We prove that uniformly small shortrange perturbations do not close the bulk gap above the ground state of frustrationfree quantum spin systems that satisfy a standard local topological quantum order condition. In contrast with earlier results, we do not require a positive lower bound for finitesystem spectral gaps uniform in the system size. To obtain this result, we extend the Bravyi–Hastings–Michalakis strategy so it can be applied to perturbations of the GNS Hamiltonian of the infinitesystem ground state.
We study the stability with respect to a broad class of perturbations of gapped groundstate phases of quantum spin systems defined by frustrationfree 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 frustrationfree 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 finitevolume 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 nonzero spectral gap above the ground state.
more » « less Award ID(s):
 1813149
 NSFPAR ID:
 10287153
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Annales Henri Poincaré
 Volume:
 23
 Issue:
 2
 ISSN:
 14240637
 Page Range / eLocation ID:
 p. 393511
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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