Fermi arc surface state and topological switch in the gyromagnetic metamaterials

A landmark feature of the Weyl system is that it possesses the Fermi arc surface states. In this work, we demonstrate that the Fermi arc surface states connect the vacuum state and the Weyl points of gyromagnetic metamaterials (GMs). The nonzero Chern numbers and Berry phases show the nontrivial topological property of the GMs in momentum space. Full-wave simulations demonstrate that the chiral surface waves on the boundary between the GMs and vacuum state can achieve robustness against sharp corners of step-type configurations. Remarkably, the topological switch can be realized by adopting the Fermi arc surface states between two different GMs. We theoretically prove that the physical mechanism of realizing topological switch is caused by different gap Chern numbers of the material system. Moreover, the direction of the topological switch can be operated by manipulating the gyromagnetic parameters of the GMs in the ‘button’ region. Our work may provide more flexibility for the flexible and robust topological devices.

In physics, each Weyl point of Weyl semimetals carries a quantized topological charge and acts as the Berry curvature's source (or drain) [21,22]. The Weyl points carrying opposite topological charges (chiralities) are always born or annihilated in pairs. Hence, the Weyl points are extremely stable [15]. The typical characteristic of the Weyl semimetals is that the nonclosed Fermi arc surface states exist between Weyl point projection at different positions [23][24][25]. The Fermi arc surface states are topologically protected, and their lengths can be used to measure the topological strength of the system [26,27]. The Fermi arc surface states can bring new capabilities to manipulate wave propagation in classical systems, including electromagnetic, acoustic, and elastic waves [28][29][30][31][32].
Recently, topological switches based on the topological boundary modes in photonic crystals and phononic crystals have been investigated [3,[33][34][35][36]. Based on the topological switch effect, some interesting applications can be realized in the topological system, such as dynamically controllable topological optical waveguide [3], dynamically turned on/off continuous-wave laser [34], optical communications and quantum information processing [35]. Generally, photonic and phononic crystals have complex structures. By contrast, the electromagnetic continuum media can provide topological protection in its fundamental modes by breaking certain symmetry conditions, temporally or spatially [20,28]. For instance, the gyromagnetic metamaterials (GMs) are kinds of electromagnetic continuum media. The time-reversal symmetry of homogeneous GMs is broken under an external magnetic field. In the experiment, the commonly used medium is the yttrium iron garnet to realize the GMs [37][38][39]. The GMs have been extensively studied in the fields of topological transitions [37], non-Hermitian triply degenerate points [38], and optical gyromagnetic properties [39]. Recently, Liu et al experimentally confirmed that Weyl semimetals can be realized in the gyromagnetic system and demonstrated the topological Chern vectors in the three-dimensional space [40]. Hence, an important question arises: can the topological switch realize based on the Fermi arc surface states in the GMs? If so, what interesting phenomena will it bring? These questions need further study.
In this work, we demonstrate the existence of Fermi arc surface states in the GMs. The nontrivial Fermi arc surface states arise from the photonic Weyl points of the GMs. We establish the theoretical formulas of the transverse and longitudinal modes generating the Weyl points. The nonzero Chern numbers and Berry phases in wave vector space demonstrate the nontrivial topological property of the GMs. The numerically simulated results show that the chiral surface waves propagate robustly against the step transition of the structures. Based on the Fermi arc surface states, we propose the topological switch between two GMs with opposite gyromagnetic parameters. We reveal that the physical mechanism of realizing topological switch is caused by different gap Chern numbers of the material system. Remarkably, the topological switch can be realized direction reconfigurable by manipulating the gyromagnetic parameters of the GMs in the 'button' region.

