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

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 & alpha;) 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 & alpha; 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 & alpha;1 in gapped ground states, provided that the perturbations do not close the spectral gap. The power-law exponent & alpha;1 is tight if & alpha; > 2D and interactions are two-body, where we have & alpha;1 = & alpha;. 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.