Tunable multi-magnon Floquet topological edge states

TL;DR

The paper addresses how to realize robust edge states in magnon insulators by using a periodic modulation of the Dzyaloshinskii–Moriya interaction (DMI) to create Floquet topological phases. It analyzes a 2D square-lattice XXZ Heisenberg model where single-magnon states hybridize with two-magnon bound states (TMBS) through the DMI, and shows that a Floquet drive can induce a band inversion between the single-magnon band and TMBS, opening a quasienergy gap and producing bands with nontrivial Chern numbers. The results demonstrate edge modes in a ribbon geometry that traverse the Floquet gap and show that the edge-state chirality can be tuned by the relative phase between DMI drives along $x$ and $y$ directions. These findings point to experimentally accessible routes for magnonic topological devices, including THz DMI modulation in van der Waals magnets and programmable quantum simulators.

Abstract

We show that periodically time-modulating the Dzyaloshinskii-Moriya interaction (DMI) in a two-dimensional magnon insulator may induce a topological phase transition that results in the presence of robust edge modes. To this end, we study a square lattice of spins interacting via an XXZ Heisenberg model with a ferromagnetic longitudinal coupling and antiferromagnetic transverse coupling, as well as the aforementioned time-modulated DMI. The topologically protected edge states of this system are composed of coherent superpositions of single-magnon excitations and two magnon bound states. Furthermore, we show that the chirality of the edge states can be controlled by adjusting the relative phase for the drive on the DMI associated with nearest neighbors in the x and y directions.

Figures (7)

• Figure 1: (a) Square lattice of spins with nearest-neighbor anisotropic spin-spin couplings ($J_z$, $J_\perp$), a time-modulated Dzyaloshinsky-Moriya interaction (DMI), $D(t)$, and a longitudinal Zeeman field $B$. (b) Energies for the single-magnon states (black $\downarrow$), two-magnon states (orange, above $\downarrow\downarrow$), and three-magnon states (green, above $\downarrow\downarrow\downarrow$) that are energies of the of the static Hamiltonian $H_\mathrm{S}$ [Eq. \ref{['eq:H_S']}]. The DMI is modulated with frequency $\omega$ so that it resonantly couples the single-magnon band to the two-magnon bound state (TMBS), but the modulation frequency is detuned by $\Delta$ from resonance with the three-magnon states. The spectrum is obtained by exact diagonalization (see Appendix \ref{['sec:exact_diagonalization']}) in a $7\times 7$ lattice. The energy $E-E_0$ for each state is shown here relative to the ground state energy $E_0$ [Eq. \ref{['eq:ground_state_energy']}] using $J_\perp=J_z/5$, $B=2J_z$.
• Figure 2: (a) States with two spin flips and their associated energies in the Ising limit $J_\perp=0$. (b) Hopping of a spin flip produced by the transverse exchange interactions in states with two spin flips.
• Figure 3: (a) Effective Lieb lattice describing the low-energy subspace composed of single- and two-adjacent spin flips. The arrows indicate the direction in which the phase $\phi=\pi/2$ is acquired. (b) Some second-order processes contributing to matrix elements in Eq. \ref{['eq:H_eff_2']}. The virtual three-magnon states mediate an effective coupling between the TMBSs.
• Figure 4: (a) Bulk band structure with static DMI. In the undriven system, the Chern numbers are $(0,1,-1)$. Driving the DMI with frequency $\omega$ resonantly couples the single-magnon band to the TMBS bands. (b) Bulk quasienergies of the Hamiltonian given by Eq. \ref{['eq:H_eff_3']} (in the rotating frame). The color represents single-magnon (black) and TMBS character (orange/gray). The quasienergies for the system with open boundary conditions and the corresponding edge states are shown in Fig. \ref{['fig:fig6']}.
• Figure 5: Exact diagonalization of the static Hamiltonian $H_\mathrm{S}$ with $J_\perp = J_z/5$ and two values of the magnetic field $B$. (a) In the absence of a magnetic field ($B=0$), the frequency $\omega$ required to resonantly couple the single-magnon and TMBS bands also resonantly couples the TMBSs to the two-magnon continuum. (b) A magnetic field $B$ can be used to avoid these undesired resonances.