harmonic bath is fully entangled because of influence functional terms that couple the variables at different time points. The iterative decomposition of the path integral, which exploits the finite length of influence functional memory, involves a tensor propagator whose size grows exponentially with the memory length. The present Communication disentangles the path integral by recursively spreading the temporal entanglement over longer path segments, while decreasing its contribution. Eventually, the entangled term becomes sufficiently small and may be neglected, leading to iterative propagation of the RDM through simple multiplication of matrices whose size is equal to that of the bare system. It is found that the temporal entanglement length is practically equal to the bath-induced memory length. The small matrix decomposition of the path integral (SMatPI) is stable and very efficient, extending the applicability of numerically exact real-time path integral methods to multi-state systems.
A small matrix decomposition of the path integral expression (SMatPI) that yields the reduced density matrix of a system interacting with a dissipative harmonic bath is obtained by recursively spreading the entangled influence functional terms over longer time intervals while simultaneously decreasing their magnitude until these terms become negligible. This allows summation over the path integral variables one by one through multiplication of small matrices with dimension equal to that of the bare system. The theoretical framework of the decomposition is described using a diagrammatic approach. Analytical and numerical calculations show that the necessary time length for the temporal entanglement to become negligible is practically the same as the bath-induced memory. The properties and structure of the propagator matrices are discussed, and applications to multistate systems are presented.
The small matrix decomposition of the real-time path integral (SMatPI) allows for numerically exact and efficient propagation of the reduced density matrix (RDM) for system-bath Hamiltonians. Its high efficiency lies in the small size of the SMatPI matrices employed in the iterative algorithm, whose size is equal to that of the full RDM. By avoiding the storage and multiplication of large tensors, the SMatPI algorithm is applicable in multistate systems under a variety of conditions. The main computational effort is the evaluation of path sums within the entangled memory length to construct the SMatPI matrices. A number of methods are available for this task, each with its own favorable parameter regime, but calculations with strong system-bath coupling and long memory at low temperatures remain out of reach. The present paper evaluates the path sums by binning the paths (in forward time only) based on their amplitudes, which depend on the number and type of kinks they contain. The algorithm is very efficient, leading to a dramatic acceleration of path sums and significantly extending the accessible memory length in the most challenging regimes.
This work shows that the recently discovered operator contraction identity for solving the discrete path integral of the harmonic oscillator can be applied equally to fermions in any dimension. This then yields an exactly solvable model for studying the sign problem where the path integral Monte Carlo energy at any time step for any number of fermions is known analytically or can be computed numerically. It is found that the sign problem is primarily a property of the free fermion propagator, but repulsive/attractive pairwise interactions can shift the sign problem to larger/smaller imaginary time but do not make it more severe than the non-interacting case. More surprisingly, one can prove analytically that the first closed-shell state in D dimension, with n = D + 1 fermions, has no sign problem at large imaginary time. Direct numerical simulations confirm that this is also true for higher closed-shell states in two and three dimensions. Fourth-order and newly found variable-bead algorithms are used to compute ground state energies of quantum dots with up to 110 electrons and are compared to results obtained by modern neural networks.