Influence of topological constraints and topological excitations: Decomposition formulas for calculating homotopy groups of symmetry-broken phases

A symmetry broken phase of a system with internal degrees of freedom often features a complex order parameter, which generates a rich variety of topological excitations and imposes topological constraints on their interaction (topological influence); yet the very complexity of the order parameter makes it difficult to treat topological excitations and topological influence systematically. To overcome this problem, we develop a general method to calculate homotopy groups and derive decomposition formulas which express homotopy groups of the order parameter manifold $G/H$ in terms of those of the symmetry $G$ of a system and those of the remaining symmetry $H$ of the state. By applying these formulas to general monopoles and three-dimensional skyrmions, we show that their textures are obtained through substitution of the corresponding $\mathfrak{su}\left(2\right)$ subalgebra for the $\mathfrak{su}\left(2\right)$ spin. We also show that a discrete symmetry of $H$ is necessary for the presence of topological influence and find topological influence on a skyrmion characterized by a non-Abelian permutation group of three elements in the ground state of an SU(3)-Heisenberg model.

Figure 1

Coroot lattice $L_G^c$ of $G=\mathrm{SU}\left(3\right)$ generated by the three coroots ${\mathbf{\alpha }}_{RG}^c,{\mathbf{\alpha }}_{GB}^c$, and ${\mathbf{\alpha }}_{BR}^c$ in Eq. (14). Since ${\mathbf{\alpha }}_{RG}^c+{\mathbf{\alpha }}_{GB}^c+{\mathbf{\alpha }}_{BR}^c=0, L_G^c$ is isomorphic to the triangular lattice. We take the system of coordinates such that ${\mathbf{\alpha }}_{RG}^c=(1,0),{\mathbf{\alpha }}_{GB}^c=(-1/2,\sqrt{3}/2)$, and ${\mathbf{\alpha }}_{BR}^c=(-1/2,-\sqrt{3}/2)$.

Figure 2

Schematic illustration of the symmetry transformation $g_{\mathbf{\alpha }}^{\left(3\right)}(\psi ,\theta ,\varphi )$ defined in Eq. (23). The red arrow indicates the generalized $\mathfrak{su}\left(2\right)$-spin vector parallel to the unit vector $\widehat{\mathit{r}}(\theta ,\varphi ):=(sin\theta cos\varphi ,sin\theta sin\varphi ,cos\theta )$ and $g_{\mathbf{\alpha }}^{\left(3\right)}(\psi ,\theta ,\varphi )$ describes the spin rotation about $\widehat{\mathit{r}}(\theta ,\varphi )$ through angle $2\psi$.

Figure 3

(a) Schematic illustration of an inclusion map $i:H\to G$. Given a map $g$ from $D^m$ to $H$, the composition $i\circ g$ gives a map from $D^m$ to $G$, where $D^m$ is an $m$-dimensional disk defined in (26). (b) Schematic illustration of the cokernel $\mathrm{Coker}{i}_{*m}$, the kernel $\mathrm{Ker}{i}_{*m}$, and the image $\mathrm{Im}{i}_{*m}$ of $i_{*,m}$. The kernel $\mathrm{Ker}{i}_{*m-1}$ is a subgroup of ${\pi }_{m-1}\left(H\right)$, whose elements are mapped to the identity element $e$ of ${\pi }_{m-1}\left(G\right)$. The image $\mathrm{Im}{i}_{*m}$ is a subgroup of ${\pi }_m\left(G\right)$, whose elements are obtained through $i_{*m}$. The cokernel $\mathrm{Coker}{i}_{*m}$ is the quotient group ${\pi }_m\left(G\right)/\mathrm{Im}{i}_{*m}$, representing the elements in ${\pi }_m\left(G\right)$ that cannot be obtained through $i_{*m}$.

Figure 4

(a) Construction of the texture $O^{\left[a\right]}$ from the map $a$ defined in Eq. (31) for $m=2$, where $a$ is a map from a two-dimensional disk $D^2$ to $G$ that maps the boundary (red dashed line) of $D^2$ to the identity element $e$. The texture $O^{\left[a\right]}$ is a map from $D^2$ to $G/H$ that maps the boundary (blue dotted line) of $D^2$ to the reference order parameter $O_0$. (b) Construction of the texture $O^b$ from the map $b$ defined in Eq. (37) for $m=2$. Here $b$ is a map from a circle $S^1$ to $H$ and $b_s$ is a continuous deformation from $b$ to the uniform texture $e$ subject to the condition (36), where the points on the red dashed line are mapped to $e$. The texture $O^b$ is a map from $D^2$ to $G/H$ which maps the points on the blue dotted line to $O_0$.

