Magnonic quantum Hall effect and Wiedemann-Franz law

We present a quantum Hall effect of magnons in two-dimensional clean insulating magnets at finite temperature. Through the Aharonov-Casher effect, a magnon moving in an electric field acquires a geometric phase and forms Landau levels in an electric field gradient of sawtooth form. At low temperatures, the lowest energy band being almost flat carries a Chern number associated with a Berry curvature. Appropriately defining the thermal conductance for bosons, we find that the magnon Hall conductances get quantized and show a universal thermomagnetic behavior, i.e., are independent of materials, and obey a Wiedemann-Franz law for magnon transport. We consider magnons with quadratic and linear (Dirac-like) dispersions. Finally, we show that our predictions are within experimental reach for ferromagnets and skyrmion lattices with current device and measurement techniques.

Figure 1

Schematic representation of magnonic Hall effects in a two-dimensional clean insulating magnet where magnetic field or temperature gradients along the $x$ direction produce a transverse Hall current of magnon (red dot) along the $y$ direction. Landau levels and cyclotron motion of the magnons can be induced by extrinsic electric field gradients or intrinsic spin orbit interactions, giving rise to quantized Hall conductances for magnon and thermal currents. The ratio of these Hall conductances satisfies a Wiedemann-Franz law at low temperatures.

Figure 2

Schematic representation of a skew-harmonic electromagnetic scalar potential $\phi$ periodically arranged. Through the AC effect, the magnon $\left({\mu }_{\mathrm{m}}\right)$ moving in the potential experiences an electric vector potential ${\mathbf{A}}_{\mathrm{m}}$ analogous to the symmetric gauge giving rise to cyclotron motion of the magnon corresponding to Landau levels. Such a skew-harmonic potential may be realized by periodically arranging STM tips.

Figure 3

Plots of the magnonic band structure, rescaled energy $E/t_x$, in the isotropic limit $t_x=t_y$ as a function of the rescaled wave vector $k_y{a}_y/\pi$ obtained by numerically solving the tight-binding model [Eq. (8)] for the value ${\theta }_0=2\pi /5$, showing the first and partially the second Landau levels. The periodicity of the vector potential ${\mathbf{A}}_{\mathrm{m}\mathrm{q}}^{\prime }=(\mathcal{E}{R}_q/c)(0,\{x/R_q\},0)$ is (a) $q\gg 1$, (b) $q=6$, (c) $q=4$, and (d) $q=3$. (a): Standard QHE band structure with a well-developed gap and a chiral in-gap edge state (one for each edge). (b): Qualitatively still the same as in (a); with two chiral edge states (green) connecting the second and third Landau level. (c) and (d): There are well-defined edge modes for fixed values of $k_y$, but they coexist with extended bulk modes at different momenta. Similar plots are obtained for the anisotropic case, see Appendix pp2.

Figure 4

Plot of the ratio ${(g{\mu }_{\mathrm{B}}/k_{\mathrm{B}})}^2(K^{yx}/G^{yx}T)$ as a function of $E_0^{*}/k_{\mathrm{B}}T$. At low temperatures $E_0^{*}/k_{\mathrm{B}}T\ge 5$, the ratio becomes constant and the magnonic WF law [Eq. (16)] is realized. See also Fig. 5.

Figure 5

Plots of (a) $h{K}^{yx}/k_{\mathrm{B}}^{2}T$ and (b) $h{\kappa }^{yx}/k_{\mathrm{B}}^{2}T$ as a function of $E_0^{*}/k_{\mathrm{B}}T$ with assuming ${\nu }_0=1$. The deviation remains substantial even at low temperatures. Inset: Plots of the ratio (a′) ${(g{\mu }_{\mathrm{B}}/k_{\mathrm{B}})}^2(K^{yx}/G^{yx}T)$ and (b′) ${(g{\mu }_{\mathrm{B}}/k_{\mathrm{B}})}^2({\kappa }^{yx}/G^{yx}T)$. In contrast to (a′), the ratio (b′) does not reduce to a constant even at low temperatures.

Figure 6

Plots of the magnonic band structure, rescaled energy $E/t_x$, for an anisotropic case $J_x\ne J_y$ as a function of the rescaled wave vector $k_y{a}_y/\pi$ obtained by numerically solving the tight-binding model [Eq. (8)] for the values ${\theta }_0=2\pi /5,t_y=0.1t_x$, showing the first and second Landau levels. The periodicity of the vector potential ${\mathbf{A}}_{\mathrm{m}\mathrm{q}}^{\prime }=(\mathcal{E}{R}_q/c)(0,\{x/R_q\},0)$ is (a) $q\gg 1$, (b) $q=6$, (c) $q=4$, and (d) $q=3$. (a) and (b): There is a well-developed gap with a corresponding edge state. (c) and (d): There are well-defined edge modes for fixed values of $k_y$, but they coexist with extended bulk modes at different momenta as in the isotropic case $t_x=t_y$.