量子格点系统中带隙相的自守等价性
Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems
摘要：
量子自旋系统的带隙基态在物理学文献中被称为“同相”，如果存在一族哈密顿量H（s），有限范围的相互作用连续地依赖于$s \in [0,1]$，使得对于每个$s$，H（s）在其基态之上具有非零的带隙，并且两个初始态分别是H（0）和H（1）的基态。在这项工作中，我们给出了精确的条件下，任何两个带隙基态的一个给定的量子自旋系统，“属于同一阶段”是自同构等价的，并表明这种等价性可以实现为一个流产生的$s$依赖的相互作用，衰减速度比任何幂律（事实上，几乎指数）。流构造使用Hastings的“准绝热演化”技术，其中我们给出了一个证明扩展到无限维希尔伯特空间。此外，我们还得到了关于具有准局域结构的量子系统动力学扰动效应的局域性的一般结果，并证明了连接处于同一相位的带隙基态的{ \em谱流}满足Lieb-Robinson界.结果表明，在热力学极限下，谱流收敛于无限自旋系统的准定域观测量代数的自同构的余圈。这证明了在模型$H(s), 0\leq s\leq 1$曲线上基态相结构是保持不变的。
英文摘要：
Gapped ground states of quantum spin systems have been referred to in the physics literature as being `in the same phase' if there exists a family of Hamiltonians H(s), with finite range interactions depending continuously on $s \in [0,1]$, such that for each $s$, H(s) has a non-vanishing gap above its ground state and with the two initial states being the ground states of H(0) and H(1), respectively. In this work, we give precise conditions under which any two gapped ground states of a given quantum spin system that 'belong to the same phase' are automorphically equivalent and show that this equivalence can be implemented as a flow generated by an $s$-dependent interaction which decays faster than any power law (in fact, almost exponentially). The flow is constructed using Hastings' 'quasi-adiabatic evolution' technique, of which we give a proof extended to infinite-dimensional Hilbert spaces. In addition, we derive a general result about the locality properties of the effect of perturbations of the dynamics for quantum systems with a quasi-local structure and prove that the flow, which we call the {\em spectral flow}, connecting the gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a co-cycle of automorphisms of the algebra of quasi-local observables of the infinite spin system. This proves that the ground state phase structure is preserved along the curve of models $H(s), 0\leq s\leq 1$.
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