Abstract We show that some gapped quantum many-body systems have a ground state degeneracy that is stable to long-range (e.g. power-law) perturbations, in the sense that any ground state energy splitting induced by such perturbations is exponentially small in the system size. More specifically, we consider an Ising symmetry-breaking Hamiltonian with several exactly degenerate ground states and an energy gap, and we then perturb the system with Ising symmetric long-range interactions. For these models we prove (a) the stability of the gap, and (b) that the residual splitting of the low-energy states below the gap is exponentially small in the system size. Our proof relies on a convergent polymer expansion that is adapted to handle the long-range interactions in our model. We also discuss applications of our result to several models of physical interest, including the Kitaev p-wave wire model perturbed by power-law density–density interactions with an exponent greater than 1.
We study the robustness of topological ground state degeneracy to long-range interactions in quantum many-body systems. We focus on slowly decaying two-body interactions that scale like a power-law
1/rα where
α is smaller than the spatial dimension; such interactions are beyond the reach of known stability theorems which only apply to short-range or rapidly decaying long-range perturbations. Our main result is a computation of the ground state splitting of several toy models, which are variants of the 1D Ising model
H=−∑iσizσi+1z+λ∑ij∣i−j∣−ασixσjx with
λ>0 and
α<1. In one variant, the power-law interactions are replaced by all-to-all interactions,
4Lαλ∑ijσixσjx, where
L is the system size, while the other variant has true power-law interactions but is built out of quantum rotors rather than Ising spins. These models are also closely connected to the Kitaev p-wave wire model with power-law density-density interactions. In these examples, we find that the splitting
δ scales like a stretched exponential
δ∼exp(−CL21+α).
Our computations are based on path integral techniques similar to the instanton method introduced by Coleman. We also study another toy model with long-range interactions that can be analyzed without path integral techniques and that shows similar behavior.
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.
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.
We prove that weak long-range many-body interactions preserve the bulk gap of a free fermion system and that a quasi-local spectral flow transports the free ground state to ground states of the interacting system. Let
Φh be a free fermionic interaction, possibly including pairing terms, whose infinite-volume Majorana coefficient operator
h has no spectrum in
(−g,g), and let
V be an even self-adjoint many-body interaction. For sufficiently small
∣λ∣, we construct pure ground states
ωλ of the interacting bulk system
Φλ=Φh+λV and quasi-local automorphisms
β0,λ such that
ωλ=ω0∘β0,λ, where
ω0 is the free ground state of
Φh. Each
ωλ has a GNS Hamiltonian with gap at least
g−C∗∣λ∣∥V∥∗, where
∥⋅∥∗ is the relevant interaction norm.
Our result allows polynomial decay of both the free coefficients and the many-body interaction. For two-site interactions on a uniformly discrete point set
Γ⊂Rd, a sufficient condition is decay as
(1+∣x−y∣)−ϖ with
ϖ>2d, for both the hopping and pairing coefficients and the interaction terms. The spectral-flow generator preserves the spatial exponent of the polynomial interaction norm, and its automorphisms satisfy locality bounds at that exponent. If
h and
V have stretched-exponential decay, so does the spectral-flow generator.
The proof constructs the spectral flow and establishes the gap through an excitation-number comparison, using polynomial propagation bounds for evolved interactions proved in a companion paper.
We bound the error in ground-state expectations of local observables caused by truncating the spatial tails of a gapped quantum lattice Hamiltonian. We show that the error is controlled by the interaction strength discarded near each site, rather than by the extensive norm of the omitted Hamiltonian. If the gap remains open along a path that removes the interaction tail, the resulting bounds are uniform in system size and extend, under suitable assumptions, to thermodynamic-limit ground states. The convergence rate reflects the decay of the interaction tail: it is algebraic for power-law interactions, superpolynomial for superpolynomial interactions, and exponential at any strictly smaller rate for exponentially decaying interactions. For superpolynomial interactions, we also derive a direct infinite-volume estimate using automorphic equivalence. The results extend to isolated low-energy sectors and parity-even fermionic systems. For two-body interactions decaying as
r−p in
d dimensions, we prove an error bound
O(R−(p−d)) for
p>2d, where
R is the truncation range. We construct a gapped non-translation-invariant example that saturates this scaling, showing that the bound is optimal for the general class considered. Our results quantify when finite-range truncations faithfully reproduce the local physics of gapped long-range systems.
It has been proved that in gapped ground states of locally interacting lattice quantum systems with a finite local Hilbert space, the effect of local perturbations decays exponentially with distance.However, in systems with power-law-(1/r α ) decaying interactions, no analogous statement has been shown and there are serious mathematical obstacles to proving it with existing methods.In this paper, we prove that when α exceeds the spatial dimension D, the effect of local perturbations on local properties a distance r away is upper bounded by a power law 1/r α 1 in gapped ground states, provided that the perturbations do not close the spectral gap.The power-law exponent α 1 is tight if α > 2D and interactions are two-body, where we have α 1 = α.The proof is enabled by a method that avoids the use of quasiadiabatic continuation and incorporates techniques of complex analysis.This method also improves bounds on ground-state correlation decay, even in short-range interacting systems.Our work generalizes the fundamental notion that local perturbations have local effects to power-law interacting systems, with broad implications for numerical simulations and experiments.