Linked Weyl surfaces and Weyl arcs in photonic metamaterials

Topology in higher dimensions In condensed-matter systems, the band structure of a material has often been equated with functionality. However, consideration of the topology of the band structure now provides a route to developing a functionality that goes far beyond the expected properties of the materials. Using electromagnetic metamaterials as building blocks, Ma et al. realized a five-dimensional generalization of a topological Weyl semimetal. Along with the three real momentum dimensions, these included two bi-anisotropy material parameters as synthetic dimensions to demonstrate both linked Weyl surfaces and Yang monopoles. The metamaterial platform provides a powerful route to explore the exotic physics associated with higher-order topological phenomena. Science, abi7803, this issue p. 572 A five-dimensional generalization of a topological Weyl semimetal is realized with electromagnetic metamaterials. Generalization of the concept of band topology from lower-dimensional to higher-dimensional (n > 3) physical systems is expected to introduce new bulk and boundary topological effects. However, theoretically predicted topological singularities in five-dimensional systems—Weyl surfaces and Yang monopoles—have either not been demonstrated in realistic physical systems or are limited to purely synthetic dimensions. We constructed a system possessing Yang monopoles and Weyl surfaces based on metamaterials with engineered electromagnetic properties, leading to the observation of several intriguing bulk and surface phenomena, such as linking of Weyl surfaces and surface Weyl arcs, via selected three-dimensional subspaces. The demonstrated photonic Weyl surfaces and Weyl arcs leverage the concept of higher-dimension topology to control the propagation of electromagnetic waves in artificially engineered photonic media.

G apless topological phases with various band crossings in crystals play an important role in topological physics, as they host a suite of fascinating bulk and surface transport phenomena that include one-way propagation of energy and novel relativistic behaviors (1). As one of the most studied two-dimensional (2D) systems, graphene hosts Dirac points in the momentum space, which support massless quasiparticles and lie at the transition between different topological insulating phases (2,3). The same band dispersion, when extended into three dimensions, generalizes into Weyl points, which serve as the monopoles of Berry curvature-the momentum-space counterpart of magnetic field (4)(5)(6)(7)(8). Besides Weyl points, there exist other gapless topological phases in 3D systems, such as nodal lines and Dirac points. All the aforementioned 3D gapless phases greatly enrich the observed topology-related phenomena, such as Fermi arc surface states, drumhead surface states, chiral zero modes, and quantum oscillations (9)(10)(11). These topological states have potential applications ranging from spin electronic devices (12) to quantum information technology (13).
Recently, the topological properties of higherdimensional systems (n > 3) have drawn con-siderable attention, including the 4D quantum Hall effect (14)(15)(16)(17)(18)(19), Yang monopoles (20)(21)(22), and 2D Weyl surfaces in 5D systems (23)(24)(25)(26)(27). Such systems are expected to possess properties that their lower-dimensional counterparts do not support, that is, those associated with a nonzero second Chern number (C 2 ). It has been theoretically verified (23)(24)(25) that Weyl surfaces-the higher-dimensional extension of the traditional 0D Weyl point that is characterized by a U(1) second Chern number C 2form a Hopf link with a topological linking number equal to C 2 . However, no realizable systems have been proposed yet. Here, we experimentally demonstrate metamaterials exhibiting linked 2D Weyl surfaces and nontrivial Weyl arcs, as a result of the nontrivial C 2 . In our systems, in addition to the 3D momentum space, we consider two more synthetic dimensions, two bianisotropic terms, as the synthetic fourth and fifth momentum vector components. We investigate the key signatures such as linking between Weyl surfaces in selected 3D subspaces through judicious design of the metamaterials. Furthermore, a 1D Weyl arc on a 3D Fermi hypersurface at the 4D boundary of a 5D system possessing Yang monopoles or Weyl surfaces is highlighted to verify the distinctive phenomenon protected by C 2 in the higher-dimensional system.
