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Molecular Physics

Efficient matrix factorisation of the modular path integral for extended systems

Sohang Kundu, Nancy Makri

Published Aug 7, 2020• 21 citations
Publisher

Abstract

The modular path integral (MPI) formulation offers a numerically exact, versatile, and highly efficient approach to the quantum dynamics of extended systems characterised by a single-file arrangement of units with short-range interactions, such as hydrocarbons, molecular aggregates with exciton couplings, or spin chains. Rather than propagating the many-particle wavefunction or density matrix in time in the usual stepwise fashion, the MPI scheme proceeds by sequentially integrating over each unit after linking its quantum paths to those of its neighbour and leads to linear scaling with system length. This paper shows that the linking matrix can be further decomposed into a diagonal matrix and a product of low-dimensional matrices, which can be applied sequentially. This factorisation changes the cost scaling from L2 to Llog⁡L, where L is the number of quantum paths, leading to dramatic savings that are analogous to those attained by the fast Fourier transform algorithm and allowing application of the MPI procedure to systems with larger units and to longer propagation times. Applications to spin chains and to electron-vibration dynamics in a large molecular aggregate illustrate the efficiency of the algorithm.

DOI
10.1080/00268976.2020.1797200

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The modular path integral (MPI) formulation offers a numerically exact, versatile, and highly efficient approach to the quantum dynamics of extended systems characterised by a single-file arrangement of units with short-range interactions, such as hydrocarbons, molecular aggregates with exciton couplings, or spin chains. Rather than propagating the many-particle wavefunction or density matrix in time in the usual stepwise fashion, the MPI scheme proceeds by sequentially integrating over each unit after linking its quantum paths to those of its neighbour and leads to linear scaling with system length. This paper shows that the linking matrix can be further decomposed into a diagonal matrix and a product of low-dimensional matrices, which can be applied sequentially. This factorisation changes the cost scaling from L2 to Llog⁡L, where L is the number of quantum paths, leading to dramatic savings that are analogous to those attained by the fast Fourier transform algorithm and allowing application of the MPI procedure to systems with larger units and to longer propagation times. Applications to spin chains and to electron-vibration dynamics in a large molecular aggregate illustrate the efficiency of the algorithm.
, 101101 (2018)], it was shown that the locality of interactions in many systems of interest allows a decomposition of the path integral and its evaluation via sequential linking of the paths of relatively small "modules" (e.g., chemical groups or monomers). The present paper describes the modular path integral methodology for simulating dynamical properties by propagating the density matrix in real time. The procedure is first presented for the simple topology of a single-file arrangement of units interacting via nearest neighbor couplings and subsequently extended to the calculation of two-particle correlations in arrays that may also contain some long-range interactions, to the treatment of systems with side chains or cyclic structures, to the simulation of internal dynamics in long organic molecules, and to the modifications required for coupling of one or several units of a system to dissipative environments. Illustrative applications to the dynamics of interacting two-level-systems are presented.
The modular decomposition of the path integral is a linear-scaling, numerically exact algorithm for calculating dynamical properties of extended systems composed of multilevel units with local couplings. In a recent article, we generalized the method to wavefunction propagation in aggregates characterized by non-diagonal couplings between adjacent units. Here, we extend the method to the calculation of reduced density matrices in aggregates where each unit includes an arbitrary number of coupled harmonic bath modes, which may describe intramolecular normal mode vibrations, at finite temperature. The effects of harmonic modes are included through influence functional factors, which involve analytical expressions that we derive. Representative applications to spin arrays described by the Heisenberg Hamiltonian with dissipative interactions and to J-aggregates of perylene bisimide, where all coupled normal modes are treated explicitly, are presented.
The modular decomposition of the path integral, which leads to linear scaling with the system length, is extended to Hamiltonians with intermonomer couplings that are not diagonalizable in any single-particle basis. An optimal factorization of the time evolution operator is identified, which minimizes the number of path integral variables while ensuring high accuracy and preservation of detailed balance. The modular path integral decomposition is described, along with a highly efficient tensor factorization of the path linking process. The algorithm is illustrated with applications to a model of coupled spins and a Frenkel exciton chain.
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.
This paper reports the release of PathSum, a new software suite of state-of-the-art path integral methods for studying the dynamics of single or extended systems coupled to harmonic environments. The package includes two modules, suitable for system-bath problems and extended systems comprising many coupled system-bath units, and is offered in C++ and Fortran implementations. The system-bath module offers the recently developed small matrix path integral (SMatPI) and the well-established iterative quasi-adiabatic propagator path integral (i-QuAPI) method for iteration of the reduced density matrix of the system. In the SMatPI module, the dynamics within the entanglement interval can be computed using QuAPI, the blip sum, time evolving matrix product operators, or the quantum-classical path integral method. These methods have distinct convergence characteristics and their combination allows a user to access a variety of regimes. The extended system module provides the user with two algorithms of the modular path integral method, applicable to quantum spin chains or excitonic molecular aggregates. An overview of the methods and code structure is provided, along with guidance on method selection and representative examples.