The discovery of topological phases has recently led to a paradigm shift in condensed matter physics, and facilitated breakthroughs in engineered photonics and acoustic metamaterials. Topological insulators (TIs) enable the generation of electronic, photonic, and acoustic modes exhibiting wave propagation that is resilient to disorder, irrespective of manufacturing precision or unpredictable defects induced by the operational environment, known as topological protection. While originally limited to a dimensionality of the protected states that is one dimension lower than the host TI material, the recent discovery of higher-order topological insulators (HOTIs) provides the potential to overcome this dimensionality limitations by offering topological protection over an extended range of dimensionalities. Here we demonstrate 2D photonic HOTI (PHOTI) with topological states two dimensions lower than the one of the host system. We consider a photonic metacrystal of distorted Kagome lattice geometry that exhibits topological bulk polarization, leading to the emergence of 1D topological edge states and of higher order 0D states confined to the corners of the structure. Interestingly, in addition to corner states due to the nearest neighbour interactions and protected by generalized chiral symmetry 1 , we discover and take advantage of a new class of topological corner states sustained by long-range interactions, available in wave-based systems, such as in photonics. Our findings demonstrate that photonic HOTIs possess richer physics compared to their condensed matter counterparts, offering opportunities for engineering novel designer electromagnetic states with unique topological robustness.Main text: Topological systems exhibit unique and often counterintuitive properties, such as robust electronic transport and wave propagation, which promise to revolutionizing technologies across different fields, from quantum electronics and quantum photonics, topological phenomena unlocked novel approaches for quantum computing interfaces 32, 33 and robust lasing 34,35,36 , while in a broad range of classical fields, including optics, mechanics and acoustics, they offer an unprecedented degree of control via synthetic degrees of freedom and robustness, which manifests itself as resilience to defects and disorder 11While most topological systems studied so far have been characterized by the presence of topological states with dimensionality one order lower than the one of the system, recently a new class of topological systems, so called higher-order topological insulators (HOTIs), have been introduced 38,39,40,41 . As opposed to conventional topological insulators, HOTIs support topological states 39,40,42,43,44,45,46,47 two and more dimensions lower than the system itself, referred to as higher-order topological (HOT) states. One example of such systems is given by quadrupole topological insulators 38,48 , which have been recently implemented in mechanical 49
