Locality bounds for quantum dynamics at low energy

TL;DR

This paper establishes bounds on the speed of quantum dynamics at low energies in local Hamiltonian systems, showing that particle velocities scale with energy according to specific power laws, which has implications for understanding quantum slowdowns.

Contribution

It provides the first rigorous bounds on low-energy quantum dynamics velocities in local Hamiltonians, extending from single-particle quantum walks to many-body systems.

Findings

Butterfly velocity at low energies scales as E^{(2k-1)/2k} for certain Hamiltonians.

Derived bounds on particle velocities in many-body Hubbard-like models.

Results align with dimensional analysis expectations.

Abstract

We discuss the generic slowing down of quantum dynamics in low energy density states of spatially local Hamiltonians. Beginning with quantum walks of a single particle, we prove that for certain classes of Hamiltonians (deformations of lattice-regularized $H\propto p^{2k}$), the ``butterfly velocity" of particle motion at low energies has an upper bound that must scale as $E^{(2k-1)/2k}$, as expected from dimensional analysis. We generalize these results to obtain bounds on the typical velocities of particles in many-body systems with repulsive interactions, where for certain families of Hubbard-like models we obtain similar scaling.