In condensed systems, Weyl quasiparticles have a massless nature and exhibit various relativistic chiral phenomena such as Klein tunneling, chiral anomaly, and Fermi arc surface states. However, in photonic systems, Weyl points (WPs) are protected by the D2d symmetry, often leading to multiple chiral WPs at the same energy level, which makes generating chirality-related effects challenging. To overcome this hinderance, a perturbation that breaks mirror symmetry in the metallic saddle structure was introduced. This perturbation effectively separates the energies of distinct chiral WPs, enabling the experimental measurement of the spectral intensity for each Weyl band and the assessment of chirality imbalance among the WPs. By maintaining time-reversal symmetry, the present study offers an approach for investigating the imbalance in the chirality of pseudo-fermionic fields in photonic materials.
Topological materials have garnered significant attention in the field of condensed matter systems in recent years. Particularly, topological insulators and topological Weyl semimetals have emerged as highly intriguing topics, which makes traditional definitions and interpretations of metals and insulators challenging. One crucial discovery is the dispersion relation observed in the three-dimensional energy band structure of certain materials. This dispersion relation arises from the linear crossing of two energy bands and satisfies the relativistic Weyl equation near the crossing point.1 Such a band degenerate point is called a Weyl point (WP), and its dispersion relation exhibits the quasiparticle excitation characteristics of chiral Weyl fermions.2,3 Additionally, Dirac semimetals4,5 and nodal chain semimetals6–8 exist according to the distribution characteristics of the cross-point. In the surface energy band structure of the Weyl material, the electron dispersion relation is expressed as a segment of open arcs connecting the projection positions of different chiral WPs, also known as Fermi arc surface states.9,10 Any point on the arc is topologically nontrivial and protected topologically, and they have stable and low-loss transport properties such as negative magnetoresistance, giant magnetoresistance effects, chiral anomalies, and very high carrier mobility, which are of great interest for future low-energy lossless devices or photonic devices.11–13
Analogous to the topological energy band structures in electronic systems,14,15 the degeneracy features are extended to photonic16–19 and phononic systems,20,21 opening up research directions in topological photonics and topological acoustics. In 2018, the University of Birmingham introduced an electromagnetic metamaterial, namely, a metallic saddle-atom structure,22,23 comprising a pair of resonant rings with opposite opening directions. Within the energy bandgap, this structure features four WPs of the same frequency, referred to as ideal WPs. Each WP carries an integer “charge,” known as the Chern number, corresponding to its chirality. Similar to magnetic monopoles, WPs exist only in pairs with opposite charges. Isolated ideal WPs exhibit absolute topological stability, meaning that any perturbation will solely affect the position of the cross-point in momentum-energy space. WP degeneracy can be eliminated only by bringing different chiral WPs together at the same position, leading to their cancelation.
This Letter differs from previous research, as it focuses on examining the effect of mirror symmetry-breaking in ideal Weyl materials. In particular, we investigated the effect of introducing a small deformation to saddle-shaped metallic atoms within electromagnetic crystals. This deformation is applied to break the mirror symmetry in the x–z and y–z directions while maintaining D2d symmetry. The frequency of the WPs produces a chirality-dependent separation, leading to separation of the density of states (DOS) in the Weyl energy band. The DOS can be used in photonic systems to describe the degree of transport intensity in the material. The results show that (± representing the different chirality of WPs), a phenomenon referred to as chirality imbalance in Weyl semimetals. The distribution of DOS curves, illustrating the chiral separation, is determined through theoretical calculations. To experimentally validate our theoretical findings, we applied fast Fourier transform (FFT) to the field diagram, revealing a strong agreement between the theory and the experiment. Additionally, we conducted experimental examinations of the Fermi arc surface states on the ⟨001⟩ crystal plane. Our findings provide evidence of chirality imbalance in photonic systems.24–27 This imbalance opens up an avenue for studying various effects of chiral physics, including Klein tunneling, chiral responses, and others. The results present an approach to investigate the properties of chiral materials in photonic Weyl materials, offering a perspective on the study of chiral materials in photonic Weyl systems.
Protected by time-reversal symmetry and D2d symmetry [Fig. 1(a)], this metallic saddle-shaped structure possesses two pairs of distinct chiral Weyl points. The WPs are distributed along the four diagonal directions of the square lattice, possessing the same frequency [as shown by the purple plane in Fig. 1(a)]. These four WPs exhibit an identical DOS distribution, and their degeneracy originates from the energy band crossing between the transverse and longitudinal modes in the Weyl metamaterial. Their stability is safeguarded by the second-order rotational symmetry along the diagonal directions within the D2d operation.
