Tutorial: From Topology to Hall Effects -- Implications of Berry Phase Physics

TL;DR

This tutorial explains the Berry phase's fundamental role in quantum mechanics, its connection to topological properties, and how it underpins phenomena like Hall effects and transport in topological insulators.

Contribution

It provides a comprehensive introduction linking Berry phase mathematics to topological quantum effects and transport phenomena, highlighting their physical implications.

Findings

Berry phase is gauge-invariant during cyclic evolutions

Topological effects like quantum and Hall effects arise from Berry phase

Transport properties of topological insulators are rooted in Berry phase

Abstract

The Berry phase is a fundamental concept in quantum mechanics with profound implications for understanding topological properties of quantum systems. This tutorial provides a comprehensive introduction to the Berry phase, beginning with the essential mathematical framework required to grasp its significance. We explore the intrinsic link between the emergence of a non-trivial Berry phase and the presence of topological characteristics in quantum systems, showing the connection between the Berry phase and the band structure as well as the phase's gauge-invariant nature during cyclic evolutions. The tutorial delves into various topological effects arising from the Berry phase, such as the quantum, anomalous, and spin Hall effects, which exemplify how these quantum phases manifest in observable phenomena. We then extend our discussion to cover the transport properties of topological insulators, elucidating their unique behaviour rooted in Berry phase physics. This tutorial aims at equipping its readers with a robust understanding of the basic theory around the Berry phase and its pivotal role in the realm of topological quantum phenomena.