  We study the temperature dependence of energy diffusion in two chaotic gapped
quantum spin chains, a tilted-field Ising model and an XZ model, using an open
system approach. We introduce an energy imbalance by coupling the chain to
thermal baths at its boundary and study the non-equilibrium steady states of
the resulting Lindblad dynamics using a matrix product operator ansatz for the
density matrix. We define an effective local temperature profile by comparing
local reduced density matrices in the steady state to those of a uniform
thermal state. We then measure the energy current for a variety of driving
temperatures and extract the temperature dependence of the energy diffusion
constant. For the Ising model, we are able to study temperatures well below the
energy gap and find a regime of dilute excitations where rare three-body
collisions control energy diffusion. A kinetic model correctly predicts the
observed exponential increase of the energy diffusion constant at low
temperatures. For the XZ model, we are only able to access intermediate to high
temperatures relative to the energy gap and we show that the data is well
described by an expansion around the infinite temperature limit. We also
discuss the limitations of the particular driving scheme and suggest that lower
temperatures can be accessed using larger baths.
