Charge density waves in multiple-$Q$ spin states

The spin texture characterized by the 1$Q$ sinusoidal wave is represented by

where $\bm{r}_{i}$ is the position vector at site $i$ and $\mathcal{N}_{i}$ represents the normalization to satisfy $|\bm{S}_{i}|=1$ at each site. Here we take the ordering vector $\bm{Q}_{1}=(\pi/3,0)$ (the lattice constant is set to unity). The spin configuration is shown in the left panel of Fig. 2(a), where the spins are aligned in an up-up-up-down-down-down way along the $x$ direction, and hence, there is no net magnetization. This spin state shows dominant Bragg peaks at $\pm\bm{Q}_{1}$ in the spin structure factor, and in addition, higher harmonics at $\pm 3\bm{Q}_{1}$, as shown in the middle left panel of Fig. 2(a). Here, the spin structure factor for the localized spins is calculated as

In this situation, we can predict the CDW modulation by using Eq. (III.1). Specifically, by plugging $\pm\bm{Q}_{1}$ and $\pm 3\bm{Q}_{1}$ into $\bm{q}_{1}$ and $\bm{q}_{2}$ in Eq. (III.1), we find that only the $\bm{q}=\pm 2\bm{Q}_{1}$ components of $n_{\bm{q}}$ become nonzero in the first Brillouin zone. Note that there are three combinations that contribute to $n_{2\bm{Q}_{1}}$: $(\bm{q}_{1},\bm{q}_{2})=(\bm{Q}_{1},\bm{Q}_{1})$, $(-\bm{Q}_{1},3\bm{Q}_{1})$, and $(3\bm{Q}_{1},-\bm{Q}_{1})$ ($\pm 3\bm{Q}_{1}$ are equivalent as they are on the zone boundary). In Fig. 3(a), the solid lines show the chemical potential $\mu$ dependence of $|n_{2\bm{Q}_{1}}|$ calculated by Eq. (III.1) for the $\bm{k}$-space mesh of $1440\times 1440$ ($N=1440^{2}$). We choose a small value of $J=0.05$, as the expression in Eq. (III.1) is derived in the limit of the weak spin-charge coupling. The result for $\mu<0$ is obtained from that for $\mu>0$ owing to particle-hole symmetry of the model; $\mu=0$ corresponds to the half filling. As shown in Fig. 3(a), $|n_{2\bm{Q}_{1}}|$ shows a nonmonotonic behavior against $\mu$. The sign of $n_{2\bm{Q}_{1}}$, which corresponds to the phase of the CDW, changes at particular values of $\mu$; $n_{2\bm{Q}_{1}}>0$ for $0.35\lesssim\mu\lesssim 1.42$ and $3.13\lesssim\mu\lesssim 4$, while $n_{2\bm{Q}_{1}}<0$ for the other regions.

Independently, we can evaluate the CDW modulation by the direct numerical diagonalization of the Kondo lattice model in Eq. (1). Substituting the $1Q$ sinusoidal state in Eq. (5) into Eq. (1), the local charge density modulation is calculated as $\Delta n_{i}=\sum_{\sigma}\langle c^{\dagger}_{i\sigma}c_{i\sigma}\rangle-n^{\rm ave}$, where $n^{\rm ave}$ is the average charge density. The real-space distribution of $\Delta n_{i}$ at $\mu=3$ is shown in the middle right panel of Fig. 2(a). The data are calculated for the system size $N=96^{2}$ (the figure shows a part of the whole system with $24^{2}$ sites). The characteristic wave vectors of this CDW is extracted from the charge structure factor defined by

Note that $|n_{\bm{q}}|=\sqrt{N^{c}(\bm{q})/N}$. The result is plotted in the right panel of Fig. 2(a). We find that the Bragg peaks appear only at $\pm 2\bm{Q}_{1}$, as predicted by the perturbative formula in Eq. (III.1).