We start from a 3D photonic system based on a uniaxial metamaterial with both permittivity and permeability along the axis given by e z ¼ m z ¼ 1 À w 2 p =w 2 . The system possesses two Dirac points located at K T k D ; w p in the momentum space (28,29), where k D = w p . By introducing a purely antisymmetric bianisotropic matrix, g zx = − g xz , g zy = − g yz , the Dirac point can be promoted into a 5D Yang monopole (20) with the following effective Hamiltonian [see section I in the supplementary text section of the supplementary materials (30)] determined by the properties of the metamaterial, and G → ¼ Às 0 t 1 ; s 3 t 2 ; s 0 t 3 ; ½ s 2 t 2 ; s 1 t 2 , with s i and t i being the Pauli matrices. We can consider the 3D Dirac point as the projection of a 5D Yang monopole onto the 3D momentum space. The Hamiltonian H Y satisfies TP symmetry (T is time-reversal symmetry, and P is space-inversion symmetry) with T = is 2 t 0 K (K is the complex conjugation) and (TP) 2 = −1 and has a globally doubly degenerate band structure as shown in Fig. 1A. By defining a U(2) Berry connection, one can calculate the Yang monopole's non-Abelian second Chern number The Weyl surfaces can be obtained by introducing a P breaking term, which deforms the Yang monopole, as shown in Fig. 1, B and C. By introducing a single perturbation term a·G mn to the original Hamiltonian H Y , one obtains Here, k 3 plays a special role owing to the presence of term w k0 = v t k 3 . The four eigenfrequencies are are the five unsorted momentum vector components. The degeneracy points of these dispersion spectra form two 2D manifolds: an S 2 ellipsoid M 1 : . These degeneracy points can be seen clearly in the dispersion along k 3 for selected k m (k n ) and k i (k j ) momentum vector components, as shown in Fig. 1B. The degeneracy M 1 points are lifted with nonzero k m (k n ), and the degeneracy M 2 points are lifted with nonzero k i (k j ), meaning thatM 1 and M 2 are both 2D manifolds in the 5D space.
The perturbation term G mn for different combinations of m and n in Eq. 2 can be realized by introducing specific types of symmetry breaking terms into the Yang monopole system [see section I in (30)]. It should be noted that different choices of [m,n] only rotate the Weyl surfaces' orientation in the 5D space but do not affect the system's overall topological properties. Indeed, the Weyl surfaces described by Eq. 2 always consist of three twofold degeneracy nodal surfaces, and each point on the nodal surfaces serves as a Weyl point in the corresponding three-codimension subspace orthogonal to the nodal surface [see section II in (30)]. This selective rotation can help us study the key characteristics of the linked Weyl surfaces system in the momentum space by avoiding the difficulty of investigating the system in the synthetic dimensions k 4 and k 5 , which are fixed for a given metamaterial design. As shown in Fig. 1, D to F, in the 3D momentum space, for the choice of perturbation along G 45 , the Weyl surfaces appear as two overlapped Weyl points wrapped by a degeneracy S 2 ellipsoid (Fig. 1D); for the choice of the perturbation along G 15 , the Weyl surfaces appear as linked nodal lines (Fig. 1E); and for the perturbation along G 12 , the Weyl surfaces take the form of two Weyl points located on opposite sides of a degeneracy T 2 plane (Fig. 1F). Notably, the appearance of the Weyl surfaces in the 3D subspace formed by k 4 , k 2 , and k 3 in a system with DH = a· G 45 is the same as that of a system with DH = a· G 15 in the 3D real momentum space. This correspondence indicates that the above configurations (Fig. 1, D to F) in the 3D momentum space with different choices of [m,n] simply correspond to the different intersections of the same Weyl surfaces in the 5D space. Note that the Abelian second Chern numberC A 2 of these 2D manifold Weyl surfaces is the same as the non-Abelian one in the original Yang monopole system: C A 2 j WS ¼ C NA 2 j YM ¼ T1 . This global topological invariant describes the linking number of these two Weyl surfaces (24,25), which corresponds to the nodal link shown in Fig. 1E with two dimensions hidden [see section III in (30)].
