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QUANTUM MARKOVIAN SEMIGROUPS ON QUANTUM SPIN SYSTEMS: GLAUBER DYNAMICS

  • Choi, Veni ;
  • Ko, Chul-Ki ;
  • Park, Yong-Moon
  • Published : 2008.07.31

Abstract

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

Keywords

KMS symmetric quantum Markovian semigroups;quantum spin systems;diagonal subspace;Glauber dynamics

References

  1. L. Accardi and S. Koyzyrev, Lectures on quantum interacting particle systems, Quantum interacting particle systems (Trento, 2000), 1-195, QP-PQ: Quantum Probab. White Noise Anal., 14, World Sci. Publ., River Edge, NJ, 2002 https://doi.org/10.1142/9789812776853_0001
  2. S. Albeverio and R. Hoegh-Krohn, Dirichlet forms and Markovian semigroups on $C^{\ast}$-algebras, Comm. Math. Phys. 56 (1997), 173-187 https://doi.org/10.1007/BF01611502
  3. C. Bahn, C. K. Ko, and Y. M. Park, Dirichlet forms and symmetric Markovian semigroups on CCR algebras with respect to quasi-free states, J. Math. Phys. 44 (2003), no. 2, 723-753 https://doi.org/10.1063/1.1532770
  4. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics. 2, Equilibrium states. Models in quantum statistical mechanics. Second edition. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1997
  5. R. Carbone, F. Fagnola, and S. Hachicha, Generic quantum Markov semigroups: the Gaussian quage invariant case, preprint
  6. F. Cipriani, Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebras, J. Funct. Anal. 147 (1997), no. 2, 259-300 https://doi.org/10.1006/jfan.1996.3063
  7. Y. M. Park, Construction of Dirichlet forms on standard forms of von Neumann algebras, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), no. 1, 1-14 https://doi.org/10.1142/S0219025700000029
  8. K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhauser, Basel, 1992
  9. D. Goderis and C. Maes, Constructing quantum dissipations and their reversible states from classical interacting spin systems, Ann. Inst. H. Poincare Phys. Theor. 55 (1991), no. 3, 805-828

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