Study of Low Dimensional Quantum Systems with Infinite Degree of Freedom by Non-Commutative Functional Analysis

点击翻译按钮获取中文摘要

(i)We proved the non-commutative central limit theorem of Goderis-Verbeure-Vets for states with uniform exponential decay of correlation. Our results can be applied to the KMS state of one-dimensional quantum spin chains for finite range translationally invariant interactions. Our limit theorem is valid for a class of quasilocal(non local) observables.(ii)We obtained a fluctuation theorem for quantum systems with continuous spectrum. For a Gibbs state we impose an external field for a finite time and we have shown that the energy transition probability satisfies an inequality similar to Gallavotti-Cohen's fluctuation theorem. We use the modular theory of von Neumann algebras in crucial steps.(iii)We consider characterization of certain non-equilibrium steady states of infinite quantum systems. If the time evolution satisfies the L1 asymptotic abelian condition, the non-equilibrium steady states are the KMS states for the Zubarev Hamiltonian. However, for the XY model in which the L1 asymptotic abelian condition is not valid, the non-equilibrium steady states fail to be KMS states for any continuous time evolution. Nevertheless, the states are characterized by the variational principle of free energy for a formal Zubarev Hamiltonian.(iv)We obtained the complete set of ground states for the one-dimensional XXZ model with lsing-like anisotropy.