On Lieb–Robinson bounds for the double bracket flow

Abstract

We consider the possibility of developing a Lieb–Robinson bound for the double bracket flow (Brockett, 1991; Chu and Driessel, 1990). This is a differential equation ${\partial_{B}H(B)\mkern 1.0mu{=}\mkern2mu[[V,H(B)]},{H(B)]}$ 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 exponentially with the control parameter $B$.