Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
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DOI：
10.1007/s00023-021-01086-5
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发表时间：
2020-10
期刊：
影响因子：
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通讯作者：
B. Nachtergaele;Robert Sims;Amanda Young
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文献类型：
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作者：
B. Nachtergaele;Robert Sims;Amanda Young
我们研究了由无挫折哈密顿量定义的量子自旋系统的间隙基态相相对于一类广泛扰动的稳定性。这项工作的核心结果是使用Bravyi-Hastings-Michalakis （BHM）策略证明了在局部拓扑量子有序（LTQO）条件下，体隙在远距离衰减速度比拉伸指数更快的扰动下是稳定的。与以前的工作相比，我们扩展了可处理的无挫折量子自旋模型的类别，包括具有更一般边界条件的模型和具有离散对称破缺的模型。详细的估计使我们能够为间隙的正下界的有效性制定充分的条件，该下界在系统大小中是均匀的，并且在某种程度上是明确的。我们根据Michalakis和Zwolak的方法对BHM策略进行了调查，并引入了一些改变，以适应更一般的而不仅仅是周期边界条件和更一般的晶格。我们通过引入不可分辨半径来表达LTQO的基本条件。利用均匀有限体积的结果，我们接着研究热力学极限。我们首先研究了唯一极限基态的情况，然后也考虑了具有离散对称的自发破缺的模型。在后一种情况下，LTQO不能适用于所有的本地可观察对象。然而，对于保持对称性的扰动，我们显示了间隙和破缺对称相结构的稳定性。我们证明了与每个纯态相关的GNS哈密顿量在基态以上具有非零谱隙。
We study the stability with respect to a broad class of perturbations of gapped ground-state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi–Hastings–Michalakis (BHM) strategy that under a condition of local topological quantum order (LTQO), the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work, we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of an indistinguishability radius, which we introduce. Using the uniform finite-volume results, we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state.