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 quantum systems, 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.