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Infinite Dimensional Analysis Quantum Probability and Related Topics

MARKOVIAN KMS-STATES FOR ONE-DIMENSIONAL SPIN CHAINS

Luigi Accardi, Volkmar Liebscher

Published Dec 1, 1999• 31 citations
Publisher

Abstract

We characterize a class of quantum Markov states in terms of a locality property of their modular automorphism group or, equivalently, of their φ-conditional expectations and we give an explicit description of the structure of these states. This study is meant as a starting point for the investigation of the structure of Markovian KMS-states of quantum spin chains as well as of multidimensional quantum spin lattices.

DOI
10.1142/s0219025799000369

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We characterize a class of quantum Markov states in terms of a locality property of their modular automorphism group or, equivalently, of their φ-conditional expectations and we give an explicit description of the structure of these states. This study is meant as a starting point for the investigation of the structure of Markovian KMS-states of quantum spin chains as well as of multidimensional quantum spin lattices.
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A KMS-symmetric quantum Markov semigroup is completely determined by the quadratic form in induces on the GNS Hilbert space. The quadratic forms that arise in this way are called quantum Dirichlet forms. In this article we give a complete description of bounded quantum Dirichlet forms in terms of derivations. In particular, we exhibit an explicit characterization of the bounded quantum Dirichlet forms on type I factors. We also show that any unbounded quantum Dirichlet form can be represented in terms of derivations. The key technical insight is that the modular group always restricts to a strongly continuous group on the domain of a quantum Dirichlet form such that the square root of the analytic generator is accretive with respect to the form norm.
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