Band structures and Weyl points of the GMs
The relative permittivity and permeability tensors of the GMs [39] are given by where ϵ z = 1 − ω 2 p /ω 2 possesses Drude's dispersion with ω p being the plasma frequency. The GMs described by equation (1) can be realized by utilizing the periodically layered structure [41]. Generally, the multi-layered structure can be treated as an effective homogeneous medium under the long-wavelength approximation. In particular, the electromagnetic parameters of this medium can be obtained from the effective medium theory. Typical examples could be multi-layered metamaterials with effective negative parameters [41], hyperbolic dispersions [7], and chirality [23]. The benefit of the layered structure is that its macroscopic responses can be conveniently tuned by changing the property of each layer. Therefore, the electromagnetic constitutive parameters requested can be realizable.
Following [20,30], the wave solution in the interior of the GMs can be obtained by solving the extended eigenvalue equation, i.e.Ĥ where µ 0 and ϵ 0 are the permeability and permittivity in the vacuum. H, E, and J are the magnetic field, electric field, and current density vectors, respectively. In equation (2), the two 9×9 parameter tensors are given byĤ where κ is the skew-symmetric tensor of normalized k, ϵ 1 = diag(ϵ t , ϵ t , 1), χ = diag(0, 0, ω p ), and c is the speed of light in the vacuum. For simplicity, ω is normalized by ω p and k is normalized by k p (k p = ω p /c). By simplifying the equations, the master equation determining the eigenstates of the GMs is found to be Considering the specific forms of the relative permittivity and permeability tensors, equation (4) can be expressed by the matrix form, which in detail reads  Here, respectively. Now we are going to find out the Weyl degeneracy in the GMs. The typical band structures of the GMs (ϵ t = 2, µ t = 1, g = 0.8, and ω p = √ 2) are shown in figure 1. It can be intuitively seen that there are four Weyl points along the k z axis, as illustrated in figure 1(a). The locations of two pairs of Weyl points in momentum space are (k x , k y , k z ) = (0, 0, ±2.68) and (0, 0, ±0.89). As a twofold linearly degenerate point, the band dispersions around the Weyl point have a linear dispersion relation with the wave vectors along each direction. Analytically, along the z (k x = k y = 0) axis, the dispersions of longitudinal and transverse modes of the GMs can be expressed as The location of Weyl points in the GMs can be addressed through combining equations (6) and (7), which are The black curves in figure 1(a) represent the equifrequency contour with ω = ω p / √ 2 = 1, and the details are illustrated in figure 2 of section 3. Figures 1(b) and (c) show the band structure containing a single Weyl point in the k x − k z space (k y = 0) and k x − k y space (k z = k w = 2.68), respectively. It is evident that the Weyl point is a twofold linearly degenerate point. For better illustration, the dispersion curves along k z (k x = 0, k y = 0) and k y (k x = 0, k z = k w ) are exhibited in figures 1(d) and (e), which validate our analysis. The bold purple (brown) lines represent the dispersion of these longitudinal (transverse) modes, as illustrated in figure 1(d). In particular, the dispersion of longitudinal mode forming Weyl points is flat (ω = ω p ). It is quite interesting that the Weyl points shown here (figure 1(a)) are much different from the canonical type-I and type-II kinds, which adopt the intermediate form and behave like the critical state between the two typical Weyl degeneracies [20].

