Engineering chiral density waves and topological band structures by multiple-$Q$ superpositions of collinear up-up-down-down orders

Magnetic orders characterized by multiple ordering vectors harbor noncollinear and noncoplanar spin textures and can be a source of unusual electronic properties through the spin Berry phase mechanism. We theoretically show that such multiple-$Q$ states are stabilized in itinerant magnets in the form of superpositions of collinear up-up-down-down (UUDD) spin states, which accompany the density waves of vector and scalar chirality. The result is drawn by examining the ground state of the Kondo lattice model with classical localized moments, especially when the Fermi surface is tuned to be partially nested by the symmetry-related commensurate vectors. We unveil the instability toward a double-$Q$ UUDD state with vector chirality density waves on the square lattice and a triple-$Q$ UUDD state with scalar chirality density waves on the triangular lattice, using the perturbative theory and variational calculations. The former double-$Q$ state is also confirmed by large-scale Langevin dynamics simulations. We also show that, for a sufficiently large exchange coupling, the chirality density waves can induce rich nontrivial topology of electronic structures, such as the massless Dirac semimetal, Chern insulator with quantized topological Hall response, and peculiar edge states which depend on the phase of chirality density waves at the edges.

Figure 1

Schematic pictures of (a) helical and (b) UUDD magnetic structures in the one-dimensional representation, (c) collinear single-$Q$ $\left(1Q\right)$ UUDD, (d) $1Q$ UUDD with different ordering vectors from (c) (see the text in detail), (e) coplanar double-$Q$ $\left(2Q\right)$ UUDD consisting of a superposition of (c), (f) $2Q$ UUDD consisting of a superposition of (d) on the square lattice, and (g) noncoplanar triple-$Q$ $\left(3Q\right)$ UUDD magnetic structures on the triangular lattice. The arrows denote the directions of localized moments. The inset of (g) shows the directions of magnetic moments in the $3Q$ UUDD state; each spin points along the local [111] directions in the cubic representation. In all cases, a global spin rotation does not alter the consequences. (h) The square lattice with diagonal bonds, which is topologically equivalent to the triangular lattice in (g). In (e) and (f) [(g) and (h)], the red and blue plaquettes show the positive and negative vector (scalar) chirality defined in Eq. (12) [Eq. (13)].

Figure 2

The Fermi surface for (a) the square lattice model at $t_2=0$ and $\mu =-\sqrt{2}$ and (c) triangular lattice model at $t_2=0.5$ and $\mu =2$. The triangular lattice is regarded as a topologically equivalent square lattice with diagonal bonds, as shown in Fig. 1. ${\mathbf{Q}}_{\nu }$ $(\nu =1$, 2, and 3) are the vectors connecting the Fermi surfaces. (b), (d) The contour plots of the bare susceptibility corresponding to (a) and (c), respectively. The bare susceptibility exhibits maxima at ${\mathbf{Q}}_{\nu }$ and the symmetry-related wave numbers.

Figure 3

The fourth-order contributions to the free energy $F_{\nu }^{\left(4\right)}$ and $F_{\mathrm{helical}}^{\left(4\right)}$ in Eqs. (14) and (15), respectively, divided by $J^4$, as functions of the chemical potential $\mu$ for the ground-state candidates at the RKKY level on (a), (b) square, and (c), (d) triangular lattices. The parameters are (a) $t_2=0,{\mathbf{Q}}_1=(\pi /2,0)$, and ${\mathbf{Q}}_2=(0,\pi /2)$, (b) $t_2=0,{\mathbf{Q}}_1=(\pi /2,\pi )$, and ${\mathbf{Q}}_2=(\pi ,-\pi /2)$, (c) $t_2=0,{\mathbf{Q}}_1=(\pi /2,0),{\mathbf{Q}}_2=(0,-\pi /2)$, and ${\mathbf{Q}}_3=(-\pi /2,\pi /2)$, and (d) $t_2=0.5,{\mathbf{Q}}_1=(\pi /2,0),{\mathbf{Q}}_2=(0,-\pi /2)$, and ${\mathbf{Q}}_3=(-\pi /2,\pi /2)$. The data are calculated at $T=0.03$. The vertical dashed lines point to the optimal chemical potential where the bare susceptibility has maxima at the corresponding wave numbers: (a) $\mu =-\sqrt{2}(1+\sqrt{2})$, (b) $-\sqrt{2}$, (c) $-2(1+\sqrt{2})$, and (d) 2.

