1. Introduction

Topological phase is a new phase of matter characterized by integer quantities known as topological invariants. They are insulating in the bulk but possess conducting currents on the surface. The existence of such surface states is protected by topology, which is concerned with the properties of a geometric object that are preserved under continuous deformations. The very first example of the topological phase is the quantum Hall (QH) phase [1], which is a two-dimensional (2D) electron gas under an external magnetic field. The time-reversal (TR) symmetry is broken in the QH phase becuase of the presence of magnetic field. A different class of the 2D topological phase that preserves the TR symmetry is the quantum spin Hall (QSH) phase [2–4], in which the spin-orbit coupling is responsible for the topological characters and no megntic field is required. The QH and QSH phases are characterized by the Chern number or TKNN invariant [5] and the $Z_2$ invariant [2] or spin Chern number [6], respectively. Theoretical concepts developed in the QSH phases are generalized to three dimensions (3D), leading to the more general class of 3D topological insulators [7,8].

In the band gap of a QSH phase, gapless edge states exist for each spin, and the group velocity direction of the edge states is locked by the spin [9]. The spin-momentum locking enables topologically protected edge states that propagate unidirectionally without back scattering [10]. As the edge states are protected by the bulk topology, they are insensitive to small perturbations that do not change the topology. In 3D gapped topological phases, that is, 3D topological insulators, gapless surface states appear inside the band gap between two topologically distinct bands as in 2D topological phases [11,12], which can be realized in both TR broken [13,14] and TR invariant systems [15–17]. On the other hand, 3D gapless topological phases, also known as topological semimetals [18,19], are a new class of topological phase of matter different from the topological insulators [20–22], which do not have 2D counterparts. The 3D gapless topological phases are characterized by Weyl degeneracies, which are degeneracies between topologically inequivalent bands. The main signature of 3D gapless topological phases is the appearance of Weyl points, which can exist in systems that lack TR symmetry, inversion symmetry, or both. The Weyl points are understood as the monopoles of Berry curvature in the momentum space that carry quantized topological charges, which are equal to the topological invariants of the system. An important feature of the Weyl points is the existence of Fermi arcs that connect the Weyl points, which correspond to the topologically protected surface states that are robust against disorder. A useful perspective on the Weyl semimetals is to view them as the transitional state between a topological insulator and a trivial insulator [19].

The topological phases in electronic strucures have been extended to photonic systems [23–26], leading to the discovery of photonic QH phases [27–31], photonic QSH phases [32–36], photonic 3D topological insulators [37–39], and photonic topological semimetals [40–48]. The key aspect to construct a topological phase is having a Kramers pair in the system, which are doubly degenerate eigenstates under TR symmetry [49]. The Kramers theorem, however, is usually valid for a TR invariant system with spin 1/2 [10] and cannot readily apply to the photonic system with spin 1 [50,51], unless additional symmetry has been imposed. Nevertheless, photons have spin properties as a result of circular polarization [52]. A spin-like quantity called pseudospin can be formed by the linear combination of electric and magnetic fields when a certain degenerate condition between the electric and magnetic parameters is satisfied [32]. As a result, the photonic system can be described by an effective Hamiltonian consisting of two subsystems for the pseudospin states [32–34], and the photonic Kramers doublet can be formed in the system. In the presence of chirality or bianisotropy that emulates the spin-orbit coupling, a topological phase can be constructed in the photonic system [53–56].

