Topological characteristic of Weyl degeneracies in a reciprocal chiral metamaterials system

Being a research hotspot in the field of topological semimetals, Weyl points (WPs) are monopoles of Berry curvature in momentum space. In this paper, we report the existence of photonic Weyl degeneracies in a reciprocal chiral metamaterials system. Due to the flat dispersion relation of the bulk plasmon modes, Weyl degeneracies here lie right on the critical transition between the type-I and type-II WPs. The photonic ‘Fermi arc’ connects the projection of pairs of WPs at the interface between the metamaterials and vacuum. Despite the bulk equi-frequency surfaces have changed dramatically, the ‘Fermi arc’ always exists. In addition, numerical simulations of topologically protected ‘Fermi arc’ surface states show that the surface waves are not scattered or reflected by the presence of sharp corners. Notably, such metamaterials host either type-I, type-II WPs or triple degenerate points (TDPs) depending on the nonlocal response. Our work provides an ideal photonic platform for studying the closely relation between WPs and other exotic states.


Introduction
Topological semimetals [1] are characterized by surface states induced by the topology of the bulk band structure and represent a new class of topological matter [2]. Recently, Dirac semimetals [3], Weyl semimetals [4], triple degenerate points (TDPs) semimetals [5], and node-line semimetals [6,7] have been predicted theoretically and observed experimentally. Among them, the Dirac semimetals [8] bridge Weyl semimetals, topological insulators, and conventional insulators. The Dirac points (DPs) as the fourfold degenerate points, are protected by the time-reversal and spatial inversion symmetry [9,10]. Hence, they are unstable against perturbations and require extra constraints to stabilize [11]. Systems with broken either spatial inversion or time-reversal symmetry can potentially host Weyl points (WPs) [12][13][14]. They are topologically robust against small perturbation, and may disappear only by annihilation with other WP of opposite chirality [15,16]. In addition, TDPs possess linear dispersion and effective integer spin [17,18]. They can be viewed as an intermediate phase between fourfold degenerate DPs and twofold degenerate WPs [19]. TDPs have been predicted in many materials with threefold linear band crossing points [20,21]. Moreover, different from the isolated points in the Dirac, Weyl and TDPs semimetals, for the node-line semimetals, the band crossing points can form one-dimensional nodal-lines in three-dimensional momentum space [22][23][24]. The physical properties of the three types nodal points and the nodal-lines in topological semimetals have attracted much attention recently in virtue of new physics and potential applications behind them [25][26][27].
As the simplest class of three-dimensional topological systems [28][29][30], Weyl semimetals have been extensively studied owing to their exotic behaviors, such as chiral anomaly [31], topologically protected 'Fermi arc' [32], and quantized circular photo-galvanic effects [33]. In recent years, WPs have been studied in electronic and classical wave systems. Usually, the study of WPs physics has more flexible geometric control in classical wave systems than in electronic systems [34]. In momentum space, serving as sources and sinks of the Berry curvature, the topological charges carried by the WPs can be defined by the topological invariant [35]. There are two types of WPs. The Fermi surface of the type-I WPs is point-like at the Weyl frequency [36], while the type-II WPs host a conical Fermi surface [37]. Both the type-I and type-II WPs have been investigated extensively in condensed matter systems [38], electromagnetic systems [39], and acoustic systems [40].
Most of the previous studies on WPs are based on photonic or phononic crystals, where periodicity plays an important role [41,42]. In particular, photonic WPs have been studied in magnetized plasma material [43] and gyromagnetic metamaterials [44]. However, the two systems without time-reversal symmetry require magneto-optical effects under a large static magnetic field. It is not conducive to the experimental implementation. On the other hand, magneto-optical effects are generally weak at optical frequencies [45]. Therefore, it is an important research area to study the topological photonic systems without magneto-optical materials. From the experimental point of view, chiral effect exists in a series of natural materials. The development of metamaterials enables us to synthesize strong chiral media [46,47]. Many studies use the effective medium theory to construct chiral metamaterials supporting WPs [28,34]. At optical frequency, the loss of these chiral structures can not be ignored. However, they can work in terahertz or far-infrared band by adjusting the structure parameters [11]. The chiral systems allow the study of the properties of topological semimetals in a wider frequency domain [47]. Therefore, it is necessary to study other different chiral systems hosting the WPs.
