  Given a finite-range, translation-invariant commuting system Hamiltonians on
a spin chain, we show that the Davies semigroup describing the reduced dynamics
resulting from the joint Hamiltonian evolution of a spin chain weakly coupled
to a large heat bath thermalizes rapidly at any temperature. More precisely, we
prove that the relative entropy between any evolved state and the equilibrium
Gibbs state contracts exponentially fast with an exponent that scales
logarithmically with the length of the chain. Our theorem extends a seminal
result of Holley and Stroock to the quantum setting, up to a logarithmic
overhead, as well as provides an exponential improvement over the non-closure
of the gap proved by Brandao and Kastoryano. This has wide-ranging applications
to the study of many-body in and out-of-equilibrium quantum systems. Our proof
relies upon a recently derived strong decay of correlations for Gibbs states of
one dimensional, translation-invariant local Hamiltonians, and tools from the
theory of operator spaces.
