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Quadrupole topological phases, exhibiting protected boundary states that are themselves topological insulators of lower dimensions, have recently been of great interest. Extensions of these ideas from current tight binding models to continuum theories for realistic materials require the identification of quantized invariants describing the bulk quadrupole order. Here we identify the analog of quadrupole order in Maxwells equations for a photonic crystal (PhC) and identify quadrupole topological photonic crystals formed through a band inversion process. Unlike prior studies relying on threaded flux, our quadrupole moment is quantized purely by crystalline symmetries, which we confirm using three independent methods: analysis of symmetry eigenvalues, numerical calculations of the nested Wannier bands, and the expectation value of the quadrupole operator. Furthermore, through the bulk-edge correspondence of Wannier bands, we reveal the boundary manifestations of nontrivial quadrupole phases as quantized polarizations at edges and bound states at corners. Finally, we relate the nontrivial corner states to the emergent phenomena of quantized fractional corner charges and a filling anomaly as first predicted in electronic systems. Our work paves the way to further explore higher-order topological phases in nanophotonic systems and our method of inducing quadrupole phase transitions is also applicable to other wave systems, such as electrons, phonons and polaritons.
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Topological manipulation of waves is at the heart of the cutting-edge metamaterial researches. Quadrupole topological insulators were recently discovered in two-dimensional (2D) flux-threading lattices which exhibit higher-order topological wave trapping at both the edges and corners. Photonic crystals (PhCs), lying at the boundary between continuous media and discrete lattices, however, are incompatible with the present quadrupole topological theory. Here, we unveil quadrupole topological PhCs triggered by a twisting degree-of-freedom. Using a topologically trivial PhC as the motherboard, we show that twisting induces quadrupole topological PhCs without flux-threading. The twisting-induced crystalline symmetry enriches the Wannier polarizations and lead to the anomalous quadrupole topology. Versatile edge and corner phenomena are observed by controlling the twisting angles in a lateral heterostructure of 2D PhCs. Our study paves the way toward topological twist-photonics as well as the quadrupole topology in the quasi-continuum regime for phonons and polaritons.
We clarify theoretically that the topological ring-cavity (TRC) modes propagating along the interface between two honeycomb-type photonic crystals distinct in topology can be exploited for achieving stable single-mode lasing, with the maximal intensity larger than a whispering-gallery-mode counterpart by order of magnitude. Especially, we show that the TRC modes located at the bulk bandgap center benefit maximally from the gain profile since they are most concentrated and uniform along the ring cavity, and that, inheriting from the Dirac-like dispersion of topological interface states, they are separated in frequency from each other and from other photonic modes, both favoring intrinsically single-mode lasing. A TRC mode running in a specific direction with desired orbital angular momentum can be stimulated selectively by injecting circularly polarized light. The TRC laser proposed in the present work can be fabricated by means of advanced semiconductor nanotechnologies, which generates chiral laser beams ideal for novel photonic functions.
When the crystalline symmetries that protect a higher-order topological phase are not preserved at the boundaries of the sample, gapless hinge modes or in-gap corner states cannot be stabilized. Therefore, careful engineering of the sample termination is required. Similarly, magnetic textures, whose quantum fluctuations determine the supported magnonic excitations, tend to relax to new configurations that may also break crystalline symmetries when boundaries are introduced. Here we uncover that antiskyrmion crystals provide an experimentally accessible platform to realize a magnonic topological quadrupole insulator, whose hallmark signature are robust magnonic corner states. Furthermore, we show that tuning an applied magnetic field can trigger the self-assembly of antiskyrmions carrying a fractional topological charge along the sample edges. Crucially, these fractional antiskyrmions restore the symmetries needed to enforce the emergence of the magnonic corner states. Using the machinery of nested Wilson loops, adapted to magnonic systems supported by noncollinear magnetic textures, we demonstrate the quantization of the bulk quadrupole moment, edge dipole moments, and corner charges.
Quadrupole topological insulator is a symmetry-protected higher-order topological phase with intriguing topology of Wannier bands, which, however, has not yet been realized in plasmonic metamaterials. Here, we propose a lattice of plasmon-polaritonic nanocavities which can realize quadrupole topological insulators by exploiting the geometry-dependent sign-reversal of the couplings between the daisy-like nanocavities. The designed system exhibits various topological and trivial phases as characterized by the nested Wannier bands and the topological quadrupole moment which can be controlled by the distances between the nanocavities. Our study opens a pathway toward plasmonic topological metamaterials with quadrupole topology.
High-order topological phases, such as those with nontrivial quadrupole moments, protect edge states that are themselves topological insulators in lower dimensions. So far, most quadrupole phases of light are explored in linear optical systems, which are protected by spatial symmetries or synthetic symmetries. Here we present Floquet quadrupole phases in driven nonlinear photonic crystals (PhCs) that are protected by space-time screw symmetries. We start by illustrating space-time symmetries by tracking the trajectory of instantaneous optical axes of the driven media. Our Floquet quadrupole phase is then confirmed in two independent ways: symmetry indices at high-symmetry momentum points and calculations of the nested Wannier bands. Our work presents a general framework to analyze symmetries in driven optical materials and paves the way to further exploring symmetry-protected topological phases in Floquet systems and their optoelectronic applications.
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