Topological invariants and Fermi arc surface states
Conceptually, the nontrivial property of the GMs can be interpreted through the well-known topological invariants (Chern number and Berry phase). In particular, the Chern number C is an integer number defined as the surface integral of the Berry curvature [1] where U(k) = [E, H] T represents the eigenstates of the GMs. Again from equation (4), the dispersion curvature of the GMs can be obtained. Here,the GMs' dispersion with ω = ω p / √ 2 = 1 is shown in figure 2(a), i.e. the orange and purple-colored surfaces. There are two band gap regions of the three-dimensional equifrequency surfaces in wave vector space (figure 2(a)). It is the prerequisite for forming Fermi arc surface states in the GMs. Moreover, the positive k z and negative k z band gap regions have the same gap Chern number because the Chern number of the middle purple equifrequency surface is zero (figure 2(a)). On the other hand, we calculate the Berry curvature at every point on the two-dimensional equifrequency surface (k y , k z ) of the GMs, as shown in figure 2(b). The blue and purple arrows represent the inward and outward Berry curvatures, respectively. The length of the blue (purple) arrows denotes the amplitudes of the Berry curvatures (figure 2(b)). The Berry phases of ±2π and 0 can be obtained by integrating the Berry curvatures (figure 2(b)), which corresponds to the Chern number of ±1 and 0 (figure 2(a)), respectively. Next, we will study the Fermi arc surface states supported by the boundary between the GMs and the vacuum.
The green cone illustrates the vacuum state in figure 3(a). On the other hand, the orange/purple/cyan surfaces represent the bulk states of the GMs. Notably, when the angular frequency ω is less than ω p , there are band gaps between the vacuum state and the GMs, as illustrated in figure 3(a). Now, we investigate the Fermi arc surface states supported by the boundary between the vacuum state and the GMs (band gap regions in figure 3(a)). The half-spaces x < 0 and x > 0 are occupied by the GMs and vacuum state respectively, as shown in figure 3(b). According to Maxwell's equations, the eigenmodes on each side of the boundary (x = 0) can be given by nontrivial solutions of the E and H.
In the vacuum state, the two independent eigenmodes can be described as where k x1 = k 2 y + k 2 z − ω 2 is the attenuation constant inside the vacuum. On the other hand, two independent eigenmodes of the GMs can be given by The Fermi arc surface states at the boundary (x = 0) can be formulated according to Maxwell boundary conditions, i.e. the tangential magnetic field and electric field components are continuous at the boundary where A 1 , A 2 , B 1 , and B 2 represent the field amplitudes on each side of the boundary, respectively. Then, the existence of nontrivial solution of equations (14) and (15) request that the determinant of the 4 × 4 matrix M (equations (14) and (15)) should vanish, i.e.
Equation (16) is the characteristic equation of the Fermi arc surface states of the GMs. Based on equation (16), we show the Fermi arc surface states connecting the Weyl point and vacuum state, as depicted by the magenta curve in figure 3(b). Moreover, the Fermi arc surface states are valid in the frequency region with ω < ω p . It actually falls into the band gap region in figure 3(a). In the following section, we will study the robustness against defects of the Fermi arc surface states. In contrast to electronic materials, the vacuum does not behave as an insulator of photons. Thus, there are photonic bands distribution for the vacuum states [38]. However, the propagation is ceased for the waves possessing a wavenumber exceeding k 0 . This phenomenon occurs in momentum space below the light cone, even though the vacuum state is topologically trivial. Consequently, it is still imagined that some particular electromagnetism states can be excited between the GMs and the vacuum. Based on equation (16), the Fermi arc surface states between the GMs and vacuum state in wave vector space can be obtained, as shown by the black and purple lines in figures 4(a) and (d), respectively. It can be seen that there is only one propagation mode in the common band gap, i.e. the blue regions in figures 4(a) and (d). The topologically protected surface states are guaranteed by the transition of the Chern number between these two materials in the common band gap regions, as illustrated by the blue regions in figures 4(a) and (d). Moreover, the common band gaps and Fermi arc surface states in figures 4(a) and (d) have opposite gap Chern numbers and group velocities when the applied magnetic field of the GMs is flipped, respectively. In a photonic system, the skin depth of the Fermi arc surface states is the key parameter to describe the confinement of the electromagnetic wave. On the vacuum side, the skin depth can be evaluated by , which is plotted in figures 4(b) and (e). In the k z ∈ [1.0, 1.9] region, we can find that the skin depth decreases with the increased k z . It means that the Fermi arc surface states in the common band gap (figures 4(a) and (d)) possess better field localization characteristics with enhanced k z . In figures 4(c) and (f), we show the mode profiles |E| (points A and B) on the Fermi arc surface states. In these cases, the electric fields are localized at the interface between the GMs and the vacuum, and no radiation mode is excited. Therefore, these Fermi arc surface states possess field localization characteristics. Moreover, for the GMs and vacuum state, the asymmetry of the electric field intensity is because the skin depths of the Fermi arc surface states in the two media are different, as shown in figures 4(c) and (f).

