Stability of Gapped One-Dimensional Quantum Spin Systems

Abstract: An important class of models studied in quantum statistical mechanics consists of frustration-free interactions whose Hamiltonians exhibit a gap in the spectrum above their ground state energies. The presence of a spectral gap in a quantum spin system has significant implications for the physical properties that the model demonstrates and plays a particularly strong role in the study of topological phases of matter. Given the exponential growth of the dimension of the configuration space with system size, the question of whether a model possesses a spectral gap, and a derivation of a lower bound on that quantity, is generally difficult. While the existence of a gap has been verified for several interesting models, the question of stability of these models was, until recent years, limited to a restrictive class of models or required strong assumptions on the perturbations. Among such models with a uniform spectral gap is the well-studied Heisenberg XXZ spin chain, which interpolates between major physical regimes. In this talk, we discuss recent techniques which dramatically extend the class of models to which stability applies for general perturbations. We will present a result for the stability of frustration-free, one-dimensional models in the form of an estimate uniform in the system size, together with conditions ensuring the existence of a pure perturbed ground state whose GNS Hamiltonian remains gapped. As an application, we demonstrate the stability of the XXZ model for perturbations respecting a local gauge symmetry and discuss challenges to additional conjectured stability results.