Quantum mechanics

Root mean square

Square (algebra)

Mathematical optimization

Hamiltonian (control theory)

Square root

Physics

Mathematics

Optics

Topological insulator

Resonator

Condensed matter physics

Geometry

Combinatorics

Topology (electrical circuits)

Hermitian matrix

Optics Express

We investigate the topological skin effect in a ring resonator array which can be mapped into the square root of a Su-Schrieffer-Heeger (SSH) model with non-Hermitian asymmetric coupling. The asymmetric coupling is realized by integrating the same amount of gain and loss into the two half perimeters of linking rings that effectively couple two adjacent site rings. Such a square-root topological insulator inherits the properties from its parent Hamiltonian, which has the same phase transition points and exhibits non-Bloch features as well. We show the band closing points for open chain are different from that of periodic chain as a result of the skin effect. Moreover, the square-root insulator supports multiple topological edge modes as the number of band gaps is doubled compared to the original Hamiltonian. The full-wave simulations agree well with the theoretical analyses based on a tight-binding model. The study provides a promising approach to investigate the skin effect by utilizing ring resonators and may find potential applications in light trapping, lasers, and filters.

Square-root non-Bloch topological insulators in non-Hermitian ring resonators