In the present study, we investigate the photonic topological phases in Tellegen metamaterials characterized by the magnetoelectric tensors with real-valued antisymmetric components. [57–60]. Bulk modes of the underlying medium are represented by two decoupled quadratic equations with a symmetric pattern between electric and magnetic parameters. When the ’spin’-degenerate condition [32,34,38] is satisfied, the bulk modes are featured with a Weyl cone displaced a distance from the origin in the frequency-wave vector space. The electromagnetic duality allows for the photonic system to be decoupled as two subsystems of the hybrid modes defined as the linear combinations of electric and magnetic fields. By introducing the pseudospin states as the basis for the hybrid modes, the photonic system can be described by a pair of spin-orbit Hamiltonians with spin 1 [54–56,61,62] that respect the fermionic-like pseudo time-reversal symmetry. The topological properties of the photonic system are determined by the nonzero spin Chern numbers calculated from the eigenfields of the Hamiltonians. Surface modes at the interface between two Tellgen metamaterials with opposite sign of the magnetoelectric parameter exist at their common gap in the wave vector space, which are analytically formulated by algebraic equations. In particular, there exist two types of surface modes tangent to or wrapping around the Weyl cones that form a pair of bended surface sheets and a pair of twisted surface sheets. At the Weyl frequency, the surface mode contains a typical Fermi arc as a line segment that connects two Weyl points, and two open Fermi arcs as semi-infinite lines that emanate from the Weyl points. These arcs concatenate to yield an infinite straight line as the Fermi-arc like edge state. Finally, the topological features of the Tellegen metamaterials are illustrated with the robust transport of surface modes at an irregular boundary, which are able to bend around sharp corners without backscattering.

2. Basic equations

2.1 Bulk modes

Consider a general bianisotropic medium characterized by the constitutive relations:

The existence of a nontrivial solution of ${\bf E}$ and ${\bf H}$ requires that the determinant of the 6 $\times$ 6 matrix in Eq. (3) be zero, which gives the characteristic equation of the bulk modes as

Note that the features of bulk modes may change with the frequency in a dispersive medium (which is usually the case of metamaterials), depending on the choice of frequency range. In the neighborhood of a reference frequency $\omega _\text {ref}$, ${\varepsilon _n}$ ($n=t,z$) can be approximated as ${\varepsilon _n} \approx {\varepsilon _{n0}} + {\left. {\frac {{d{\varepsilon _n}}}{{d\omega }}} \right |_{\omega = {\omega _\text {ref}}}}\left ( {\omega - {\omega _\text {ref}}} \right ) \equiv {\varepsilon _{n0}} + {\tilde \varepsilon _n}\delta \omega /{\omega _\text {ref}}$, where ${\tilde \varepsilon }_n$ is positive definite [61]. A similar approximation is valid for $\mu _n$ ($n=t,z$). We further assume that the magnetoelectric parameter $\xi _{xy}$ varies smoothly around $\omega _{\rm ref}$ and can be treated as a constant in the analysis [32,54–56].

2.2 Spin-orbit Hamiltonians

The electromagnetic duality of Maxwell’s equations ensures that the matrix in Eq. (3) holds a symmetric pattern when the ’spin’-degenerate condition $\underline {\varepsilon }=\underline {\mu }$ [32,34,38] is satisfied. This allows us to rewrite Eq. (3) as

2.3 Topological invariants

The topological properties of the spin-orbit Hamiltonians ${\cal H}_\pm$ can be characterized by the topological invariants using the eigenfields. For this purpose, we calculate the Berry flux over a sphere that encloses ${\bf {k}}_w$ in the wave vector space. The eigensystem for the Hamiltonian ${\cal H}_\pm$ in Eq. (9):

The nonzero $C_\sigma$ ($\sigma =\pm 1$) characterize the topological properties of the system, where $\sigma$ refers to the helicity (or handedness) of the pseudospin states. In particular, the surface or edge states at the interface between two distinct topological phases are topologically protected, which means that their existence is guaranteed by the difference in band topology on two sides of the interface. In this system, the total Chern number $C=\sum _\sigma {{C_\sigma }}=0$ and the spin Chern number $C_{\rm spin}=\sum _\sigma {{\sigma C_\sigma }}=4$, which are consistent with the quantum spin Hall effect of light [52]. The topological invariants remain unchanged under arbitrary continuous deformations of the system. The topological properties in the isotropic case will be retained when a certain anisotropy is included in the system. For a more general anisotropic case, the exact calculation of topological invariants can be obtained by the numerical integration of Berry curvatures [71].

