A tensor network representation of path integrals: Implementation and analysis

Abstract

Tensors with finite correlation afford very compact tensor network representations. A novel tensor network-based decomposition of real-time path integral simulations involving Feynman-Vernon influence functional is introduced. In this tensor network path integral (TNPI) technique, the finite temporarily non-local interactions introduced by the influence functional can be captured very efficiently using matrix product state representation for the path-dependent Green's function (PDGF). We illustrate this particular TNPI method through various realistic examples, including a charge transfer reaction and an exciton transfer in a dimer. We also show how it is readily applied to systems with greater than two states. PDGF-TNPI utilizes the symmetries of the problem, leading to accelerated convergence and dramatic reductions of computational effort. We also introduce an approximate method that speeds up propagation beyond the non-local memory length. Furthermore, the structure imposed by the tensor network representation of the PDGF naturally suggests other factorizations that make simulations for extended systems more efficient. These factorizations would be the subject of future explorations. The flexibility of the PDGF-TNPI framework makes it a promising new addition to the family of path integral methods for non-equilibrium quantum dynamics.

A Pairwise Connected Tensor Network Representation of Path Integrals

“…The bath is characterized by an Ohmic spectral density with an exponential cutoff, Eq. (20) with ω c = 4 and ξ = 0.12 [24] equilibrated at an inverse temperature of β = 0.1. As discussed in Sec.…”

“…PCTNPI provides an alternative to the MPS representation [21,24], serving as a small step in further elucidating the deep relation between tensor networks and path integrals. While no path filtration scheme has been developed, PCTNPI is already quite useable.…”

“…Algorithms based on MPS representations of the augmented reduced density tensor [21] or of the pathdependent Green's function [24] can be thought of as particular optimized re-factorizations of the PCTNPI network. We have demonstrated the viability of evaluating PCTNPI in a brute force manner compared to other methods.…”

“…Having discussed the tensor network, now let us turn to the job of contracting it. Typically, many tensor networks are constructed using singular value decomposition (SVD) and evaluated via the truncation of the singular values [21,24]. The PCTNPI network is constructed without resorting to any SVD calculations and consequently "exact."…”

“…While many recent developments have helped improve the efficiency of simulations [16][17][18][19][20], each of them utilize very different and deep insights into the structure of path integrals. It has recently been shown that the MPS representation can be very effectively utilized to reformulate real-time path integrals involving the influence functional leveraging the finite nature of the non-local memory [21][22][23][24]. While the MPS structure is the simplest tensor network that can be used, the 1D topology is probably not optimal when the non-Markovian memory spans a large number of time-steps and suffers from growing bond dimensions.…”