1. Introduction

The discovery of quantum Hall effect has opened up an avenue for understanding the states of matter from a topological perspective [1–4]. In recent decades, research on novel topological phases has expanded beyond the condensed matter systems and into photonic systems, leading to the discoveries of various topological phases including photonic quantum Hall effect [5–9], topological insulators [10–13], Dirac/Weyl semimetals [14–22], nodal line/Chain semimetals [23–27] and high order topological phases [28–31]. The unique one-way disorder-immune surface states and some other topological phenomena are valuable complements to traditional way of controlling light in classical optics, creating new opportunities for device applications [32,33].

Dirac semimetals and nodal line semimetals represent two distinct gapless topological phases in three-dimensional systems. In Dirac semimetals, there exist four-fold linear degeneracies, or Dirac points, in the momentum space. The Dirac semimetals serve as basis for various topological phases. For example, two Weyl points of opposite chirality can be obtained by breaking either time reversal symmetry (T) or parity inversion symmetry (P) of a Dirac point [34]. The nodal line semimetals exhibit closed line degeneracies, and nodal chain semimetals are special types of nodal line semimetals with line degeneracies chained together at specific points in the momentum space [35,36]. Metamaterials offer a versatile platform for realizing various photonic topological phases, of which the topological properties can be achieved through the design of tailored meta-structures and described by simple effective medium theory [25,37,38].

While Dirac semimetals and nodal chain semimetals have received considerable attention in recent studies, the coexistence of Dirac degeneracies and nodal chain degeneracies has not been reported in photonics. In this letter, we propose a novel topological phase featuring coexistence of Dirac points and nodal chains. The configuration of the co-existing nodal chains and the Dirac points is schematically illustrated in Fig. 1, wherein the green and red circles denote the nodal line degeneracies lying in the ${k_z} = 0$ plane and the ${k_x} ={\pm} {k_y}$ planes, respectively, formed between different pairs of bands. The red nodal lines and the green one are chained together in the ${\mathrm{\Gamma} \mathrm{M}}({\mathrm{M^{\prime}}} )$ direction, and the four red nodal lines meet at the red and green spheres, which represent a novel type of Dirac point.

 

Fig. 1. The schematic illustration of topological phase with coexistence of Dirac points and nodal chains in the three-dimensional momentum space. The green nodal line is centered on high symmetry point $\mathrm{\Gamma }$ in the ${k_z} = 0$ plane and the red nodal lines are vertically located in the ${k_x} ={\pm} {k_y}$ planes, those that are chained together. The red and blue balls indicate Dirac points with opposite charges, respectively. The right panel shows the Dirac points lying in the chain points of red nodal lines.

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2. Photonic metacrystal model

The topological configuration depicted in Fig. 1 can be achieved by using a carefully designed metacrystal, as illustrated in Fig. 2(a). The designed simulation model is shown in Code 1 (Ref. [39]). The unit cell of metacrystal comprises two symmetric H-shaped metallic coils embedded in a hosting material with a dielectric constant of 4.2. The metallic components are placed on a diagonal line and oriented perpendicular to each other, as shown in Fig. 2(a). The photonic metacrystal is constructed in a simple tetragonal lattice with periods of ${p_x} = {p_y} = 4mm,\; {p_z} = 4.2mm$. The crystal possesses nonsymmorphic space group $P4/mbm$ (No: 127) [40], which includes some symmetry operations useful for our subsequent analysis, such as the rotation operation ${C_{2z}}:({x,y,z} )\to ({ - x, - y,z} )$, the screw rotation operations ${C_{2x}}:({x,y,z} )\to \left( {\frac{1}{2} + x,\frac{1}{2} - y, - z} \right)$, ${C_{2y}}:({x,y,z} )\to \left( {\frac{1}{2} - x,\frac{1}{2} + y, - z} \right)$, the glide mirror operation $\widetilde {M^{\prime}}:({x,y,z} )\to \left( {\frac{1}{2} + y,\frac{1}{2} + x,z} \right)$, and the mirror operation $M^{\prime\prime}:({x,y,z} )\to \left( {\frac{1}{2} - y,\frac{1}{2} - x,z} \right)$. In the microwave region, the metallic components are regarded as perfect electric conductor (PEC). By using the “Eigenmode Solver” module of the CST MICROWAVE STUDIO we can obtain the band structure of the metacrystal. The tetragonal Brillouin zone (BZ) and the surface BZ are shown in Fig. 2(b), with the reduced BZ enclosed by the red lines and the high symmetry points indicated by black dots. The Dirac points located on the M-A line are indicated by red and blue spheres. Figure 2(c) shows the band structure along high symmetry lines, where the band crossing along $\mathrm{\Gamma }$-X and $\mathrm{\Gamma }$-M give rise to the green nodal line degeneracies and the band crossings along M-A line result in the formation of the Dirac points.

