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The Journal of Chemical Physics

Modular path integral for discrete systems with non-diagonal couplings

Sohang Kundu, Nancy Makri

Published Aug 16, 2019• 24 citations
Publisher

Abstract

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.

DOI
10.1063/1.5108692

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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.
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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.
, 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.
We describe an efficient path integral scheme for calculating the propagator of an arbitrary quantum system, as well as that of a stochastic system in special cases where the Fokker–Planck equation obeys strict detailed balance. The basic idea is to split the respective Hamiltonian into two exactly solvable parts and then to employ a symmetric decomposition of the time evolution operator, which is exact up to a high order in the time step. The resulting single step propagator allows rather large time steps in a path integral and leads to convergence with fewer time slices. Because it involves no system-specific reference system, the algorithm is amenable to all known numerical schemes available for evaluating quantum path integrals. In this way one obtains a highly accurate method, which is simultaneously fast, stable, and computationally simple. Numerical applications to the real time quantum dynamics in a double well and to the stochastic dynamics of a bistable system coupled to a harmonic mode show our method to be superior over the approach developed by the Makri group in their quasiadiabatic propagator representation, to say nothing about the propagation scheme based on the standard Trotter splitting.
Path-integral techniques are a powerful tool for obtaining exact solutions for the non-Markovian dynamics of open quantum systems. However, the exponential scaling of the tensor size with the quantum memory length limits their applicability to systems with long memory times. Here we provide a general optimization of tensor multiplication schemes for systems with pair correlations and finite memory times. This optimization effectively reduces the tensor sizes by using a matrix representation of tensors combined with singular value decomposition to filter out negligible contributions. This approach dramatically reduces both computational time and memory usage of the traditional tensor-multiplication schemes. While more memory-efficient representations exist, this approach enables a consistent extrapolation scheme for the rapid estimation of the exact value. As a demonstration, we apply it to the Trotter decomposition with linked cluster expansion technique and use it to investigate a quantum dot-microcavity system at large coupling strengths. We also apply the optimization in a case where the memory time is very long—specifically in a system containing two spatially separated quantum dots in a common phonon bath.