The eigenvalues of the second-order rotational operators for the electromagnetic longitudinal and transverse modes are +1 and −1, respectively. The observed frequency agreement among WPs of the same chirality is a direct consequence of the C2 symmetry and time-reversal symmetry along the z-direction. Furthermore, the identical frequency characteristics exhibited by different chiral WPs are preserved by the mirror symmetry about the x–z and y–z planes within the D2d operation. To disrupt the identical frequency properties of the WPs, a structural deformation was introduced, as depicted in Fig. 1(b). This involves modifying the distances, l1 and l2, between the columns aligned along the diagonal and antidiagonal directions, respectively. Particularly, the length l2 of the square structure was reduced in the antidiagonal direction (from Γ→ ). When viewed from the top, the structure undergoes degeneration thereby transforming the structure into a diamond-shaped configuration. This structural transformation breaks the mirror symmetries Mx and My within the structure, leading to WPs with a chiral frequency separation [represented by the blue and red planes below Fig. 1(b)]. Consequently, a separation exists in the DOS. The extent of the structural deformation can be quantified from the θ value, which characterizes the degree of breaking. The magnitude of θ directly influences the difference in the frequency between distinct chiral WPs and affects the bandgap width.28
We also investigated the energy band structure of the superconformal material after deformation. We designed samples with the following parameters in CST STUDIO SUITE: ax = ay=1.7, hc = 1.524, and h0 = 0.508 mm, width of the metal connecting plate w = 0.325, rout = 0.3, and rin = 0.2 mm, length of the two diagonals l1 = 0.9, l2 = 0.84, and θ = 86°. To obtain the experimental results and simplify the calculation, the metal structure inside the unit cell was set as a loss-free perfect electric conductor (PEC) and the rest is set as printed circuit board (PCB) with a dielectric constant of ε = 3.3. The energy band structure of the highly symmetric path in the first Brillouin zone can be obtained by using the eigenmode solver of CST [Fig. 2(a)], where kx-ky-kz is the Bloch wave vector in three directions (the data were normalized to π/ax − π/ay − π/az), and two different chiral WPs appear in the diagonal and antidiagonal directions in red dot and blue dot. The positions of the two external points are f1 = 21.4 GHz, −kx = ky = 0.467 and f2 = 21.8 GHz, kx = ky = 0.467. As for the frequency of the WPs, an evident blue shift was observed along the diagonal of the longer side, and a red shift along the diagonal of the shorter side. This phenomenon arises because the symmetry operation solely disrupts the mirror symmetry in the x-y direction, while preserving the rotational symmetries of C2 (along the z-direction) and (along the Γ and Γ′ directions) within the D2d operation in the unit cell. These results confirmed the presence of the WPs.
When measuring the electric field intensity distribution between the air field and the sample, an air noise intensity approximately one-tenth of the experimental intensity was observed. Using the correction equation, two DOS curves associated with chirality (+ red, − blue) were derived [as shown on Fig. 2(a)]. The vertical axis represents the intensity on a normalized logarithmic scale, while the horizontal axis represents the frequency, corresponding to the bandgap region. As the linear distribution of WPs is confined to narrow bands, the present study focused on a frequency range of 3 GHz. Within this range, the intensity reversal within the bandgap suggests the presence of chirality imbalance. At a frequency of 21 GHz (below the low-frequency WPs), the band 2 energy band exhibits an elliptical curve, with WP1 (WP3) being larger than WP2 (WP4). As the frequency gradually increases, the size of the Weyl cone reverses. At a frequency of 22 GHz (above the high-frequency WPs), the band 3 energy band displays an iso-frequency elliptical curve, where WP2 (WP4) surpasses the iso-frequency curve of WP1 (WP3).
To investigate the chiral segregation of DOS in the WPs, measurements of the intensity were performed from iso-frequency maps of distinct chiral WPs, and momentum-resolved spectral intensity was determined at these points. During the experiment, a layered structural sample measuring 50 × 100 × 10 was created [Fig. 3(a)]. A dipole antenna was positioned at the center of the sample's lower surface to stimulate the electric field. Additionally, a stripped coaxial cable, devoid of its metal casing, was affixed to a stepper motor that scanned the upper surface of the sample as it moved (supplementary material 2). A 30 × 30 lattice width in the central region was scanned to exclude any influence from the Fermi arc on the intensity extraction. After applying FFT, an iso-frequency map of the Weyl energy band was prepared [Fig. 3(b)]. Additional iso-frequency surfaces can be found in supplementary material 3. To enhance measurement accuracy, the intensity in the designated region was determined by considering a length of 0.17×π/ai (where i represents x or y) both above and below each WP, and the DOS of the WPs was considered proportional to , where S denotes the area of a square, P represents the spectral intensity of each point in the area, and =1–2–3–4 represents the four quadrants in the coordinate system (supplementary material 3). Figure 3(c) shows the DOS curves for different frequencies, unveiling a conspicuous segregation between positive and negative chiral WPs. Noteworthy, the simulation showed an intensity shift when traversing low-frequency WPs and high-frequency WPs. Conversely, in the experiment, the low-frequency signal is accompanied by heightened noise levels, which makes the distinction of energy band information from the noise challenging and thus resulting in the absence of evident intensity changes. At a frequency of 21.3 GHz on the iso-frequency diagram, all four WPs exhibit evident and nearly uniform intensities. This uniformity is further reflected in the DOS spectral line, wherein the intensity profile remains largely constant [(PWP2+PWP4) − (PWP1+PWP3) ≈ 0] [Fig. 3(c)]. However, upon crossing the high Weyl frequency (21.8 GHz), discernible sets of intensity curves emerge on the iso-frequency diagram [(PWP2+PWP4) − (PWP1+PWP3) > 0]. Figure 3(d) illustrates the direct manifestation of DOS segregation on the iso-frequency diagram, thereby substantiating the chirality imbalance and aligning with the earlier observations shown in Figs. 2(c) and 2(d).
