Symmetry analysis of translational symmetry broken density waves: application to hexagonal lattices in two dimensions

TL;DR

This paper introduces a symmetry classification for electronic density waves that break translational symmetry, focusing on hexagonal lattices, and explores their electronic properties and topological features.

Contribution

It develops a symmetry classification based on extended point groups for density waves, distinguishing time-reversal invariant and breaking orders, and applies it to 2D hexagonal lattices.

Findings

Classified density waves using extended point groups.

Analyzed electronic properties like band degeneracies and topology.

Applied the framework to triangular and honeycomb lattices.

Abstract

In this work we introduce a symmetry classification for electronic density waves which break translational symmetry due to commensurate wave vector modulations. The symmetry classification builds on the concept of extended point groups: symmetry groups which contain, in addition to the lattice point group, translations that do not map the enlarged unit cell of the density wave to itself, and become "non-symmorphic"-like elements. Multi-dimensional representations of the extended point group are associated with degenerate wave vectors. Electronic properties such as (nodal) band degeneracies and topological character can be straightforwardly addressed, and often follow directly. To further flesh out the idea of symmetry, the classification is constructed so as to manifestly distinguish time-reversal invariant charge (i.e., site and bond) order, and time-reversal breaking flux order. For the purpose of the present work we particularize to spin-rotation invariant density waves. As a first example of the application of the classification we consider the density waves of a simple single- and two-orbital square lattice model. The main objective, however, is to apply the classification to two-dimensional ($2D$) hexagonal lattices, specifically the triangular and the honeycomb lattices. The multi-component density waves corresponding to the commensurate $M$-point ordering vectors are worked out in detail. To show that our results generally apply to $2D$ hexagonal lattices, we develop a general low-energy $SU(3)$ theory of (spinless) saddle point electrons.