Magnetic field–temperature phase diagrams for multiple-$Q$ magnetic ordering: Exact steepest descent approach to long-range interacting spin systems

Multiple-$Q$ magnetic orderings represent magnetic textures composed of superpositions of multiple spin density waves or spin spirals, as represented by two-dimensional skyrmion crystals and three-dimensional hedgehog lattices. Such magnetic orderings have been observed in various magnetic materials in recent years, and attracted enormous attention, especially from the viewpoint of topology and emergent electromagnetic fields originating from noncoplanar magnetic structures. Although they often exhibit successive phase transitions among different multiple-$Q$ states while changing temperature and an external magnetic field, it is not straightforward to elucidate the phase diagrams, mainly due to the lack of concise theoretical tools as well as appropriate microscopic models. Here, we provide a theoretical framework for a class of effective spin models with long-range magnetic interactions mediated by conduction electrons in magnetic metals. Our framework is based on the steepest descent method with a set of self-consistent equations that leads to exact solutions in the thermodynamic limit, and has many advantages over existing methods such as biased variational calculations and numerical Monte Carlo simulations. We develop two methods that complement each other in terms of the computational cost and the range of applications. As a demonstration, applying the framework to the models with instabilities toward triple- and hextuple-$Q$ magnetic orderings, we clarify the magnetic field–temperature phase diagrams with a variety of multiple-$Q$ phases. We find that the models exhibit interesting reentrant phase transitions where the multiple-$Q$ phases appear only at finite temperature and/or nonzero magnetic field. Furthermore, we show that the multiple-$Q$ states can be topologically nontrivial stacked skyrmion crystals or hedgehog lattices, which exhibit large net spin scalar chirality associated with nonzero skyrmion number. The results demonstrate that our framework could be a versatile tool for studying magnetic and topological phase transitions and related quantum phenomena in actual magnetic metals hosting multiple-$Q$ magnetic orderings.

Figure 1

Pictorial representation of the coupling constants for the symmetric and antisymmetric interactions in the $3Q$ model. The blue ellipsoids at $\pm {\mathbf{Q}}_{\eta }$ represent $J_{\eta }^{\alpha \alpha }$: The lengths along the principal axes [100], [010], and [001] denote the amplitudes of $J_{\eta }^{xx}, J_{\eta }^{yy}$, and $J_{\eta }^{zz}$, respectively. The red arrows at $\pm {\mathbf{Q}}_{\eta }$ represent $\pm {\mathbf{D}}_{\eta }$. The labeled numbers indicate $\eta$. The gray cube is a guide to the eye.

Figure 2

Ground-state phase diagram for the $3Q$ model at zero magnetic field. The phase diagram includes the $1Q$ phase where one of three $|{\tilde{\mathbf{M}}}_{\eta }|$ is nonzero, the $2Q$ phase where two of $|{\tilde{\mathbf{M}}}_{\eta }|$ are nonzero at different values, and the isotropic $3Q$ phase where $|{\tilde{\mathbf{M}}}_1|=|{\tilde{\mathbf{M}}}_2|=|{\tilde{\mathbf{M}}}_3|>0$. The white dashed line between the $3Q$ and $2Q$ phases represents the first-order phase transition, while the black solid line between the $2Q$ and $1Q$ phases represents the second-order phase transition.

Figure 3

Ground-state spin configurations stabilized in the $3Q$ model at zero magnetic field for (a) the $3Q$ state at $(D,\mathrm{\Delta })=(0.15,0.3)$, (c) the $2Q$ state at $(D,\mathrm{\Delta })=(0.25,0.2)$, and (d) the $1Q$ state at $(D,\mathrm{\Delta })=(0.35,0.1)$. The color of the arrows denotes the [111] component of spins $S_{\mathbf{r}}^{\left[111\right]}=(S_{\mathbf{r}}^x+S_{\mathbf{r}}^y+S_{\mathbf{r}}^z)/\sqrt{3}$ in (a), while it represents the $x$ component of spins $S_{\mathbf{r}}^x$ in (c) and (d); see the color bars in (a) and (c). (b) Positions of the magnetic hedgehogs (magenta spheres) and the magnetic antihedgehogs (cyan spheres) in the $3Q$ spin state in (a). The dashed lines are guides to the eye. Insets of (a), (c), and (d) show distributions of $|{\tilde{\mathbf{M}}}_{\eta }|$ for each spin state.

