Topological Magnons

One of the most well-known models in topological physics is the Haldane model, which builds on the tight-binding model of graphene. In this model, when spin-orbit coupling is introduced, an imaginary term is added between next-nearest neighbors, breaking time-reversal symmetry. This term acts as an antisymmetric off-diagonal component in the Hamiltonian. Upon diagonalizing the Hamiltonian, a gap opens at the Dirac point, accompanied by surrounding spikes in the Berry curvature.

Honeycomb ferromagnets are magnetic analogs of graphene, not only due to their similar crystal structures but also because their linear spin wave theory calculations closely mirror the tight-binding model in graphene. This resemblance makes it possible to realize topological magnons in these materials. The simplest method (and an analog to the Haldane term) is to introduce a Dzyaloshinskii-Moriya (DM) interaction between next-nearest neighbors. This also opens a gap at the Dirac point, with non-trivial topological properties.

Using neutron scattering, we have observed the magnon dispersion in the honeycomb ferromagnet CrI3. A gap at the Dirac point was detected, which can be explained by incorporating a DM interaction term into the Heisenberg Hamiltonian.

Kitaev interactions, first proposed by Alexei Kitaev, describe a set of bond-dependent anisotropic exchange interactions. In a honeycomb lattice with edge-sharing octahedra, the Kitaev interaction's easy axis is typically perpendicular to the rhombus plane formed by the nearest neighbor bond in the honeycomb plane and the shared octahedral edge. While this interaction might seem unusual, it is permitted by the crystal symmetry and leads to many non-trivial outcomes, such as a spin-liquid ground state with Majorana excitations in spin-1/2 honeycomb magnets and topological magnon excitations in honeycomb ferromagnets.

Interestingly, adding a Kitaev term to the Heisenberg Hamiltonian, without including the DM interaction, can reproduce the neutron scattering results observed in CrI3. However, due to the highly anisotropic nature of the Kitaev interaction, the magnon dispersion changes significantly with different moment directions. Leveraging this characteristic, we performed neutron scattering experiments under an applied magnetic field that altered the moment directions. The results show that the magnon dispersion changed minimally with the applied field, suggesting that Kitaev interactions are not the primary cause of the topological gap in CrI3.

Magnons and phonons, as well-defined quasiparticle excitations, can be accurately described by their respective theories, which provide information about their eigenvectors. This allows for the calculation of the Berry curvatures of both excitations. When a bilinear magnon-phonon coupling theory is introduced, it becomes straightforward to compute the topological properties of the hybridized modes, known as magnon polarons. Unlike topological magnons, which require a non-Bravais lattice (to generate more than one magnon mode), topological magnon polarons can occur in simpler structures, such as a 2D square lattice, making them more accessible. Additionally, while magnons are sensitive to external fields, phonons are not, offering tunable topological properties.

One promising candidate for hosting topological magnon polarons is the transition metal phosphorus trichalcogenide family, particularly FeP(S,Se)3. In a successful collaboration between Rice University and Texas Tech, we used Raman scattering to probe the magnon and phonon dynamics in both bulk and monolayer FePSe3. The results were striking: a magnon-phonon gap was observed near the bottom of the magnon dispersion, which could be tuned by adjusting the magnetic field. Further calculations revealed that the symmetry of the spin and lattice systems theoretically guarantees a magnetic-field-controlled topological phase transition, which was confirmed by the presence of non-zero Chern numbers derived from the coupled spin-lattice model.