Quantum Markovian semigroups on quantum spin systems: Glauber dynamics

We study a class of KMS-symmetric quantum Markovian semigroups on a quantum spin system (A, τ, ω), where A is a quasi-local algebra, τ is a strongly continuous one parameter group of *-automorphisms of A and ω is a Gibbs state on A. The semigroups can be considered as the extension of semigroups on the nontrivial abelian subalgebra. Let H be a Hubert space corresponding to the GNS representation constructed from ω. Using the general construction method of Dirichlet form developed in [8], we construct the symmetric Markovian semigroup {T_t}_t≥0 on H.The semigroup {T_t}_t≥0 acts separately on two subspaces Hd and H_0d of H where H_d is the diagonal subspace and H _od is the off-diagonal subspace, H= H_d ⊕ H _od. The restriction of the semigroup {Tt}_t≥0 on H _d is Glauber dynamics, and for any η € H_0d, T_tη decays to zero exponentially fast as t approaches to the infinity.