# Hall conductivity as the topological invariant in phase space in the presence of interactions and non-uniform magnetic field

## TL;DR

This paper extends the topological phase space invariant representation of quantum Hall conductivity to include electron interactions, demonstrating the conjecture within a 2+1D tight-binding model using perturbation theory.

## Contribution

It proves that the topological invariant expression for Hall conductivity remains valid with interactions by substituting the Green function with the full interacting Green function.

## Key findings

- The topological invariant form of Hall conductivity holds with interactions.
- The proof is conducted within a 2+1D tight-binding model using perturbation theory.
- The representation applies to non-uniform magnetic fields and includes interaction effects.

## Abstract

The quantum Hall conductivity in the presence of constant magnetic field may be represented as the topological TKNN invariant. Recently the generalization of this expression has been proposed for the non - uniform magnetic field. \rev{The quantum Hall conductivity is represented as the topological invariant in phase space in terms of the Wigner transformed two - point Green function.} This representation has been derived when the inter - electron interactions were neglected. It is natural to suppose, that in the presence of interactions the Hall conductivity is still given by the same expression, in which the non - interacting Green function is substituted by the complete two - point Green function \rev{ including the interaction contributions}. We prove this conjecture within the framework of the $2+1$ D tight - binding model of rather general type using the ordinary perturbation theory.