Figure 5

Schematic illustration of a monopole in a ferromagnet with topological charge $m=+1$. Each arrow indicates the direction of the spin. The texture of the monopole is described by a hedgehog configuration of the spin, which is obtained from successive applications of spin rotation $exp\left(i\theta S_2\right)$ about the $y$ axis through angle $\theta$ followed by spin rotation $exp\left(i\varphi S_3\right)$ about the $z$ axis through angle $\varphi$, where $\theta$ and $\varphi$ are the polar angle and the azimuth angle, respectively.

Figure 6

Schematic illustration of the map $f$ defined in Eq. (46). The entire region shows the connected component $G_0$ of $G$ and the white regions represent the connected components of $H$, where $H_0$ shows the connected component including the identity element $e$, and $\sigma , \tau$, and $\sigma \tau$ denote the elements of $G_0$ in different connected components in $H$. The map $f$ is obtained from the composition ${\gamma }_{\sigma }{\gamma }_{\tau }{\left({\gamma }_{\sigma \tau }\right)}^{-1}$ of the three paths ${\gamma }_{\sigma }, {\gamma }_{\tau }$, and ${\gamma }_{\sigma \tau }$.

Figure 7

Schematic illustration of the three-color density-wave state on a triangular lattice. The red (R), green (G), and blue (B) disks show the internal states $|R\rangle ,|G\rangle$, and $|B\rangle$ defined in Eq. (79), respectively. The sites in states $|R\rangle , |G\rangle$, and $|B\rangle$ constitute the sublattices $L_R, L_G$, and $L_B$, respectively.

Figure 8

Schematic illustrations of an RG vortex in the 3-CDW state for a triangular lattice. It is a disclination with the Frank angle $\pi /3$ around a site belonging to the sublattice $L_B$. Here, the Frank angle describes the angle over which the lattice sites are missing. The texture of the RG-vortex described in Eqs. (93), (94), and (95) is shown in (c), and (a) and (b) illustrate how the RG-spin vortex is obtained as the Frank angle vanishes. In (a)–(c), each arrow indicates the expectation value of $(S_{RG,2},S_{RG,3})$, where the red dashed (green) arrows correspond to the sublattice $L_R$ ($L_G$). In (d)–(f), the red circle, green square, and blue triangule at each site indicate the sublattices $L_R, L_G$, and $L_B$, respectively. Across the disclination, the sublattices $L_R$ and $L_G$ are exchanged.

Figure 9

Schematic illustrations of an RG skyrmion. Each arrow shows the expectation value of the generalized $\mathfrak{su}\left(2\right)$-spin vector ${〈{\mathit{S}}_{\mathrm{RG}}〉}_{}(\theta ,\varphi )$, where the color represents the third component ${〈{\mathit{S}}_{\mathrm{RG},3}〉}_{}(\theta ,\varphi )$. The absence of an arrow on a site indicates that the expectation value vanishes there. When the two-dimensional plane is compactified into a sphere, in which the points at infinity are mapped onto the north pole, the texture on the sublattice $L_R$ ($L_G$) describes a hedgehog texture of ${〈{\mathit{S}}_{\mathrm{RG}}〉}_{}$ with winding number $+1$ ($-1$).

Figure 10

Three types of skyrmions in a 3-CDW state. (a) An RG skyrmion has topological charge $\mathbf{\alpha }={\mathbf{\alpha }}_{RG}^c$ and a hedgehog texture of ${\mathit{S}}_{RG}$ with winding number $+1$ ($-1$) on the sublattice $L_R$ ($L_G$). (b) A GB skyrmion has topological charge $\mathbf{\alpha }={\mathbf{\alpha }}_{GB}^c$ and a hedgehog texture of ${\mathit{S}}_{GB}$ with winding number $+1$ ($-1$) on the sublattice $L_G$ ($L_B$). (c) A BR skyrmion has topological charge $\mathbf{\alpha }={\mathbf{\alpha }}_{BR}^c$ and a hedgehog texture of ${\mathit{S}}_{BR}$ with winding number $+1$ ($-1$) on the sublattice $L_B$ ($L_R$).

Figure 11

Topological charges of skyrmions (a)–(c) before and (d)–(f) after making a complete circuit of the RG, GB, and BR vortices, respectively. The RG, GB, and GR vortices act on skyrmions by inverting them about the line perpendicular to ${\mathbf{\alpha }}_{RG}, {\mathbf{\alpha }}_{GB}$, and ${\mathbf{\alpha }}_{BR}$, respectively.

Figure 12

Topological influence of two vortices with spin topological charges $\sigma$ and $\tau$ on a skyrmion with topological charge $n$. The skyrmion goes around the vortex with $\sigma$ clockwise (a), then around the vortex with $\tau$ clockwise (b), then around the vortex with $\sigma$ anticlockwise (c), and finally around the vortex with $\tau$ anticlockwise (d). Through these processes, the topological charge of a skyrmion changes from $n$ to ${\lambda }_2^{\rho }\left(n\right)$ with $\rho ={\tau }^{-1}{\sigma }^{-1}\tau \sigma$.