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

(i)We证明了Goderis-Verbeure-Vets关于关联一致指数衰减态的非对易中心极限定理。我们的结果可以应用于有限程非线性相互作用下一维量子自旋链的KMS态。我们的极限定理对一类拟局部（非局部）可观量是有效的。（ii）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场，我们证明了能量跃迁概率满足类似于Gallavotti-Cohen涨落定理的不等式。我们使用冯诺依曼代数的模块化理论的关键步骤。（iii）我们考虑无限量子系统的某些非平衡定态的特征。如果时间演化满足L1渐近阿贝尔条件，则非平衡定态是Zubarev哈密顿量的KMS态。然而，对于XY模型，其中的L1渐近阿贝尔条件是不成立的，非平衡定态不能为KMS态的任何连续时间的演变。然而，状态的特点是由一个正式的Zubarev哈密顿的自由能变分原理。（iv）得到了一维类lsing-like各向异性XXZ模型的完整基态。

(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.