Objective  As a massless relativistic fermion, Weyl fermions play a crucial role in quantum theory and the standard model. To mimic the physical properties of Weyl fermions, constructing Weyl points in momentum space needs breaking the inversion or time-reversal symmetry, and those Weyl points are topologically protected. Weyl points possess unique characteristics, including positive and negative chiralities corresponding to the sources and sinks of the Berry curvature, respectively. Consequently, Weyl points are regarded as magnetic monopoles in momentum space. Weyl points have also attracted significant attention due to their scattering and transport properties. For example, Weyl semimetal exhibits chiral zero modes and corresponding chiral magnetic effects in condensed matter physics. It is worth mentioning that Weyl points are widely acknowledged as singularities in the reflection phase, but there has been relatively little study on the reflection eigenmodes of the interface between a Weyl metamaterial and air.

  Methods  This article utilizes the ideal Weyl metamaterial as a research platform, which offers a relatively large frequency range for exploring the fundamental properties of Weyl points. By applying the effective media theory, the constitutive relation of saddle-shaped metallic structures (Fig.1) can be described concisely. Additionally, the band structure of Weyl metamaterial can be calculated using simulation software.

  Results and Discussions   For a totally reflected interface, the wave vectors of the incident and reflected state space are different. Before solving for the reflection eigenmodes, it is necessary to define the basis separately for the two different state spaces of the incident and reflected electromagnetic fields. Since the electric field is a polar vector and the magnetic field is an axial vector, the mirror operation introduces different responses in the directions perpendicular and parallel to the interface. By applying the mirror operation, we can connect the incident and reflected state spaces, allowing us to solve for the eigenmodes using conventional methods and determine their matrix representation. Given this definition, energy conservation ensures that the reflection coefficient matrix must be unitary, while Lorentz reciprocity guarantees that the reflection coefficient matrix must be symmetric. Such a unique reflection coefficient matrix must have real eigenstates, resulting in both reflection eigenmodes being linearly polarized. Using this method to analyze the reflection eigenmodes of Weyl metamaterials, it is found that all reflection eigenstates are linearly polarized. The two eigen electromagnetic fields are perpendicular to each other, forming a cross shape. As the scanning path changes continuously in the Brillouin zone, the orientation of the cross shape also varies. When the scanning path surrounds the Weyl points in momentum space, the eigen field (cross shape) undergoes an additional phase shift of \mathrm\pi /2 . A quadratic Möbius strip can describe this feature.

  Conclusions  In the case of total reflection at an interface, the incident state space and the reflected state space are bridged via a mirror operator. Combining the interface reflection operator with the mirror operator allow us to define the eigenstates of the total reflection interface. When the incident basis is chosen as linear polarized, this unique definition results in the reflection eigenstates of the total reflection interface being linear polarization modes protected by energy conservation and Lorentz reciprocity. Taking the Weyl metamaterial as an example, even if the interface reflection between the Weyl metamaterial and air exhibits polarization conversion characteristics, given this unique definition, the eigenmodes of the interface reflection can be obtained as linear polarized. Since the two linear polarization eigenstates are perpendicular to each other, a rotation angle could be defined to characterize the change in the rotation angle of the eigenmode. In addition, when scanning a loop path around the Weyl points in momentum space, the eigen field acquires an additional phase shift of \mathrm\pi /2 , which can be described using a quadratic Möbius strip.