Large Chern-Number Quantum Anomalous Hall Effect from Canted Antiferromagnetic Order in $d$-Electron System on Kagome Lattice

TL;DR

The paper addresses the quantum anomalous Hall effect in a multi-orbital $d$-electron kagome lattice driven by a noncoplanar, canted spin texture that yields scalar spin chirality $\chi = \mathbf{S}_i\cdot(\mathbf{S}_j\times\mathbf{S}_k)$. It builds a tight-binding model with five $d$-orbitals, derives a low-energy massive Dirac Hamiltonian around the Dirac point $\mathbf{K}$ with a canting-induced Zeeman term, and computes Berry-curvature induced Chern numbers, finding a maximal $C=\pm5$ when onsite energies are degenerate. The two spin sectors carry opposite Chern numbers, and the plateau can be split into lower-$|C|$ peaks by onsite-energy differences or robust under anisotropy of Slater-Koster integrals. This work shows a natural, high-Chern-number QAHE in $d$-electron kagome systems without SOC, with potential applications in topological electronics and quantum information.

Abstract

Electrons of $d$-symmetry interacting with a localized non-collinear antiferromagnetic spin order on a kagome lattice are considered. Even in the absence of an external magnetic field, spin-orbit coupling or relativistic effects, the spin texture produces a non-trivial intrinsic Berry curvature. This opens the route for a quantum anomalous Hall effect in the $d$-system. For spin orders with an out-of-plane component, the scalar spin chirality is finite, and the integration of the Berry curvature over the Brillouin zone may yield integer Hall conductivities in units of $e^2/h$. This canted configuration gives rise to the maximal possible Chern number, $C=\pm 5$ when the Fermi level is within nontrivial gap. The effect is best understood for -- but not limited to -- isotropic $d$-electron hopping and degenerate $d$-levels. In this case, analytic expressions are available and point to a topological origin for the manifestation of the maximal $C$. Numerical calculations show that these findings are robust to some anisotropy in the hopping integrals and to moderate splittings of the $d$ levels. The $C=\pm 5$ plateau can be split into Chern peaks with smaller integers by varying the onsite energies. The topological phase transition between Hall plateaus of opposite $C$ can be driven by flipping the out-of-plane component of the spin order, alluding to the potential of this system to quantum information.

Figures (14)

• Figure 1: (a) Coplanar non-collinear spin order with three sub-lattices carrying spins at 120° from each other. (b) Non-coplanar version obtained by adding an out-of-plane component to the spin order. The spins on the three sublattices are related to each other by 120° rotations around the center of the white triangle in the unit cell.
• Figure 2: (a) First Brillouin zone of the Kagome lattice, with points $\Gamma$, $K$ and $M$. (b),(c) Band structures along the path $\Gamma-K-M-\Gamma$ for the hamiltonian (\ref{['Hamiltonian']}), with isotropic hopping, and spin splitting arising due to non-collinear spin order. (b) Coplanar non-collinear spin order, with a large trivial gap around $E=0$. (c) Non-coplanar spin order, with a large trivial gap and four small nontrivial gaps. Parameters: $E_1=E_2=E_3=0\,$eV, $V_{dd\pi}=V_{dd\delta}=V_{dd\sigma}=-0.25\,$eV, and $\mathbf{M}(\mathbf{r})=(M_x(\mathbf{r}),M_y(\mathbf{r}),M_z(\mathbf{r}))=M_s(\sin\theta_{\mathbf{r}}\cos\phi_{\mathbf{r}},\sin\theta_{\mathbf{r}}\sin\phi_{\mathbf{r}},\cos\theta_{\mathbf{r}})$, with $M_s=1\,$eV, azimuthal angles $\phi_{\mathbf{r}}=120^{\circ}$, and polar angles $\theta_{\mathbf{r}}=90^{\circ}$ and $\theta_{\mathbf{r}}=53^{\circ}$, in (b) and (c) respectively.
• Figure 3: Heat maps of the dimensionless Berry curvature $\Omega(k_x, k_y)$ for the parameter set of Fig. \ref{['fig:BandStructure']}(c). The Fermi energy lies within the non-trivial gaps of the band structure, and is located in the spin-up sector for (a) and (b), and in the spin-down sector for (c) and (d).
• Figure 4: Dimensionless Hall conductivity $\sigma_{xy}/(e^2/h)$ as a function of the Fermi energy $E_F$ for $T=0.1\,$meV$=1.1\,$K for 120° canted (non-coplanar) antiferromagnetic order. Parameters as in Fig. \ref{['fig:BandStructure']}(c), with $M_z=0.6\,$eV (red) and $M_z=-0.6\,$eV (blue), thus illustrating the symmetry. For some $E_F$, plateaus are found at the maximal Chern number $C=\pm 5$. The dots correspond to numerical data while the solid lines are guide to the eyes. The arrows mark the spin sectors.
• Figure 5: Band structure (a) and Hall conductivity (b) at $T=0.1\,$meV $=1.1\,$K for a kagome $d$-electron system with 120° canted antiferromagnetic order, with parameters (in eV) $E_1=0$, $E_2=0.05$, $E_3=0.10$, $V_{dd\pi}=-0.25$, $V_{dd\delta}=-0.35$, $V_{dd\sigma}=-0.15$, and $M_s=1$, $M_z=\pm 0.6$ for the red/blue curves. Although far from the isotropic case, these parameters still give the maximum value $C=\pm 5$.