Finite temperature quantum walk of a single particle

We study the dynamics of a single quantum particle in a d-dimensional space, generated by a local, time-independent Hamiltonian. We prove that any state projected onto low energy eigenstates must propagate slowly, with a β-dependent effective velocity that vanishes as β is taken to infinity. The β-dependence of this effective velocity matches the conjectured scaling of the analogous butterfly velocity which has been recently studied in the context of

quantum many-body chaos. We provide results for dynamics generated by Hamiltonians on a lattice and by Hamiltonians in the continuum.