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

非交换泛函分析低维无限自由度量子系统研究

基本信息

批准号：
12440049
负责人：
MATSUI Taku
金额：
$ 4.22万
依托单位：
Kyushu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2003
项目状态：
已结题

项目摘要

(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.

(i)我们证明了相关性均匀指数衰减状态的 Goderis-Verbeure-Vets 非交换中心极限定理。我们的结果可以应用于一维量子自旋链的 KMS 状态，以实现有限范围平移不变相互作用。我们的极限定理对于一类准局域（非局域）可观测量是有效的。(ii)我们得到了连续谱量子系统的涨落定理。对于吉布斯态，我们在有限时间内施加一个外部场，并且我们已经证明能量转移概率满足类似于加拉沃蒂-科恩涨落定理的不等式。我们在关键步骤中使用了冯·诺依曼代数的模理论。(iii)我们考虑了无限量子系统的某些非平衡稳态的表征。如果时间演化满足 L1 渐近阿贝尔条件，则非平衡稳态是 Zubarev 哈密顿量的 KMS 状态。然而，对于 L1 渐近阿贝尔条件无效的 XY 模型，对于任何连续时间演化，非平衡稳态都不能成为 KMS 状态。然而，这些态的特征在于形式Zubarev哈密顿量的自由能变分原理。(iv)我们获得了具有类lsing各向异性的一维XXZ模型的完整基态集。