A Multisite Decomposition of the Tensor Network Path Integrals

Amartya Bose and Peter L. Walters, J. Chem. Phys., arXiv:2109.09723, submitted (2021)

Abstract
Tensor network decompositions of path integrals for simulating open quantum systems have recently been proven to be useful. In this work, we extend the tensor network path integral (TNPI) framework to efficiently simulate extended systems coupled with local vibrational and phononic modes. The Feynman-Vernon influence functional is a very popular approach used to account for the effect of a bath on the dynamics of the system. In order to facilitate the incorporation of the influence functional into a multisite framework (MS-TNPI), we combine a matrix product state decomposition of the reduced density tensor of the system along the sites with a corresponding tensor network representation of the time axis to construct an efficient 2D tensor network. The 2D MS-TNPI network, when finally contracted, yields the time-dependent reduced density tensor of the extended system as a matrix product state. The decomposition and algorithm presented are independent of the nature of the system Hamiltonian. We also outline an iteration scheme to take the simulation beyond the non-Markovian memory length introduced by the dissipative baths. Applications to spin chains coupled to local harmonic baths is presented; we consider interactions defined by the Ising, XXZ and the Heisenberg models. We demonstrate that the presence of dissipative environments can often dissipate the entanglement between the sites as measured by the bond dimension of the reduced density matrix product state. The MS-TNPI method would be useful for studying a variety of extended quantum systems coupled with vibrational baths or phononic modes.