Three-dimensional (3D) topological nodal points [1][2][3][4][5][6], such as Weyl and Dirac nodes have attracted wide-spread interest across multiple disciplines and diverse material systems. Unlike nodal points that contain little structural variations, nodal lines [7][8][9] can have numerous topological configurations in the momentum space, forming nodal rings [10][11][12][13], nodal chains [14][15][16][17][18] and potentially nodal links [19][20][21] and nodal knots [22,23]. However, nodal lines have much less development for the lack of ideal material platforms [24][25][26]. In condensed matter for example, nodal lines are often fragile to spin-orbitcoupling, locating off the Fermi level, coexisting with energy-degenerate trivial bands and dispersing strongly in energy of the line degeneracy. Here, overcoming all above difficulties, we theoretically predict and experimentally observe nodal chains in a metallic-mesh photonic crystal having frequency-isolated linear bandtouching rings chained across the entire Brillouin zone (BZ). These nodal chains are protected by mirror symmetries and have a frequency variation less than 1%. We used angle-resolved transmission (ART) to probe the projected bulk dispersions and performed Fourier-transformed field scan (FTFS) to map out the surface dispersions, which is a quadratic touching between two drumhead surface bands. Our results established an ideal nodal-line material for further studies of topological line-degeneracies with nontrivial connectivities, as well as the consequent wave dynamics richer than 2D Dirac and 3D Weyl materials.Chain Hamiltonian Nodal line is the extrusion of a Dirac cone, arguably the most intriguing 2D band structure, into 3D momentum space. They share the same local Hamiltonian H(k) = k x σ x + k y σ z that can be protected by the PT symmetry forbidding the mass term of σ y in the whole BZ, where P is parity inversion and T is time-reversal symmetry. A single nodal line usually form a closed ring, due to the periodicity of the BZ. Surprisingly, it was recently proposed that nodal rings can be chained together as shown in Fig. 1a. Other than PT , the critical chain point requires an extra symmetry to Berry phase π 0 π a Nodal chain b Chain Hamiltonian H = k x σ x + (k y k z + m z )σ z k y = 0 k x = 0 m z = 0 m z > 0 m z < 0 Mirror plane k z =0 FIG. 1: Nodal-chain Hamiltonian and stability. a, Illustration of the simplest chain structure between two rings. The Berry phase around the chain point is 0, in contrast to the π Berry phase of nodal lines. b, The chain point is the crossing between two nodal lines defined between three zero planes.The third plane in yellow represents the mirror plane protecting the chain point. When the mirror symmetry is broken by the mass term mz, the chain point splits.be stabilized. For example, such symmetry can be glide or mirror planes. This is clear in the chain Hamiltonian H(k) = k x σ x + (k y k z + m z )σ z that we propose here and plot in Fig. 1b. When m z = 0, it defines two nodal lines crossing at the origi...

Within standard ΛCDM cosmology, Population III (Pop III) star formation in minihalos of mass M halo 5 × 10 5 M provides the first stellar sources of Lyman α (Lyα) photons. The Experiment to Detect the Global Epoch of Reionization Signature (EDGES) has measured a strong absorption signal of the redshifted 21 cm radiation from neutral hydrogen at z ≈ 17, requiring efficient formation of massive stars before then. In this paper, we investigate whether star formation in minihalos plays a significant role in establishing the early Lyα background required to produce the EDGES absorption feature. We find that Pop III stars are important in providing the necessary Lyα-flux at high redshifts, and derive a best-fitting average Pop III stellar mass of ∼ 750 M per minihalo, corresponding to a star formation efficiency of 0.1%. Further, it is important to include baryon-dark matter streaming velocities in the calculation, to limit the efficiency of Pop III star formation in minihalos. Without this effect, the cosmic dawn coupling between 21 cm spin temperature and that of the gas would occur at redshifts higher than what is implied by EDGES.

Metasurfaces have been used to realize optical functions such as focusing and beam steering. They use sub-wavelength nanostructures to control the local amplitude and phase of light. Here we show that such control could also enable a new function of artificial neural inference. We demonstrate that metasurfaces can directly recognize objects by focusing light from an object to different spatial locations that correspond to the class of the object.