Here, the WPs can exist in the reciprocal chiral metamaterials. The system preserves time-reversal symmetry, while inversion symmetry is broken due to chiral effects. Specifically, a DP splits into pairs of WPs with opposite chirality and the total number of WPs must be a multiple of four. According to the intrinsic electric field of the chiral metamaterials, we give the expressions of the longitudinal and transverse modes forming WPs. It proves the newly generated degenerate points are indeed WPs due to the fact that the band dispersions are linear along all three directions around the crossing point. This is different from other studies of the quadratic WPs [48,49], which exhibit dispersions quadratic in two directions and linear in the third one. Furthermore, compared with reference [34], we introduce Drude's dispersion in the z direction of both permittivity and permeability tensors. It can increase the number of WPs formed. Notably, two longitudinal modes possess flat dispersion relations. Hence, the Weyl degeneracy points are exactly at the critical transition between type-I and type-II WPs. Moreover, as an intuitive manifestation of the nontrivial topological properties of the Weyl systems, there are 'Fermi arc' connected projections of pairs of WPs. COMSOL multiphysics simulations show that the surface waves can bypass sharp defects and realize robust transmission without reflection and scattering. In addition, taking the nonlocal effects into account, the reciprocal chiral metamaterials host either type-I, type-II WPs, even exist TDPs. In particular, when the signs of the values corresponding to the nonlocal parameters are opposite, the type-I and type-II WPs can coexist in the system. Our work provides a platform for the study of topological semimetals in the electromagnetic metamaterials.
Combining × E = iωB and × H = −iωD, and we choose to use the electric field E = (E x , E y , E z ) T to describe the bulk state equation of metamaterials: The relation between corresponding eigen electric field and eigen magnetic field is expressed as where 0 and μ 0 are the permittivity and permeability of vacuum, respectively, vacuum, and k = [k x , k y , k z ] is the propagation wave vector. For the sake of simplicity, we set ω is normalized to ω mp and the wave vectors k are normalized to k 0 . In addition, we assume t = μ t = const. According to equation (2), the evolution of the band structures under different electromagnetic parameters is obtained, as shown in figure 1. We start to analyze the dispersion relation of the electromagnetic duality material with = μ, γ = 0, and ω ep = ω mp in figure 1(a). There are two fourfold degenerate DPs symmetrically distributed on the k z axis. The DPs are not topologically protected due to the net Chern number is zero. The two DPs split into four TDPs at ω ep = ω mp , as illustrated in figure 1(b). Here, the illustration shows the specific band structures of TDPs. The neutral TDPs was formed by the linear crossing between one longitudinal mode and two transverse modes [8,11]. Moreover, figures 2(a) and (b) show their band dispersions in the k x -k y plane. It can be seen intuitively that three different bands intersect at the TDP 1 and TDP 2 .
To further construct a reciprocal metamaterial that hosts WPs, the magnetoelectric coupling tensors are considered (ξ and η), i.e. chiral coupling term γ = 0. In this case, the system preserves the time-reversal symmetry, while inversion symmetry is broken. Thus, DP further splits into pairs of WPs. The Weyl metamaterials we proposed here will have a minimum of four WPs, as shown in figures 1(c) and (d). Due to the presence of Drude's dispersion in the z direction of both permittivity and permeability tensors, two bulk plasmon modes exist in the Weyl metamaterials, i.e. a longitudinal electric mode which is induced by z , and the other is the longitudinal magnetic mode induced by μ z . Therefore, we can easily foresee that the WPs in the proposed system are exactly at the critical transition between the type-I and type-II WPs. The dispersion relations along z axis for several sets of specific parameters that satisfy the above conditions are given in figure 1.
