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

Bruno Nachtergaele, Robert Sims+1 more|arXiv (Cornell University)|Oct 29, 2020

Quantum many-body systems124 references25 citations

TL;DR

This paper establishes quasi-locality bounds for quantum lattice systems, proving stability of the spectral gap in frustration-free spin models with gapped ground states under perturbations decaying faster than a stretched exponential. Using the Bravyi-Hastings-Michalakis (BHM) strategy, it extends previous results by introducing a refined Local Topological Quantum Order (LTQO) condition via the indistinguishability radius, enabling stability analysis for models with general boundary conditions and discrete symmetry breaking.

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, 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 the notion of 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.

Motivation & Objective

• . To extend the stability of the spectral gap in frustration-free quantum spin systems beyond periodic boundary conditions.
• . To establish gap stability for models exhibiting discrete symmetry breaking, where standard LTQO fails.
• . To generalize the BHM strategy to accommodate non-commuting, non-periodic, and symmetry-broken systems.
• . To introduce and formalize the indistinguishability radius as a tool for quantifying LTQO in finite and infinite systems.
• . To prove uniform, system-size-independent lower bounds on the spectral gap in the thermodynamic limit.

Proposed method

• . Applies the Bravyi-Hastings-Michalakis (BHM) strategy to analyze spectral gap stability under perturbations.
• . Introduces the indistinguishability radius as a new measure of Local Topological Quantum Order (LTQO), quantifying how far local observables can be distinguished from their infinite-volume limits.
• . Uses spectral flow and quasi-locality estimates to control the effect of perturbations on the ground state space.
• . Employs anchored interactions and form-bounded perturbation theory to handle long-range decay in interactions.
• . Establishes uniform finite-volume bounds and takes the thermodynamic limit via uniform sequences of systems.
• . Proves injectivity of the MPS map and uses matrix product state (MPS) techniques to estimate indistinguishability radii.

Experimental results

Research questions

• RQ1 . Can the spectral gap of a gapped, frustration-free quantum spin system be proven stable under perturbations that decay faster than a stretched exponential?
• RQ2 . How can the LTQO condition be reformulated to apply to systems with general boundary conditions and without periodicity?
• RQ3 . What is the role of the indistinguishability radius in quantifying the stability of topological order under perturbations?
• RQ4 . How does gap stability behave in the presence of spontaneous discrete symmetry breaking?
• RQ5 . Can uniform, system-size-independent lower bounds on the spectral gap be established in the thermodynamic limit?

Key findings

• . The spectral gap remains uniformly bounded below for all sufficiently small perturbations s when the interaction decays faster than a stretched exponential.
• . The indistinguishability radius is bounded from below by a function of the distance to the boundary, ensuring stability of the gap in finite systems.
• . For systems with unique ground states, the GNS Hamiltonian associated with the infinite-volume state has a non-zero spectral gap above the ground state.
• . In models with discrete symmetry breaking, the gap remains stable under symmetry-preserving perturbations, and the GNS Hamiltonian for each pure state has a non-zero spectral gap.
• . The paper proves that the LTQO condition, formulated via the indistinguishability radius, is sufficient for spectral gap stability even in the absence of periodic boundary conditions.
• . Explicit quantitative bounds are derived for the indistinguishability radius in matrix product state (MPS) systems, showing exponential decay in the distance from the region of interest.