Lieb-Robinson Bounds and the Exponential Clustering Theorem

One of the folk theorems in quantum lattice models claims the equivalence of the existence of a nonvanishing spectral gap and exponential decay of spatial correlations in the ground state. It has been known for some time that there are exceptions to one direction of this equivalence. There are models with a unique ground state with exponential decay of correlations but without a gap in the excitation spectrum above the ground state. For a simple example see Example 2 in [2, p 596]. In this paper we provide a rigorous proof of the other implication: a spectral gap implies exponential decay in the ground state. In relativistic quantum field theory it has been known for a long time that the existence of a mass (energy) gap implies exponential decay of spatial correlations. For example, in [1] Fredenhagen proves a general theorem applicable to arbitrary strictly local theories. It is the strict locality, i.e., the fact that space-like separated observables commute, not the relativistic invariance per se, which plays a crucial role in the proof of exponential decay. Non-relativistic models of statistical mechanics do not have strict locality, but there is a finite speed of propagation up to exponentially small corrections. This was first proven by Lieb and Robinson [13]. It is not a surprise that the Lieb-Robinson bound can replace the strict locality property. In particular, Wreszinski relied on it to prove a Goldstone Theorem in non-relativistic quantum statistical mechanics [9]. Precisely how to apply the Lieb-Robinson result to work around non-locality is not entirely obvious. Only recently, Hastings used it to derive exponential clustering for lattice models with a gap [12], and to