# Absence of a Charge Diffusion Pole at Finite Energies in an Exactly Solvable Interacting Flat Band Model in d-dimensions

## TL;DR

This paper investigates charge diffusion in a scale-invariant, flat-band model at the Mott transition, revealing that the diffusion pole vanishes at finite energy, making the diffusion constant ill-defined.

## Contribution

It demonstrates that in a scale-invariant flat-band system, the charge diffusion pole is absent at finite energies, challenging conventional diffusion concepts.

## Key findings

- Diffusion pole is pushed to zero energy in the flat-band limit.
- The diffusion constant becomes ill-defined due to scale invariance.
- A scale related to band velocity and density of states exists but is irrelevant to diffusion.

## Abstract

Motivated by recent bounds for charge diffusion in critical matter, we investigate the question: What sets the scale for charge diffusion in a scale-invariant system? To make our statements precise, we analyze the diffusion pole in an exactly solvable model for a Mott transition in the presence of a long-range interaction term. To achieve scale invariance, we limit our discussion to the flat-band regime. We find in this limit that the diffusion pole which would normally obtain at finite energy is pushed to zero energy resulting in a vanishing of the diffusion constant. This occurs even in the presence of interactions in certain limits, indicating the robustness of this result to the inclusion of a scale in the problem. Consequently, scale-invariance precludes any reasonable definition of the diffusion constant. Nonetheless, we do find that a scale can be defined, all be it, irrelevant to diffusion, which is the product of the squared band velocity and the density of states.