Discrete symmetries of low-dimensional Dirac models: A selective review with a focus on condensed-matter realisations

TL;DR

This review explores the discrete symmetry properties of low-dimensional Dirac models, emphasizing their relevance in condensed matter systems like graphene, and uncovers a previously unnoticed duplicity in defining symmetry operations.

Contribution

It provides a detailed analysis of symmetry properties in (2+1) and (1+1)-dimensional Dirac theories, highlighting the necessity of flavor degrees of freedom and revealing a new aspect of symmetry operation definitions.

Findings

Two possible definitions for flavor-coupling discrete symmetries in (2+1)D Dirac theory.

Physical implications of symmetry duplicity in condensed matter systems.

Enhanced understanding of symmetry behavior in low-dimensional quantum models.

Abstract

The most fundamental characteristics of a physical system can often be deduced from its behaviour under discrete symmetry transformations such as time reversal, parity and chirality. Here we review basic symmetry properties of the relativistic quantum theories for free electrons in (2+1)- and (1+1)-dimensional spacetime. Additional flavour degrees of freedom are necessary to properly define symmetry operations in (2+1) dimensions and are generally present in physical realisations of such systems, e.g., in single sheets of graphite. We find that there exist two possibilities for defining any flavour-coupling discrete symmetry operation of the two-flavour (2+1)-dimensional Dirac theory. Physical implications of this previously unnoticed duplicity are discussed.