Locality of Gapped Ground States in Systems with Power-Law-Decaying Interactions

Abstract

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.