In order to quantitatively test the perturbative argument, we compare the $\mu$ dependences of $|n_{2\bm{Q}_{1}}|$ between the results by Eq. (III.1) and the direct diagonalization at $J=0.05$ in Fig. 3(a). We find good agreement in the whole range of $\mu$ including the sign change of $n_{2\bm{Q}_{1}}$, which validates the perturbative argument. Figure 3(b) shows further comparison for larger $J$. The results by the diagonalization gradually deviate from that by the perturbative calculations while increasing $J$, whereas we do not find any additional CDW modulations at other $\bm{q}$ within this range (see Sec. III.5 for larger $J$).

The $2\bm{Q}_{1}$ CDW modulation for the $\bm{Q}_{1}$ sinusoidal SDW is also intuitively understood from the real-space picture. When the itinerant electrons move on the up-up-up-down-down-down spin texture, the effective hopping amplitude is modulated in a different way for neighboring parallel (up-up or down-down) spins and antiparallel (up-down or down-up) spins. This modulation of the kinetic energy results in the CDW modulation. As the period of the modulation of the kinetic energy is a half of that of the SDW, the period of the charge modulation is also halved, which results in the $2\bm{Q}_{1}$ CDW. (Detailed analysis will be given for the $2Q$ cases in Sec. III.4.)

Similar results will apply to the $1Q$ SDW states with spin spirals, at least, when the spirals are elliptically modulated as

with $a_{y}\neq a_{z}$; note that Eq. (8) with $a_{y}=0$ and $a_{z}\neq 0$ reduces to the 1$Q$ sinusoidal spin state in Eq. (5). Meanwhile, the CDW modulation vanishes for the 1$Q$ circular spiral state with $a_{y}=a_{z}$, since $n_{2\bm{Q}_{1}}$ in Eq. (III.1) becomes zero owing to the cancellation between $S^{y}_{\bm{Q}_{1}}S^{y}_{\bm{Q}_{1}}$ and $S^{z}_{\bm{Q}_{1}}S^{z}_{\bm{Q}_{1}}$. This is reasonable from the real-space picture: the effective hopping is renormalized but remains spatially uniform because of the uniform twist of spins.

We further perform the comparison for the 1$Q$ sinusoidal spin configuration with a net magnetization given by

We take $\tilde{M}^{z}=0.7$ so that the real-space spin configuration becomes the up-up-up-up-up-down spin configuration, as shown in the left panel of Fig. 2(b). Note that $\tilde{M}^{z}$ is not the actual value of the net magnetization. In contrast to the case without the magnetization, this spin state shows additional peaks at $\bm{q}=\bm{0}$ and $\pm 2\bm{Q}_{1}$ in the spin structure factor, as shown in the middle left panel of Fig. 2(b). Reflecting the additional Fourier components, the perturbative formula in Eq. (III.1) predicts additional CDW modulations at $\bm{q}=\pm\bm{Q}_{1}$ and $\pm 3\bm{Q}_{1}$ owing to nonzero spin products of $S^{z}_{\pm\bm{Q}_{1}}S^{z}_{\bm{0}}$ and $S^{z}_{\pm\bm{Q}_{1}}S^{z}_{\pm 2\bm{Q}_{1}}$. This is indeed confirmed by the direct diagonalization, as shown in the real-space charge modulation and the charge structure factor in the middle right and right panels of Fig. 2(b), respectively.

We begin with a superposition of the 1$Q$ spiral wave along the $\bm{Q}_{1}$ direction and the 1$Q$ sinusoidal wave along the $\bm{Q}_{2}$ direction. This is called the $2Q$ chiral stripe state, whose spin configuration is given by [64]

