On Lieb-Robinson Bounds for the Double Bracket Flow

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 $B$.

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