Nonconvergence conjecture for the double bracket flow in infinite spin systems

Consider a $d$-dimensional hypercubic lattice with translation-invariant Hamiltonians $H$ and $V$. For a sequence of lattices of increasing size, let $H(B)$ be the solution of the double bracket flow with initial Hamiltonian $H(0)=H$. Nonconvergence conjecture. There exist such $H$ and $V$ for which $H(B)$ does not converge to a translationally invariant Hamiltonian in the infinite system-size limit for any real $B\neq 0$. This failure of convergence may occur even when each site has a two-dimensional Hilbert space, $V$ is the sum of Pauli $Z$ operators over all sites, and $H=V+\epsilon\Delta$, where $\Delta$ is translationally invariant and $\epsilon$ is an arbitrarily small real scalar. The conjecture concerns whether a Lieb–Robinson-type locality bound can hold uniformly in system size for the double bracket flow; the preceding discussion establishes no such result and presents this claim only as a motivation for further investigation.