Rapid thermalization of spin chain commuting Hamiltonians

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We prove that spin chains weakly coupled to a large heat bath thermalize rapidly at any temperature for finite-range, translation-invariant commuting Hamiltonians, reaching equilibrium in a time which scales logarithmically with the system size. From a physical point of view, our result rigorously establishes the absence of dissipative phase transitions for Davies evolutions over translation-invariant spin chains. The result has also implications in the understanding of Symmetry Protected Topological phases for open quantum systems.
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