Some Basic Aspects of Fractional Quantum Numbers

TL;DR

This paper reviews how fractional quantum numbers arise in physical systems through topological mechanisms, illustrating with examples like domain walls, chiral fermions, and phenomena in quantum Hall systems.

Contribution

It provides a comprehensive overview of the mechanisms behind fractional quantum numbers, connecting simple models to complex topological and quantum effects.

Findings

Charge fractionalization occurs via topological states and zero modes.

Fractional angular momentum and statistics are observed in two-dimensional systems.

Topological mechanisms enable the realization of chiral fermions on domain walls.

Abstract

I review why and how physical states with fractional quantum numbers can occur, emphasizing basic mechanisms in simple contexts. The general mechanism of charge fractionalization is the passage from states created by local action of fields to states having a topological character, which permits mixing between local and topological charges. The primeval case of charge fractionalization for domain walls, in polyacetylene and more generally, can be demonstrated convincingly using Schrieffer's intuitive counting argument, and derived formally from analysis of zero modes and vacuum polarization. An important generalization realizes chiral fermions on high-dimensional domain walls, in particular for liquid He3 in the A phase. In two spatial dimensions, fractionalization of angular momentum and quantum statistics occurs, for reasons that can be elucidated both abstractly, and specifically in the context of the quantum Hall effect.