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The discovery of topological insulators has reformed modern materials science, promising to be a platform for tabletop relativistic physics, electronic transport without scattering, and stable quantum computation. Topological invariants are…

This monograph introduces key concepts and problems in the new research area of Periodic Geometry and Topology for materials applications.Periodic structures such as solid crystalline materials or textiles were previously classified in…

Predictive theory to geometrically engineer devices and materials in continuum systems to have desired topological-like effects is developed here by bridging the gap between quantum and continuum mechanical descriptions. A platonic crystal,…

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…

We report the observation of a symmetry-protected topological time crystal, which is implemented with an array of programmable superconducting qubits. Unlike the time crystals reported in previous experiments, where spontaneous breaking of…

With the rapid development of topological states in crystals, the study of topological states has been extended to quasicrystals in recent years. In this review, we summarize the recent progress of topological states in quasicrystals,…

Topological insulators are materials that conduct on the surface and insulate in their interior due to non-trivial topological order. The edge states on the interface between topological (non-trivial) and conventional (trivial) insulators…

The goal of this paper is to advance the development of wave-guiding subwavelength crystals by developing designs whose properties are stable with respect to imperfections in their construction. In particular, we make use of a locally…

We map the topological properties of a one dimensional superlattice to the optical properties of an electronic system. We find that the nonlinear-optical response is optimized for electrons that live in the transitional morphology between…

Topological photonic crystals (PCs) can support robust edge modes to transport electromagnetic energy in an efficient manner. Such edge modes are the eigenmodes of the PDE operator for a joint optical structure formed by connecting together…

Chirality manifests across multiple scales, yielding unique phenomena that break mirror symmetry. In chiral materials, unexpectedly large spin-filtering or photogalvanic effects have been observed even in materials composed of light…

Topological phases of matter is an exotic phenomena in modern condense matter physics, which has attracted much attention due to the unique boundary states and transport properties. Recently, this topological concept in electronic materials…

Electromagnetic topological insulators have been explored extensively due to the robust edge states they support. In this work, we propose a topological electromagnetic system based on a line defect in topologically nontrivial photonic…

Due to the absence of periodic length scale, electronic states and their topological properties in quasicrystals have been barely understood. Here, we focus on one dimensional quasicrystal and reveal that their electronic critical states…

Tensor networks impose a notion of geometry on the entanglement of a quantum system. In some cases, this geometry is found to reproduce key properties of holographic dualities, and subsequently much work has focused on using tensor networks…

Liquid crystals generally support orientational singularities of the director field known as topological defects. These latter modifiy transport properties in their vicinity as if the geometry was non-Euclidean. We present a state of the…

The strict geometric rules that define aperiodic tilings lead to the unique spectral and transport properties of quasicrystals, but also limit our ability to design them. In this Letter, we explore a novel example of a continuously tunable…

Atomic metasurfaces (AMs) provide a powerful nanophotonic platform for integrating topological effects into quantum many-body systems. In this Letter, we investigate the quantum optical and topological properties of a two-dimensional Kagome…

Band topology and related spin (or pseudo-spin) physics of photons provide us with a new dimension for manipulating light, which is potentially useful for information communication and data storage. Especially, the quantum spin Hall effect…

Three dimensional topological insulator crystals consist of an insulating bulk enclosed by metallic surfaces, and detailed theoretical predictions about the surface state band topology and spin texture are available. While several…