Hopf-chain networks evolved from triple points

Exotic links and chains attract interest across various disciplines including mathematics, biology, chemistry, and physics. Here, we propose that topological Hopf-chain networks, consisting of one-, two-, and three-dimensional (3D) Hopf chains, can be found in the momentum space. These networks can be evolved from a 3D triple-points phase by varying symmetries. Moreover, we identify that the Hopf-chain networks exist in a family of crystals $\mathrm{S}{\mathrm{c}}_{3}X\mathrm{C}\left(X=\mathrm{Al},\mathrm{Ga},\mathrm{In},\mathrm{Tl}\right)$. The crystals are 3D triple-points metals, and transit to topological metals with Hopf-chain networks under strains. These Hopf networks will exhibit unique magnetotransport properties.

Figure 1

Hopf link, Hopf chains, and Hopf-chain networks. (a) Hopf link. (b) A 1D Hopf chain with a unit consisting of one Hopf link. (c) A 1D Hopf chain with a unit consisting of two Hopf links. (d), (e) 2D Hopf-chain networks formed by two mutually vertical chains along $X$ and $Y$ axes. The difference between the two networks is that the two vertical chains in (e) share a blue ring while those in (d) intersect each other. (f) A 3D Hopf-chain network, consisting of a 2D network like that in (e) and a 1D chain like that in (b) along $Z$ direction, where the 1D chain crosses over the 2D network without any contacts. The unit length of each periodic structure is labeled as light-blue dashed lines.

Figure 2

Transition from a triple-points phase to a 1D Hopf chain. (a) A triple-points phase, where two (black) triple points $T$ are connected by a (red) straight nodal line along $k_z$. (b), (c) 1D Hopf chains where two nodal rings link each other. The two mutually vertical rings locate on two mirror planes $k_x=0$ and $k_y=0$, respectively, and the middle ring is centered at Γ point. The difference between (b) and (c) is that the two rings lie on different mirror planes. The middle red ring in (b) is on the mirror plane $k_y=0$ while the middle blue ring in (c) is on the mirror plane $k_x=0$. The phase transition from (a) to (b) or (c) is induced by the elimination of $C_4$ rotation symmetry. (d)–(f) Band structures corresponding to the topological phases in (a)–(c) along $k_z$, respectively. The double-degenerate band in (d) splits into two bands in (e) or (f), which results in the triple point $T$ in (d) splitting into two crossing points $W_1$ and $W_2$ in (e) or (f). In (e) the splitting red band is above on the light-blue band, while the case in (f) is inverse. $X\pm$ and $Y\pm$ in the parentheses in (d)–(f) represent the eigenvalues of mirror planes $k_x=0, k_y=0$ being $\pm 1$, respectively.

Figure 3

Transition from a 3D triple-points phase to 2D/3D Hopf-chain networks. (a) A 3D triple-points phase with seven triple points, which is a 3D version of the phase in Fig. 2. (b) A mixture phase of a 2D Hopf-chain network [like that in Fig. 1], a nodal ring, and two triple points. (c) A mixture phase of a 2D Hopf-chain network [like that in Fig. 1], and four triple points. (d) A 3D Hopf-chain networks. (e)–(h) Band structures corresponding to the phases in (a)–(d), respectively, in which $X\pm , Y\pm$, and $Z\pm$ in the parentheses represent the eigenvalues of mirror planes $k_x=0, k_y=0$, and $k_z=0$ being $\pm 1$. All the phases and corresponding band structures are generated by Eq. (1) with different parameters. The phase in (a) is protected by three normal $C_4$ rotation axes along $k_{x/y/z}$ and three mirror planes. When the $C_4$ rotation axes along $k_{x/y}$ are broken, it evolves into the phases in (b) and (c). When the $C_4$ rotation axis along $k_z$ is further broken, the phases (b) and (c) evolve into the phase in (d). The coordinate axes in and out of the parentheses in (d) represent the axes for the phase transiting from (b) and (c), respectively.

Figure 4

Material realizations of the triple-points and 2D/3D Hopf-chain networks in $\mathrm{S}{\mathrm{c}}_{3}X\mathrm{C}$. (a) Crystal structure of $\mathrm{S}{\mathrm{c}}_{3}X\mathrm{C}\left(X=\mathrm{Al},\mathrm{Ga},\mathrm{In},\mathrm{Tl}\right)$. (b) BZ with high-symmetry points marked. (c) Band structure of $\mathrm{S}{\mathrm{c}}_3\mathrm{InC}$ in the case of no SOC; the insets show the band structures near the triple points in the case of considering SOC.