where $b$ is a parameter that controls the mixture of the second component with $\bm{Q}_{2}$; we here take $b=0.8$. The real-space spin configuration is shown in the left panel of Fig. 4(a), which consists of a periodic array of vortices and antivortices in the $xy$-spin components and the stripe modulation in the $z$-spin component. Owing to the noncoplanar spin texture, this state accompanies a density wave of the scalar spin chirality defined by the triple product of three neighboring spins, $\bm{S}_{i}\cdot(\bm{S}_{j}\times\bm{S}_{k})$, along the $\bm{Q}_{2}$ direction, which is the reason why this spin state is called the $2Q$ chiral stripe. There is no net scalar chirality, and hence, no topological Hall effect occurs in this state. The spin structure factor exhibits the Bragg peaks at $\pm\bm{Q}_{1}$ and $\pm\bm{Q}_{2}$ with different intensities and at the higher harmonics at $\pm\bm{Q}_{1}\pm 2\bm{Q}_{2}$, as shown in the middle left panel of Fig. 4(a). The nonzero intensities at $\pm\bm{Q}_{1}\pm 2\bm{Q}_{2}$ are attributed to the factor $\sqrt{1-b^{2}\sin^{2}\bm{Q}_{2}\cdot\bm{r}_{i}}$ in Eq. (13), which includes the contribution of $\cos 2\bm{Q}_{2}\cdot\bm{r}_{i}$. This 2$Q$ chiral stripe has been widely found in itinerant magnets on various lattices, e.g., the Kondo lattice model on the square [64, 66, 58], triangular [66, 67, 68], and cubic [69] lattices, and the $d$-$p$ model on the square lattice [70].

The right two panels of Fig. 4(a) represent the real-space charge distribution and the charge structure factor in the 2$Q$ chiral stripe state. The local charge density oscillates only along the $\bm{Q}_{2}$ direction, and indeed, the charge structure factor exhibits the peaks only at $\pm\bm{Q}_{2}$. At first glance, the result appears to contradict with the perturbative argument for the given peaks in the spin structure factor at $\pm\bm{Q}_{1}$, $\pm\bm{Q}_{2}$, and $\pm\bm{Q}_{1}\pm 2\bm{Q}_{2}$, but it is understood as follows. Equation (13) is represented by a superposition of the circular spiral wave with $|S^{y}_{\bm{Q}_{1}}|=|S^{z}_{\bm{Q}_{1}}|$ along the $\bm{Q}_{1}$ direction and the sinusoidal wave along the $\bm{Q}_{2}$ direction. The latter gives rise to nonzero $n_{2\bm{Q}_{2}}$, whereas the former does not lead to any CDW, as discussed in Sec. III.2.1. Thus, the CDW with $\pm\bm{Q}_{2}$ is compatible with the perturbative formula in Eq. (III.1).

Next, we consider the 2$Q$ coplanar state, which is characterized by a superposition of two sinusoidal waves. The spin configuration is given by

The real-space picture of the localized spins is shown in the left panel of Fig. 4(b). This state also consists of a periodic array of vortices and antivortices like the 2$Q$ chiral stripe state, but all the spins are coplanar with no $z$-spin component. Owing to the normalization condition of $|\bm{S}_{i}|=1$, the spin structure factor shows the peaks at higher harmonics $\pm\bm{Q}_{\nu}\pm 2\bm{Q}_{\nu}$ and $\pm 2\bm{Q}_{\nu}\pm 3\bm{Q}_{\nu}$ ($\nu=1,2$), in addition to $\pm\bm{Q}_{\nu}$ and $\pm 3\bm{Q}_{\nu}$ expected for each sinusoidal wave, as shown in the middle left panel of Fig. 4(b). This state has been discussed in itinerant magnets at zero field [71] and in frustrated and itinerant magnets under an external magnetic field [66, 58, 72, 73].

In the 2$Q$ coplanar state, the local charge density is modulated in both the $x$ and $y$ directions unlike the $2Q$ chiral stripe, as shown in the middle right panel of Fig. 4(b). The charge structure factor exhibits the peaks at $\pm 2\bm{Q}_{1}\pm 2\bm{Q}_{2}$, in addition to $\pm 2\bm{Q}_{1}$ and $\pm 2\bm{Q}_{2}$ which are expected from each sinusoidal wave. This is because there are nonzero contributions from, e.g., $\bm{S}_{\bm{Q}_{1}}\cdot\bm{S}_{\bm{Q}_{1}+2\bm{Q}_{2}}$ and $\bm{S}_{3\bm{Q}_{2}}\cdot\bm{S}_{2\bm{Q}_{1}-\bm{Q}_{2}}$, to $n_{\bm{q}}$ with $\bm{q}=2\bm{Q}_{1}+2\bm{Q}_{2}$ in Eq. (III.1).