2.4 Pseudo time-reversal symmetry

Because of the nonreciprocity of the Tellegen medium (cf. Sec. 2.1), the Hamiltonian for Maxwell’s equations is not TR invariant under $T_b$, that is, $\left ({T_b \otimes I}\right ){{\cal H}_m \left ( {\bf k} \right )}\left ({T_b \otimes I}\right )^{ - 1} \ne {{\cal H}_m\left (-{\bf k} \right )}$, where ${\cal H}_m$ is the $6\times 6$ matrix in Eq. (3), ${T_b} = {\sigma _z}K$ (with $T_b^2=1$) is the bosonic TR operator for photons [23], with $K$ being the complex conjugation, and $\otimes$ denotes the tensor product. The combined Hamiltonian formed by two spin-orbit Hamiltonians $\cal H_\pm$ [cf. Eq. (9)], is TR invariant under $T_p$:

2.5 Surface modes

Let the $xz$ plane be an interface between two Tellegen metamaterials with opposite sign of the magnetoelectric parameter, characterized by $\varepsilon _n$, $\mu _n$ ($n=t,z$), and $\pm \xi _{xy}$ (cf. Sec. 2.1), at which the surface modes may exist. According to Maxwell’s boundary conditions: the continuity of tangential electric and magnetic field components at the interface, the characteristic equation of surface modes can be analytically formulated by using the eigenfields of bulk modes on two sides of the interface, which is given by (see Supplementary Material C)

3. Results and discussion

3.1 Bulk modes

Figure 1 shows the equifrequency surfaces of the bulk modes in the wave vector space for the Tellegen metamaterial based on Eq. (5). In the present study, we assume that $\varepsilon _t\mu _t\ge 0$, $\varepsilon _t\mu _z\ge 0$, and $\varepsilon _z\mu _t\ge 0$ so that the bulk modes are described by the elliptic equations. This condition is crucial to form the Weyl system in the Tellegen metamaterial, which will be discussed later (cf. Sec. 3.3). Regarding the relative magnitude of magnetoelectric parameter (to the geometric mean of transverse permittivity and permeability components), the bulk modes can be classified into two phases:

 

Fig. 1. Equifrequency surfaces of the bulk modes in the wave vector space for the Tellegen metamaterial with (a) $\varepsilon _t=1.5$, $\varepsilon _z=2$, $\mu _t=1.5$, $\mu _z=1$, and $\xi _{xy}=\pm 1$ (b) $\varepsilon _t=1$, $\varepsilon _z=3.5$, $\mu _t=1$, $\mu _z=2$, and $\xi _{xy}=\pm 1.5$.

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In the isotropic case, where $\varepsilon _t=\varepsilon _z\equiv \varepsilon$ and $\mu _t=\mu _z\equiv \mu$, the bulks in both phases (I) and (II) are represented by two identical ellipsoids (cf. Sec. 2.1). If the ’spin’-degenerate condition $\varepsilon =\mu$ is further satisfied, the bulk modes are represented by two identical spheres. Recall that the effective Hamiltonian for the present problem consists of two subsystems of the hybrid modes. Each subsystem is described by the spin-orbit Hamiltonian with spin 1 (cf. Sec. 2.2) and characterized by nonzero topological invariants (cf. Sec. 2.3). In this regard, the Tellegen metamaterial is considered a photonic analogue of the topological phase.

3.2 Surface modes

Figure 2 shows the surface modes at the interface between two Tellegen metamaterials with opposite sign of the magnetoelectric parameter $\xi _{xy}$ in the $k_x$–$k_z$ plane based on Eq. (15). The bulk modes at $k_y=0$ are also shown in the plots. For clarity, we discuss the surface modes in the isotropic case where $\varepsilon _n=\varepsilon$ and $\mu _n=\mu$ ($n=t,z$), and the analytical expressions for the surface modes are available. Regarding the relative magnitude of magnetoelectric parameter (to the geometric mean of permittivity and permeability), the surface modes can be classified into two types:

 

Fig. 2. Surface modes at the interface between two Tellegen metamaterial with opposite sign of the magnetoelectric parameter (a) $\varepsilon _n=\mu _n=1.5$ and $\xi _{xy}=\pm 1$ (b) $\varepsilon _n=\mu _n=1$ and $\xi _{xy}=\pm 1.5$ ($n=t,z$). Gray solid and dashed contours are bulk modes at $k_y=0$ for positive and negative $\xi _{xy}$, respectively. Orange and green lines are type (I) and type (II) surface modes, respectively. In (b), blue and red dots are chosen points for surface wave simulations (cf. Figure 4).

