Magnonic Floquet Hofstadter butterfly

Abstract

We introduce the magnonic Floquet Hofstadter butterfly in the two-dimensional (2D) insulating honeycomb ferromagnet using a combination of spin wave theory and quantum field theory. We show that hopping magnetic dipole moment (i.e. magnon quasiparticles) in the 2D insulating honeycomb ferromagnet accumulates the Aharonov–Casher phase when irradiated by an oscillating space- and time-dependent electric field. We study two different cases. In the first case, we consider the effects of only space-dependent electric field. In this case, we realize the magnonic Hofstadter spectrum with similar fractal structure as graphene subject to a perpendicular magnetic field, but with no spin degeneracy due to broken time-reversal symmetry by the ferromagnetic order. In addition, the magnonic Dirac points and Landau levels occur at finite energy as expected in a bosonic system. Remarkably, this discrepancy does not affect the topological invariant of the system. Consequently, the magnonic Chern number assumes odd values and the magnon Hall conductance gets quantized by odd integers. In the second case, we study the effects of both space- and time-dependent electric field. In this case, the theoretical framework is studied by the Floquet formalism. We show that the magnonic Floquet Hofstadter spectrum emerges entirely from the oscillating space- and time-dependent electric field, which is in stark contrast to electronic Floquet Hofstadter spectrum, where irradiation by circularly polarized light and a perpendicular magnetic field are applied independently. We study the deformation of the fractal structure at different laser frequencies and amplitudes, and analyze the topological phase transitions associated with gap openings in the magnonic Floquet Hofstadter butterfly.