Vortices, skyrmions, and chirality waves in frustrated Mott insulators with a quenched periodic array of impurities

Finite-$Q$ magnetic instabilities are rather common in frustrated magnets. When the magnetic susceptibility is maximized at multiple-$Q$ vectors related through lattice symmetry operations, exotic magnetic orderings such as vortex and skyrmion crystals may follow. Here, we show that a periodic array of nonmagnetic impurities, which can be realized through charge density wave ordering, leads to a rich phase diagram featuring a plethora of chiral magnetic phases, especially when there is a simple relation between the reciprocal vectors of the impurity superlattice and the magnetic $Q$ vectors. We also investigate the effect of changing the impurity concentration or disturbing the impurity array with small quenched randomness. Alternative realizations of impurity superlattices are briefly discussed.

Figure 1

Schematic picture for a local configuration in a magnetic field around a nonmagnetic impurity (at the center). The straight solid (dashed) lines represent the remaining (removed) ferromagnetic exchange $J_1$ interactions. The arc lines represent the antiferromagnetic exchange $J_3$.

Figure 2

(a) Phase diagram for a perfect periodic array of impurities $\left(a_{\text{imp}}=8\right)$ with $J_3/\left|J_1\right|\approx 0.854$ $\left(Q=2\pi /4\right)$. The inset shows the impurity superlattice. (b) Phase diagram of ${\mathcal{H}}_{\text{pure}}$. (c) $T$ dependence of the net scalar chirality at $H/J_3=0.18$, (d) magnetization curve and its $H$ derivative at $T/J_3=0.05$, and (e) $H$ dependence of ${\left({\chi }^{\text{tot}}\right)}^2$ at $T/J_3=0.05$, for the model including a periodic array of impurities. (f), (g) $T$ dependence of the specific heat at (f) $H/J_3=3.6$ and (g) $H/J_3=0.9$.

Figure 3

Snapshots of the spin (left) and the chirality (middle) configurations in the ordered phases for $J_3/\left|J_1\right|\approx 0.854$ $\left(Q=2\pi /4\right)$ when periodic impurities (triangles) with $a_{\text{imp}}=8$ are present. The spin and the chirality structure factors are shown on the right: (a) ferrochiral $3Q^M\text{−}6Q^{\chi }$ vortex crystal $(H/J_3=4.0$ and $T/J_3=0.05)$, (b) ferrochiral $3Q^M$ vortex crystal $(H/J_3=1.9$ and $T/J_3=0.41)$, (c) ferrochiral $3Q^M\text{−}1Q^{\chi }$ spiral $(H/J_3=3.0$ and $T/J_3=0.05)$, and (d) ferrichiral $3Q^M\text{−}2Q^{\chi }$ spiral I $(H/J_3=1.5$ and $T/J_3=0.05)$. The circles with solid (dashed) lines indicate the dominant (subdominant) $\mathbf{q}\ne 0$ peak(s). Note that the $\mathbf{q}=0$ component is removed from $S_s^{zz}\left(\mathbf{q}\right)$. The hexagon with a solid (dashed) line shows the first Brillouin zone (of the impurity superlattice). To obtain the smooth spin configurations, we integrate out short-wavelength fluctuations by averaging over 50–500 MCS.

Figure 4

Snapshots of the spin (left) and the chirality (middle) configurations in the ordered phases for $J_3/\left|J_1\right|\approx 0.854$ $\left(Q=2\pi /4\right)$ when periodic impurities (triangles) with $a_{\text{imp}}=8$ are present. The spin and the chirality structure factors are shown on the right: (a) ferrichiral $3Q^M\text{−}2Q^{\chi }$ spiral II $(H/J_3=2.2$ and $T/J_3=0.35)$, (b) antiferrochiral single-$Q^M$ spiral $(H/J_3=1.0$ and $T/J_3=0.35)$, (c) ferrochiral $3Q^M\text{−}2Q^{\chi }$ spiral $(H/J_3=0.1$ and $T/J_3=0.05)$, and (d) vertical single-$Q^M$ spiral $(H/J_3=0.2$ and $T/J_3=0.35)$. The circles with solid (dashed) lines indicate the dominant (subdominant) $\mathbf{q}\ne 0$ peak(s). Note that the $\mathbf{q}=0$ component is removed from $S_s^{zz}\left(\mathbf{q}\right)$. The hexagon with a solid (dashed) line shows the first Brillouin zone (of the impurity superlattice). To obtain the smooth spin configurations, we integrate out short-wavelength fluctuations by averaging over 50–500 MCS.