Chiral surface wave and topological switch
At the vacuum-GMs boundary, the ellipticities of the Fermi arc surface states can be calculated by the boundary condition [9]. The electric field can be calculated by the linear superposition of two eigenmodes expressed in equations (10)-(13) where C 4 , C 5 , C 6 , and C 7 are constants. The ellipticities of the Fermi arc surface states lie in the plane perpendicular to the boundary because the energy flow is along the boundary. Then, the ellipticities are calculated by the vertical and tangential components of the electric field E at a point very close to the interface [7]. Numerically, the electric field components in equation (17) can be decomposed into the right circular polarization (C RCP ) and left circular polarization (C LCP ) states, which are where E vac(gyr) t represents the complex amplitude of the tangential electric field (0, E vac(gyr) y , E vac(gyr) z ) T . Then, the ellipticities of the Fermi arc surface states can be evaluated by Based on equation (20), we get the ellipticities of the Fermi arc surface states on the vacuum side, as shown in figures 5(a) and (d). It is clear that the Fermi arc surface states with elliptical polarizations exist on the boundary (GMs and vacuum state) because the electromagnetic duality symmetry (ϵ ̸ = µ) is broken in the medium system [7]. The ellipticities of the two Fermi arc surface states are distinguished by the LEP and REP, as a consequence of the flipped gyromagnetic parameters. It demonstrates the chirality of the Fermi arc surface states. In figures 5(b) and (e), we show the time snapshots of the electric field |E| (the COMSOL Multiphysics software obtains this). In the simulation, the dipoles are used as the sources located on the boundary between the GMs and the vacuum. In figures 5(b) and (e), the surface wave can propagate without any reflection against the sharp corners of step-type configurations (S 2 /S 1 = S 4 /S 3 = 1), as promised by the topological protection in the nontrivial system. The propagation direction of the chiral surface waves is reversed by flipping the gyromagnetic parameters, which demonstrates the nontrivial feature of the surface waves, as illustrated in figures 5(c) and (f). Now we will study the topological protection in the GMs-GMs edge system. According to the bulk-edge correspondence, the number of Fermi arc surface states in the common band gap region is equal to the gap Chern number. Therefore, there are different numbers of Fermi arc surface states in the common band gap regions in figures 6(a) and (d). Here, based on the Fermi arc surface states shown in figures 6(a) and (d), we show the topological switch by using the Fermi arc surface states on the GMs-GMs interface, as illustrated in figures 6(b) and (e). The selective excitation of the surface states is enabled by changing the gyromagnetic parameters in the 'button' region (as the khaki area, i.e. Region II). Remarkably, the physical mechanism of realizing the topological switch in figure 6 is caused by different gap Chern numbers (figures 6(a) and (d)) of the material system. The topological switch can be achieved between the regions with opposite gyromagnetic parameters. In particular, the power flow of the chiral surface waves is not influenced (S 6 /S 5 = S 8 /S 7 = 1) after experiencing the dual-channel junction configuration in the topological switch. Considering that the 'button' region can be freely defined with the homogeneous GMs material, this topological switch may be flexibly implemented in various scenarios.