Figure 4

Magnetic field–temperature phase diagrams of the $3Q$ model with $(D,\mathrm{\Delta })=(0.25,0.2)$ for the magnetic field directions (a) $\mathbf{B}\parallel \left[100\right]$, (b) $\mathbf{B}\parallel \left[110\right]$, and (c) $\mathbf{B}\parallel \left[111\right]$. The white dashed and black solid lines represent first- and second-order phase transitions, respectively, while the white dotted lines represent phase transitions whose order is undetermined.

Figure 5

Magnetic field–temperature phase diagrams of the $3Q$ model with $(D,\mathrm{\Delta })=(0.15,0.3)$ for the magnetic field directions (a) $\mathbf{B}\parallel \left[100\right]$, (b) $\mathbf{B}\parallel \left[110\right]$, and (c) $\mathbf{B}\parallel \left[111\right]$. The notations are common to those in Fig. 4.

Figure 6

Temperature dependences of (a) the order parameters $|{\tilde{\mathbf{M}}}_{\eta }|$ and (b) the specific heat $C$ and the spin scalar chirality $\chi$ for the $3Q$ model with $(D,\mathrm{\Delta })=(0.25,0.2)$ and $B=0.1$ ($\mathbf{B}\parallel \left[100\right]$). The spin scalar chirality is multiplied by a factor of ${10}^4$ for better visibility.

Figure 7

Magnetic field dependences of (a), (c), (e) the order parameters $|{\tilde{\mathbf{M}}}_{\eta }|$ and (b), (d), (f) the magnetization $m$ and the spin scalar chirality $\chi$ for the $3Q$ model with $(D,\mathrm{\Delta })=(0.15,0.3)$ and $T=0.5$. The magnetic field directions are (a), (b) $\mathbf{B}\parallel \left[100\right]$, (c), (d) $\mathbf{B}\parallel \left[110\right]$, and (e), (f) $\mathbf{B}\parallel \left[111\right]$. The spin scalar chirality is multiplied by a factor of 500, respectively, for better visibility.

Figure 8

Pictorial representation of the coupling constants for the symmetric and antisymmetric interactions in the $6Q$ model. The notations are common to those in Fig. 1.

Figure 9

Ground-state phase diagram for the $6Q$ model at zero magnetic field. The phase diagram includes the $1Q$ phase where one of $|{\tilde{\mathbf{M}}}_{\eta }|$ is nonzero, the $3Q$ phase where three of $|{\tilde{\mathbf{M}}}_{\eta }|$ are nonzero, and the $6Q$ phase where all six $|{\tilde{\mathbf{M}}}_{\eta }|$ are nonzero. In the $3Q$ and $6Q$ phases, all the nonzero $|{\tilde{\mathbf{M}}}_{\eta }|$ take the same value, while the value depends on $D$ and $\mathrm{\Delta }$. The white dashed lines separating these phases represent the first-order phase transitions, and the black dot locates the triple point where the three first-order transition lines meet.

Figure 10

Ground-state spin configurations stabilized in the $6Q$ model at zero magnetic field for (a) the $6Q$ state at $(D,\mathrm{\Delta })=(0.2,0.3)$, (c) the $3Q$ state at $(D,\mathrm{\Delta })=(0.3,0.3)$, and (d) the $1Q$ state at $(D,\mathrm{\Delta })=(0.1,0.0)$. The color of the arrows denotes the [111] component of spins, $S_{\mathbf{r}}^{\left[111\right]}$, according to the color bar in (a). (b) Positions of the hedgehogs (magenta spheres) and the antihedgehogs (cyan spheres) in the $6Q$ state shown in (a). The dashed lines are guides to the eye. Insets of (a), (c), and (d) show distributions of $|{\tilde{\mathbf{M}}}_{\eta }|$ for each state. (e) Spin configuration on a (111) slice of (c). The black dashed rhombus indicates the 2D magnetic unit cell. (f) Distribution of the solid angle $\mathrm{\Omega }$ spanned by neighboring three spins calculated from the spin configuration in (e).