 

Fig. 2. (a) Unit cell of metacrystal structure with periods ${p_x} = {p_y} = 4mm,{p_z} = 4.2mm$. The metallic coils colored in yellow with thickness of $4mm$ are embedded in dielectric materials with dielectric constant of 4.2 and the spacer layer with thickness of $0.2mm$ and the same dielectric constant is shown below the axis. The details are presented in the Supplement 1. (b) The first (bulk) BZ of tetragonal lattice is boxed by black lines and reduced (bulk) BZ by red lines. The upper and left light yellow plane shows the surface BZ in z- and x-direction, respectively. The blue and red balls located at $({ \pm \pi /{p_x}, \pm \pi /{p_y},{k_0}} )$ and $({ \pm \pi /{p_x}, \pm \pi /{p_y}, - {k_0}} )$ indicate the position of the Dirac points with opposite topological charges, where ${k_0} = 0.21\pi /{p_z}$ represents the magnitude of the z-component of wavevectors of Dirac points. (c) The bulk bands spectrum along high symmetry lines. The band crossing along M-A line around 8.1 GHz is the Dirac point.

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3. Dirac points

A synthetic time reversal symmetry and a synthetic parity inversion symmetry are proposed to elucidate the formation of Dirac points along M-A line. The synthetic time reversal symmetry is given by the composite operations ${S_i} = {C_{2i}}\ast T,(i = x,y)$, the production of time reversal operation and screw rotation operations. Application of operation ${S_x}$ twice on an arbitrary photonic Bloch state $|\psi\rangle$ of the system is equivalent to a spatial translation acting on $|\psi\rangle $, i.e. $S_x^2|{\psi\rangle = {{({{C_{2x}}T} )}^2}} |\psi\rangle = {t_{100}}|\psi\rangle $, where ${t_{100}}$ is the spatial translation of one lattice in x-direction. From Bloch’s theorem, we have $S_x^2|{\psi\rangle = {e^{i{k_x}{p_x}}}} |\psi\rangle $. The ${k_x} = \pi /{p_x}$ plane is invariant under operation ${S_x}$, which yields,

Based on Kramers’ theorem, Eq. (1) ensures that all the photonic states are doubly degenerate in the ${k_x} = \pi /{p_x}$ plane. A similar analysis can be applied to ${S_y}$ which ensures double degeneracy in the ${k_y} = \pi /{p_y}$ plane. The product of two composite operations gives ${S_x}{S_y} = {S_y}{S_x} ={-} {C_{2z}}$ along M-A line. This indicates that the doubly degenerate states resulting from ${S_i}$ possess the same eigenvalue under ${C_{2z}}$ rotation. Therefore, the two-fold rotation ${C_{2z}}$ plays the same role as a synthetic parity inversion symmetry with eigenvalues ${C_{2z}} ={\pm} 1$, which indicate the parities of photonic Bloch states along M-A line. The Dirac points are formed by degenerate pairs with opposite parities. The topological charges of the Dirac points, denoted by red and blue spheres respectively, are ${N_{DP}} ={\pm} 1$, as shown in Fig. 2(b) [41–43].

The two screw rotation operations are schematically illustrated in Fig. 3(a). The bulk band dispersion at plane of ${k_x} = \pi /{p_x}$ are presented in Fig. 3(b), which show cone-shaped dispersion around the Dirac points in a clean frequency range. Figure 3(c) shows the band crossing in the vicinity of Dirac point (8.1 GHz) in three perpendicular directions. According to the bulk-edge correspondence principle [3,4], the (100) surface states exhibit ${Z_2}$ topology due to Dirac points [42,43], which one surface state exist in $|{{k_z}} |> {k_0}$ region and no surface state in $|{{k_z}} |< {k_0}$ corresponding to ${Z_2}$ value of 1 and 0, respectively. The surface states are calculated by supercell constructed by stacking the metacrystal unit with 15 layers along x-direction followed by air layers with thickness of $15{p_x}$, where periodic boundary condition is applied to three directions with ${k_x} = 0$ in the CST MIRCOWAVE STUDIO. The band spectrums with ${k_y} = \pi /{p_y}$ along ${k_z}$ direction and ${k_z} = 0.1\pi /{p_z},\; \; 0.5\pi /{p_z}$ along ${k_y}$ direction are shown in Fig. 3(d-f), respectively, where the red lines indicate the surface states and black lines represent the projected bulk states.