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The different characters for the two types of surface modes come from the relative maganitude of the magnetoelectric parameter. This feature becomes more evident in the frequency-wave vector space when the fequency dispersion of material parameters is taken into account (cf. Sec. 3.3). Note that the bulk modes on two sides of the interface are mirror reflections of each other about the $k_x$ in the presence of the magnetoelectric parameter $\xi _{xy}$ [cf. Eq. (5)]. The surface modes are located in the common gap of the bulk modes for two Tellegen metamaterials in the wave vector space, that is, outside two intersecting or separate ellispses. All the surface modes are tangent to the bulk modes [19,20], including the line segments and semi-infinite lines. This feature follows from the fact that the surface modes must convert seamlessly into the bulk modes as they approach their termination points [72]. The evanescent depth of the surface mode grows until at the point where the surface mode merges with the bulk mode [19]. The bulk modes on either side of the interface are topologically nontrivial with nonzero topological invariants (cf. Sec. 2.3). The surface modes correspond to the topological phase transition between two distinct topological phases in the momentum space [54,73], their existence being guaranteed by the bulk-edge correspondence.

3.3 Photonic Weyl system

Let the frequency dependence of the Tellegen medium be characterized by the Lorentz-type dispersive models: $\varepsilon = \varepsilon _\infty - \omega _p^2/\left ( \omega ^2 - \omega _0^2 \right )$ and $\mu = \mu _\infty - \Omega _\mu \omega ^2/\left (\omega ^2-\omega _0^2\right )$, which are usually employed in the study of metamaterials [74]. Here, $\omega _0$ is the the resonance frequency of the resonators and $\omega _p$ is the effective plasma frequency of the medium. The magnetoelectric parameter is given by $\xi _{xy} = \Omega _\xi \omega \omega _{p}/\left ( \omega ^2 - \omega _0^2 \right )$ [34]. This model guarantees that the energy density in the underlying medium is positive definite (see Supplementary Material D).

Figure 3(a) shows the dispersion of bulk modes for two Tellegen metamaterials with opposite sign of the magnetoelectric parameter $\xi _{xy}$ in the frequency-wave vector space. The bulk modes for the two metamaterials are represented by a pair of conic surfaces symmetrically displaced on the $k_z$ axis. In the present configuration, the material parameters are arranged such that $\varepsilon$ and $\mu$ become zero at a particular frequency $\omega _1 = \sqrt {\omega _0^2 + \omega _p^2/{\varepsilon _{\infty }}}$, where the bulk modes of each medium are reduced to a single point: $\left (k_t,k_z\right )=\left (0,\xi _{xy} \omega _1/c\right )$. This is a condition that forms the point-like degeneracy in the bulk modes. In this situation, the dispersion of bulk modes resembles the linear crossing of valence and conduction bands in the Weyl semimetal [75], with the crossing points known as the Weyl points and the associated frequency as the Weyl frequency. As the bulk modes at the Weyl frequency are represented by point degeneracies, the underling medium is regraded as a photonic analogue of the type-I Weyl semimetal [19].

 

Fig. 3. (a) Bulk modes and (b) surface modes in the frequency-wave vector space for the Tellegen metamaterial with $\varepsilon _{n\infty }=\mu _{n\infty }=1$, $\Omega _{\mu n}=0.735$ ($n=t,z$), $\Omega _\xi =\pm 0.4$ and $\omega _0/\omega _p=0.6$. Wave vector components are scaled by $k_p=\omega _p/c$. In (a), red flat plane is the longitudinal mode. In (b), bulk modes at $k_y=0$ are outlined in gray mesh. Red dots are the Weyl points. Black lines are the Fermi arcs.