In three-dimensional momentum space, WPs are monopoles of the Berry curvature. The topological property of the WPs can be characterized by the topological invariant known as Chern number [51]:  The expression for the position of linear degenerate WPs in momentum space is: The number of WPs in the reciprocal chiral metamaterials proposed here is related to the relative size of the parameters t , μ t and γ. The minimum number (four) of WPs is γ = √ t μ t , as shown in figure 1(d). However, there are eight WPs in the system when γ = √ t μ t in figure 1(c). Here, the condition γ = √ t μ t is considered to be a transition point of the Weyl metamaterials. Moreover, a degenerate line exists in momentum space as illustrated in figure 1(d). In particular, what caused this degenerate line and why only at γ = √ t μ t , the system can have a minimum of four WPs, we can understand from the following eigen mode analysis.
In three-dimensional momentum space, the dispersion of WPs is linear in all direction. Owing to the chiral metamaterials have rotational invariance in the x-y plane, so the dispersion relation around the WP can be analyzed in the wave vector space k x -k z or k y -k z . The band structures of the negative chirality WP 1 (blue dot) in figure 1(d) is given here, as shown in figures 2(c) and (d). It is clearly to see that the band dispersion near the WP 1 along any momentum direction remains linear. They all possess conical structures around the crossing points (WPs). Then, for the purpose of further understand the evolution of the band structure under different electromagnetic parameters, and how WPs were formed, further eigen mode analysis is needed.

The eigen mode analysis of the degenerate points
As mentioned before, we introduce Drude's dispersion of both permittivity and permeability tensors in the z direction. From z = 1 − ω 2 ep /ω 2 = 0 and μ z = 1 − ω 2 mp /ω 2 = 0, we can get two bulk plasmon modes, i.e. a longitudinal electric mode is represented by ω = ω ep , and a longitudinal magnetic mode is expressed by ω = ω mp . The two longitudinal bulk modes are independent of the wave vector k z . Therefore, the dispersion relation of the two longitudinal modes is flat, represented by an orange dotted line and a black dashed horizontal line, as shown in figure 3. To deduce the specific expression of the transverse modes forming the WPs, we rewrite the equation (2) in matrix form as ⎛ When the determinant of the matrix of equation (4) is equal to zero, the nontrivial solution of E = (E x , E y , E z ) T exists, and the eigen equation is expressed by where k 4 t = k 4 x + k 4 y , m = −γ 2 + t μ t , and n = (k 2 z − mω 2 )( z μ t + t μ z ). Along the z axis, i.e. k x = k y = 0, the transverse modes can be written as Here, we set t = μ t = 1. As depicted in figure 3, the red and green lines represent the dispersion of the transverse modes by equations (6) and (7), respectively. The purple line indicates a degeneracy of two transverse modes. Next, we start with an electromagnetic duality metamaterial model (with = μ and γ = 0) to discuss each topological phase mentioned above. Furthermore, we set the longitudinal electric and magnetic modes have the same dispersion, that is, ω ep = ω mp . Under these circumstances, the metamaterial system protects the existence of DPs, as shown in figure 3(a), the locations of the DPs are (0, 0, ±1). Here, the net Chern number of the fourfold degenerate DP is zero, so it is not topologically protected. Any perturbation that breaks the symmetry of time-reversal or spatial inversion will lift the degeneracy of DP. Then, we consider the case where the dispersion of the longitudinal electric and magnetic modes are different, such as ω ep = 2ω mp , as depicted in figure 3(b). Two DPs decouple and split into two pairs of TDPs, which located at (0, 0, ±1) and (0, 0, ±2), respectively. The number of topological charges carried by the newly obtained TDPs in the system is zero, too. Hence, these TDPs are topologically neutral.
Neutral TDPs can further split into WPs by breaking either time-reversal or spatial inversion symmetry. Here, to construct a metamaterial hosting the WPs, we break the spatial inversion symmetry by introducing chiral effects, i.e. γ = 0. Specifically, after breaking spatial inversion symmetry, a DP splits into pairs of WPs of opposite chirality, and the total number of WPs will be a multiple of four, as indicated in figures 3(c) and (d).
