  Numerical and analytic results have been used to characterize quantum
transport in spin chains, showing the existence of both ballistic and diffusive
motion. Experiments have shown that heat transfer is surprisingly always
diffusive. The scattering from phonons and impurities have been postulated to
be the two factors critical in causing the diffusive transport. In this work,
we evaluate the transport process by incorporating a bath of phonons and
impurities in order to understand the role played by each of the factors. While
methods like time-dependent density matrix renormalization group (tDMRG) can be
used to simulate isolated spin chains, the coupling with phonons make
simulations significantly more challenging. The recently developed multisite
tensor network path integral (MS-TNPI) method builds a framework for simulating
the dynamics in extended open quantum systems by combining ideas from tDMRG and
Feynman-Vernon influence functional. This MS-TNPI is used to characterize
dynamics in open, extended quantum systems. Simulations are done with the
commonly used sub-Ohmic, Ohmic and super-Ohmic spectral densities describing
the phononic bath. We show that while the transport in presence of impurities
eventually becomes diffusive, the exact details are dependent on the specifics
of the interactions and amount of impurities. In contrast, the presence of a
bath makes the transport diffusive irrespective of the parameters
characterizing the bath.
