Lieb-Robinson bounds for Bose-Hubbard Hamiltonians: A review with a simplified proof

Abstract

We review recent progress on state-dependent Lieb--Robinson bounds for Bose--Hubbard Hamiltonians. In particular, Kuwahara, Vu, and Saito established that, for general bounded-density initial states, the Lieb--Robinson velocity is bounded by $t^{d-1}$ for large times, where $d$ denotes the lattice dimension. We present a shorter proof of the weaker, but still polynomial velocity bound $t^{d+\epsilon }$.

32 pages, 1 figure. final version to appear in PAFA special issue dedicated to Barry Simon on the occasion of his 80th birthday