  Tensor network decompositions of path integrals for simulating open quantum
systems have recently been proven to be useful. However, these methods scale
exponentially with the system size. This makes it challenging to simulate the
non-equilibrium dynamics of extended quantum systems coupled with local
dissipative environments. In this work, we extend the tensor network path
integral (TNPI) framework to efficiently simulate such extended systems. The
Feynman-Vernon influence functional is a popular approach used to account for
the effect of environments 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 (MPS) 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 contracted, yields the
time-dependent reduced density tensor of the extended system as an MPS. The
algorithm presented is independent of the system Hamiltonian. We outline an
iteration scheme to take the simulation beyond the non-Markovian memory
introduced by solvents. Applications to spin chains coupled to local harmonic
baths are presented; we consider the Ising, XXZ and the Heisenberg models,
demonstrating that the presence of local environments can often dissipate the
entanglement between the sites. We discuss three factors causing the system to
transition from a coherent oscillatory dynamics to a fully incoherent dynamics.
The MS-TNPI method is useful for studying a variety of extended quantum systems
coupled with solvents.
