Theory of nematic charge orders in kagome metals

Kagome metals $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ ($A=\mathrm{K}$,Rb,Cs) exhibit an exotic charge order (CO), involving three order parameters, with broken translation and time-reversal symmetries compatible with the presence of orbital currents. The properties of this phase are still intensely debated, and it is unclear if the origin of the CO is mainly due to electron-electron or electron-phonon interactions. Most of the experimental studies confirm the nematicity of this state, a feature that might be enhanced by electronic correlations. However, it is still unclear whether the nematic CO becomes stable at a temperature equal to ($T_{\text{nem}}=T_{\text{C}}$) or lower than ($T_{\text{nem}}<T_{\text{C}}$) that of the CO itself. Here, we systematically characterize several CO configurations, some proposed for the new member of the family ${\mathrm{ScV}}_6{\mathrm{Sn}}_6$, by combining phenomenological Ginzburg-Landau theories, valid irrespective of the specific ordering mechanism, with mean-field analysis. We find a few configurations for the CO that are in agreement with most of the experimental findings to date and that are described by different Ginzburg-Landau potentials. We propose to use resonant ultrasound spectroscopy to experimentally characterize the order parameters of the CO, such as the number of their components and their relative amplitude, and we provide an analysis of the corresponding elastic tensors. This might help to explain which mean-field configuration found in our study is the most representative for describing the CO state of kagome metals, and it can provide information regarding the nematicity onset temperature $T_{\text{nem}}$ with respect to $T_{\text{C}}$.

Figure 1

Hexagonal charge-bond order. (a)–(c) Real-space representations of the Hex CBO, where solid (dashed) lines connecting two nearest-neighbor atoms imply a positive (negative) sign for the real part of the order parameter $\text{Re}\left[{\mathrm{\Delta }}_i\right]$. The arrows indicate the sign of the imaginary part $\text{Im}\left[{\mathrm{\Delta }}_i\right]$, which is positive (negative) if we move along (opposite to) the direction of the arrow. (a) $\mathrm{p}6$ wallpaper group pattern and the corresponding generating symmetry operations (in green), which consist of two translations ${\mathbf{R}}_1:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)$ and ${\mathbf{R}}_2:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)$ (which act as the identity), a rotation of $2\pi /6$ around the center of the hexagonal pattern ${\mathrm{C}}_6:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_3,{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_6,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5)$, a mirror symmetry with respect to the dashed line shown in the image $\text{MP}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_6^{*},{\mathrm{\Delta }}_5^{*},{\mathrm{\Delta }}_4^{*})$ and $\text{TRS}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_4^{*},{\mathrm{\Delta }}_5^{*},{\mathrm{\Delta }}_6^{*})$. These transformations have to be supplied with the condition $({\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_1,-{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3)$, implying that the six parameters are not independent, but they can be reduced to three order parameters. (b) $\mathrm{p}31\mathrm{m}$ wallpaper group pattern with symmetries ${\mathrm{C}}_6:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_6^{*},{\mathrm{\Delta }}_4^{*},{\mathrm{\Delta }}_5^{*}), \text{MP}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_6^{*},{\mathrm{\Delta }}_5^{*},{\mathrm{\Delta }}_4^{*})$, and $\text{TRS}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_4^{*},{\mathrm{\Delta }}_5^{*},{\mathrm{\Delta }}_6^{*})$, together with $({\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3)$. (c) $\mathrm{p}6$ configuration with symmetry transformations ${\mathrm{C}}_6:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_3,{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_6,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5), \text{MP}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_6^{*},{\mathrm{\Delta }}_5^{*},{\mathrm{\Delta }}_4^{*}), \text{TRS}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_4^{*},{\mathrm{\Delta }}_5^{*},{\mathrm{\Delta }}_6^{*})$ and $({\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_1^{*},-{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3^{*})$. (d)–(f) Real-space representation of the zero-temperature mean-field ($U=1.5, V=0.8$) occupation per site $\langle n_i\rangle$, the bond correlation pattern $|\langle c_i^{†}{c}_{j\text{n.n.}i}\rangle |$, and of the current distribution corresponding to the order parameter configuration displayed in the panel above. Dark red corresponds to a large occupation (strong bond), while dark blue corresponds to a small occupation (weak bond). The size of the arrows suggests the magnitude of the current flowing through that bond. (d)–(f) ${\psi }_j=0.3, {\varphi }_1={\varphi }_2={\varphi }_3=\pi /2$.