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Note that in the absence of magnetoelectric coupling ($\xi _{xy}=0$), the bulk modes are featured with the Dirac cone with fourfold degeneracy at the Dirac point: $(k_x,k_z,\omega )=(0,0,\omega _1)$ located at the center of the wave vector space [cf. Eq. (6)]. In the presence of magnetoelectric coupling ($\xi _{xy}\ne 0$), the TR symmetry is broken and the fourfold degeneracy is lifted. As a result, the bulk modes are featured with a Weyl cone with twofold degeneracy at the Weyl point [cf. Eq. (5)]. The broken TR symmetry is associated with the nonreciprocity intrinsic to the Tellegen medium (cf. Sec. 2.1). The topological charges carried by the Weyl points are consistent with the nonzero topological invariants $C_\pm =\pm 2$ of the present system (cf. Sec. 2.3). Here, the topological charges $\pm 2$ are associated with the unconventional spin-1 Weyl points with threefold linear degeneracy [76–78]. The net chirality vanishes in the Weyl semimetal, which agrees with the fact that the total Chern number is zero (cf. Sec. 2.3).

In another aspect, the bulk mode at the Weyl frequency $\omega _1$, where $\varepsilon =\mu =0$, is represented by a flat plane, which is known as the longitudinal mode, in contrast to the transverse mode for $\varepsilon \ne 0$ and $\mu \ne 0$. Note that the two conic surfaces intersect above the frequency ${\omega _2} = \frac {{\sqrt {{\varepsilon _\infty }\omega _0^2 + \omega _p^2} \sqrt {2{\varepsilon _\infty }{\mu _\infty }\omega _0^2 + \omega _p^2\Omega _\xi ^2 + \sqrt {4{\varepsilon _\infty }{\mu _\infty }\omega _0^2\omega _p^2\Omega _\xi ^2 + \omega _p^4\Omega _\xi ^4} } }}{{\sqrt 2 {\varepsilon _\infty }\sqrt {{\mu _\infty }} {\omega _0}}}$, at which $|\xi _{xy}| =\sqrt {\varepsilon \mu }$. This is the frequency that separates phase (I) and phase (II) of the bulk modes (cf. Sec. 3.1) as well as type (i) and type (ii) surface modes (cf. Sec. 3.2). Because of the frequency dispersion of the magnetoelectric parameter $\xi _{xy}$, the two conic surfaces for the bulk modes are tilted to a certain extent toward opposite directions.

Figure 3(b) shows the dispersion of surface modes at the interface between two Tellegen metamaterials with opposite sign of the magnetoelectric parameter $\xi _{xy}$ in the frequency-wave vector space. For comparison, the bulk modes (with $k_y=0$) at constant frequencies are outlined in gray mesh. Different from the surface modes in topological insulators that exist in the frequency (energy) band gap, the surface modes in gapless topological semimetals are defined in the region free of bulk modes at the same frequency (energy) [19]. Recall that there exist two types of surface modes in the present problem: the line segments and semi-infinite lines (cf. Sec. 3.2). The former appears in both phases (I) and (II), while the latter exists only in phase (II). Because of the frequency dependence of material parameters, the dispersion of surface modes is shown to be bended or twisted. The type (i) surface modes form a pair of bended surface sheets tangent to the Weyl cones in the frequency-wave vector space [cf. orange surface in Fig. 3(b)], while the type (ii) surface modes form a pair of twisted surface sheets wrapping around the Weyl cones [cf. green surfaces in Fig. 3(b)]. The latter feature is similar, though not identical, to the helicoid edge states in topological semimetals [45,76,79,80]. In the present configuration, the twisted nature of surface sheets composed of type (ii) surface modes can be understood from the varying slope of surface modes in the $k_z-k_x$ plane: $m = \sqrt {\frac {{{\varepsilon }{\mu }}}{{\xi _{xy}^2 - \varepsilon \mu }}}$ (cf. Sec. 3.2), which changes substantially with the frequency and becomes zero at the Weyl frequency. The slope $m$ goes to infinity as the frequency approaches $\omega _2$, at which the Weyl cones intersect.

