Abstract
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.
Highlights
We discuss in some detail how the BHM strategy can be adapted to the situation with discrete symmetry breaking of three types: (S1) local symmetries such as spin flip, discrete spin rotation, time-reversal etc; (S2) breaking of lattice translations to a subgroup leading to periodic ground states; (S3) other lattice symmetries such as reflections and lattice rotations. In each of these cases we show that, if the unperturbed model has the symmetry and it is spontaneously broken in the ground states, the spectral gap and the symmetry breaking are stable under perturbations that possess the symmetry
Our emphasis is on general ideas that work for large classes of systems and we illustrate the application of these ideas by presenting detailed arguments that cover the known results and, allow us to provide a number of generalizations and new results that can be obtained using the same principles. In support of this goal, we describe the more or less standard mathematical framework for studying quantum spin systems and discuss the basic notions that feature in the stability properties of the spectral gap above the ground state(s)
Theorem 6.8, Corollary 6.9, and Theorem 6.12 specify conditions under which we have a uniform positive lower bound for the spectral gap of a family of perturbed Hamiltonians defined on a sequence of finite volumes Λn ↑ Γ
Summary
The Toric Code model [55] was the first test case for proving this type of stability in the presence of topological order It has a unique frustration-free ground state on the infinite lattice Z2 [4], but finite systems have multiple ground states and the ground-state degeneracy is strongly dependent on the boundary conditions. Frohlich and Pizzo introduced a method that handles a class of unbounded one-dimensional lattice Hamiltonians with ease as long as the unperturbed ground state is unique and given by a product state The latter restriction excludes non-trivial order, topological or otherwise, and, naturally, any version of an LTQO condition is automatically satisfied. We will present a direct approach to the bulk gap in the infinite system that bypasses this difficulty in a forthcoming paper [81]
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