Spin-Space Groups and Magnon Band Topology

Abstract

Band topology is both constrained and enriched by the presence of symmetry. The importance of anti-unitary symmetries such as time reversal was recognized early on leading to the classification of topological band structures based on the ten-fold way. Since then, lattice point group and nonsymmorphic symmetries have been seen to lead to a vast range of possible topologically nontrivial band structures many of which are realized in materials. In this paper we show that band topology is further enriched in many physically realizable instances where magnetic and lattice degrees of freedom are wholly or partially decoupled. The appropriate symmetry groups to describe general magnetic systems are the spin-space groups. Here we describe cases where spin-space groups are essential to understand the band topology in magnetic materials. We then focus on magnon band topology where the theory of spin-space groups has its simplest realization. We consider magnetic Hamiltonians with various types of coupling including Heisenberg and Kitaev couplings revealing a hierarchy of enhanced magnetic symmetry groups depending on the nature of the lattice and the couplings. We describe, in detail, the associated representation theory and compatibility relations thus characterizing symmetry-enforced constraints on the magnon bands revealing a proliferation of nodal points, lines, planes and volumes.

Dirac magnons, nodal lines, and nodal plane in elemental gadolinium

“…These experiments and calculations were carried out on Gd, which has near-perfect isotropic Heisenberg exchange. However, because of the intrinsic connection between symmetry, degeneracy, and topology [13,14,[59][60][61] similar topological features can be expected in more anisotropic ferromagnetic HCP metals such as Tb [62,63], Dy [62,64], and hexagonal Co [65]. (However, for Co one must consider the effects of itinerancy on the possible Dirac spectrum [66] and continuum scattering likely eliminates the observability of Dirac magnons in HCP Co [67,68].)…”

“…It was subsequently realized that related physics can occur in systems with bosonic quasiparticles including among others phonons [4], photons [5,6], and more recently, magnons [7][8][9][10][11][12]. The interesting topological features of magnon bands are often associated with band degeneracies that can be understood as a consequence of symmetries describable by spin-space groups [13,14]. Magnon band structures can realize analogs of e.g.…”

“…On the HCP lattice, they appear with both the simplest single nearest neighbor ferromagnetic exchange interaction, or with any number of further neighbor exchanges-so long as they are all Heisenberg exchanges and the ground state remains ferromagnetic (this was first noted by Brinkman in 1967 [13] and the topological consequences have been explored in Ref. [14]). Thus, although the further neighbor exchange interactions are important for understanding the wiggles in Gd's magnon dispersion, they are not important for understanding the topology.…”

Krein-Hermitian Schrieffer-Wolff transformation and band touchings in bosonic BdG Hamiltonians

“…Non-Hermitian Hamiltonians have become a central field of study in condensed-matter physics in the last decades [1]. Important examples of systems governed by non-Hermitian Hamiltonians are quadratic bosonic Bogoliubov-de Gennes (BdG) Hamiltonians [2,3], which have been the subject of renewed interest recently in the context of topological systems [4][5][6][7][8][9][10][11][12][13], and so-called "P T -symmetric" non-Hermitian Hamiltonians, which initially attracted interest because they could give rise to real spectra despite not being Hermitian [14,15]. Note that in the context of non-Hermitian Hamiltonians, "P T symmetry" has come to refer to any antilinear symmetry of a specific form [10], and not necessarily to space-time inversion symmetry.…”