The Fermi arc, a crucial characteristic of Weyl semimetals, is preserved by the bulk state of the semimetal. The shape of the Fermi arc state can be predicted by truncating the infinite periodicity in the z-direction. The experimental setup involved a ten-layer cyclic structure in CST along the z-direction, where PEC boundaries were applied to the zmax and zmin surfaces to simulate open boundaries in real space, located a distance of 4.5 mm (equivalent to one cycle length). The x–y plane followed periodic boundary conditions, while ky was fixed at 0.467 and the value of kx= . With this setup, we derived the projected energy bands passing through different chiral WPs, comprising two components: a bulk band originating from the Weyl cone and a Fermi surface state resulting from the finite period. To simplify the calculation, we utilized a single-cell structure in the simulation, selecting 11 discrete points along the kz direction and projecting them onto the ky = 0.467 direction. By applying this procedure, the light gray region (representing the bulk state) was determined in the middle of the figure. Notably, the structures of the upper and lower connected surfaces were not identical. Therefore, in this experiment, the zmax and zmin surfaces were measured, and a lattice size of 98 × 48 in the sample area was scanned. The Brillouin zone was set to ky = −0.467, and the value of kx = . The results of (zmax) and (zmin), displayed on the right side of Figs. 4(a) and 4(b), respectively, suggest the separation of the two distinct chiral WPs at frequencies f1 = 21.8 and f2 = 21.4 GHz, consistent with the simulation outcomes (supplementary material 4). On the two different surfaces, surface Fermi arcs extended in diverse directions, connecting the two WPs. Together, these Fermi arcs constituted the spiral surface state observed in Weyl semimetals.
Figure 4(c) shows the results of the projected cross section along the diagonal direction. The simulated energy band results reveal the same frequency, f1 = 21.8 GHz, and exhibit a symmetrical shape of the WPs. The orange curve represents the calculated surface Fermi arc. It is worth noting that the Fermi arc exhibits a twofold degeneracy, and the eigen electric field is localized on both the upper and lower surfaces, mainly because the two surfaces of ⟨001⟩ are equivalent when kx = ky.
In this study, breaking of the mirror symmetry reveals the splitting of frequencies in WPs with different chirality from the central position f0. Particularly, the “+” chiral WPs exhibit a blue-shift to f1, while the “−” chiral WPs display a red shift to f2, depending on their chirality. The distribution of the DOS was investigated both theoretically and experimentally. Notably, we observed that the DOS of “−” chiral WPs surpasses that of “+” chiral WPs when the frequency is below that of the “−” chiral WPs. Additionally, a flip in the DOS occurs across the “+” chiral WPs. The DOS plays a crucial role in describing the transport properties of electron systems. By establishing a connection between WPs and chirality transport, the findings of this study provide insights into the phenomenon of imbalanced chirality in real space within photonic Weyl semimetals. This research opens up avenues for further investigations into the chiral physics of pseudo-fermionic fields in photonic materials.
See the supplementary material that includes the theoretical formula for calculating the density of states (DOS) and the experimental measurement and post-processing methods.
This work was supported by the Natural Science Foundation of China (NSFC) (Grant No. 12074279), the Major Program of Natural Science Research of Jiangsu Higher Education Institutions (Grant No.18KJA140003), and the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.
Conflict of Interest
The authors have no conflicts to disclose.
Xiaoxi Zhou: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Supervision (equal); Writing – original draft (lead); Writing – review & editing (lead). Shanshan Li: Conceptualization (supporting); Data curation (supporting); Formal analysis (supporting); Investigation (supporting); Methodology (supporting). Chuandeng Hu: Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Writing – review & editing (supporting). Gang Wang: Formal analysis (supporting); Funding acquisition (supporting); Methodology (supporting); Supervision (supporting); Writing – review & editing (supporting). Bo Hou: Conceptualization (equal); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead).
The data that support the findings of this study are available from the corresponding authors upon reasonable request.