image: a, Structure of the PhC, where the two insets display the iso-frequency surfaces in the xz plane of the biaxial dielectrics in layer A (orange) and layer B (green). b, Band structure along high symmetry lines in the k_y=0 plane. The blue and magenta lines represent the bands with odd and even mirror-parities, respectively. The red dots correspond to the nodal lines along which two bands with opposite mirror-parities intersect. The orange and purple dots denote the threefold and fourfold degenerate nexus points, respectively. c, Sketch map of two red nodal rings (corresponding to the two lowest red dots in b) crossing a Kramers-like nodal line (blue line) at two nexus points in the first (bulk) Brillouin zone of the PhC view more
Credit: by Zhongfei Xiong, Ruo-Yang Zhang2, Rui Yu, C. T. Chan, Yuntian Chen
Discovering and synthesizing symmetry-protected topological (SPT) band degeneracies, including nodal points and nodal lines (NLs), is a rapidly growing frontier in the field of topological materials. Interestingly, since the crystallographic space groups impose fewer constraints on the energy bands than the continuous Poincaré group, more exotic multifold band crossings were found in lattice systems, which have no counterparts in high-energy physics. In PhCs, the topology of band structures is usually thought to be adequately described by spinless space groups, provided that special internal symmetries, such as electromagnetic (EM) duality, are not imposed on the EM materials. However, in dielectric PhCs, there are always two gapless bands emerging from the origin of light cone (ω=|k|=0), irrespective of the space group representations at that point. It was recently demonstrated that this intrinsic singularity of EM fields permits higher minimal connectivity for the lowest photonic bands than for their electronic counterparts without spin-orbit coupling and may further enforce unique photonic band crossings even in symmorphic lattices.
In a new paper published in Light Science & Application, a team of scientists, led by Professor C. T. Chan from the Hong Kong University of Science and Technology and Associate Professor Yuntian Chen from Huazhong University of Science and Technology, and co-workers have discovered a new kind of hidden symmetry in PhCs, due to the special characters of Maxwell equations. Based on this hidden symmetry, they found that in an AB-layer-stacked photonic crystal composed of anisotropic dielectrics, the unique photonic band connectivity leads to a new kind of symmetry-enforced triply degenerate points at the nexuses of two nodal rings and a Kramers-like nodal line. More interestingly, the nexus points of three NLs behaver as a new kind of magnetic monopole terminating Berry flux strings in the momentum space, and show novel spin-1 canonical diffraction.
In general, the stationary Maxwell's equations can be written as a generalized eigenvalue problem . Since all space group transformations leave the curl matrix N ?(r) invariant, a PhC respects a space group symmetry A ? only if its constitutive tensor obeys A ??M ?(r)A ?^(-1)=M ?(r). However, a generic symmetry A ? of Maxwell's equations (1) operates on the Hamiltonian H ?(r)=M ?(r)^(-1) N ?(r) of EM fields, namely, requiring ?A H ?(r) A ?^(-1)=H ?(r), and not on N ?(r) and M ?(r) separately. This fact implies that the conventional space groups alone are insufficient to determine the symmetry properties as well as the band connectivities of photonic systems.
In this work, they proposed a simple layer-stacked photonic structure consisting of anisotropic dielectrics to exemplify such hidden symmetries of Maxwell's equations beyond space groups. They show that a hidden symmetry, more specifically, a generalized fractional screw rotation symmetry, together with time reversal symmetry guarantees the emergence of Kramers-like straight NLs passing through the Brillouin zone centre and results in unusual photonic band connectivities. Furthermore, they demonstrated the lowest Kramers-like NL can almost always intersect with two other SPT nodal rings at two triply degenerate nexus points (NPs), which can be seen as a new kind of magnetic monopole connecting Berry flux strings in momentum space. By breaking the hidden symmetry, the two NPs are lifted and type-II and type-III nodal rings are achieved in the PhC for the first time.
In addition, the peculiar anisotropic band structure near the NPs, especially the spin-1 conical dispersion of the iso-frequency surfaces, lead to novel transport phenomena.
Unlike the usual conical diffraction of light scattered at an ordinary linearly crossing point on the nodal lines characterized by spin-1/2 dynamics, , the diffraction at the triple NPs appears strikingly different spin-1 wave behavior described by a Schrödinger equation with the 2D spin-1 Hamiltonian. The authors showed that such unconventional spin-1 conical diffraction can be used to generate optical vortices with a maximum topological charge of 2.
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