Research On Topological States In Electronic Circuit

In the past decade,topological insulators have attracted extensive attention and discussion in the field of condensed matter physics due to their excellent characteristics of strong robustness to the disorder of the system.At the same time,various platforms for studying topological states are constantly being developed,including photonic,acoustic,ultracold atoms,and electrical circuit.Among them,the circuit system has quickly set off an upsurge for the research and realization of topological states of matter due to its extraordinary advantages such as mature technology,low cost,and easy construction.This paper firstly reviews the development and research status of topolectrical circuits since the introduction of electronic circuits in 2015.Then,the construction and research theory of the topolectrical circuits mapped by the tight-binding model are briefly introduced.This paper then mainly presents two novel topological states and physical properties that have not yet been realized in circuit systems,namely,the finite-energy topological bound corner states,square-root non-Hermitian topological edge states and unconventional complex-real transitions of eigenenergies.The main research contents are as follows:(1)Based on the current research status of high-order topological states in circuit system,we designed a circuit system based on the kagome model to simulate and detect the finiteadmittance topological bounded corner states.The long-range interactions simulated by capacitors are introduced in this circuit model.By solving for the non-zero bulk polarization of the circuit system,we demonstrate that the introduction of long-range coupling can induce a phase transition in this second-order topolectrical circuit.Therefore,by calculating the admittance spectra under open boundary conditions,we find that the novel topological bounded corner states separate from the boundary in pairs and appear in the vicinity of zero-energy corner states with increasing coupling strength.However,these bounded states correspond to the non-zero admittance,and their topological properties cannot be directly detected by measuring the two-point impedance.Therefore,we add the same grounding capacitance to each node in the original circuit so that the bounded state with finite admittance is moved to the position of zero admittance.The strong resonance peak in the impedance spectrum of the disordered circuit implies the topological characteristics of the topological bounded corner states.Our research proposes another alternative method for exploring and verifying the finite energy corner states.(2)Based on the research status of square-root topological insulators,we investigate physical properties characterized by complex-real transitions of eigenenergies and topological properties characterized by robust edge states in a circuit version of the square-root non-Hermitian topological insulator.This unconventional complex-real transition is definitely present in nonHermitian systems without PT symmetry,and corresponds exactly to the existence and disappearance of a flat band with all imaginary-valued eigenenergies in the system.Furthermore,we find that the degeneracy of this particular flat band is exactly equal to twice the number of unit cells in the system,and reveal that its appearance is directly related to the negative hopping in the nonreciprocal coupling term.The existence of robust non-zero real energy topological edge states was hinted by a strong impedance resonance peak displayed on the boundary in a disordered circuit system.We also introduced the non-Bloch theorem to renovate the broken bulk-boundary correspondence confirmed by theoretical analysis and numerical calculations,and help to complete the construction of the entire phase diagram.Specifically,we find that such complex-real transitions do not depend on the breaking of the PT symmetry of the system,and occur precisely in topologically nontrivial regions.And,the topological edge states in the systems with complex eigenenergies can be used to fabricate topological lasing and spontaneous topological pumping.