Dynamic scaling relation in quantum many-body systems

TL;DR

This paper establishes a dynamic scaling relation for quantum surface roughness and mean-square displacement in many-body systems, linking their growth behaviors and classifying diffusive systems within the Edwards-Wilkinson universality class.

Contribution

It derives a new dynamic scaling relation connecting quantum surface roughness and transport exponents, supported by numerical verification in multiple models.

Findings

Quantum surface roughness and displacement follow Family-Vicsek scaling.

Diffusive systems are classified within the Edwards-Wilkinson universality class.

The scaling relation aids in assessing quantum transport experimentally.

Abstract

In delocalized systems, particle number fluctuations, also known as quantum surface roughness, and the mean-square displacement exhibit a temporal power-law growth followed by a saturation to a system-size-dependent value. We use simple scaling arguments to show that these quantities satisfy the Family-Vicsek scaling law and derive a dynamic scaling relation between the dynamical exponents, assuming that the saturation times of both quantities scale similarly with the system size. This relation clarifies the mechanism behind quantum surface roughness growth and suggests that diffusive quantum many-body systems belong to the Edwards-Wilkinson universality class. Moreover, it provides a convenient way to assess quantum transport in cold-atom experiments. We numerically verify our results by studying two non-interacting models and one interacting model having regimes with distinct dynamical exponents.