$\mathbb{Z}_2$ Vortex Crystals in Tetrahedral Antiferromagnets: Fractional Charges and Topological Magnons

Abstract

We report the formation of a $\mathbb{Z}_2$ vortex crystal in the tetrahedral antiferromagnetic order on a triangular lattice. The noncoplanar tetrahedral state consists of four sublattices with spins oriented along the faces of a tetrahedron in spin space. The long-range order characterized by a $\mathbb{Z}_2$ topology arises due to the Dzyaloshinskii-Moriya interaction and appears at zero temperature and without external fields. Each vortex carries a half-integer electric charge relative to the uniform background in itinerant magnets, enabling the emergence of anyonic excitations. Its magnetic excitations include magnetically active gyrotropic and breathing modes, which -- under an external magnetic field -- carry nontrivial Chern numbers that stabilize chiral magnon edge states.

Figures (4)

• Figure 1: The $\mathbb{Z}_2$ vortex crystal in the noncoplanar four-sublattice tetrahedral AFM order consists of interpenetrating triple-$q$ orders. (a) The full spin configuration near the $\mathbb{Z}_2$ vortex, (b) its decomposition into one of the four sublattices, and (c) the tetrahedral representation of the spin structure, where each tetrahedron corresponds to four lattice sites. A magnetic unit cell of the $\mathbb{Z}_2$ vortex crystal is plotted in (b) and (c). Faces of tetrahedra are orthogonal to spin vectors of each sublattice in (c), which are colored magenta, cyan, black, and yellow. The rotation centers of the tetrahedra (encircled in blue) are characterized by a nontrivial $\mathbb{Z}_2$ vorticity. The sublattice spin texture exhibits singular defects at $\mathbb{Z}_2$ vortex cores as highlighted in the inset of (b). The inset of (c) shows the four-sublattice order on a triangular lattice with the numbers corresponding to the sublattice index. The spin texture was obtained by Monte Carlo simulations on a $50\times 50$ spin lattice with periodic boundary conditions. The parameters are set to $B_1/J_1=1$, $J_2/J_1=B_2/B_1=0.5$, $D_1/J_1=0.5$, and $b/J_1=0$.
• Figure 2: (a) Classical ground state phase diagram of $H_\text{BQ}$ in Eq. \ref{['eq:H_BQ']} spanned by the biquadratic interaction $B_1/J_1$ and the relative strength of next nearest coupling $\lambda=J_2/J_1=B_2/B_1$. This phase diagram is obtained by Monte Carlo simulations on a $30\times 30$ triangular lattice of spins. The color indicates the average angle $\overline{\theta}$ between nearest neighbor spins. (b) Magnon band structure of the tetrahedral AFM order. The parameters are set as $B_1/J_1=1$ and $\lambda=0.5$.
• Figure 3: (a) The magnetic phase diagram for the model described in Eq. \ref{['eq:model']}, showing the stability region for the $\mathbb{Z}_2$ VC phase as a function of the relative strength of interfacial DM interactions ($D_1/J_1$) and external magnetic fields ($b/J_1$). This phase diagram is obtained by Monte Carlo simulations on a $120\times 120$ triangular lattice of spins. The color scale indicates the ratio between the total number of $\mathbb{Z}_2$ vortices ($\nu$) and the difference in the total skyrmion number compared to the uniform tetrahedral state ($\Delta N_\textrm{sk}=N_\textrm{sk}-N_\textrm{sk}^\textrm{tetra}$). Black dashed lines are included as visual guides. The tetrahedral phase denotes the uniform tetrahedral state, where $\nu=0$ (light gray region). The helical state with $\nu=0$ is also obtained as a metastable configuration (magenta circles). The gray region indicates $|\Delta N_\textrm{sk}|/\nu>1.0$. The parameters are set as $B_1/J_1=1$ and $\lambda=0.5$. (b) The half-electric charge bound to a $\mathbb{Z}_2$ vortex in the Kondo lattice model, computed for a hexagon of length $L$. The red dashed line plots the average of two consecutive points. The magnetic unit cell of the $\mathbb{Z}_2$ VC in Fig. \ref{['fig1']} is used for the calculation of the Kondo lattice model.
• Figure 4: Magnetic activity and topological magnons in the $\mathbb{Z}_2$ VC phase. (a) Magnon band structure and (b) imaginary part of the dynamical susceptibility $\textrm{Im}\,\chi(\omega)$ at $b/J_1=0$. The blue (red) line shows the response of in-plane (out-of-plane) magnetization to in-plane (out-of-plane) fields. (c) Magnon band structure and (d) local density of states (LDOS) at the edge of a semi-infinite lattice at $b/J_1=0.5$. The LDOS is computed for regions enclosed by red dashed lines in (c). Encircled numbers indicate Chern numbers of magnon bulk bands. In (a)-(c), filled circles indicate magnetically active magnon modes at the $\Gamma$ point, corresponding to CCW (blue), CW (red), and breathing+polygon deformation mode (green) from the lowest band upward. Parameters are consistent with those of Fig. \ref{['fig1']} except for the applied magnetic field.