On Lieb-Robinson Bounds for the Double Bracket Flow

TL;DR

This paper explores Lieb-Robinson bounds for the double bracket flow, showing bounds hold for free fermion systems with exponential range growth, but convergence of the flow remains uncertain.

Contribution

It establishes Lieb-Robinson bounds for the double bracket flow in free fermion systems, highlighting exponential growth of the interaction range with the control parameter.

Findings

Lieb-Robinson bounds are proven for free fermion systems under the double bracket flow.

The interaction range increases exponentially with the flow parameter B.

Flow convergence to a limit is not guaranteed in the infinite volume limit.

Abstract

We consider the possibility of developing a Lieb-Robinson bound for the double bracket flow. This is a differential equation $$\partial_B H(B)=[[V,H(B)],H(B)]$$ which may be used to diagonalize Hamiltonians. Here, $V$ is fixed and $H(0)=H$. We argue (but do not prove) that $H(B)$ need not converge to a limit for nonzero real $B$ in the infinite volume limit, even assuming several conditions on $H(0)$. However, we prove Lieb-Robinson bounds for all $B$ for the double-bracket flow for free fermion systems, but the range increases \emph{exponentially} with the control parameter $B$.