Local Perturbations Perturb -Exponentially- Locally

Abstract

We elaborate on the principle that for gapped quantum spin systems with local interaction "local perturbations [in the Hamiltonian] perturb locally [the ground state]". This principle was established in [Bachmann et al. 2012], relying on the `spectral flow technique' or `quasi-adiabatic continuation' [Hastings 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. 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 ground state.

v1 --> v2: introduction revised, main text rearranged, typos corrected, matches published version

References in corpus (6)

• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Quasi-adiabatic Continuation for Disordered Systems: Applications to Correlations, Lieb-Schultz-Mattis, and Hall Conductance
• Much Ado About Something: Why Lieb-Robinson bounds are useful
• Quasi-Adiabatic Continuation in Gapped Spin and Fermion Systems: Goldstone's Theorem and Flux Periodicity
• Approximating the ground state of gapped quantum spin systems

Cited by in corpus (14)

• Floquet time crystals in clock models
• Quasi-Locality Bounds for Quantum Lattice Systems. Part I. Lieb-Robinson Bounds, Quasi-Local Maps, and Spectral Flow Automorphisms
• Shortcuts to Adiabaticity in Krylov Space
• Rational indices for quantum ground state sectors
• Local stability of ground states in locally gapped and weakly interacting quantum spin systems
• An exponentially local spectral flow for possibly non-self-adjoint perturbations of non-interacting quantum spins, inspired by KAM theory
• From decay of correlations to locality and stability of the Gibbs state
• Stability of invertible, frustration-free ground states against large perturbations
• Slow propagation in some disordered quantum spin chains
• Locality of gapped ground states in systems with power-law decaying interactions
• Fighting Exponentially Small Gaps by Counterdiabatic Driving
• Stability of thermal equilibrium in long-range quantum systems
• The robustness condition for general disordered discrete time crystals, and subspace-thermal DTCs from phase transitions between different n-tuple DTCs
• A Shielding Property for Thermal Equilibrium States on the Quantum Ising Model