It has been recently shown that in the Heisenberg (anti)ferromagnet on the honeycomb lattice, the magnons (spin wave quasipacticles) realize a massless two-dimensional (2D) Dirac-like Hamiltonian. It was shown that the Dirac magnon Hamiltonian preserves time-reversal symmetry defined with the sublattice pseudo spins and the Dirac points are robust against magnon-magnon interactions. The Dirac points also occur at nonzero energy. In this paper, we propose a simple realization of nontrivial topology (magnon edge states) in this system. We show that the Dirac points are gapped when the inversion symmetry of the lattice is broken by introducing a next-nearest neighbour Dzyaloshinskii-Moriya (DM) interaction. Thus, the system realizes magnon edge states similar to the Haldane model for quantum anomalous Hall effect in electronic systems. However, in contrast to electronic spin current where dissipation can be very large due to Ohmic heating, noninteracting topological magnons can propagate for a long time without dissipation as magnons are uncharged particles. We observe the same magnon edge states for the XY model on the honeycomb lattice. Remarkably, in this case the model maps to interacting hardcore bosons on the honeycomb lattice. Quantum magnetic systems with nontrivial magnon edge states are called topological magnon insulators. They have been studied theoretically on the kagome lattice and recently observed experimentally on the kagome magnet Cu(1-3, bdc) with three magnon bulk bands. Our results for the honeycomb lattice suggests an experimental procedure to search for honeycomb topological magnon insulators within a class of 2D quantum magnets and ultracold atoms trapped in honeycomb optical lattices. In 3D lattices, Dirac and Weyl points were recently studied theoretically, however, the criteria that give rise to them were not well-understood. We argue that the low-energy Hamiltonian near the Weyl points should break time-reversal symmetry of the pseudo spins. Thus, recovering the same criteria in electronic systems.

Quite recently, magnon Hall effect of spin excitations has been observed experimentally on the kagome and pyrochlore lattices. Thermal Hall conductivity κ xy , changes sign as a function of magnetic field or temperature on the kagome lattice, and κ xy changes sign upon reversing the sign of the magnetic field on the pyrochlore lattice. Motivated by these recent exciting experimental observations, we theoretically propose a simple realization of magnon Hall effect in a two-band model on the honeycomb lattice. The magnon Hall effect of spin excitations arises in the usual way via the breaking of inversion symmetry of the lattice, however, by a next-nearest-neighbour DzyaloshinskyMoriya (DM) interaction. We find that κ xy has a fixed sign for all parameter regimes considered. These results are in contrast to the Lieb, kagome and pyrochlore lattices. We further show that the low-temperature dependence on the magnon Hall conductivity follows a T 2 law, as opposed to the kagome and pyrochlore lattices. These results suggest an experimental procedure to measure thermal Hall conductivity within a class of 2D honeycomb quantum magnets and ultracold atoms trapped in honeycomb optical lattice.Introduction.-In recent years, the understanding of the topological nature of phonons and magnons in quantum materials has been at the pinnacle of intense investigation. These materials are believed to be applicable to many technological systems such as thermal devices and spintronics. The most fascinating property of these materials is the observation of thermal Hall effect, which occurs at finite temperature. Phonon Hall effect has been observed experimentally in Tb 3 Ga 5 O 12 [1], and the topological properties have been studied in terms of the Berry curvature of the system in different lattice geometries [2,3]. Recently, the topological properties of magnons in quantum magnets [4][5][6][7][8] have become a subject of interest because of the possibility of thermal Hall effect characterized by a nonzero thermal Hall conductivity κ xy , at finite temperature. Thermal Hall effect in quantum magnets was first predicted theoretically by Katsura-Nagaosa-Lee [4] on the kagome and pyrochlore ferromagnets with a nearest-neighbour (NN) Dzyaloshinsky-Moriya (DM) interaction [9]. It was later discovered experimentally by Onose et al [5] in the ferromagnetic insulator Lu 2 V 2 O 7 on three-dimensional (3D) pyrochlore lattice. Subsequently, Matsumoto and Murakami [6] relates κ xy directly to the Berry curvature of the magnon bulk bands reminiscent of Hall conductivity in electronic systems [10]. This result shows that at nonzero temperature, κ xy = 0 provided that the magnon bulk bands have a nontrivial gap at the Dirac points, i.e., the points where two bands touch in the Brillouin zone.

We introduce the concept of Floquet topological magnons-a mechanism by which a synthetic tunable Dzyaloshinskii-Moriya interaction (DMI) can be generated in quantum magnets using circularly polarized electric (laser) field. The resulting effect is that Dirac magnons and nodal-line magnons in two-dimensional and three-dimensional quantum magnets can be tuned to magnon Chern insulators and Weyl magnons respectively under circularly polarized laser field. The Floquet formalism also yields a tunable intrinsic DMI in insulating quantum magnets without an inversion center. We demonstrate that the Floquet topological magnons possess a finite thermal Hall conductivity tunable by the laser field.