Notice that the surface modes flip to the other side of the $k_z$ axis as the frequency moves across the Weyl frequency $\omega _1$, where the dispersion of surface sheets experiences a smooth transition. At the Weyl frequency, the edge state that connects the two Weyl points forms the so-called Fermi arc. The arc beginning on a Weyl point of chirality has to terminate on a Weyl point of the opposite chirality [20]. The type (i) surface mode at the Weyl frequency (a line segment) is considered the Fermi arc-like edge state. In particular, the type (ii) edge state at the Weyl frequency emanates from one Weyl point and extents to infinity without terminating on the other Weyl point, which forms an open Fermi arc [45,79]. In the present configuration, the typical Fermi arc (line segment) and two open Fermi arcs (semi-infinite lines) concatenate to yield an infinite straight line: $k_x=0$ [cf. black line in Fig. 3(b)]. In particular, the two Weyl points bridge the type (i) surface modes and the two disconnected type (ii) surface modes, which implies that they carry a nontrivial Chern number [19].

Finally, the topological features of the Tellegen metamaterial are illustrated with the propagation of surface waves at an irregular boundary [53–56,62,70,71,81]. For this purpose, a dipole source is placed at the interface between two Tellegen metamaterials with opposite sign of the magnetoelectric parameter to excite the surface modes in the their common gap (outside the bulk modes in the wave vector space), so that the waves are evanescent away from the interface on both sides. The surface waves are simulated using the finite element method for Maxwell’s equations in bianisotropic media by incorporating a general form of the magnetoelectric tensors in the constitutive relations [54,55,82]. In Fig. 4, a pair of surface modes are excited at $k_x/k_0=2.5$ [cf. blue and red dots in Fig. 2(b)] with right- or left-handed circular polarizations (see Supplementary Material E), which correspond to the opposite helicity of topological edge states. The surface waves propagate unidirectionally to the right or left along an irregular boundary with sharp corners. In particular, the surface waves counterpropagate at the boundary for different handednesses of circular polarization. This feature is consistent with the characteristic of surface modes in the present configuration [cf. Figure 2(b)], in which there exist a positive $k_z$ (blue dot) and a negative $k_z$ (red dot) for a fixed $k_x$. The surface waves are able to bend around sharp corners without backscattering, which demonstrates that the edge states are topologically protected.

 

Fig. 4. Surface wave propagation at the interface between two Tellegen metamaterials with opposite sign of the magnetoelectric parameter, where $\varepsilon _n=\mu _n=1$ ($n=t,z$), $\xi _{xy}=\pm 1.5$, and $k_x/k_0=2.5$ for (a) right-handed and (b) left-handed circular polarization. Green dot is the position of dipole source. Circular arrow denotes the handedness of circular polarization. Red and blue colors correspond to positive and negative values of Re[$E_z$], respectively, and $y$ and $z$ coordinates are scaled by $l_0=12.4\lambda _0$, with $\lambda _0=2\pi /k_0$.

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4. Concluding remarks

In conclusion, we have investigated the photonic topological phases in Tellegen metamaterials characterized by the antisymmetric magnetoelectric tensors with real-valued quantities. The photonic system is described by a pair of spin-orbit Hamiltonians with spin 1 in terms of the pseudospin states, and the topological properties are determined by the nonzero spin Chern numbers. Surface modes exist at the interface between two Tellegen metamaterials with opposite sign of the magnetoelectric parameter, which depict the typical features of topological edge states between two distinct topological phases. The underlying medium is regarded as a photonic analogue of the type-I Weyl semimetal featured with the Weyl cone and the associated Fermi arcs. Topological features of the Tellegen metamaterials are illustrated with the robust transport of surface modes at an irregular boundary.