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Photonic crystals have been demonstrated as a versatile platform for the study of topological phenomena. The recent discovery of higher-order topological insulators introduces new aspects of topological photonic crystals which are yet to be explored.Here, we propose a dielectric photonic crystal with unconventional higher-order band topology. Besides the conventional spectral features of gapped edge states and in-gap corner states, topological band theory predicts that the corner boundary of the higherorder topological insulator hosts a 2 3 fractional charge. We demonstrate that in the photonic crystal such a fractional charge can be verified from the local density-of-states of photons, through the concept of local "spectral charge" as an analog of the local electric charge due to band filling anomaly in electronic systems. Furthermore, we show that by introducing a disclination in the proposed photonic crystal, localized states and a 2 3 fractional spectral charge emerge around the disclination core, as the manifestation of the bulk-disclination correspondence. The predicted effects can be readily observed in the state-of-the-art experiments and may lead to potential applications in integrated and quantum photonics.

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A Dirac-like cone is formed by utilizing the flat bands associated with localized modes in an acoustic crystal (AC) composed of a square array of core-shell-structure cylinders in a water host. Although the triply-degeneracy seems to arise from two almost-overlapping flat bands touching another curved band, the enlarged view of the band structure around the degenerate point reveals that there are actually two linear bands intersecting each other at the Brillouin zone center, with another flat band passing through the same crossing point. The linearity of dispersion relations is achieved by tuning the geometrical parameters of the cylindrical scatterers. A perturbation method is used to not only accurately predict the linear slopes of the dispersions, but also confirm the linearity of the bands from first principles. An effective medium theory based on coherent potential approximation is developed, and it shows that a slab made of the AC carries a near-zero refractive index around the Dirac-like point. Full-wave simulations are performed to unambiguously demonstrate the wave manipulating properties of the AC structures such as perfect transmission, unidirectional transmission and wave front shaping.

We report on the experimental observation of a gapped nontrivial Euler topology in an acoustic metamaterial. The topological invariant in this case is the Euler class, which characterizes the twodimensional geometry of the acoustic Bloch wave function of multiple bands in the Brillouin zone-a feature that cannot be captured by conventional topological classification schemes. Starting from a carefully designed acoustic system, we retrieve a rich phase diagram that includes the gapped Euler phase on top of non-Abelian topological nodal phases. In particular we find that the gapped phase is characterized by a meron (half-skyrmion) type topology in which the cancellation of the orbital Zak phases by Zak phases of the bands contributes in forming the characterizing winding number of the acoustic bands. Using pump-probe techniques, we are able to extract the wave functions of the acoustic Bloch bands and demonstrate the non-trivial Euler topology via a direct observation of the meron geometry of these bands in the Brillouin zone, in addition to the detection of the acoustic bulk and edge dispersions. We finally observe a topological edge state in the gapped Euler phase which is a direct manifestation of the underlying cancellation of the Zak phases. These experimental features reveal the unprecedented topological Euler phase as well as setting a benchmark for probing acoustic wave function geometry in the Brillouin zone.