Figure 5

(a) Phase diagram of the model with a periodic array of impurities with $a_{\text{imp}}=4$ and $J_3/\left|J_1\right|\approx 0.854$ $\left(Q=2\pi /4\right)$. The inset shows the impurity configuration. (b), (c) $T$ dependence of the specific heat at (b) $H/J_3=3.3$ and (c) $H/J_3=0.5$.

Figure 6

Snapshots of the spin configuration (left), the chirality texture (middle), and the spin and the chirality structure factors (right) in the model with periodic impurities (triangles) with $a_{\text{imp}}=4$ for $J_3/\left|J_1\right|\approx 0.854$ $\left(Q=2\pi /4\right)$: (a) nonchiral $2Q^M\text{−}3Q^{\chi }$ vortex crystal $\left(H/J_3=3.9\right)$ and (b) nonchiral $3Q^M\text{−}2Q^{\chi }$ spiral $\left(H/J_3=1.6\right)$. The circles with solid (dashed) lines indicate the dominant (subdominant) peak(s) at $\mathbf{q}\ne 0$. The $\mathbf{q}=0$ component is removed from $S_s^{zz}\left(\mathbf{q}\right)$. The hexagon with a solid (dashed) line shows the first Brillouin zone (of the impurity superlattice). We average over 50–500 MCS to integrate out short-wavelength fluctuations.

Figure 7

$H$ dependencies of (a) the energy per site, (b) magnetization, $M$, (c) $S_s^{\perp }\left({\mathbf{Q}}_{\nu }\right)/N$, and (d) $S_s^{zz}\left({\mathbf{Q}}_{\nu }\right)/N$ $\left(\nu =1–3\right)$ for $Q=2\pi /4$ and $a_{\mathrm{imp}}=8$. “V” and “MC” represent the results obtained by $T=0$ variational calculations and Monte Carlo simulations at $T/J_3=0.05$.

Figure 8

$H$ dependence of ${\left({\chi }^{\text{tot}}\right)}^2$ for $J_3/\left|J_1\right|\approx 0.394$ $\left(Q=2\pi /6\right)$ at $T/J_3\approx 0.1268$ in the model with or without periodic impurities: $a_{\text{imp}}=\infty$ (no impurities), 12, and 11.

Figure 9

Snapshots of the spin configuration (left), the chirality texture (middle), and the spin and the chirality structure factors (right) in the model with perfect periodic impurities (triangles) with $a_{\text{imp}}=11$ for $J_3/\left|J_1\right|\approx 0.394$ $\left(Q=2\pi /6\right)$ at $H/J_3\sim 0.634$. The $\mathbf{q}=0$ component is removed from $S_s^{zz}\left(\mathbf{q}\right)$. We average over 500 MCS to integrate out short-wavelength fluctuations.

Figure 10

Allowed positions of a nonmagnetic impurity. Partial randomness is included by locating the nonmagnetic impurities either at a regular superlattice site (the open circle at the origin) or at one of its neighbors (see the text). The filled circles, triangles, and squares denote the NN, second-NN, and the third-NN sites, respectively. The average distance between closest impurities is ${\overline{a}}_{\text{imp}}=8$.

Figure 11

The impurity structure factors (a) for the regular superlattice site and with a randomly distribution up to (b) NN sites, (c) second-NN sites, and (d) third-NN sites.

Figure 12

Snapshots of the spin configuration (left), the chirality texture (middle), and the spin and the chirality structure factors (right) for periodic impurities slightly disordered up to NNs (triangles) with ${\overline{a}}_{\text{imp}}=8,H/J_3=1.4,T/J_3=0.05$, and $J_3/\left|J_1\right|\approx 0.854$ $\left(Q=2\pi /4\right)$. The dashed circles represent the positions without randomness. The $\mathbf{q}=0$ component has been removed from $S_s^{zz}\left(\mathbf{q}\right)$. We average over 500 MCS to integrate out short-wavelength fluctuations.

Figure 13

(a) $H$ dependence of ${\left({\chi }^{\text{tot}}\right)}^2$ for the model with partial randomness of the impurity array $[J_3/|J_1|\approx 0.854(Q=2\pi /4)$ and $T/J_3=0.05]$. (b), (c) Enlarged views of (a).