 

Fig. 3. (a) The schematic illustration of two screw symmetry operations ${C_{2x}}$ and ${C_{2y}}$ from top-down view, respectively. (b) The bulk bands dispersion on ${k_x} = \pi /{p_x}$ plane with cone shape around the Dirac points. (c) The bands crossing around the Dirac point (8.1 GHz) along three perpendicular directions. (d-f) The (100) surface states denoted by red lines and projected bulk states indicated by black lines along three specific lines, ${k_y} = \pi /{p_y}$ along ${k_z}$ direction and ${k_z} = 0.1\pi /{p_z},\; \; 0.5\pi /{p_z}$ along ${k_y}$ direction in the surface BZ, are illustrated in Fig. 3(d-f), respectively. The red lines in the insets shows the positions of the three lines relative to the Dirac points in the surface BZ.

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4. Nodal chains

The green nodal line degeneracies are formed by two bands possessing opposite eigenstates of the mirror symmetry operation ${M_z}$. The parts of red nodal line in the first BZ are connected to other parts outside of first BZ due to the periodicity of the BZ, as schematically shown in Fig. 1. The red nodal line degeneracies in the ${k_x} = {k_y}$ and ${k_x} ={-} {k_y}$ planes are protected by the glide mirror operation $\widetilde {M^{\prime}}$ and mirror operation $M^{\prime\prime}$, respectively. For a photonic Bloch state adiabatically evolving along a circle around the chain point, it acquires a Berry phase of 0, in contrast to the $\pi $ Berry phase for that around a nodal line. The equi-frequency contour (7.9 GHz) slightly below the nodal chain degeneracies is shown in Fig. 4(a), which display the tori in perpendicular planes joining with each other. To investigate the topological properties of the surface states of nodal chains, we project the system into z-direction. Four representative lines ${k_x} = {k_y}\tan \theta $ ($\theta = \pi /3\; $), ${k_y} = 0$, ${k_y} = 0.25\pi /{p_y}$ and ${k_y} = 0.55\pi /{p_y}$ in the projected surface BZ, labeled with I, II, III and IV, respectively, are illustrated in Fig. 4(b), where the lines I, II and III intersect with the projected green nodal line and the line IV is located outside of the projected green nodal line. For the simulation of the surface states, a supercell is constructed which comprise 15 unit-cells along z-direction, where periodic boundary condition is applied to the three directions with ${k_z} = 0$. The band dispersions, with red lines denoting surface states and black lines representing projected bulk states, are plotted along lines I, II, III and IV in Fig. 4(c-f), respectively. As the system projected into the z-direction, the red nodal lines shrink into line segments and the green nodal line keep intact, which correspond to red line segments and green circles shown in Fig. 4(b), respectively. Therefore, the nodal chains projected into z-direction exhibit properties resembling the green nodal line. The three lines I, II and III intersected with the projected green nodal line have surface states connected to the projected green nodal line, while the line IV located outside has none of surface states.

 

Fig. 4. (a) The copper-colored shape is the equi-frequency contour at frequency of 7.9 GHz (slightly below the nodal line degeneracies frequency) in the three-dimensional momentum space. (b) The surface BZ as the system projected into z-direction. The four dash lines I, II, III and IV correspond to four different cuts: ${k_x} = {k_y}tan\theta $, ${k_y} = 0$, ${k_y} = 0.25({\pi /{p_y}} )$ and ${k_y} = 0.55({\pi /{p_y}} )$ at the surface BZ, respectively. The red segments and green circle are the projected red and green nodal lines. (c-f) The projected bulk (black) and (001) surface states (red) along lines I, II, III and IV are presented in Fig. 4(c-f), respectively. The three lines I, II and III intersected with the projected green nodal line have surface states connected to the projected green nodal line. The line IV is located outside the projected green nodal line without surface states.

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5. Conclusion

In conclusion, we have proposed a novel topological phase featuring coexistence of Dirac points and nodal chains. We further designed a realistic photonic metacrystal for experimental implementation of this topological phase. The features of bulk and (001) surface states of the nodal chains are analyzed and the nontrivial ${Z_2}$ topology of the Dirac points is presented via (100) surface states. The Dirac points and nodal chains reside in a clear frequency interval, which can support negative refraction and vortex beam generation as well as other photonic application related to the unique properties of Dirac points or nodal chains [44–49]. Meanwhile, our work offers a platform for exploring systems simultaneously hosting multiple topological phases, and for investigating their interactions.