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.

我们研究了由无挫折哈密顿量定义的量子自旋系统的有隙基态相位的广泛扰动的稳定性。这项工作的核心结果是使用 Bravyi-Hastings-Michalakis (BHM) 策略证明，在局部拓扑量子阶 (LTQO) 条件下，体隙在远距离衰减比拉伸指数更快的扰动下是稳定的. 与之前的工作相比，我们扩展了可以处理的无挫折量子自旋模型类别，包括具有更一般边界条件的模型和具有离散对称破坏的模型。详细的估计使我们能够为差距的正下限的有效性制定充分的条件，这些条件在系统规模上是一致的，并且在某种程度上是明确的。我们按照 Michalakis 和 Zwolak 的方法提供了 BHM 策略的调查，引入了改变以适应更一般的，而不仅仅是周期性边界条件和更一般的晶格。我们通过引入的不可区分性半径来表达称为 LTQO 的基本条件。使用均匀有限体积结果，我们然后继续研究热力学极限。我们首先研究了独特的极限基态的情况，然后还考虑了离散对称性自发破坏的模型。在后一种情况下，LTQO 不能适用于所有本地可观察量。然而，对于保持对称性的扰动，我们展示了间隙的稳定性和破坏对称相位的结构。