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

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 "nonsymmorphic"-like elements. Multidimensional 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 this 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 multicomponent 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.

Multi-Qhexagonal spin density waves and dynamically generated spin-orbit coupling: Time-reversal invariant analog of the chiral spin density wave

“…In previous work we have analyzed hexagonal lattice Mpoint order in the singlet channel using extended point group symmetry [13] and found a set of charge density (s) waves, in addition to a set of time-reversal odd charge-current density (d) waves. Both sets of orders correspond to nesting instabilities.…”

“…The uniaxial order corresponds to an L = 4 cubic α phase, similar to the L = 2 α phase of Eq. (13). The α phase is interesting in its own right since it corresponds to a quantum spin Hall (QSH) phase [3,46].…”

“…Specifically, we study density wave formation, i.e., condensation of particle-hole pairs, at finite commensurate wave vectors associated with nested van Hove singularities. Our study comprises a classification of density waves in the spin channel, referred to as triplet density waves [12], building on previous work which has focused on the singlet channel [13]. One of our main results is to show that multicomponent density waves with nonzero (commensurate) wave vector in 2D can be mapped to and classified as condensates of particle-hole pairs with nonzero angular momentum in three dimensions (3D).…”

“…[13] for details). To construct the scalar triplet state, we pair each component with a distinct spin component, leading to…”

“…The analysis of hexagonal lattice nesting instabilities in the presence of spin is a straightforward extension of the analysis without spin degree of freedom [13]. The goal is to construct an algebra of the van Hove electrons and use that algebra to determine the nesting instabilities.…”