Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States

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.

Stability of the bulk gap for frustration-free topologically ordered quantum lattice systems

“…Both these maps depend on the choice of γ through their weight functions, w γ and W γ respectively, but we suppress this in the notation. Arguing as in [26, Section VI.E.1], see also [25,Section 4.3.2], we find that for all A ∈ A (3.17)…”

“…As explained in detail in [25,Section 8], if both the initial Hamiltonian and the perturbation (see below) have a local gauge symmetry, only observables A that commute with this symmetry need to satisfy (2.21). Other discrete symmetries can be treated similarly (see [25,Section 8]). Therefore, the stability results proved here (Theorems 2.7 and 2.8) will also hold for symmetry-protected topological phases.…”

“…Previously, it was shown how certain cases can be handled by considering sequences of finite systems with suitable boundary conditions. For example, such an approach may work if the edge states are absent in the model considered with periodic boundary conditions [21,25]. In general, however, we may not know a suitable boundary condition that 'gaps out' the boundary modes.…”

“…We follow the strategy of Bravyi, Hastings, and Michalakis [7,8,21] and use the techniques we developed in [25,26] to handle the infinite system setting. From a certain perspective, and apart from the technical aspects to deal with unbounded Hamiltonians, the infinite system setting allows for a simplification in the statement of conditions and the main result.…”

Block-diagonalization of infinite-volume lattice Hamiltonians with unbounded interactions

“…The motivation of our analysis comes from recent studies of Hamiltonians of "topological insulators" appearing in the characterization of "topological phases", see e.g. [BN,BH,BHM,NSY2,NSY3]. However, the scope of our techniques is actually more general as shown in the present paper focused on boson systems.…”