Rapid thermalization of spin chain commuting Hamiltonians

Abstract

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. Our main result is a generalization to the quantum setting of a seminal result of Holley and Stroock for classical spin chains and represents an exponential improvement over bounds based on the non-closure of the spectral gap. From a physical point of view, our result rigorously establishes the absence of dissipative phase transition for Davies evolutions over translation-invariant spin chains. The result also applies in the case of Symmetry Protected Topological phases where the evolution is respecting the symmetry of the phase. This has wide-ranging applications to the study of many-body in and out-of-equilibrium quantum systems.

Quantum many-body systems in thermal equilibrium

“…On the other hand, a bound on the so-called log-Sobolev contant [168] implies that instead it takes a time logarithmic in the system size to thermalize. This was recently proven for 1D chains [169,170] and in [171,172] for other models of dissipation. These results show that the dissipative processes at hand can also be seen as an efficient quantum algorithms preparing thermal states, since in principle they can be simulated efficiently with a quantum computer [152].…”