Magnons and thermal Hall effect in an inversion symmetry broken Antiferromagnet
[Kawano-Onose-Hotta, Comm.Phys. 2 27(2019), Kawano-Hotta, PRB99 054422 (2019)] still on-going

Rashba-Dresselhaus effects of magnons
Rashba-Dresselhaus effects in conducting electrons (semiconductors) are the already established phenomena and have produced numerous numbers of scientific papers with a scope to find pathways for device applications. However, in many cases, controlling the spin textures in momentum space requires an electric current (at least as a secondary role), and the Joule heating had been a bottleneck for microfabrication. An alternative recent trend is to make use of the topological insulators where the bulk sample remains insulating. Recently, there are numbers of studies on the antiferromagnetic Weyl semimetals such as MnSi.
We instead consider a simple ubiquitous and "boring" insulating magnets with antiferromagnetic ordering realized in noncentrosymmetric materials, such as Ba₂MnGe₂O₇ and Ba₂CoGe₂O₇. Here, the Dzyaloshinskii-Moriya (DM) interaction, which is the antisymmetric exchange interaction between adjacent spin moments play a crucial role in some interesting physics to happen. Previous studies on insulating magnets were mostly dealing with ferromagnets and its analogues, focusing on thermal Hall effect. The reason why nothing more was discussed in insulating magnets is that the spin-orbit coupling was indispensable for Rashba and Dresselhaus effects to happen, whereas, in ferro/ferrimagnets, there are no "spin degrees of freedom" to couple to orbital motions of magnons. The magnon is a quasi-particle of magnets carrying spin moment. In ferromagnets there is only one species of magnon, whereas in antiferromagnets we have two which belong to antiparallelly aligned magnetic moments of two sublattices. Our findings are based on the picture that one can make good (but not fully one-to-one) correspondence between the electronic spin degrees of freedom with the two sublattice degrees of freedom in antiferromagnets.

The two species of magnons, a and b, carry spin moments pointing in the opposite directions. The DM interaction which breaks the inversion symmetry of the crystal, couples the orbital motion of these two species of magnons. The energy eigen state at each point on a magnon band consists of the linear combinations of these magnons with opposite spins, and ratio of this combination determines and varies the direction of the spin moment in momentum space.
One can construct an SU(2) algebra representing the symmetry of the four dimensional space of magnons consisting of particle-hole and a/b sublattice degrees of freedom. The DM interaction reduces this SU(2) symmetry down to U(1) and the magnetic field further reduces this symmetry down to {e}, which gives the rich spin texture given in the figure. This texture varies very easily by the rotating magnetic field. Also, since the spin moments that are carried by the magnons of wave number k and -k differ point in the nearly opposite direction, generating these two sets of magnons with the same excitation energy by the microwave enables the pure spin current to flow. By detecting this current, one can determine the antiferromagnetic structure in real space.

SU(2) magnon, Berry curvature, thermal Hall effect
This work discovers the new class of magnon Hall effect, "Anomalous thermal Hall effect of magnons", which is the analogue of the electronic Anomalous Hall effect. Both come from the mixing of two bands (magnon bands/electron bands) with different symmetries in the presence of the SU(2) gauge field embedded in the Hamiltonian, which generate a finite Berry curvature in momentum space, and bends the propagation of particles in real space.
It is well known that the ferromagnets show magnon Hall effect since 2010 [Onose-Katsura-Nagaosa]. Representative mechanism making use of the DM interaction is three fold: (1) generate a U(1) gauge field, i.e. (Peierls phase) to the hopping of magnons, (2) these phases are integrated and have finite contribution to the Berry curvature in momentum space, not being hindered by the symmetry of the crystal, (3) the contribution to the thermal Hall coefficients from the Berry curvature remains finite by the breaking of time reversal symmetry.
Intuitively, the U(1) gauge field which is a vector potential of magnons means the existence of magnetic flux that bends the propagation of magnons and contribute to the Hall effect. The above three rules should be fulfilled in order to have a magnon Hall effect. For (1) to happen, the DM vector should have a finite element in the direction parallel to the ordered magnetic moments, and align in a staggered manner along the bond (meaning that the inversion symmetry is kept). Regarding (3) for the square and triangular magnets, there is always a symmetry operation that connects the two fluxes with the same amplitude but pointing in the opposite direction on the lattices. In such case the contribution from these fluxes cancel out and the Hall coefficients become zero.
According to the above rules, the antiferromagnets are never allowed to have a finite Hall coefficient. However, this rule is not applied to our antiferromagnets, because the mechanism differs. In our case, the DM vector is aligned uniformly along the bonds and is pointing always perpendicular to the magnetic moments, thus does not generate a U(1) gauge field. Instead, our DM vector generates a SU(2) gauge field. This gauge field gives an opposite sign to the magnons propagating leftward and rightward from one site to another, and since this hopping transforms the a-magnon to b-magnon or vise versa, one can see that it has a very good correspondence with the role of Rashba spin-orbit coupling on electron motion. Since the Rashba electrons show anomalous Hall effects, this thermal Hall effect is regarded as a magnon analog of the anomalous Hall effect.
In this paper, we also showed that there emerges a topological Z2 phase by the application of the small magnetic field, which is detected by the existence of the edge state. The Z2 topological number is also defined in magnonic systems for the first time.