The Crystallographic Spin Point Groups and their Representations

TL;DR

This work provides a complete crystallographic theory for spin point groups, extending magnetic point group concepts to cases with intermediate and weak spin–orbit coupling. By carefully enumerating 598 nontrivial spin point groups and detailing their spin-only pairings, the authors derive the full (co-)representation theory, showing that unitary nontrivial SPGs share irreps with their spatial partners while non-unitary SPGs correspond to black-and-white groups via Dimmock indicators. They also develop explicit co-irrep tables for coplanar, collinear, and non-magnetic total groups, and demonstrate how these symmetries constrain electronic bands and magnon spectra in altermagnets and Kitaev–Heisenberg magnets. The results provide practical tools to predict degeneracies and band topology from SPGs, and reveal new spin-only group extensions beyond Litvin–Opechowski, with implications for spin-space groups and reciprocal-space degeneracies.

Abstract

The spin point groups are finite groups whose elements act on both real space and spin space. Among these groups are the magnetic point groups in the case where the real and spin space operations are locked to one another. The magnetic point groups are central to magnetic crystallography for strong spin-orbit coupled systems and the spin point groups generalize these to the intermediate and weak spin-orbit coupled cases. The spin point groups were introduced in the 1960's in the context of condensed matter physics and enumerated shortly thereafter. In this paper, we complete the theory ofcrystallographic spin point groups by presenting an account of these groups and their representation theory. Our main findings are that the so-called nontrivial spin point groups (numbering $598$ groups) have co-irreps corresponding exactly to the (co-)-irreps of regular or black and white groups and we tabulate this correspondence for each nontrivial group. However a total spin group, comprising the product of a nontrivial group and a spin-only group, has new co-irreps in cases where there is continuous rotational freedom. We provide explicit co-irrep tables for all these instances. We also discuss new forms of spin-only group extending the Litvin-Opechowski classes. To exhibit the usefulness of these groups to physically relevant problems we discuss a number of examples from electronic band structures of altermagnets to magnons.

Figures (53)

• Figure 1: The isomorphisms between $\mathbf{S},$$\mathbf{\tilde{S}},$$\mathbf{G}$ and $\mathbf{G}^{\mathrm{BW}}$ that ensure the construction of equivalent co-irreps.
• Figure 2: Crystal structure of rutile MX$_2$ where the magnetic M ions (blue) are on the tetragonal unit cell vertices and the X ions (translucent) crucially are arranged so that the local environments around the two magnetic sublattices are related by a $C_{4z}$ operation and a translation. The $\uparrow / \downarrow$ sublattice structure is also shown, and the left figure indicates the $t_{1}$ hopping, while the right figure shows two inequivalent $t_{3}$ hoppings
• Figure 3: The two lowest energy bands of the toy model described in the main text. The colours indicate the spin state. The total spin is a good quantum number and it exhibits a d wave pattern in momentum space.
• Figure 4: Plot of a unit cell of the hyperhoneycomb lattice with nearest neighbour bonds coloured according to the Kitaev coupling on each bond.
• Figure :