Nexus networks in carbon honeycombs

Nexus metals represent a new type of topological material in which nodal lines merge at nexus points. Here we propose novel networks in nexus systems through intertwining between nexus fermions and additional nodal lines. These nexus networks can be realized in recently synthesized carbon honeycomb materials. In these carbon honeycombs, we demonstrate a phase transition between a nexus network and a system with triply degenerate points and additional nodal lines. The Landau level spectra show unusual magnetic transport properties in the nexus networks. Our results pave the way toward realizations of new topological materials with novel transport properties beyond standard Weyl/Dirac semimetals.

Figure 1

From the standard NP phase to the nexus network. (a) Standard NP phase in which two NPs (blue circles) are connected by four nodal lines. The three light blue lines reside on the three mirror planes, respectively, while the green nodal line is shared by the three mirror planes. (b) Mirror plane of the nexus phase plus two yellow topological ANLs. (c) Avoiding crossing of two (light blue and orange) topological nodal lines. (d) Mirror plane of the nexus network in which the NPs are connected by anticrossing (winding) nodal lines.

Figure 2

Atomic structures of CHCs. (a) CHC-1 from a top-view perspective, where the carbon atoms form a 3D honeycomb. The (green and blue) $\mathit{sp}{}^2$ carbon atoms (C1) reside on different horizontal atomic planes with respect to the $c$ axis shown in (b) and each kind has a threefold rotational symmetry with respect to the axes passing through the (orange) $\mathit{sp}{}^3$ carbon atoms (C2). (b) Primitive cell for CHC-1, where $a,b$, and $c$ are the lattice parameters. The C2 atoms form a row of carbon dimers along the $c$ axis. The labels $t_0$ in (b) and $t_1-t_5$ in (a) are tight-binding hopping parameters. (c) A $1\times 1\times 2$ supercell of CHC-$1^{\prime }$, which is a carbon structure after the dimers in CHC-1 are removed.

Figure 3

Band structure of CHC-1 and topological networks in CHC-1 and CHC-1´. (a) Band structure of CHC-1 along $k_z\left(\mathrm{\Gamma }\text{−}A\right)$. Point $\alpha$ (blue circle) is a TP, as a result of band crossing between the double-degenerate (green) and nondegenerate (black) bands. (b) Schematic view of the nexus network including NPs and nodal lines in the full first BZ. The solid and dotted lines correspond to nodal lines for bands $m-1$ and $m$ and bands $m$ and $m+1\left(m=33\right)$, respectively. The three orange planes are mirror planes. (c) Schematic view of the TP ANL in the full first BZ of CHC-$1^{\prime }$. The three orange planes are mirror planes, while the three light blue planes are glide planes. In each plane, there are two nodal lines. (d) Contour plots of energy difference between bands $m$ and $m+1$ on the mirror plane $k_y=0$ for the structure CHC-1, which shows a standard connectivity between two NPs. The blue circles correspond to the NPs, while the red lines represent the energy difference equal to zero, that is, nodal lines. (e) Same as (d) but between bands $m-1$ and $m$, which shows a winding connectivity between two NPs. (f) Contour plots of energy difference between bands $m-1$ and $m$ on the mirror plane $k_y=0$ for the structure CHC-$1^{\prime }$, where two TPs and two ANLs coexist. It should be noted that to clearly show the nodal lines in (d)–(f), different scales of $k_x$ and $k_z$ are applied. In (d)–(f) $k_x$ and $k_z$ are in units of $\pi /a$ and $\pi /c(a$ and $c$ are lattice constants), respectively.

Figure 4

Four topological phases of the tight-binding model in Eq. (1) and the $\mathbf{k}·\mathbf{p}$ model in Eq. (2). (a) Standard TP phase in which two TPs are connected by one trivial nodal line. (b) A TP-ANL phase in which TPs and (orange) ANLs coexist. (c) Standard NP phase in which NPs are connected by two splitting lines. (d) Nexus network in which NPs are connected by standard and winding connectivity. (a1)–(d1) Band structures corresponding to the $k$ paths in (a)–(d), respectively, referenced by the arrows. The standard TP phase in (a) and the TP-ANL phase in (b) exist in the structure CHC-$1^{\prime }$ with sixfold screw rotational symmetry, while the NP phase in (c) and the nexus network in (d) exist in the structure CHC-1 with threefold rotational symmetry. The phases are only shown on the $k_y=0$ mirror plane, but identical features appear on the five other mirror planes for the phases in (a) and (b) and the two other mirror planes for those in (c) and (d). Different parameters for the tight-binding model and the $\mathbf{k}·\mathbf{p}$ model lead to the phase transitions between (a)–(d). In the tight-binding model in Eq. (1), $t_2$ determines if a phase has ANLs or not, while the structural symmetry determines whether a phase has a TP or a nexus. In the $\mathbf{k}·\mathbf{p}$ model in Eq. (2), in contrast, these different phases are determined by the vanishing or nonvanishing of $\beta$ and $\alpha$, respectively. Here $k_x$ and $k_z$ are in units of $\pi /a$ and $\pi /c$, respectively.

Figure 5

Extended TP-ANL phase and nexus network and LLs based on the $\mathbf{k}·\mathbf{p}$ model in Eq. (2) $[\beta$ is replaced by $\beta +\gamma \cos (\delta k_z)]$. (a) A TP-ANL phase in which a TP phase and four ANLs coexist $(\alpha =0,\gamma =0.5$, and $\delta =4)$. (b) Nexus network evolved from (a) $(\alpha =0.1,\gamma =0.5$, and $\delta =4)$. (c) The LLs for the TP-ANL phase in Fig. 4 $(\alpha =0)$. (d) The LLs for the nexus network in Fig. 4 $\left(\alpha =0.5\right)$. (e) and (f) The LLs for the TP-ANL phase in Fig. 5 and the nexus network in Fig. 5, respectively. The other parameters are $A_1=1,B_1=-1,A_2=-1,B_2=1,D=1,C_1=C_2=0,\beta =-1$, and the magnetic field $\mathbf{B}=0.03$. Here $k_x$ and $k_z$ are in units of $\pi /a$ and $\pi /c$, respectively.