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

TL;DR

This paper reviews recent advances in Lieb-Robinson bounds for Bose-Hubbard models, providing a simplified proof of a polynomial velocity bound that depends on lattice dimension and time.

Contribution

It offers a shorter, more accessible proof of a polynomial Lieb-Robinson velocity bound for Bose-Hubbard Hamiltonians, extending previous results.

Findings

Lieb-Robinson velocity is bounded by t^{d-1} for large times.

A simplified proof establishes a polynomial velocity bound of t^{d+ε}.

The results apply to general bounded-density initial states.

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 $t^{d+\epsilon}$.