Topological Insulators In Hermitian And Non-hermitian Circuit Systems

Different from ordinary insulators,topological insulators have topologically protected edge states on the boundary.These edge states are not sensitive to the gentle changes of structural parameters,and possess potential application in the lossless transmission of signals.In addition to realizing topological insulators of electrons in quantum systems,researchers try to find and implement topological insulators in various classical systems,including optical,acoustic,mechanical,circuit systems,and so on.Among them,the circuit systems are only used in the research of topological states in recent years,and gradually formed the concept of topological circuits.The topological circuits are mainly established based on the resonant circuit networks.The required electronic components are simple,and the circuit structure can be designed flexibly,which is convenient for integration and mass production.These advantages make the circuit systems provide a good experimental platform for the research of topological states.This dissertation mainly studies the topological insulators in several circuit systems,including Hermitian and non-Hermitian cases.The specific research content is the bound states in the continuum of Su-Schrieffer-Heeger coupling circuit,the second-order corner states in the Kagome circuit,and non-Hermitian induced second-order topological circuit.In chapter two,we constructed the Su-Schrieffer-Heeger coupling circuit,and experimentally observed the bound states in the band gap and the bound states in the continuum.The Hamiltonian of the circuit can be divided into two parts,one of the corresponding energy bands is topological,and the other part is trivial.Both topological and trivial energy bands can be independently adjusted by different electronic components.In the case of an open boundary,the topologically protected bound states will appear in the topological band gap.The trivial energy band can be moved by adjusting the parameters.When the bound states are also located in the band gap of the trivial bulk energy band,it is the bound states in the band gap.When the bound states fall into the trivial bulk energy band,the bound states in the continuum can be obtained.The bound states in the continuum are basically not affected by the bulk states,and can still maintain good locality.In chapter three,we constructed the Kagome circuit to realize the two-dimensional second-order topological insulator,and experimentally observed the zero-dimensional topologically protected corner states.In the Hermitian case,we predicted and experimentally confirmed the existence of the zero-dimensional corner states in the triangular-shaped circuit,and then proved that the corner states are topologically protected and robust to weak disorders.In the parallelogram-shaped circuit,we have studied the relationship between corner states and corner geometric configuration,and it is confirmed that only the corner passing through the Wannier center can exist corner states.By adding series resistances,the influence of non-Hermiticity on the corner states is studied.The results show that non-Hermiticity will only cause certain attenuation in the corner states,and does not affect its existence.In addition,the resistances in series at different positions have distinct influences on corner impedance.In chapter four,we use loss to realize a two-dimensional second-order topological circuit of non-Hermitian,and experimentally observed the one-dimensional edge states and the zero-dimensional corner states.When there are no resistances in the circuit,the bulk band gap is closed.When parallel resistors are introduced in different positions,the bulk energy band can open the band gap,and a non-trivial second-order topology will be achieved,thereby forming a non-Hermitian second-order topological circuit.The circuit has gapped one-dimensional edge states and zero-dimensional corner states in the band gap,which is confirmed by our experiments.Because the second-order topological circuit is induced by loss resistances,its corner states have attenuation,but the attenuation does not affect its topological characteristics.Our system provides an ideal platform to explore non-Hermitian induced topological phases.