Effective dissipation rate in a Liouvillian graph picture of high-temperature quantum hydrodynamics

TL;DR

This paper presents a graph-based framework to understand how high-temperature quantum spin systems exhibit classical hydrodynamics, linking microscopic unitary dynamics to emergent dissipative behavior.

Contribution

It introduces the Liouvillean graph concept to quantitatively connect microscopic quantum dynamics with macroscopic dissipative hydrodynamics.

Findings

Liouvillean graph decomposes into subgraphs of constant diameter

Operator weight spreads to long operators at a rate determined by subgraph dynamics

Provides a model with Hilbert space dimension linear in system size

Abstract

At high temperature, generic strongly interacting spin systems are expected to display hydrodynamics: local transport of conserved quantities, governed by classical partial differential equations like the diffusion equation. I argue that the emergence of this dissipative long-wavelength dynamics from the system's unitary microscopic dynamics is controlled by the structure of the \textit{Liouvillean graph} of the system's Hamiltonian, that is, the graph induced on Pauli strings by commutation with that Hamiltonian. The Liouvillean graph decomposes naturally into subgraphs of Pauli strings of constant diameter, and the coherent dynamics of these subgraphs determines the rate at which operator weight spreads to long operators. This argument provides a quantitative theory of the emergence of a dissipative effective dynamics from unitary microscopic dynamics; it also leads to an effective model with Hilbert space dimension linear in system size and exponential in the UV cutoff for diffusion.