Magnonic topological insulators in antiferromagnets

Extending the notion of symmetry protected topological phases to insulating antiferromagnets (AFs) described in terms of opposite magnetic dipole moments associated with the magnetic $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ order, we establish a bosonic counterpart of topological insulators in semiconductors. Making use of the Aharonov-Casher effect, induced by electric field gradients, we propose a magnonic analog of the quantum spin Hall effect (magnonic QSHE) for edge states that carry helical magnons. We show that such up and down magnons form the same Landau levels and perform cyclotron motion with the same frequency but propagate in opposite direction. The insulating AF becomes characterized by a topological ${\mathbb{Z}}_2$ number consisting of the Chern integer associated with each helical magnon edge state. Focusing on the topological Hall phase for magnons, we study bulk magnon effects such as magnonic spin, thermal, Nernst, and Ettinghausen effects, as well as the thermomagnetic properties of helical magnon transport both in topologically trivial and nontrivial bulk AFs and establish the magnonic Wiedemann-Franz law. We show that our predictions are within experimental reach with current device and measurement techniques.

Figure 1

(Left) Schematic representation of spin excitations in (a) a FM with the uniform ground-state magnetization and (b) an AF with classical magnetic $\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$ order. (Right) Schematic representation of edge magnon states in a two-dimensional topological (a) FM and (b) AF. (a) Chiral edge magnon state where magnons with a magnetic dipole moment $g{\mu }_{\mathrm{B}}{\mathbf{e}}_z$ propagate along the edge of a finite sample in a given direction. (b) Helical edge magnon state where up and down magnons $\left(\sigma =\pm 1\right)$ with opposite magnetic dipole moments $\sigma g{\mu }_{\mathrm{B}}{\mathbf{e}}_z$ propagate along the edge in opposite directions. The AF thus forms a bosonic analog of a TI characterized by two edge modes with opposite chiralities and can be identified with two independent copies (with opposite magnetic moments) of single-layer FMs shown in (a).

Figure 2

Plots of the magnonic energy spectrum, rescaled energy $E/t_x$, as a function of the rescaled wave vector $k_y{a}_y/\pi$ obtained by numerically solving the tight-binding model (24) for a strip with width of 115 lattice sites, for $t_x=t_y=1$ and ${\theta }_1=2\pi /5$, showing the first and partially the second Landau levels (black). The up magnon edge states $\left(\sigma =1\right)$ are in red while down magnon edge states $\left(\sigma =-1\right)$ are in blue. The periodicity of the vector potential is (a) $q\gg 1$, (b) $q=7$, and (c) $q=4$. (a) shows a band structure characteristic for a TI with a well-developed gap in an almost flat band and with helical edge states formed by the up and down magnons. (b) Qualitatively the same as in (a). (c) For each given value of $k_y$ there are well-defined edge states with a gap in the spectrum, but they coexist with bulk states at different momenta.