With increasing values of γ, the green (red) branch of the band rotates clockwise (counterclockwise) in the ω-k z plane. At γ = 1, as shown in figure 3(d), a red and a green band dispersion overlap again to form the degenerate line (purple) that we mentioned in figure 1(d). Therefore, only the intersection point formed by the other two transverse modes and the two longitudinal modes are WPs, the number of which is four. At the plasma frequency ω = ω mp , there exists one pair of WPs along the k z axis in momentum space located at (0, 0, ±2), and (0, 0, ±4) for another pair of WPs at ω = ω ep in figure 3(d). Therefore, the number of WPs in the bulk band in figure 1 is consistent with the eigen mode analysis results in this section.

The 'Fermi arc' exists at the interface between the reciprocal chiral metamaterials and vacuum
As a direct physical consequence, 'Fermi arcs' terminate at a WPs' projections on the surface. The existence of nontrivial 'Fermi arc' surface states is an important signature of the topological nature of Weyl semimetals.
Here, we use the minimum number of WPs in the proposed reciprocal chiral metamaterial as a prototype to study the topological connectivity characteristics of 'Fermi arc'. The photonic 'Fermi arc' exists at the interface (y-z plane, i.e. x = 0) between vacuum (x > 0) and the chiral Weyl metamaterials (x < 0). The method proposed by Dyakonov [52] can be used to calculate the 'Fermi arc'. Via scanning in-plane propagation vector components [k y , k z ], the 'Fermi arc' can be obtained [53].
One pair of negative WPs represented by two blue dots (WP 1 ) at the small wave vector k are shown in figure 4. Correspondingly, as the chiral partners of the two WP 1 , the two red points WP 2 at the large wave vector denote the other pair of positive WPs. The vacuum state is shown in figure 4(a) by the green light cone. The brown solid lines represent the bulk states of reciprocal chiral metamaterial. The pairs of WP 1 and WP 2 are at different frequencies. Along the k z axis, it shows the projection of the WP 1 (0, 0, ±2) and WP 2 (0, 0, ±4) on the varying-frequency k y -k z plane, as depicted in figure 4(b). Given the dispersion along the k y direction, k z is fixed at WP 1 and WP 2 , respectively, as shown in figures 4(c) and (d).
Figures 4(e)-(h) show four equi-frequency contours containing both bulk and surface states on the k y -k z plane. The 'Fermi arc' is represented by the black solid curve at (e) ω = 1.00; (f) ω = 1.01; (g) ω = 1.98; and (h) ω = 2.00, as illustrated in figure 4. At the Weyl degeneracy frequency ω = 1 (or ω = 2), as shown in figure 4(e) (or figure 4(h)), it can be clearly seen that there is a 'Fermi arc' between the two WPs, and the middle part of it merges into the vacuum. In addition, slightly shifted the Weyl frequency (figures 4(f) and (g)), the bulk equi-frequency surfaces have changed dramatically. However, the 'Fermi arc' always remains connected and exhibits a very similar configuration as that in Weyl frequency. The phenomenon serves as a characteristic of the topological nature of the Weyl system. The 'Fermi arcs' produced by WPs may give rise to nontrivial topological matter with unprecedented properties.  The corresponding robust propagation of topologically protected 'Fermi arc' surface states can be further confirmed through COMSOL multiphysics simulation. Here, to study the excitation and propagation of the surface waves, three different values of propagation constants k z are adopted, as indicated by points A, B, and C in figure 5(a). The values of the propagation constant k z at points A, B, and C are (b) k z = 0.9, (c) k z = 1.5, and (d) k z = 1.98, at ω = 0.99, respectively. The field distributions at an interface between the chiral Weyl metamaterial and vacuum are shown in figures 5(b)-(e). In addition, the black stars represent electric dipole sources to excite electromagnetic waves.
