QUANTUM MARKOVIAN SEMIGROUPS ON QUANTUM SPIN SYSTEMS: GLAUBER DYNAMICS

Title & Authors
QUANTUM MARKOVIAN SEMIGROUPS ON QUANTUM SPIN SYSTEMS: GLAUBER DYNAMICS
Choi, Veni; Ko, Chul-Ki; Park, Yong-Moon;

Abstract
We study a class of KMS-symmetric quantum Markovian semigroups on a quantum spin system ($\small{\mathcal{A},{\tau},{\omega}}$), where $\small{\mathcal{A}}$ is a quasi-local algebra, $\small{\tau}$ is a strongly continuous one parameter group of *-automorphisms of $\small{\mathcal{A}}$ and $\small{\omega}$ is a Gibbs state on $\small{\mathcal{A}}$. The semigroups can be considered as the extension of semi groups on the nontrivial abelian subalgebra. Let $\small{\mathcal{H}}$ be a Hilbert space corresponding to the GNS representation con structed from $\small{\omega}$. Using the general construction method of Dirichlet form developed in [8], we construct the symmetric Markovian semigroup $\small{\{T_t\}{_t_\geq_0}}$ on $\small{\mathcal{H}}$. The semigroup $\small{\{T_t\}{_t_\geq_0}}$ acts separately on two subspaces $\small{\mathcal{H}_d}$ and $\small{\mathcal{H}_{od}}$ of $\small{\mathcal{H}}$, where $\small{\mathcal{H}_d}$ is the diagonal subspace and $\small{\mathcal{H}_{od}}$ is the off-diagonal subspace, $\small{\mathcal{H}=\mathcal{H}_d\;{\bigoplus}\;\mathcal{H}_{od}}$. The restriction of the semigroup $\small{\{T_t\}{_t_\geq_0}}$ on $\small{\mathcal{H}_d}$ is Glauber dynamics, and for any $\small{{\eta}{\in}\mathcal{H}_{od}}$, $\small{T_t{\eta}}$, decays to zero exponentially fast as t approaches to the infinity.
Keywords
KMS symmetric quantum Markovian semigroups;quantum spin systems;diagonal subspace;Glauber dynamics;
Language
English
Cited by
1.
Linear and Nonlinear Dissipative Dynamics, Reports on Mathematical Physics, 2016, 77, 3, 377
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