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

Abstract

The most fundamental characteristic 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 some of the basic symmetry properties of the relativistic quantum theories for free electrons in - and -dimensional spacetime. Additional flavour degrees of freedom are necessary to properly define symmetry operations in dimensions, and are generally present in physical realizations of such systems, for example in single sheets of graphite. We find that there exist two possibilities for defining any flavour-coupling discrete symmetry operation of the two-flavour -dimensional Dirac theory. Some physical implications of this previously unnoticed duplicity are discussed.