Odd-parity magnons in the Haldane-Hubbard model from topological exciton condensation

Odd-wave magnets are the counterparts to even-wave altermagnets realizing odd-parity spin splitting. Normally discussed for noncollinear systems, they have recently been shown to appear in collinear magnetic states in the presence of loop currents. Here we study collective excitations of the paramagnetic and magnetic phase of the seminal Haldane-Hubbard model. We identify the existence topological excitons in the paramagnetic phase, and their condensation as the driving mechanism into the collinear Néel state. The latter realizes an odd-wave magnet with odd-parity magnons displaying a characteristic $f$-wave splitting. We further uncover that an electron bandgap closing ensures magnon bandgap closing causing a change in odd-parity magnon topology, as well as a drastically enlarged spin splitting. Our results establish the presence of topological excitons and odd-parity magnons in the Haldane-Hubbard, with potential realizations in Floquet-driven materials and cold atomic gases.