Exotic magnetic orderings in the kagome Kondo-lattice model

We consider the Kondo-lattice model on the kagome lattice and study its weak-coupling instabilities at band filling fractions for which the Fermi surface has singularities. These singularities include Dirac points, quadratic Fermi points in contact with a flat band, and van Hove saddle points. By combining a controlled analytical approach with large-scale numerical simulations, we demonstrate that the weak-coupling instabilities of the Kondo-lattice model lead to exotic magnetic orderings. In particular, some of these magnetic orderings produce a spontaneous quantum anomalous Hall state.

Figure 1

Kondo-lattice model (KLM) on the kagome lattice. The model can be viewed as consisting of two subsystems represented here by two fictitious kagome layers. Conduction electrons hop on the top layer according to a nearest-neighbor transfer integral $t$. The bottom layer consists of localized spins (magenta arrows) at every kagome site. The electron spins (green arrows) are coupled to the localized moments through an on-site exchange constant $J$. The numbers 1, 2, 3 denote the three inequivalent sublattices of kagome.

Figure 2

(a) Band structure of the tight-binding model on the kagome lattice. Here and in the following figures, the band energy ${\epsilon }_{\mathbf{k}}$ is measured in units of NN hopping $t$. (b) Brillouin zone (BZ) of the kagome lattice. At filling fraction $f=1/3$, the Fermi surface shrinks to two inequivalent Dirac points $K_{1,2}$ at the corners of the BZ. These two Dirac points are connected by wave vectors ${\mathbf{K}}_{\pm }=(\pm 4\pi /3,0)$. (c) The dispersion has three saddle points $M_{1,2,3}$ at the edge of the BZ, giving rise to a logarithmically divergent DOS at filling fractions $f=1/4$ and $5/12$. The corresponding Fermi surface is a regular hexagon inscribed within the BZ. Pairs of parallel edges of this Fermi surface are perfectly nested by wave vectors ${\mathbf{Q}}_1=(2\pi ,0)$ and ${\mathbf{Q}}_{2,3}=(-\pi ,\pm \sqrt{3}\pi )$.

Figure 3

(a)–(c) Schematic diagram showing the triple-$\mathbf{Q}$ scalar order (or collinear spin order) corresponding to the three irreducible representations $A_1,B_2$, and $B_1$. The blue (red) circles denote on-site scalar variable ${\zeta }_i=+1 (-1$), while empty site corresponds to ${\zeta }_i=0$. (d) shows the collinear and orthogonal configurations of the triple-$\mathbf{Q}$ vector order parameters.

Figure 4

(a) The triple-$\mathbf{Q}$ noncoplanar (orthogonal) spin order on kagome lattice that transforms according to the $A_2$ irreducible representation of the symmetry group. Spins at the three inequivalent sublattices are indicated by different colors. This $A_2$ symmetry state is the ground state at filling fraction $f=1/4$ (for $J\lesssim 2.2t$) and $f=7/12$ (for $J\lesssim 1.3t$). The corresponding band structure is shown in panel (b). The exchange coupling $J=0.22t$ is used in the calculation. The dashed lines indicate the Fermi levels at filling fractions $f=1/4$ and $7/12$.

Figure 5

The chiral structure of spin orders at van Hove fillings in the kagome lattice. There is a vector associated with each triangle. The four different orientations of the vectors (indicated by four different colors) point to the corners of a regular tetrahedron. The vectors associated with each triangle correspond to the total spins ${\mathbf{M}}_{▵}={\mathbf{S}}_1+{\mathbf{S}}_2+{\mathbf{S}}_3$ in the $A_2$ symmetry order at $f=1/4$ filling [Fig. 4], and to the vector chirality ${\mathit{\chi }}_{▵}={\mathbf{S}}_1\times {\mathbf{S}}_2+{\mathbf{S}}_2\times {\mathbf{S}}_3+{\mathbf{S}}_3\times {\mathbf{S}}_1$ in the $B_2$ symmetry order at the second van Hove filling $f=5/12$ [Fig. 7].

Figure 6

Snapshots of the scalar chirality field obtained from KPM-Langevin simulations of $N=3\times {32}^2$ spins at filling fractions (a) $f=1/4$ and (b) $f=7/12$. The magnetic ground state at these two filling fractions is the $A_2$ symmetry state shown in Fig. 4. At the intermediate times shown, nonequilibrium features such as chiral domain walls and ${\mathbb{Z}}_2$ vortices are apparent, although they will eventually annihilate under the (nearly) zero-temperature KPM-Langevin dynamics. We use a moderate coupling constant of $J=0.5t$ for these simulations.

Figure 7

(a) The $B_2$ symmetry triple-$\mathbf{Q}$ noncoplanar spin order on kagome lattice. Spins at the three inequivalent sublattices are indicated by different colors. This state is the ground state at filling fraction $f=5/12$. (b) shows the corresponding band structure calculated with exchange coupling $J=0.2t$. The dashed line indicates the Fermi level at $f=5/12$ filling.

Figure 8

Snapshots of two metastable spin configurations obtained from large-scale KPM-Langevin simulations of $N=3\times {32}^2$ spins for filling fraction $f=5/12$. The color indicates the $z$ component of the vector chirality ${\mathit{\chi }}_{▵}={\mathbf{S}}_1\times {\mathbf{S}}_2+{\mathbf{S}}_2\times {\mathbf{S}}_3+{\mathbf{S}}_3\times {\mathbf{S}}_1$. The two panels show (a) ferromagnetic and (b) antiferromagnetic alignments of ${\mathit{\chi }}_{▵}$. These two ${120}^{\circ }$ coplanar orderings have nearly identical energy densities, so entropy and nonequilibrium effects determine the ordering found in our KPM-Langevin simulations.

Figure 9

(a) The $\sqrt{3}\times \sqrt{3}$ coplanar spin order on kagome lattice at filling fraction $f=1/3$. Spins at the three inequivalent sublattices are indicated by different colors. (b) The corresponding band structure with exchange coupling $J=0.2t$. The dashed line indicates filling fraction $f=1/3$.

Figure 10

(a) The $B_1$ symmetry triple-$\mathbf{Q}$ noncoplanar spin order on the kagome lattice. Spins at the three inequivalent sublattices are indicated by different colors. It is the ground state at filling fraction $f=2/3$ for weak couplings. The same magnetic order is also the ground state at filling fraction $f=1/12$ for intermediate couplings. (b) The corresponding band structure with exchange coupling $J=0.4t$. The two dashed lines indicate the Fermi levels at $f=1/12$ and $f=2/3$ filling fractions.

Figure 11

(a) The total energy (arbitrary units) as a function of the canting angle $\theta$ for the $\mathbf{Q}=0$ umbrella state. The inset shows the three spins of the umbrella state in a triangular unit cell. This curve is obtained using the following parameters: $J=0.2t,t_2=-0.2t$. (b) The corresponding band structure with exchange coupling $J=0.2t,t_2=-0.2t$ and canting angle $\theta ={10}^{\circ }$. The dashed line indicates the Fermi level at $f=2/3$ filling.