The designs of metamaterials possessing Weyl surfaces corresponding to different perturbation terms G mn are provided in section IV in (30). Here we experimentally realize the sample in fig. S2E, which corresponds to DH ≈ a· G 15 , because it reveals the key topological feature of linking between the two Weyl surfaces. The projected bulk states (PBSs) of this system are probed using the transmitted nearfield scanning configuration described in section VII in (30). The PBS exhibits very distinct features in the three different frequency regions (FR) divided by the two longitudinal modes (the two horizontal lines in the left panel of Fig. 1B), located at about 7.77 and 8.06 GHz from the simulation data; regions I, II, and III correspond to frequency ranges above, between, and below the two longitudinal modes, respec-tively (marked with different colors in Fig. 1B). At the lowest frequency range, FR-III, the outline of the PBS appears as two intersecting hyperbolic curves, as clearly shown by the effective medium theory (EMT) and full-wave simulation ( Fig. 2A). With the increase of frequency into region FR-II, the PBS transforms into a hyperbolic curve and an elliptic curve intersecting each other, with an empty density of states inside the overlapped eye-shaped region (Fig. 2B). With further increase of the frequency into region FR-I, the outline of the PBS turns into two intersecting elliptical curves (Fig. 2C). The symmetry operators G 1 G 5 and G 2 G 4 guarantee that the intersections occur on the plane k x = 0 in FR-II and k y = 0 in FR-I and FR-III, respectively. By identifying the intersection points in both simulation and EMT results, the locations of the 5D Weyl surfaces in the momentum subspace are obtained and plotted in Fig. 2D. In both the simulation and the EMT, the two lines projected by M 2 thread through the red circles projected by M 1 . This nonzero linking number between the projected M 1 and M 2 Weyl surfaces reveals the nontrivial C 2 of the Weyl surfaces in the 5D space. In FR-III and FR-II,  pronounced surface states are also observed in both EMT and experimental results, which correspond to the drumhead surface states typically present in nodal line systems. It is worth mentioning that, from a 5D perspective, the drumhead surface states are actually a special cross section of the Fermi hypersurface. Note that, owing to the limited spatial resolution in the momentum space in measurement (arising from the finite size of the sample), the intersecting points cannot be clearly identified in the measured PBS. Nonetheless, the good match in the outlines of the PBS and the drumhead surface states between the measurement and the simulation serves as strong evidence of the presence of linked nodal surfaces in the fabricated sample.
Besides the linking of the Weyl surfaces, the nontrivial C 2 in a 5D system is also manifested by the presence of 3D Fermi hypersurfaces and 1D Weyl arcs at the 4D boundaries of 5D nontrivial systems, which has been theoretically investigated in detail (25). Our effective medium model shows the presence of Fermi hypersurfaces and surface Weyl points (crossing points on the hypersurface) in both Yang monopole and Weyl surface systems, as shown in Fig. 3, A and B, where the Fermi hypersurface is formed by the evolution of the Fermi arcs with the synthetic dimensions, and the surface Weyl points correspond to the crossing points on the 3D Fermi hypersurface. The surface Weyl points at continuously varying frequencies form a 1D Weyl arc, as shown in Fig. 3, C and D. In comparison to the Yang monopole system, the medium with linked Weyl surfaces has asymmetric Weyl arcs with respect to the k y = 0 plane. For the Yang monopole case, by enforcing additional symmetry operators G 2 G 4 and G 2 G 5 and the impedance matching condition with the surrounding medium, the Weyl arc node can be restricted to the subspace with k 2 = k 4 = k 5 = 0, as shown in the inset of Fig. 3D. The Weyl arcs described above for both the Yang monopole and Weyl surfaces represent a distinctive phenomenon arising from the nontrivial C 2 .
For experimental observation of the Weyl arcs, we consider a metamaterial with Yang monopoles, whose configuration is shown in fig. S2A. The measured Fermi arc states shown in Fig. 3, E and F, are in good agreement with both EMT and full-wave simulation results using the commercial software CST Studio Suite. However, at the points (blue circles) where the surface Weyl nodes are expected, a small gap is observed (Fig. 3F). This gap closes at the Yang monopole near 7.95 GHz, gradually increases away from this frequency, and eventually disappears into the light cone. The origin of this gap is due to the impedance mismatch with the surrounding mediumair. However, the Weyl arcs, protected by the nonzero C 2 , are guaranteed to exist in the 4D synthetic space of the boundary (k y , k z , k 4 , and k 5 in our configuration) despite this impedance mismatch [see sections V and VI in (30) for numerical verification].
We have established an effective mediumbased frame to realize the linked Weyl surfaces and intriguing surface states arising from the nontrivial C 2 of the systems. By using bianisotropic terms as the synthetic fourth and fifth dimensions, and by introducing specific Pbreaking terms to deform the Yang monopole, the medium offers the possibility to systematically study the linked 5D Weyl surface system. Our work provides a singular platform to explore the topological properties of complicated electromagnetic systems in higher dimensions.