Figure 5

Bulk band structures along $Y$-Γ-$Y$ of $\mathrm{S}{\mathrm{c}}_3\mathrm{InC}$ in cases of (${\mathrm{a}}_1$) no strain, (${\mathrm{b}}_1$) 3% (5.30 GPa) tensile strain along $z$ axis, (${\mathrm{c}}_1$) 3% (6.59 GPa) compressive strain along $z$ axis, (${\mathrm{d}}_1$) 3% (7.27 GPa) compressive strain along $z$ axis and 1% (3.21 GPa) compressive strain along $y$ axis. (${\mathrm{a}}_1$)–(${\mathrm{d}}_1$) are similar to Figs. 3–3 along the same $k$ path, respectively, and thus the cases in (${\mathrm{a}}_1$)–(${\mathrm{d}}_1$) correspond to the topological phases in Figs. 3–3. (${\mathrm{a}}_2$)–(${\mathrm{d}}_2$) Surface band structures on the [100] surface corresponding to the bulk band structures in (${\mathrm{a}}_1$)–(${\mathrm{d}}_1$), respectively. (${\mathrm{a}}_3$)–(${\mathrm{d}}_3$) Surface band structures on the [001] surface corresponding to (${\mathrm{a}}_1$)–(${\mathrm{d}}_1$), respectively.

Figure 6

(a)–(d) Landau level spectra for the topological phases in Figs. 3–3, respectively. The empty green dots represent the projected triple points while the empty blue dots represent two ends of the projected nodal rings. The magnetic field $\mathit{B}$ is along $z$ axis. The color intensity is the Landau index from $n=1$ to 30.

Figure 7

Other new topological network phases generated by Eq. (1), the parameters of which are listed in Table S1. (a) A mixture phase of triple points and additional nodal lines, which has the same symmetry as the phase in Fig. 3. (b) A phase including a 2D Hopf-chain network, four triple points, and additional nodal rings. This phase is equivalent to the phase in Fig. 3 adding four red nodal rings. (c) A phase including a 3D Hopf-chain network and four red additional nodal rings. This phase is equivalent to the phase in Fig. 3 adding four red nodal rings. All the additional nodal rings and lines are located on the mirror planes $k_{x/y/z}=0$. (d)–(f) Band structures corresponding to the phases in (a)–(c), respectively. It is noted that the phase in (a) can be found in the crystals $\mathrm{S}{\mathrm{c}}_{3}X\mathrm{C}\left(X=\mathrm{Al},\mathrm{Ga},\mathrm{In},\mathrm{Tl}\right)$ by applying a hydrostatic pressure.

Figure 8

(a)–(d) Electronic band structures of $\mathrm{S}{\mathrm{c}}_{3}X\mathrm{C}\left(X=\mathrm{Al},\mathrm{Ga},\mathrm{In},\mathrm{Tl}\right)$ in case of no SOC.

Figure 9

DFT band structures for $\mathrm{S}{\mathrm{c}}_3\mathrm{InC}$ under different strains. (a) No strain. (b) 3% tensile strain along $z$ axis. (c) 3% compressive strain along $z$ axis. (d) 3% compressive strain along $z$ axis and 1% compressive strain along $y$ axis. The band structures in (a)–(d) corresponding to the band structures in Figs. 3–3, respectively. (${\mathrm{a}}_1$)–(${\mathrm{d}}_6$) Contour plots of energy difference between bands 1, 2, 3, 4 in (a)–(d). From these contour plots, the topological elements in the phases in Figs. 3–3 can be found clearly. For example, in (${\mathrm{a}}_1$), (${\mathrm{a}}_2$), the triple points on axes are shown, which are connected by white straight nodal lines; in (${\mathrm{b}}_1$), (${\mathrm{b}}_2$) or (${\mathrm{c}}_1$), (${\mathrm{c}}_2$), the mutually vertical nodal rings of the 2D Hopf-chain network are shown.

Figure 10

(a)–(d) Fermi surfaces corresponding to the topological phases in Figs. 9–9, respectively. (${\mathrm{a}}_1$)–(${\mathrm{d}}_2$) show different side views of the Fermi surfaces.

Figure 11

(a) DFT band structure of $\mathrm{S}{\mathrm{c}}_3\mathrm{InC}$ under a 3% compressive strain along $z$ axis and a 1% compressive strain along $y$ axis in the presence of SOC. (b), (c) Amplified regions in the green and red dashed boxes in (a), respectively. The numbers in (b) and (c) show the widths of the gaps between energy bands. One can find that the gaps induced by SOC are very small (about $1\sim 6\mathrm{meV}$).