# Resistivity bound for hydrodynamic bad metals

## TL;DR

This paper derives a universal upper bound on the resistivity of hydrodynamic electron fluids, explaining common temperature-dependent behaviors in various correlated materials without relying on umklapp scattering.

## Contribution

It introduces a new resistivity bound based on hydrodynamic principles, linking microscopic scattering rates to macroscopic resistivity in complex materials.

## Key findings

- Resistivity bound $
ho \,\lesssim\, A \Gamma$ derived for electron fluids.
- Explains $T^2$ and $T$ resistivity behaviors in different regimes.
- Provides a unified framework for understanding resistivity in strongly correlated systems.

## Abstract

We obtain a rigorous upper bound on the resistivity $\rho$ of an electron fluid whose electronic mean free path is short compared to the scale of spatial inhomogeneities. When such a hydrodynamic electron fluid supports a non-thermal diffusion process -- such as an imbalance mode between different bands -- we show that the resistivity bound becomes $\rho \lesssim A \, \Gamma$. The coefficient $A$ is independent of temperature and inhomogeneity lengthscale, and $\Gamma$ is a microscopic momentum-preserving scattering rate. In this way we obtain a unified and novel mechanism -- without umklapp -- for $\rho \sim T^2$ in a Fermi liquid and the crossover to $\rho \sim T$ in quantum critical regimes. This behavior is widely observed in transition metal oxides, organic metals, pnictides and heavy fermion compounds and has presented a longstanding challenge to transport theory. Our hydrodynamic bound allows phonon contributions to diffusion constants, including thermal diffusion, to directly affect the electrical resistivity.