Figure 11

Magnetic field–temperature phase diagrams of the $6Q$ model with $(D,\mathrm{\Delta })=(0.1,0.0)$ for the magnetic field directions (a) $\mathbf{B}\parallel \left[100\right]$, (b) $\mathbf{B}\parallel \left[110\right]$, and (c) $\mathbf{B}\parallel \left[111\right]$. The white dashed lines and the black solid lines represent first- and second-order phase transitions, respectively. The black dots in (a) locate the triple points.

Figure 12

Magnetic field–temperature phase diagrams of the $6Q$ model with $(D,\mathrm{\Delta })=(0.2,0.3)$ for the magnetic field directions (a) $\mathbf{B}\parallel \left[100\right]$, (b) $\mathbf{B}\parallel \left[110\right]$, and (c) $\mathbf{B}\parallel \left[111\right]$. The notations are common to those in Fig. 11.

Figure 13

Temperature dependences of (a), (c) the order parameters $|{\tilde{\mathbf{M}}}_{{\eta }_{\ell }}|$ and (b), (d) the specific heat $C$, the spin scalar chirality $\chi$, and the skyrmion number $N_{\mathrm{sk}}$ for the $6Q$ model with $(D,\mathrm{\Delta })=(0.1,0.0)$. The data in (a) and (b) are for $\mathbf{B}\parallel \left[100\right]$ with $B=0.8$, while those in (c) and (d) are for $\mathbf{B}\parallel \left[111\right]$ with $B=0.6$. The order parameters are sorted in descending order: $|{\tilde{\mathbf{M}}}_{{\eta }_1}|\ge |{\tilde{\mathbf{M}}}_{{\eta }_2}|\ge |{\tilde{\mathbf{M}}}_{{\eta }_3}|\ge |{\tilde{\mathbf{M}}}_{{\eta }_4}|\ge |{\tilde{\mathbf{M}}}_{{\eta }_5}|\ge |{\tilde{\mathbf{M}}}_{{\eta }_6}|$. The spin scalar chirality is multiplied by a factor of 10 for better visibility.

Figure 14

Magnetic field dependences of (a), (c), (d) the order parameters $|{\tilde{\mathbf{M}}}_{{\eta }_{\ell }}|$ and (b), (d), (f) the magnetization $m$, the spin scalar chirality $\chi$, and the skyrmion number $N_{\mathrm{sk}}$ for the $6Q$ model with $(D,\mathrm{\Delta })=(0.2,0.3)$ and $T=0.2$. The magnetic field directions are (a), (b) $\mathbf{B}\parallel \left[100\right]$, (c), (d) $\mathbf{B}\parallel \left[110\right]$, and (e), (f) $\mathbf{B}\parallel \left[111\right]$. The order parameters are sorted in descending order as Fig. 12. The spin scalar chiralities in (b), (d), and (f) are multiplied by a factor of 4 (solid lines) [a factor of 400 in the $6Q$ and $6Q^{\prime }$ phases (dashed lines)] for better visibility.

Figure 15

Spin configurations (left) and distribution of the solid angle $\mathrm{\Omega }$ (right) in the $6Q$ model with $(D,\mathrm{\Delta })=(0.2,0.3)$ and $T=0.2$ in $\mathbf{B}\parallel \left[100\right]$: (a) $B=0.8$ and (b) $B=1.4$, which correspond to before and after the topological transition in the $3Q$ phase, respectively. The notations are common to those in Figs. 10 and 10. The green rhombi correspond to a (100) slice of the magnetic unit cell.

Figure 16

Spin configurations (left) and distribution of the solid angle $\mathrm{\Omega }$ (right) in the $6Q$ model with $(D,\mathrm{\Delta })=(0.2,0.3)$ and $T=0.2$ in $\mathbf{B}\parallel \left[111\right]$: (a) $B=2.0$ and (b) $B=2.2$, which correspond to before and after the topological transition in the $3Q$ phase, respectively. The notations are common to those in Fig. 15.