Spontaneous formation of kagome network and Dirac half-semimetal on a triangular lattice

In spin-charge coupled systems, geometrical frustration of underlying lattice structures can give rise to nontrivial magnetic orders and electronic states. Here we explore such a possibility in the Kondo lattice model with classical localized spins on a triangular lattice by using a variational calculation and simulated annealing. We find that the system exhibits a four-sublattice collinear ferrimagnetic phase at 5/8 filling for a large Hund's-rule coupling. In this state, the system spontaneously differentiates into the up-spin kagome network and the isolated down-spin sites, which we call the kagome network formation. In the kagome network state, the system becomes Dirac half-semimetallic: The electronic structure shows a massless Dirac node at the Fermi level, and the Dirac electrons are almost fully spin polarized due to the large Hund's-rule coupling. We also study the effect of off-site Coulomb repulsion in the kagome network phase where the system is effectively regarded as a 1/3-filling spinless fermion system on the kagome lattice. We find that, at the level of the mean-field approximation, a $\sqrt{3}\times \sqrt{3}$-type charge order occurs in the kagome network state, implying the possibility of fractional charge excitations in this triangular lattice system. Moreover, we demonstrate that the kagome network formation with fully polarized Dirac electrons are controllable by an external magnetic field.

Figure 1

Ground-state phase diagrams for the model (1) on a triangular lattice as functions of (a) $n$ and $J_{\mathrm{H}}$ at $J_{\mathrm{AF}}=0.12$, and (b) $J_{\mathrm{AF}}$ and $J_{\mathrm{H}}$ at $n=5/8$. The vertical thick (thin) dashed lines at $n=1/4,1/2$, and $3/4$ $(n=1/8$ and $5/8)$ in (a) indicate gapful insulating (zero-gap semiconducting) regions. PS stands for the phase-separated region (see the text in Sec. 2b for details). The thick black line at $n=0$ represents that the (3a) ${120}^{\circ }$ coplanar ordered phase is stabilized at zero doping. In (b) we ignore the possibility of PS. (c) Ordering patterns of localized spins used in the variational calculations: (1a) FM order, (2a) and (2b) two-sublattice orders, (3a)–(3d) three-sublattice orders, and (4a)–(4g) four-sublattice orders (see the text in Sec. 2b for details). The numbers in (3a) and (4a) indicate the sublattices at which the spins are located on the two-dimensional lattice shown in each left panel.

Figure 2

(a) The band structure and (b) the density of state at $J_{\mathrm{H}}=10$ in the spontaneous KN phase. Note that the band structure does not depend on the value of $J_{\mathrm{AF}}$ up to a constant energy shift (we take $J_{\mathrm{AF}}=0$ here). The blue (white) hexagon in the inset of (a) indicates the folded (original) Brillouin zone. The band dispersions are plotted along the symmetric line in the folded Brillouin zone. The red (blue) color indicates the contributions from the majority (minority) spins. The dashed line indicates the Fermi level located just at the Dirac node at the $K^{\prime \prime }$ point.

Figure 3

Energy per site (left axis) and the spin structure factor $S\left(\mathbf{q}\right)$ at $\mathbf{q}=(0,0)$ (right axis) as functions of the cooling step in the simulated annealing for $J_{\mathrm{H}}=10,J_{\mathrm{AF}}=0.12$, and $D=0.044$ at $n=5/8$. The dashed lines indicate the values of energy and $S(0,0)$ in the limit of $T=0$ in the KN state. See the text for details. In the inset, the blue dashed hexagon on a triangular lattice represents the 12-site unit cell on which we performed the simulated annealing.

Figure 4

(a) Ground-state phase diagram obtained by the mean-field approximation as a function of $V$ and $J_{\mathrm{AF}}$. The calculation was done for $J_{\mathrm{H}}=20$ at $n=5/8$. The notations such as (1a) and (3d) are common to those in Fig. 1, which label the magnetic ordering patterns. (3d) ordered region of the phase diagram is a collinear order with $\theta =\pi$ except at $V=0$. The vertical dashed lines indicate the second-order transition. (b) Schematic pictures of the order patterns of the phases in (a): (1a-CO) four-sublattice FM $\mathbf{q}=\mathbf{0}$ charge order (CO), (3a-CO1) 12-sublattice ${120}^{\circ }$-AFM $\mathbf{q}=\mathbf{0}$ CO, (3a-CO2) 12-sublattice ${120}^{\circ }$-AFM CO with a nonzero net moment, (3d-CO1) and (3d-CO2) 12-sublattice ferrimagnetic collinear CO with nonzero net moments, and (4e-CO) 12-sublattice KN $\mathbf{q}=(2\pi /3,2\pi /\sqrt{3})$ CO with a nonzero net moment. The size of circles (length and direction of arrows) denotes the charge density (the length and direction of spins) of itinerant electrons. The blue dashed rhombuses represent the unit cells. The red lines are the up-spin networks. All the directions of localized spins are parallel to those of itinerant electrons spins. All the charge ordered phases are insulating.

Figure 5

Magnetization per site as a function of an external magnetic field $h^z$ in the $z$ direction obtained by the simulated annealing supplemented by variational calculations. The data are calculated for $J_{\mathrm{H}}=10$ and $J_{\mathrm{AF}}=0.12$ at $n=5/8$. The (4e), (4g), (3d), and (1a) phases appear for $0.04\lesssim h^z\lesssim 0.16,0.16\lesssim h^z\lesssim 0.38,0.38\lesssim h^z\lesssim 0.58$, and $0.58\lesssim h^z$, respectively. The first-order transition occurs near $h^z\sim 0.38$. Others are continuous transition. In the shaded area for $0\le h^z\lesssim 0.04$, it is hard to obtain good convergence; we plot the magnetization for the 12 sublattice chiral order [46] appearing as the solution in the simulated annealing.

Figure 6

Ground-state phase diagrams for the model (2) as functions of $n$ and $D$ at $J_{\mathrm{H}}=10$ and $J_{\mathrm{AF}}=0.12$ obtained by the variational calculation. (b) The enlarged view of (a) near $n=5/8$. The vertical thick (thin) dashed lines at $n=1/2$ and $3/4$ $\left(n=5/8\right)$ in (a) indicate gapful insulating (zero-gap semiconducting) regions. PS stands for the phase-separated region.