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.
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.
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.
We consider bulk-gapped fermion systems whose infinite-volume bulk is obtained by adding sufficiently weak even many-body interactions to a free Hamiltonian with spectral gap
g>0, allowing polynomial spatial decay of both parts. We prove quantitative local determination of low-energy states: whenever another interaction agrees with the bulk interaction inside a finite or infinite region
U, with bulk terms crossing its boundary unchanged or removed, all its states with excitation energies in
[0,3g/4] have local expectations close to those of the bulk ground state deep inside
U. The error decays with the distance to the complement of
U, independently of the interaction terms supported outside
U and of the chosen low-energy state.
Quantitative local determination has strong consequences. (i) The bulk interaction has a unique algebraic ground state. (ii) A strong "local perturbations perturb locally" (LPPL) principle holds: arbitrary distant defects change local ground-state expectations only by a spatially decaying error. (iii) Low-energy excitations in finite samples remain localized near the boundary for all times, up to spatially decaying tails, uniformly over arbitrary even boundary terms. (iv) Correlations from deep inside
U to outside
U are small, with quantitative bounds in terms of spatial separation and distance to
Uc.
To prove local determination of low-energy states, we combine quasi-adiabatic transport of the free ground state to the interacting bulk ground state with a new condition, local excitation detectability (LED). A companion paper establishes gap stability and constructs the transport for weak interactions, while we prove LED at sufficiently weak coupling. The abstract theorem is nonperturbative and applies whenever the transport and LED hypotheses hold.
We elaborate on the principle that for gapped quantum spin systems with local interaction, “local perturbations [in the Hamiltonian] perturb locally [the groundstate].” This principle was established by Bachmann et al. [Commun. Math. Phys. 309, 835–871 (2012)], relying on the “spectral flow technique” or “quasi-adiabatic continuation” [M. B. Hastings, Phys. Rev. B 69, 104431 (2004)] to obtain locality estimates with sub-exponential decay in the distance to the spatial support of the perturbation. We use ideas of Hamza et al. [J. Math. Phys. 50, 095213 (2009)] to obtain similarly a transformation between gapped eigenvectors and their perturbations that is local with exponential decay. This allows to improve locality bounds on the effect of perturbations on the low lying states in certain gapped models with a unique “bulk ground state” or “topological quantum order.” We also give some estimate on the exponential decay of correlations in models with impurities where some relevant correlations decay faster than one would naively infer from the global gap of the system, as one also expects in disordered systems with a localized groundstate.
This thesis analyses several locality properties of extended quantum lattice systems, considering both spins and fermions. We begin by assuming locality of the Hamiltonian and then discuss how this translates into locality of the time evolution and the thermal states. Starting from a local Hamiltonian, it is well known that the Heisenberg time evolution satisfies Lieb-Robinson bounds, which quantify its locality. We strengthen these bounds for long-range interacting fermions and for commuting interactions. Furthermore, we establish the first Lieb-Robinson bounds for Hamiltonians whose local terms can grow linearly away from the origin. We also prove stability of Lieb-Robinson bounds with respect to local perturbations in the Hamiltonian. For the thermal states we focus on three types of locality known as decay of correlations, the local perturbations perturb locally (LPPL) principle and local indistinguishability. As a key tool we use spectral filters, which were introduced by Hastings in the 2000s for various applications. It enable us to use Lieb-Robinson bounds to obtain information about locality properties of states. In particular, we use quantum belief propagation to prove an equivalence of decay of correlations, LPPL and local indistinguishability for Gibbs states at any positive temperature. Using the improved Lieb-Robinson bounds mentioned before, we obtain improvements for decay of correlations in the ground state, the locality of the spectral flow and LPPL for long-range interacting systems and commuting Hamiltonians. Furthermore, we adjust the filter functions to achieve exponential decay for certain applications that were previously limited to stretched exponential decay. Spectral filters are also used to rigorously justify response theory, which usually requires a gapped system to begin with. To relax this assumption, we introduce various notions of local gaps, discuss their relations and use one to justify response theory and Kubo’s formula in the absence of a global gap. In another result we use local indistinguishability as an input to prove bulk-edge correspondence at finite temperature, in the sense that the bulk magnetisation is equal to the edge current. This result has mostly been studied for non-interacting models or at zero temperature before. Finally, we present a class of weakly interacting spin systems with an on-site gap and prove that the ground state satisfies a strong form of LPPL. Moreover, we show that these systems satisfy decay of correlations and local indistinguishability uniformly in temperature.