Magnon Landau Levels and Spin Responses in Antiferromagnets*

We study emergent gauge fields produced by gradients of the Dzyaloshinskii-Moriya interaction and propose a model of antiferromagnetic topological insulator of magnons.  This model is featured with spin degenerate Landau levels, akin to the quantum spin Hall system. In the long wavelength limit, the Landau levels induced by the emergent gauge field reflect relativistic physics described by the Klein-Gordon equation. Interestingly, by tuning the induced gauge field, this model uncovers an unconventional Hofstadter's butterfly. Due to the spin degeneracy property of the Landau levels, the system naturally exhibits magnonic spin Nernst response.  Moreover, the physical picture in this model is closely tied to the topological responses in skyrmion and vortex-antivortex crystal phases of AFM insulators. Our studies show that AFM insulators exhibit rich physics associated with topological magnon excitations.