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

TL;DR

This paper systematically classifies linear band crossings at high-symmetry points in layered materials using all 320 layer groups, analyzing effects of spin-orbit coupling and time reversal symmetry on these crossings.

Contribution

It provides a comprehensive classification and effective Hamiltonians for linear band crossings in layered materials, expanding existing literature with detailed symmetry analysis.

Findings

Classified all 320 layer groups for band crossings.

Identified dispersion types: single cone, poppy-flower, fortune teller.

Analyzed effects of spin-orbit coupling and time reversal symmetry.

Abstract

All of 320 layer groups, distributed into 80 clusters - single/double ordinary/gray groups - are used to complete systematization of linear (in all directions) band crossings and corresponding effective Hamiltonians in high-symmetry Brillouin zone points of layered materials, refining and expanding in literature existing data. Two- and four-dimensional effective Hamiltonians are determined by the allowed (half)integer (co)representations of the same dimension in the crossing point and one- or two-dimensional generic allowed representations. The resulting dispersion types (having isotropic or anisotropic form) are: single cone (with double degenerate crossing point and non-degenerate branches, or 4-fold degenerate crossing point with double degenerate conical branches), poppy-flower (4-fold degenerate crossing point with two pairs of non-degenerate mutually rotated conical branches), and a fortune teller (with nodal lines). Transition to double group, enabling to include spin-orbit interaction, results in various scenarios at high symmetry points: gap closing, gap opening, cone preserving, cone splitting etc. Analogously, analyzing ordinary to gray group transitions, the role of time reversal symmetry is clarified.