Irreducible and site symmetry induced representations of single/double ordinary/gray layer groups

Abstract

Considered are eighty sets of layer groups, each set consisted of four groups: ordinary single and double, and gray single and double layer group. Structural properties of layer groups (factorization onto cyclic subgroups and existence of grading according to the sequence of halving subgroups) enable efficient symbolic computation (by POLSym code) of the relevant properties, real and complex irreducible and allowed (half-)integer (co-)representations in particular. This task includes, as the first step, classification of the irreducible domains based on the group action in Brillouin zone combined with torus topology. Also, the band (co-)representations induced from the irreducible (co-)representations of Wyckoff position stabilizers (site symmetry groups) are decomposed onto the irreducible components. These, and other layer group symmetry related theoretical data relevant for physics, layered materials in particular, are tabulated and made available through the web site.

Fully linear band crossings at high symmetry points in layers: classification and role of spin-orbit coupling and time reversal

“…Those groups with inversion symmetry included are orange coloured, while non-symmorphic groups are singled out by red in column C. In the column ID and ID' are ordinals of irreducible domains of (gray) LGs according to the Ref. [28].…”

“…Clearly, real parameters v p i and v p ij are bi-uniquely related to w p i and w p ij , respectively. For each HSP and its allowed (co)IRs [28] of dimension 2 and 4, the symmetry allowed parameters v p i and v p ij are to be found. The task is performed assuming that layer is perpendicular to the z-axis.…”

“…Action in 2D BZ is defined by the upper block. As a result, there is 10 different isogonal groups [28] composed of these 2 × 2 matrices, which, due to torus topology, yield 14 IDs of 2D BZ. IDs are the same for LG and DLG, while adding TR changes ID of noncentrosymmetric groups.…”

“…and recently made available online [28]) of the symmetry groups of HSPs obtained by the action of layer (LGs), double layer (DLGs), and corresponding gray magnetic groups (gray…”