# Introduction to Topological Phases and Electronic Interactions in (2+1) Dimensions

## TL;DR

This paper introduces topological phases in (2+1) dimensions, reviewing key concepts and models, and explores the effects of electromagnetic interactions and correlations on Dirac particles and related phenomena.

## Contribution

It provides an accessible overview of topological insulators, discusses the role of electromagnetic interactions via PQED, and analyzes symmetry breaking and quantum Hall effects in Dirac systems.

## Key findings

- Half-quantized Hall conductivity in continuum limit
- Renormalization of electron spectrum with coupling strength
- Symmetry breaking leading to Landau levels in graphene

## Abstract

A brief introduction to topological phases is provided, considering several two-band Hamiltonians in one- and two-dimensions. Relevant concepts of the topological insulator theory, such as: Berry phase, Chern number, and the quantum adiabatic theorem, are reviewed in a basic framework, which is meant to be accessible to non-specialists. We discuss the Kitaev chain, SSH, and BHZ models. The role of the electromagnetic interaction in the topological insulator theory is addressed in the light of the Pseudo quantum electrodynamics (PQED). The well known parity anomaly for massless Dirac particle is reviewed in terms of the Chern number. Within the continuum limit, a half-quantized Hall conductivity is obtained. Thereafter, by considering the lattice regularization of the Dirac theory, we show how one may obtain the well known quantum Hall conductivity for a single Dirac cone. The renormalization of the electron energy spectrum, for both small and large coupling regime, is derived. In particular, it is shown that massless Dirac particles may, only in the strong correlated limit, break either chiral or parity symmetries. For graphene, this implies the generation of Landau-like energy levels and the quantum valley Hall effect.