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Higher-order Topological Type-II Hyperbolic Lattices

Recently, higher-order topological phases have been extended from Euclidean lattices to non-Euclidean hyperbolic lattices. Though higher-order topological type-I hyperbolic lattices have been extensively studied, their counterpart, higher-order topological type-II hyperbolic lattices, have never been reported yet. Here, by mapping the celebrated Bernevig-Hughes-Zhang model onto a type-II hyperbolic lattice, we present a theoretical exploration of the first-order topological edge states and second-order topological corner states in a type-II hyperbolic lattice. Compared with the higher-order topological type-I hyperbolic lattices, we discover two unique topological phenomena that stem from the nontrivial geometrical topology of the type-II hyperbolic lattice. First, topological edge and corner states exist on both inner and outer boundaries of the type-II hyperbolic lattice and exhibit higher degeneracy than those in the type-I hyperbolic lattice with only an outer boundary. Second, the degeneracy of type-II hyperbolic corner states can be arbitrarily tuned by changing the characteristic (or inner) radius, in contrast to its type-I counterpart, which is determined by the number of sides of the tessellated polygons. Our work explores topological states in more complex hyperbolic lattices, significantly expanding the research scope of hyperbolic topological physics.