Citation: Hastings, Matthew B., and Tohru Koma. "Spectral gap and exponential decay of correlations." Communications in mathematical physics 265 (2006): 781-804.
Web: https://arxiv.org/abs/math-ph/0507008
Tags: Lieb-Robinson
This paper proves some folklore theorems in manybody quantum physics, related to the exponential decay of correlations in the presence of a spectral gap. The idea of the proof is an extremely clever combination of the Lieb-Robinson bound and Fourier analysis.
The idea is to first relate the correlation function between two operators A and B to the expectation of the commutator [A,B+], where B+ is the positive part of B (upper triangular part in eigenbasis). If B+ was localized away from A we would be done - spatially distant operators commute. The issue is that B+ is not localized at all. To resolve this, one approximates B+ by a local operator. This is done by using Fourier analysis to express B+ as a time-integral of the time evolutions B(t) of B. An exponential suppressing term is then added, approximating B+ by a time-localized integral. Since time-localization implies space-localization by the Lieb-Robinson bound, B+ can thus be approximated by an operator approximately localized around the support of B. Hence [A,B+] is approximately zero so we are done.
This paper obviously does a lot more, and there is a good deal of hard analysis. The big idea is explained beautifully in this lecture by Hastings: https://www.youtube.com/watch?v=xejkb-feh8U.