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DISCRETE SYMMETRIES OF LOW-DIMENSIONAL DIRAC MODELS: A SELECTIVE REVIEW WITH A FOCUS ON CONDENSED-MATTER REALIZATIONS

Part of: Groups and algebras in quantum theory

Published online by Cambridge University Press:  19 August 2015

R. WINKLER  and
U. ZÜLICKE
R. WINKLER
Affiliation:
Department of Physics, Northern Illinois University, DeKalb, IL 60115, USA Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany email rwinkler@niu.edu
U. ZÜLICKE*
Affiliation:
School of Chemical and Physical Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand email uli.zuelicke@vuw.ac.nz
*
✉uli.zuelicke@vuw.ac.nz
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Abstract

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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 ($2+1$)- 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 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 ($2+1$)-dimensional Dirac theory. Some physical implications of this previously unnoticed duplicity are discussed.

Keywords

relativistic quantum physicsreduced dimensionalityDirac Hamiltoniansquasi-relativity in solids

MSC classification

Primary: 81R05: Finite-dimensional groups and algebras motivated by physics and their representations
Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 1 , July 2015 , pp. 3 - 17
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© 2015 Australian Mathematical Society

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