Topological Band Systems and Finite Size Effects

TL;DR

This paper reviews topological insulators, focusing on edge states, lattice models, and finite size effects, including experimental approaches with cold atoms and theoretical analysis of gap openings.

Contribution

It provides a comprehensive overview of topological insulators, lattice models like Hofstadter, and analyzes finite size effects and gap openings in cold atom systems.

Findings

Finite size effects influence topological band structures.

Perturbation theory explains gap openings in bounded systems.

Cold atom experiments can mimic topological insulator behavior.

Abstract

The recent discoveries about topological insulators have been promoting theoretical and experimental research. In this dissertation, the basic concepts of topological insulators and the Quantum Hall Effect are reviewed focusing the discussion on edge states and their band structure. Lattice models with pierced magnetism are described and the Hofstadter model is presented for bounded systems with and without an in-site disorder. An overview of the experimental procedure based on cold atoms in optical lattices with synthetic dimensions is given. In order to understand to what extent these small systems of cold atoms mimic the behaviour of a topological insulator, an analysis of some finite size effects is provided and a deduction of the gap opening in the band structure is presented using perturbation theory.