Exploration Of The Topological States With Macroscopic Electrical Circuits

In recent years,topoletrical circuits（TECs）have emerged as a powerful platform for studying the topological states of matter.Researchers map circuit Laplacians to tight-binding models in condensed matter systems and combine traditional electrical circuits with novel topological states,paving a new way to study topological physics.Up to now,many topological states proposed in electronic systems have been realized in TECs,including both the gapped topological insulators（TIs）,such as Su-Schrieffer-Heeger model,Chern insulator,and Z₂TI,and higher-order TIs,and gapless topological semimet-als（SMs）like nodal-line,Dirac,Weyl,and HOT SMs.Due to the presence of nonlinear and non-Hermitian components and flexible connection methods in TECs,it is convenient to explore non-Hermitian,nonlinear topological states,even for the study of topological states in higher-dimensional,non-periodic lattices,and non-Euclidean spaces with this platform.Therefore,TECs can be applied to studying a variety of topological states,which not only has fundamental interests for deepening our understanding of the states of matter but also is promising for applications due to their low costs and convenient integrations.This dissertation aims to simulating and exploring various topological states based on TEC platform.Circuit experiments are used to testify some key topological band theo-ries and discover a series of novel topological states.This thesis provides a retrospect on recent progresses in TEC field,enabling readers to familiarize the current developments of TECs.Then,the fundamentals of TECs,including circuit components,basic theories,construction methods,and experimental characterization methods,are introduced.With these knowledge,the thesis firstly focuses on the first-order TIs.Imposing centrosymmet-ric deformations on the 2D Su-Schrieffer-Heeger circuit lattices,four different weak TI phases and Dirac SM phase with itinerant Dirac cones are found in this system.The work demonstrates the existence of 2D weak TIs in circuit experiments and the flat-band bound states in the bulk nodes by designing a domain wall structure.Later work concentrates on the topological states in two Kekulécircuits with molecular-zigzag and partially bearded edge terminals.It is found that edge states appear in two different regions with oppo-site parameters for the two boundary shapes.This phenomena are explained by calculat-ing topological invariants and mapping the circuit models to the Bernevig-Hughes-Zhang ones.By defining the chirality of the current within the primitive cell as pseudospins and measuring the spectra of the system,the edge-dependent quantum pseudospin Hall effect is explained and confirmed.Next,this thesis studies the higher-order topological states in the breathing kagome circuit lattice.The second-order Wannier-type corner states are confirmed in circuit experiment and characterized by Z₃Berry phase.By introducing perturbations that destroy and preserve the generalized chiral symmetry,it is shown that the corner state is protected by generalized chiral symmetry.Based on this work,long-range interactions are introduced to the breathing kagome circuit lattice.One can find rich topological states in this system,including three types of topological corner states,bound states in the continuum,and Dirac SM state.Three different types of corner states are observed in circuit experiments and explained theoretically.The thesis also focus on the circuit realization of quantum anomalous SM phase.By adjusting the mass of Wil-son fermions,the quantum anomalous SM phase,TI phase,normal SM phase coexist in this system,and their pseudospin structures in momentum space are mapped to magnetic solitons.By designing the domain wall structure,the quantum anomalous SM states and Wilson fermions are observed.Finally,it is presented a summary of all of the contents and the outlook for the future developments of TECs.This dissertation includes abundant topological states realized in TECs,including both（higher-order）TI phases and topological SM phases.These results define new topo-logical phases and elucidate their topological properties,which not only deepens the un-derstanding of these topological phases but also enables the design of new metamaterials and the discovery of new materials for other solid-state systems,such as condensed mat-ter,acoustics,optics,etc.Based on these works,robust topological states can be applied in device design,such as the generation,transmission,and processing of signals,and used in imaging and sensing devices,providing innovative ideas and methods for the next gen-eration of electronic information materials and devices.