# Microscopic Conductivity of Lattice Fermions at Equilibrium - Part II: Interacting Particles

## TL;DR

This paper extends microscopic conductivity analysis to interacting lattice fermions at equilibrium, demonstrating that fundamental laws like Ohm's and Joule's laws hold at the microscopic scale even with interactions.

## Contribution

It develops Lieb-Robinson bounds for multi-commutators to analyze response functions of interacting fermions, extending previous results from quasi-free to interacting systems.

## Key findings

- Verification of the 1st law of thermodynamics for heat production.
- Proof that Ohm's and Joule's laws hold at the microscopic scale.
- Extension of conductivity measures to interacting fermion systems.

## Abstract

We apply Lieb-Robinson bounds for multi-commutators we recently derived to study the (possibly non-linear) response of interacting fermions at thermal equilibrium to perturbations of the external electromagnetic field. This analysis leads to an extension of the results for quasi-free fermions of \cite{OhmI,OhmII} to fermion systems on the lattice with short-range interactions. More precisely, we investigate entropy production and charge transport properties of non-autonomous $C^{\ast }$-dynamical systems associated with interacting lattice fermions within bounded static potentials and in presence of an electric field that is time- and space-dependent. We verify the 1st law of thermodynamics for the heat production of the system under consideration. In linear response theory, the latter is related with Ohm and Joule's laws. These laws are proven here to hold at the microscopic scale, uniformly with respect to the size of the (microscopic) region where the electric field is applied. An important outcome is the extension of the notion of conductivity measures to interacting fermions.