  In this work we analyze the simultaneous emergence of diffusive energy
transport and local thermalization in a nonequilibrium one-dimensional quantum
system, as a result of integrability breaking. Specifically, we discuss the
local properties of the steady state induced by thermal boundary driving in a
XXZ spin chain with staggered magnetic field. By means of efficient large-scale
matrix product simulations of the equation of motion of the system, we
calculate its steady state in the long-time limit.We start by discussing the
energy transport supported by the system, finding it to be ballistic in the
integrable limit and diffusive when the staggered field is finite.
Subsequently, we examine the reduced density operators of neighboring sites and
find that for large systems they are well approximated by local thermal states
of the underlying Hamiltonian in the nonintegrable regime, even for weak
staggered fields. In the integrable limit, on the other hand, this behavior is
lost, and the identification of local temperatures is no longer possible. Our
results agree with the intuitive connection between energy diffusion and
thermalization.
