# Stability of invertible, frustration-free ground states against large perturbations

Sven Bachmann1, Wojciech De Roeck2, Brecht Donvil3,4, and Martin Fraas5

1Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
2Institute of Theoretical Physics, K.U. Leuven, 3001 Leuven, Belgium
3Institute for Complex Quantum Systems and Center for IQST, Ulm University, 89069 Ulm, Germany
4Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
5Department of Mathematics, University of California, Davis, Davis, CA, 95616, USA

### Abstract

A gapped ground state of a quantum spin system has a natural length scale set by the gap. This length scale governs the decay of correlations. A common intuition is that this length scale also controls the spatial relaxation towards the ground state away from impurities or boundaries. The aim of this article is to take a step towards a proof of this intuition. We assume that the ground state is frustration-free and invertible, i.e. it has no long-range entanglement. Moreover, we assume the property that we are aiming to prove for one specific kind of boundary condition; namely open boundary conditions. This assumption is also known as the "local topological quantum order" (LTQO) condition. With these assumptions we can prove stretched exponential decay away from boundaries or impurities, for any of the ground states of the perturbed system. In contrast to most earlier results, we do not assume that the perturbations at the boundary or the impurity are small. In particular, the perturbed system itself can have long-range entanglement.

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### Cited by

[1] Joscha Henheik, Stefan Teufel, and Tom Wessel, "Local stability of ground states in locally gapped and weakly interacting quantum spin systems", Letters in Mathematical Physics 112 1, 9 (2022).

[2] Joscha Henheik and Tom Wessel, "On adiabatic theory for extended fermionic lattice systems", arXiv:2208.12220.

[3] Angelo Lucia, Alvin Moon, and Amanda Young, "Stability of the spectral gap and ground state indistinguishability for a decorated AKLT model", arXiv:2209.01141.

The above citations are from SAO/NASA ADS (last updated successfully 2022-10-01 13:38:55). The list may be incomplete as not all publishers provide suitable and complete citation data.

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