The 2$Q$ MAX is obtained as a modulation of the $2Q$ coplanar state by adding a nonzero $z$-spin component as

This is regarded as a superposition of two spin helices: one is in the $yz$-spin component with wave vector $\bm{Q}_{1}$, and the other is in the $xz$-spin component with $\bm{Q}_{2}$. The real-space spin configuration is shown in the left panel of Fig. 4(c). Reflecting the modulation in the $z$-spin component, the 2$Q$ MAX consists of a periodic array of meron and antimeron characterized by a half skyrmion number each with opposite sign [74, 75, 76]. The cancellation of the skyrmion number between the merons and antimerons leads to no topological Hall effect. The peak positions of the spin structure factor are the same as those in the 2$Q$ coplanar state, as shown in the middle left panel of Fig. 4(c). This state has been discussed in chiral magnets [77], frustrated magnets [73], and polar itinerant magnets [78, 79].

In spite of the same set of the Bragg peaks in the spin structure factor, the CDW modulations are qualitatively different between the 2$Q$ coplanar state and the 2$Q$ MAX, as shown in the right two panels of Figs. 4(b) and 4(c), respectively. The primary difference is found in the $\pm\bm{Q}_{1}\pm\bm{Q}_{2}$ components in the charge structure factor; the 2$Q$ MAX has nonzero $n_{\pm\bm{Q}_{1}\pm\bm{Q}_{2}}$, while the 2$Q$ coplanar state does not. This is because the 2$Q$ structure in the $z$-spin component in the 2$Q$ MAX gives a contribution of $S^{z}_{\pm\bm{Q}_{1}}S^{z}_{\pm\bm{Q}_{2}}$ to $n_{\bm{q}}$ with $\bm{q}=\pm\bm{Q}_{1}\pm\bm{Q}_{2}$ in Eq. (III.1). Furthermore, it is worthwhile mentioning that the 2$Q$ MAX state exhibits the CDW modulations, although the constituent waves are the circular spiral waves rather than the sinusoidal waves. This appears to contradict with the observations in the $1Q$ case in Sec. III.2.1, but can be understood from the different amplitudes between the $xy$- and $z$-spin components; the former has the 1$Q$ component, whereas the latter has 2$Q$, which effectively make the $xy$ and $z$ spins inequivalent. Thus, the multiple-$Q$ spiral states may be accompanied by different CDW modulation from the 1$Q$ cases.

Lastly, we discuss the CDW in the 2$Q$ SkX. The spin configuration is obtained by adding a uniform $z$-spin component to the 2$Q$ MAX in Eq. (18), which is given by

We here set $\tilde{M}^{z}=0.7$. As shown in the left panel of Fig. 4(d), the skyrmion cores defined by $S^{z}_{i}\simeq-1$ form a square lattice, and hence, this state is called the square SkX. Different from the 2$Q$ MAX, this spin configuration shows a nonzero net value of the scalar spin chirality, which results in the topological Hall effect. Owing to the nonzero magnetization, there are additional peaks in the spin structure factor at $\bm{q}=\bm{0}$, $\pm\bm{Q}_{1}\pm\bm{Q}_{2}$, $\pm 2\bm{Q}_{1}$, $\pm 2\bm{Q}_{2}$, and $\pm 2\bm{Q}_{1}\pm 2\bm{Q}_{2}$, as shown in the middle left panel of Fig. 4(d) (we note that there are also peaks with weak intensity on the Brillouin zone boundary, e.g., at $3\bm{Q}_{1}+\bm{Q}_{2}$). This state has been widely discussed in itinerant magnets [58] and in localized magnets [72, 73].