Full-wave simulations of topologically protected 'Fermi arc' surface states show that the surface waves are not scattered or reflected by the presence of sharp corners. As shown in figure 5(c), for the point B, where k z is in the complete gap region, the surface wave achieves robust, unidirectional propagation. However, because the point A is not located in the common band gap region, the surface waves diffuse into vacuum at defects, as shown in figure 5(b). Moreover, the surface wave can also be scattered to the bulk state through square-shaped defects as the point C is close to the bulk state of metamaterials, as shown in figure 5(d). On the other hand, when the chirality parameter is taken as γ = −1 at k z = 1.5, the direction of propagation is switched to the right, as shown in figure 5(e). In other words, as the sign of the chirality parameter γ is reversed, the propagation of the surface wave is still robust. Therefore, the surface waves are confirmed to be topologically protected by means of numerical simulations.

Type-I, type-II Weyl points and triple degenerate points with the nonlocal effect
As discussed previously, along the z direction, the dispersion of the two longitudinal modes is flat. In this section, we prove that nonlocal parameter β can determines the type of a WP. As an illustration, we assume the nonlocal z and μ z that only depends on k z [34] The dispersion and WPs considering the nonlocal effect in reciprocal chiral metamaterials are shown in figure 6. Only the bulk state of the positive k z half-space is given here. When β > 0(β < 0), the dispersion of the longitudinal mode slopes downward (upward), generating a type-I (type-II) WPs. The other parameters are ω ep = 2, ω mp = 1, t = 1, μ t = 1, and γ = 1. Here, when β 1 = 0.01(β 2 = 0.02)/β 1 = −0.01(= β 2 ), the Weyl degeneracies becomes a type-I/type-II WP, as shown in figure 6. However, the charge of the WP will not vary with the change of the nonlocal parameter. The high/low-frequency WP always hosts positive/negative chirality.
Different types of WPs can be distinguished by group velocity signs. As indicated in figures 6(a) and (c), the group velocities around the high-frequency WP along the z direction have the opposite sign, which represents the properties of the type-I WP. On the other hand, the group velocities around the low-frequency WP along the z direction have the same sign, which is an unique feature of the type-II WP, as illustrated in figures 6(b) and (c). The two types of Weyl semimetals have significantly different physical properties in terms of magneto-transport [54], thermodynamics [55] and topological superconductivity [56], etc.
As a new type of point nodes, triply degenerate nodal points possess linear dispersions. TDPs are crossing points of three bands. Bulk band and TDPs considering the nonlocal effect in chiral metamaterials are shown in figure 7. The dispersion of the longitudinal electric mode slopes downward while β 1 = 0.75. It is represented by the orange dotted line, as shown in figure 7(b). The other relevant electromagnetic parameters are ω ep = 2, ω mp = 1, t = 1, μ t = 1 and γ = 1. On the other hand, when β 2 = 0, the longitudinal magnetic mode remains flat. It is represented by the black horizontal dashed line. As depicted in figure 7(b), the red and green lines represent the dispersion of two transverse modes. The degeneracy of two longitudinal modes and one transverse mode in momentum space produces TDPs. The positions of TDPs are (0, 0, ±2). Figures 7(c) and (d) show the band structures in the k y -k z and k x -k y plane at k x = 0 and k z = 2, respectively.

Conclusion
In conclusion, we show how an electromagnetic duality medium evolves in the transition toward a reciprocal chiral metamaterial by introducing chiral effects to break spatial inversion symmetry. We theoretically prove that a DP splits into pairs of WPs with opposite chirality. The linear crossing between one longitudinal mode and one transverse mode form photonic Weyl degeneracies. Through the analysis of intrinsic electric field of the chiral metamaterials, it is found that the dispersion relations of two longitudinal modes are flat, which are caused by the Drude's dispersion. Therefore, the WPs are exactly at the critical transition between the type-I and type-II WPs. As a topologically characteristic feature of Weyl semimetals, the photonic 'Fermi arc' connects projections of pairs of WPs. The surface waves are confirmed to be topologically protected through numerical simulations. Moreover, when the nonlocal effect is included, the Weyl degeneracies here transform into either type-I, type-II WPs or TDPs. Notably, two bands of the TDPs have linear dispersion along an arbitrary momentum direction, like that in a Weyl cone. Our work sheds new light on topological semimetals, paving the way for exploring interesting topological phases.