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The Journal of Chemical Physics

Small matrix path integral propagation for long-time quantum dynamics of multistate systems in one and two dimensions

Nancy Makri

Published Feb 17, 2026• 1 citation
Publisher

Abstract

The small matrix path integral (SMatPI) methodology allows fully quantum mechanical calculations on system-bath Hamiltonians under a variety of conditions with minimal array storage and is thus applicable to multistate systems. This work focuses on the SMatPI iteration algorithm, which can be computationally prohibitive in systems with many states. By transforming the SMatPI propagation matrices to the desired initial state, the cost of the iterative multiplication is reduced by a factor of n2, where n is the number of system states. This leads to dramatic savings when n≫10 and allows the simulation of large systems over very long propagation times. Applications are presented on models with up to 50 system states, including exciton transfer dynamics in one-dimensional aggregates and a square lattice.

DOI
10.1063/5.0315124

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The small matrix path integral (SMatPI) methodology allows fully quantum mechanical calculations on system-bath Hamiltonians under a variety of conditions with minimal array storage and is thus applicable to multistate systems. This work focuses on the SMatPI iteration algorithm, which can be computationally prohibitive in systems with many states. By transforming the SMatPI propagation matrices to the desired initial state, the cost of the iterative multiplication is reduced by a factor of n2, where n is the number of system states. This leads to dramatic savings when n≫10 and allows the simulation of large systems over very long propagation times. Applications are presented on models with up to 50 system states, including exciton transfer dynamics in one-dimensional aggregates and a square lattice.
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 small matrix path integral (SMatPI) method is an efficient numerical approach to simulating the evolution of a quantum system coupled to a harmonic bath. The method relies on a sequence of kernel matrices that defines the non-Markovian dynamics of the quantum system. In the original SMatPI method, these kernels are computed indirectly through the QuAPI method. Instead, we focus on the definition of the kernel matrices and reveal a recurrence relation in these matrices. Using such a relationship, a tree based algorithm (t-SMatPI) is developed, which is shown to be much faster than straightforward computation of the kernel matrices based on their definitions. This algorithm bypasses the step to compute the SMatPI matrices by other path integral methods and provides more understanding of the SMatPI matrices themselves. Meanwhile, it keeps the memory cost and computational cost low. Numerical experiments show that the t-SMatPI algorithm gives exactly the same result as i-QuAPI and SMatPI. In spite of this, our method may indicate some new properties of open quantum systems, and has the potential to be generalized to higher-order numerical schemes.
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.
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.