On stability of gapped states in quantum spin systems

Yoshiko Ogata

P. Lemmens, M. Fischer, M. Grove +5 more

Amanda Young

This work provides an overview of gapped quantum spin systems, including concepts, techniques, properties, and results. The basic framework and objects of interest for quantum spin systems are introduced, and the main ideas behind methods for proving spectral gaps for frustration-free models are outlined. After reviewing recent progress on several spectral gap conjectures, we discuss quasi-locality of the Heisenberg dynamics and its utility in proving properties of gapped quantum spin systems. Lieb-Robinson bounds have played a central role in establishing exponential decay of ground state correlations, an area law for one-dimensional systems, a many-body adiabatic theorem, and spectral gap stability. They also aided in the development of the quasi-adiabatic continuation, which is a useful for investigating gapped ground state phases, both of which are also discussed.

Tarun Grover

Joscha Henheik, Stefan Teufel, Tom Wessel

Abstract Based on a result by Yarotsky (J Stat Phys 118, 2005), we prove that localized but otherwise arbitrary perturbations of weakly interacting quantum spin systems with uniformly gapped on-site terms change the ground state of such a system only locally, even if they close the spectral gap. We call this astrong versionof thelocal perturbations perturb locally(LPPL) principle which is known to hold for much more general gapped systems, but only for perturbations that do not close the spectral gap of the Hamiltonian. We also extend this strong LPPL-principle to Hamiltonians that have the appropriate structure of gapped on-site terms and weak interactions only locally in some region of space. While our results are technically corollaries to a theorem of Yarotsky, we expect that the paradigm of systems with a locally gapped ground state that is completely insensitive to the form of the Hamiltonian elsewhere extends to other situations and has important physical consequences.

Samuel Herring

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. This paper discusses recent techniques which dramatically extend the class of models to which stability applies when the perturbations exhibit stretched exponential decay at long distances. We consider a class of initial models satisfying an assumption that prevents splitting of the ground state space. This assumption on the structure on the ground state space is referred to as local topological quantum order. A key tool in our analysis is the quasi-local unitary spectral flow automorphism which allows us to decompose the perturbed Hamiltonian into a perturbation model satisfying a relative form bound. We will present a result for the stability of frustration-free, one-dimensional models of this type in the form of an estimate uniform in the system size. We also present additional conditions ensuring the existence of a pure perturbed ground state whose GNS Hamiltonian remains gapped. As an application, we evaluate the stability of the Heisenberg XXZ spin-1/2 chain. Our work shows a new result on the stability of the XXZ model for perturbations respecting a local gauge symmetry and the stability above the translation invariant ground states in the thermodynamic limit. We also discuss challenges to additional conjectured stability results for the XXZ model.