Flat Landau levels and interface states in two-dimensional photonic crystals with a nodal ring

Line degeneracies in the band structures have been largely explored and exhibit exotic phenomena, particularly in three-dimensional (3D) photonic crystals. The flat Landau levels are a generic feature of nodal ring semimetals when a magnetic field perpendicular to the nodal ring plane is applied. However, solid evidence for such effects is still absent in photonic nodal ring systems due to limitations in magneto-optical materials and structural complexity in 3D systems. In this paper, we propose a two-dimensional photonic structure that exhibits nodal rings protected by mirror symmetry, demonstrable through a simple two-band tight-binding model. An interacting term that breaks the mirror symmetry can open a photonic band gap, leading to the existence of Jackiw-Rebbi-like interface states. By introducing gradient deformation to the photonic crystal ribbon, we create a uniform pseudomagnetic field, further achieving flat Landau levels and gapless interface states. Additionally, we can dynamically adjust the width of the interface state by modifying the pseudomagnetic field strength, indicating a strategy for creating a waveguide with arbitrary widths. Our results offer a simple two-dimensional platform to study nodal ring physics and its magnetic responses in a pure dielectric system, potentially providing insights for manipulating electromagnetic waves through photonic crystals.