# An upper bound on transport

## TL;DR

This paper establishes an upper bound on diffusivity in quantum systems based on operator growth velocity and local equilibration time, linking transport properties to fundamental dynamical constraints.

## Contribution

It introduces a universal bound on diffusivity derived from operator growth and equilibration times, applicable across various strongly and weakly interacting systems.

## Key findings

- Bound is obeyed in weakly and strongly interacting theories.
- In holography, the bound relates hydrodynamic and non-hydrodynamic modes.
- Connects transport coefficients to local equilibration times, explaining phenomena like linear resistivity.

## Abstract

The linear growth of operators in local quantum systems leads to an effective lightcone even if the system is non-relativistic. We show that consistency of diffusive transport with this lightcone places an upper bound on the diffusivity: $D \lesssim v^2 \tau_\text{eq}$. The operator growth velocity $v$ defines the lightcone and $\tau_\text{eq}$ is the local equilibration timescale, beyond which the dynamics of conserved densities is diffusive. We verify that the bound is obeyed in various weakly and strongly interacting theories. In holographic models this bound establishes a relation between the hydrodynamic and leading non-hydrodynamic quasinormal modes of planar black holes. Our bound relates transport data --- including the electrical resistivity and the shear viscosity --- to the local equilibration time, even in the absence of a quasiparticle description. In this way, the bound sheds light on the observed $T$-linear resistivity of many unconventional metals, the shear viscosity of the quark-gluon plasma and the spin transport of unitary fermions.