Local Perturbations Perturb -Exponentially- Locally

TL;DR

This paper improves understanding of how local perturbations affect the ground states of gapped quantum spin systems, showing effects decay exponentially with distance, which refines previous sub-exponential bounds.

Contribution

It introduces a new exponential decay bound for the locality of perturbations on gapped quantum systems, enhancing prior sub-exponential estimates using advanced mathematical techniques.

Findings

Local perturbations affect ground states exponentially less as distance increases.

The new bounds improve understanding of stability in topologically ordered systems.

Correlations decay faster in disordered systems with impurities.

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.