The additional $\bm{q}$ components of spins induce additional charge modulations, as shown in the right two panels of Fig. 4(d). The intensities become nonzero at multiples of $\bm{Q}_{1}$ and $\bm{Q}_{2}$, i.e., $m_{1}\bm{Q}_{1}+m_{2}\bm{Q}_{2}$ where $m_{1}$ and $m_{2}$ are integers, all of which are accounted for by the perturbative formula in Eq. (III.1).

We note that a similar CDW modulation is expected to appear in the collinear 2$Q$ bubble crystal without the $xy$ spin components in Eq. (22), which might be realized in CeAuSb₂ [80, 81, 82].

Figure 7(a) shows the real-space charge distribution (left panel) and the charge structure factor (right panel) in the 2$Q$ chiral stripe state, which is obtained by the direct diagonalization of $\mathcal{H}+\mathcal{H}^{\rm ASOC}$ at $J=0.05$, $\tilde{\alpha}=0.5$, and $\mu=3$. We consider the same spin configuration in Eq. (13) in order to show the effect of the spin-orbit coupling. Compared to the result in Fig. 4(a), the local charge density shows a modulation along the $x$ direction in addition to the $y$ direction, as shown in the left panel of Fig. 7(a). By performing the Fourier transformation, there are additional three peaks in the charge structure factor at $\pm 2\bm{Q}_{1}$, $\pm(\bm{Q}_{1}+\bm{Q}_{2})$, and $\pm(\bm{Q}_{1}-\bm{Q}_{2})$ in addition to $\pm 2\bm{Q}_{2}$, as shown in the right panel of Fig. 7(a). The appearance of $n_{2\bm{Q}_{1}}$ is owing to $\Pi^{yy}_{(\bm{k}\bm{Q}_{1}\bm{Q}_{1})(\sigma\sigma^{\prime}\sigma^{\prime\prime})}S^{y}_{\bm{Q}_{1}}S^{y}_{\bm{Q}_{1}}\neq\Pi^{zz}_{(\bm{k}\bm{Q}_{1}\bm{Q}_{1})(\sigma\sigma^{\prime}\sigma^{\prime\prime})}S^{z}_{\bm{Q}_{1}}S^{z}_{\bm{Q}_{1}}$ [for instance, see Eqs. (30) and (31)], while that of $n_{\bm{Q}_{1}+\bm{Q}_{2}}$ is owing to nonzero $\Pi^{yx}_{(\bm{k}\bm{Q}_{1}\bm{Q}_{2})(\sigma\sigma^{\prime}\sigma^{\prime\prime})}S^{y}_{\bm{Q}_{1}}S^{x}_{\bm{Q}_{2}}$, $\Pi^{zx}_{(\bm{k}\bm{Q}_{1}\bm{Q}_{2})(\sigma\sigma^{\prime}\sigma^{\prime\prime})}S^{z}_{\bm{Q}_{1}}S^{x}_{\bm{Q}_{2}}$ [for instance, see Eqs. (33) and (III.6)].

Figure 7(b) shows the CDW in real and momentum spaces in the 2$Q$ coplanar state with the same model parameters as in Sec. III.6.1. The spin configuration is given by Eq. (14). Although the local charge density in the left panel of Fig. 7(b) looks similar to that in Fig. 4(b), additional components at $\bm{q}=\pm\bm{Q}_{1}\pm\bm{Q}_{2}$ are found in the charge structure factor by introducing the antisymmetric spin-orbit coupling, as shown in the right panel of Fig. 7(b). This additional modulation originates from nonzero $\Pi^{yx}_{(\bm{k}\bm{Q}_{1}\bm{Q}_{2})(\sigma\sigma^{\prime}\sigma^{\prime\prime})}S^{y}_{\bm{Q}_{1}}S^{x}_{\bm{Q}_{2}}$ [for instance, see Eq. (33)].