Discussion
In general, the nature of the photonic systems is accompanied by dissipation (loss). Particularly, in the presence of non-Hermitian perturbation (loss), a zero-dimension Weyl point can transform into a one-dimension closed Weyl exceptional contour with the same topological charge [43,44]. Moreover, the size of the one-dimension Weyl exceptional contours is closely related to the magnitude of the non-Hermitian perturbation (loss) [30].
To quantitatively analyze the effect of loss on the band structures and topological characters of Weyl semimetal, we replace Drude's dispersion (ϵ z in equation (1)) in the GMs with the general function form: where γ represents the damping term (loss) [45]. The topological phase diagram and topological phase transition of the dissipation GMs are illustrated in the electromagnetic parameter space (g, γ), as shown in figure 7(a). In particular, the topological phase transition from the Weyl semimetal phase to the trivial phase of the GMs can be achieved by increasing the loss term γ. Moreover, by introducing the non-Hermitian perturbation (loss) effect, the eigenfrequency (self-energy) [46] of the GMs system changes from a pure real (figure 1) form to a complex form. At the level of physical phenomena, the zero-dimension Weyl point ( figure 7(b)) can transform into the one-dimension closed Weyl exceptional contour (figures 7(c) and (d)). By further adjusting the loss term γ and gyromagnetic parameter g of the GMs, the positions of one-dimension Weyl exceptional contours in momentum space move along the k z axis. Eventually, the Weyl exceptional contours with different topological charges will mix and annihilate each other, which leads to a zero net topological charge (corresponding to the trivial phase in figure 7(a)  shown in figure 7(d). However, for the medium Drude models, such a level of loss is unrealistic (γ = 1.91ω p in figure 7(d)) [30]. Therefore, when analyzing the Weyl semimetal phase (figure 7(a)), the influence of loss in the GMs can be ignored (equation (1)).

Conclusions
In conclusion, we study the Fermi arc surface state and topological switch in the GMs. The band structures of the GMs are obtained by the extended eigenvalue equation. The three-dimensional Weyl degeneracy is studied by employing the Drude model of the GMs. The physical mechanism of Weyl degeneracy is proved by analyzing the transverse and longitudinal modes. In addition, the nontrivial property of the GMs is demonstrated by the nonzero Chern numbers (Berry phases). Full-wave simulations verify that the topologically protected surface waves can bypass the step-type sharp corners without any reflection. Remarkably, the topological switch is suggested by adopting the surface waves between two GMs with opposite gyromagnetic parameters. It is guaranteed by the bulk-edge correspondence of the topological GMs. Different from the conventional optical switches, the topological switches based on the Fermi arc surface states in the GMs have robustness properties to structural defects. Moreover, the direction of the topological switch can be operated by manipulating the gyromagnetic parameters of the GMs in the 'button' region. We believe our research may provide new insights into the topological physics in the homogenous medium, which may be helpful in realizing the topologically nontrivial devices in practice without any periodicity restrictions.

Computing the Fermi arc surface states
In this work, we use the idea of the Dyakonov wave [47] on calculating the Fermi arc surface states at the boundary between the vacuum state and GMs ( figure 4(a)). The procedure is as follows: first, under specific angular frequency ω, plasma frequency ω p , and electromagnetic parameters (ϵ t , ϵ z , µ t , g), according to Maxwell's equations, the eigenmodes of each point (k y , k z ) in the vacuum state and GMs (equations (10)-(13)) can be given by nontrivial solutions of the electric field and magnetic field, respectively. Then, according to the tangential magnetic field and electric field components are continuous at the boundary, we arrange electric field and magnetic field tangential components at each point (k y , k z ) into 4 × 4 matrix M (equation (16)). Finally, in two-dimensional k y − k z momentum space band gap region (the blue region in figure 4(a)), for a specific k z value, we numerically search for a k y value that satisfies equation (16) (Det[M]) is equal to zero.

Numerical simulation
We use the finite-element method solver of COMSOL Multiphysics (2D wave optics frequency domain (ewfd) module) software to perform the topological surface waves simulations (figures 5(b) and (e)). Under the specific out-of-plane wave vector k z , the surface waves (in the x − y plane) at the boundary between the vacuum state and GMs are excited by an electric dipole source (black pentagram) located at the boundary.
Moreover, the boundary conditions of the vacuum state and GMs are set as scattering boundary conditions in the numerical simulation.

Data availability statement
The data generated and/or analysed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request. The data that support the findings of this study are available upon reasonable request from the authors.