Stability of slow Hamiltonian dynamics from Lieb-Robinson bounds

TL;DR

This paper proves that slow Hamiltonian dynamics in local spin systems are stable against local perturbations by analyzing Lieb-Robinson bounds, with implications for many-body localization and ergodic regions.

Contribution

It introduces a non-perturbative approach to demonstrate the stability of slow dynamics using Lieb-Robinson bounds, applicable even with significant perturbations.

Findings

Slow dynamics are stable under local perturbations.

Presence of disorder regions slows down local dynamics.

The approach applies to systems with ergodic regions and many-body localization.

Abstract

We rigorously show that a local spin system giving rise to a slow Hamiltonian dynamics is stable against generic, even time-dependent, local perturbations. The sum of these perturbations can cover a significant amount of the system's size. The stability of the slow dynamics follows from proving that the Lieb-Robinson bound for the dynamics of the total Hamiltonian is the sum of two contributions: the Lieb-Robinson bound of the unperturbed dynamics and an additional term coming from the Lieb-Robinson bound of the perturbations with respect to the unperturbed Hamiltonian. Our results are particularly relevant in the context of the study of the stability of Many-Body-Localized systems, implying that if a so called ergodic region is present in the system, to spread across a certain distance it takes a time proportional to the exponential of such distance. The non-perturbative nature of our result allows us to develop a dual description of the dynamics of a system. As a consequence we are able to prove that the presence of a region of disorder in a ergodic system implies the slowing down of the dynamics in the vicinity of that region.