Figure 2

Trihexagonal charge-bond order. (a)–(c) Real-space representations of the TrH CBO, where the same notation used in Figs. 1–1 is used. (a) $\mathrm{p}6$ wallpaper group pattern [70] and the corresponding generating symmetry operations (in green), which consist of two translations ${\mathbf{R}}_1:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to (-{\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2,-{\mathrm{\Delta }}_3^{*})$ and ${\mathbf{R}}_2:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to (-{\mathrm{\Delta }}_1^{*},-{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3)$, a rotation of $2\pi /6$ around the axis orthogonal to the plane of the lattice passing through the center of the anti-tri-hexagonal pattern ${\mathrm{C}}_6:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_3,{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2)$, a mirror symmetry with respect to a plane orthogonal to the plane of the lattice passing through the green dashed line shown in the image $\text{MP}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_1^{*})$ and the time-reversal symmetry $\text{TRS}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3^{*})$. (b) $\mathrm{p}31\mathrm{m}$ wallpaper group pattern [74] with symmetries ${\mathbf{R}}_1:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to (-{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,-{\mathrm{\Delta }}_3), {\mathbf{R}}_2:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to (-{\mathrm{\Delta }}_1,-{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3), {\mathrm{C}}_6:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*}), \text{MP}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_1^{*})$, and $\text{TRS}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3^{*})$. (c) $\mathrm{p}6$ configuration [70, 73, 75, 88] with symmetry transformations ${\mathbf{R}}_1:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to (-{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,-{\mathrm{\Delta }}_3), {\mathbf{R}}_2:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to (-{\mathrm{\Delta }}_1,-{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3), {\mathrm{C}}_6:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_3,{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2), \text{MP}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_1^{*})$, and $\text{TRS}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3^{*})$. (d)–(f) Same quantities and values as described in the caption of Fig. 1. (d) ${\psi }_j=0.3, {\varphi }_1={\varphi }_2={\varphi }_3=\pi /2$; (e), (f) ${\psi }_j=0.3, {\varphi }_1=2.4, {\varphi }_2={\varphi }_3=0.7$.

Figure 3

Star of David charge-bond order. (a)–(c) Real-space representations of the SoD CBO, where the same convention introduced in Figs. 1–1 is used. The translations act in a trivial way [${\mathbf{R}}_1:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)$ and ${\mathbf{R}}_2:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)$]. (a) $\mathrm{p}6$ wallpaper group pattern with generating symmetries ${\mathrm{C}}_6:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_3,{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_6,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5), \text{MP}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_6^{*},{\mathrm{\Delta }}_5^{*},{\mathrm{\Delta }}_4^{*}), \text{TRS}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_4^{*},{\mathrm{\Delta }}_5^{*},{\mathrm{\Delta }}_6^{*})$, and the mapping $({\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to (-{\mathrm{\Delta }}_1^{*},-{\mathrm{\Delta }}_2^{*},-{\mathrm{\Delta }}_3^{*})$. (b) $\mathrm{p}31\mathrm{m}$ wallpaper group pattern with symmetries ${\mathrm{C}}_6:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_6^{*},{\mathrm{\Delta }}_4^{*},{\mathrm{\Delta }}_5^{*}), \text{MP}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_6^{*},{\mathrm{\Delta }}_5^{*},{\mathrm{\Delta }}_4^{*})$, and $\text{TRS}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_4^{*},{\mathrm{\Delta }}_5^{*},{\mathrm{\Delta }}_6^{*})$, together with $({\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to (-{\mathrm{\Delta }}_1^{*},-{\mathrm{\Delta }}_2,-{\mathrm{\Delta }}_3^{*})$. (c) $\mathrm{p}6$ configuration with symmetry transformations ${\mathrm{C}}_6:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_3,{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_6,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5), \text{MP}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_6^{*},{\mathrm{\Delta }}_5^{*},{\mathrm{\Delta }}_4^{*}), \text{TRS}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3,{\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to ({\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_4^{*},{\mathrm{\Delta }}_5^{*},{\mathrm{\Delta }}_6^{*})$, and $({\mathrm{\Delta }}_4,{\mathrm{\Delta }}_5,{\mathrm{\Delta }}_6)\to (-{\mathrm{\Delta }}_1,-{\mathrm{\Delta }}_2^{*},-{\mathrm{\Delta }}_3)$. (d)–(f) Same quantities and values as described in the caption of Fig. 1. (d) ${\psi }_1={\psi }_3=0, {\psi }_2=0.3, {\varphi }_2=\pi /2$. (e), (f) ${\psi }_1={\psi }_3=0.35, {\psi }_2=0.20, {\varphi }_1=0, {\varphi }_2=\pi /2$, and ${\varphi }_3=\pi$.

Figure 4

$\sqrt{3}\times \sqrt{3}$ charge-bond order. (a)–(c) Real-space representations of the CBO with $\sqrt{3}\times \sqrt{3}$ unit cell. (a) Pattern belonging to the $\mathrm{p}6$ wallpaper group with real order parameters $({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)$. (b) The pattern belongs to the $\mathrm{p}6$ wallpaper group, and the lattice translations ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$ act in a trivial way on the order parameter. The sixfold rotation transforms the vector of the order parameters as ${\mathrm{C}}_6:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_3,{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2)$. The mirror plane symmetry acts as $\text{MP}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_3,{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_1)$ while the time-reversal symmetry is $\text{TRS}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3^{*})$. (c) Pattern belonging to the p3m1 wallpaper group, characterized by the transformations ${\mathrm{C}}_6:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*}), \text{MP}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_3^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_1^{*})$, and $\text{TRS}:({\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,{\mathrm{\Delta }}_3)\to ({\mathrm{\Delta }}_1^{*},{\mathrm{\Delta }}_2^{*},{\mathrm{\Delta }}_3^{*})$. (d)–(f) Same quantities and values as described in the caption of Fig. 1. (d) ${\psi }_1={\psi }_2={\psi }_3=0.3, {\varphi }_1={\varphi }_3=\pi , {\varphi }_2=0$; (e) ${\psi }_1={\psi }_3=0.26, {\psi }_2=0.34, {\varphi }_1={\varphi }_3=0, {\varphi }_2=1.72$; (f) ${\psi }_1={\psi }_3=0.26, {\psi }_2=0.34, {\varphi }_1={\varphi }_3=-1.496, {\varphi }_2=2.43$. Panels (a) and (d), which display a real order parameter configuration, are shown as a reference.