Figure 4

The grand potential at zero temperature as functions of the chemical potential $\mu$ calculated by the variational calculation for the Hamiltonian in Eq. (2) at $J=0.1$ on (a), (b) square, and (c), (d) triangular lattices. The grand potential for the $1Q,2Q$, and $3Q$ UUDD magnetically ordered states is measured from that for the helical state. The model parameters in each panel correspond to those in Fig. 3.

Figure 5

(a) Real-space configurations of localized spins and the $z$ components of vector chirality ${\left({\mathbf{\chi }}_{\mathrm{v}}^p\right)}^z$ [see Eq. (12)] in the optimized ground state obtained by the KPM-LD simulation for the model in Eq. (1) on the square lattice. The simulation is done for a ${48}^2$-site cluster at $\mu =-1.39$ $\left(n\simeq 0.522\right)$ for $t_2=0$ and $J=0.3$. Green arrows at the vertices of the square lattice represent the directions of localized spins, and the color for each square plaquette indicates the value of ${\left({\mathbf{\chi }}_{\mathrm{v}}^p\right)}^z$. (b) Enlarged picture of (a) in the dotted square. (c) Spin structure factor divided by the system size obtained from the spin configuration in (a).

Figure 6

Typical energy dispersions of (a) the $2Q$ UUDD state on the square lattice and (b) the $3Q$ UUDD state on the triangular lattice. The parameters are $t_2=0,J=1,{\mathbf{Q}}_1=(\pi /2,0)$, and ${\mathbf{Q}}_2=(0,\pi /2)$ for the $2Q$ state, while $t_2=0,J=2,{\mathbf{Q}}_1=(\pi /2,0),{\mathbf{Q}}_2=(0,-\pi /2)$, and ${\mathbf{Q}}_3=(-\pi /2,\pi /2)$ for the $3Q$ state. The dispersions are shown along the symmetric lines in the folded Brillouin zones. The lowest thick curves show the occupied bands at $n=0.125$. In (a), the massless Dirac node appears at $\mathbf{k}=(k_x,k_y)=(\pi /4,\pi /4)$ at $n=0.125$ as well as several other fillings. Meanwhile, the band gap opens at $n=0.125$ in (b), in addition to several other commensurate fillings. In (b), the values of the quantized topological Hall conductivity in unit of $e^2/h$, which are obtained when the Fermi level locates inside the gap, are also shown in the right side of the panel.

Figure 7

Schematic picture of the system with the (100) edges for (a) square lattice and (b) triangular lattice. The dashed boxes represent unit cells used for the calculations of the edge states; the unit cell contains 256 or 260 sites depending on where the edges are terminated. The red (blue) plaquettes represent positive (negative) values of (a) vector and (b) scalar chirality. (c), (d) Energy dispersions near the Fermi level at $n=0.125$ for (c) the $2Q$ and (d) $3Q$ UUDD states for the systems with open edges shown in (a) and (b), respectively. In (c) [(d)], the solid and dashed lines represent the dispersion in the systems with the $\left({\mathrm{A}}_l,{\mathrm{A}}_r\right)$ $\left[\right({\mathrm{A}}_l,{\mathrm{A}}_r\left)\right]$ and $\left({\mathrm{B}}_l,{\mathrm{B}}_r\right)$ $\left[\right({\mathrm{A}}_l,{\mathrm{B}}_r\left)\right]$ edges, while the thick red lines the edge states in the systems with the $\left({\mathrm{A}}_l,{\mathrm{A}}_r\right)$ $\left[\right({\mathrm{A}}_l,{\mathrm{A}}_r\left)\right]$.