Figure 14

Size dependence of the order parameters for each phase described in the phase diagram of Fig. 2: the chirality structure factor evaluated at $\mathbf{q}=\mathbf{0},{\mathbf{0}}^{\prime },{\mathbf{Q}}_1,{\mathbf{Q}}_2,{\mathbf{Q}}_3$ $[S_{\chi }\left(\mathbf{q}\right)=S_{\chi }^u\left(\mathbf{q}\right)+S_{\chi }^d\left(\mathbf{q}\right)$ for $\mathbf{q}=\mathbf{0},{\mathbf{Q}}_1,{\mathbf{Q}}_2$, and ${\mathbf{Q}}_3$ and $S_{\chi }\left({\mathbf{0}}^{\prime }\right)=S_{\chi }^u\left(\mathbf{0}\right)-S_{\chi }^d\left(\mathbf{0}\right)]$ and the magnetic structure factor at $\mathbf{q}={\mathbf{Q}}_1,{\mathbf{Q}}_2,{\mathbf{Q}}_3$ in (a) the ferrochiral $3Q^M\text{−}6Q^{\chi }$ vortex crystal phase $\left(H/J_3=3.8\right)$, (b) ferrochiral $3Q^M$ vortex crystal phase $\left(T/J_3=0.41\right)$, (c) ferrochiral $3Q^M\text{−}1Q^{\chi }$ spiral phase for $Q=2\pi /4$ $\left(H/J_3=3.0\right)$, and (d) ferrichiral $3Q^M\text{−}2Q^{\chi }$ spiral I phase $\left(H/J_3=1.0\right)$ for $Q=2\pi /4$ and different temperature and magnetic fields.

Figure 15

Size dependence of the order parameters for each phase described in the phase diagram of Fig. 2: the chirality structure factor evaluated at $\mathbf{q}=\mathbf{0},{\mathbf{0}}^{\prime },{\mathbf{Q}}_1,{\mathbf{Q}}_2,{\mathbf{Q}}_3$ $[S_{\chi }\left(\mathbf{q}\right)=S_{\chi }^u\left(\mathbf{q}\right)+S_{\chi }^d\left(\mathbf{q}\right)$ for $\mathbf{q}=\mathbf{0},{\mathbf{Q}}_1,{\mathbf{Q}}_2$, and ${\mathbf{Q}}_3$ and $S_{\chi }\left({\mathbf{0}}^{\prime }\right)=S_{\chi }^u\left(\mathbf{0}\right)-S_{\chi }^d\left(\mathbf{0}\right)]$ and the magnetic structure factor at $\mathbf{q}={\mathbf{Q}}_1,{\mathbf{Q}}_2,{\mathbf{Q}}_3$ in (a) ferrichiral $3Q^M\text{−}2Q^{\chi }$ spiral II phase $\left(H/J_3=2.0\right)$, (b) antiferrochiral single-$Q^M$ spiral phase $\left(H/J_3=1.0\right)$, (c) ferrochiral $3Q^M\text{−}2Q^{\chi }$ spiral phase $\left(H/J_3=0.2\right)$, and (d) vertical single-$Q^M$ spiral phase $\left(H/J_3=0.2\right)$ for $Q=2\pi /4$ and different temperature and magnetic fields.

Figure 16

Size dependence of the order parameters for two phases described in the phase diagram of Fig. 5: (a) the $xy$ and $z$ components of the spin structure factor evaluated at $\mathbf{q}={\mathbf{Q}}_1,{\mathbf{Q}}_2,{\mathbf{Q}}_3$ and the chirality structure factor evaluated at $\mathbf{q}={\mathbf{Q}}_1$ and ${\mathbf{Q}}_1^{\prime }$ $[Q_1^{\prime }=\pi /\left(2\sqrt{3}\right)]$ in the nonchiral $3Q^M\text{−}2Q^{\chi }$ spiral phase $\left(H/J_3=1.0\right)$ and (b) the $z$ $\left(xy\right)$ component of the spin structure factor evaluated at $\mathbf{q}={\mathbf{Q}}_1^{\prime \prime },{\mathbf{Q}}_2^{\prime \prime }$, and ${\mathbf{Q}}_3^{\prime \prime }$ where $Q_{\nu }^{\prime \prime }=\pi /\sqrt{3}$ $(\mathbf{q}={\mathbf{Q}}_1,{\mathbf{Q}}_2$, and ${\mathbf{Q}}_3)$ and the chirality structure factor evaluated at $\mathbf{q}={\mathbf{Q}}_1$ and ${\mathbf{Q}}_1^{\prime }$ in the nonchiral $2Q^M\text{−}3Q^{\chi }$ vortex crystal phase $\left(H/J_3=3.0\right)$.