Bound State In The Continuum In Topological LC Circuits

In recent ten years,the topological properties of physical systems have been used to classify systems,and the discovery of topological states has promoted people's understanding of electronic materials.Since the generation of topological states is only related to the Berry curvature in the energy band structure of the system and does not only exist in the electronic system,the topological energy band theory has been rapidly extended to various fields outside the electronic system,such as cold atomic system,photonic crystal,phonon crystal,mechanical system,etc.Recently,topology-related concepts have been introduced into classic electronic circuits and many breakthroughs have been made.Self-induced topological protection,topological corner states,topological nodal states and quantum spin Hall effect have all been implemented on electronic circuit platforms.Because the parameters and connection of electronic components are very convenient to adjust,and the experimental measurement methods and techniques are also very mature,electronic circuits are considered to be a very promising platform for studying topological states.This paper mainly studies the bound states in the continuum on the circuit composed of lumped parameter electronic component(capacitor and inductor).The bound state in the continuum is different from the general bound state.The general bound state exists in the band gap outside the bulk state while the bound state in the continuum exists within the bulk frequency range but still maintains perfect local property without any energy leakage.Since the system with bound states in the continuum naturally has a high Q value,the bound states in the continuum proposed in mathematics have been found in various material systems,such as piezoelectric materials,dielectric photonic crystals,optical waveguides and optical fibers,quantum dots and graphene,etc.Starting from the simplest topological state system Su-Schrieffer-Heeger(SSH)model,we constructed the circuit SSH chain composed of periodic arrangement of capacitors and inductors.By Kirchhoff's law and Bloch's theorem,the Laplacian of the circuit is corresponding to the electronic energy band structure and Hamiltonian in solid materials,and the topologically protected end states in the circuit SSH chain are verified.Then we designed and constructed a circuit composed of two circuit SSH chains on the printed circuit board(PCB).The two SSH chains are indirectly coupled through the middle inductor-capacitor(LC)chain.By measuring the impedance distribution of the circuit,we experimentally observed the phenomenon of bound states in the continuum.Then we discussed the bound state in the continuum in non-Hermitian system by adding resistors to the circuit.Finally,based on the circuit SSH chain,we constructed a topological LC circuit similar to the square lattice model and studied the circuit's bulk band and verified the expected valley boundary state.