Higher-order topological insulators are a new class of topological phases of matter, originally conceived for electrons in solids. It has been suggested that $\mathbb{Z}_N$ Berry phase (Berry phase quantized into $2π/N$) is a useful tool to characterize the symmetry protected topological states, while the experimental evidence is still elusive. Recently, topolectrical circuits have emerged as a simple yet very powerful platform for studying topological physics that are challenging to realize in condensed matter systems. Here, we present the first experimental observation of second-order corner states characterized by $\mathbb{Z}_3$ Berry phase in topolectrical circuits. We demonstrate theoretically and experimentally that the localized second-order topological states are protected by a generalized chiral symmetry of tripartite lattices, and they are pinned to "zero energy". By introducing extra capacitors within sublattices in the circuit, we are able to examine the robustness of the zero modes against both chiral-symmetry conserving and breaking disturbances. Our work paves the way for testing exotic topological band theory by electrical-circuit experiments.
Invented by Alessandro Volta and Félix Savary in the early 19th century, circuits consisting of resistor, inductor and capacitor (RLC) components are omnipresent in modern technology. The behavior of an RLC circuit is governed by its circuit Laplacian, which is analogous to the Hamiltonian describing the energetics of a physical system. We show that topological semimetal band structures can be realized as admittance bands in a periodic RLC circuit, where we employ the grounding to adjust the spectral position of the bands similar to the chemical potential in a material. Topological boundary re…
One of the core concepts from the non-Hermitian skin effect is the extended complex wavevectors (CW) in the generalized Brillouin zone (GBZ), while the origin of CW remains elusive, and further experimental demonstration of GBZ is still lacking. We show that the bulk states of an open quantum system dynamically governed by the Lindblad master equation exhibit non-Bloch evolution which results in CW. Experimentally, we present temporal topolectrical circuits to serve as simulators for the dynamics of an open system. By reconstructing the correspondence between the bulk states of an open system …
Recent theoretical studies have extended the Berry phase framework to account for higher electric multipole moments, quadrupole and octupole topological phases have been proposed. Although the two-dimensional quantized quadrupole insulators have been demonstrated experimentally, octupole topological phases have not previously been observed experimentally. Here we report on the experimental realization of classical analog of octupole topological insulator in the electric circuit system. Three-dimensional topolectrical circuits for realizing such topological phases are constructed experimentally…
We propose a scheme to realize various non-Hermitian topological phases in a topolectrical (TE) circuit network consisting of resistors, inductors, and capacitors. These phases are characterized by topologically protected exceptional points and lines. The positive and negative resistive couplings Rg in the circuit provide loss and gain factors which break the Hermiticity of the circuit Laplacian. By controlling Rg, the exceptional lines of the circuit can be modulated, e.g., from open curves to closed ellipses in the Brillouin zone. In practice, the topology of the exceptional lines can be det…
Wilson fermion (WF) is a fundamental particle in the theory of quantum chromodynamics, originally proposed by Kenneth Wilson to solve the fermion doubling problem, i.e., more fermions than expected when one puts fermionic fields on a lattice. In this Letter, we report a direct observation of the WF in circuit systems. It is found that WFs manifest as topological spin textures analogous to the half skyrmion, half-skyrmion pair, and Néel skyrmion structures, depending on their mass. Transformations of different WF states are realized by merely tuning the electric elements. Theoretical calculatio…
Quantized electric quadrupole insulators have recently been proposed as novel quantum states of matter in two spatial dimensions. Gapped otherwise, they can feature zero-dimensional topological corner mid-gap states protected by the bulk spectral gap, reflection symmetries and a spectral symmetry. Here we introduce a topolectrical circuit design for realizing such corner modes experimentally and report measurements in which the modes appear as topological boundary resonances in the corner impedance profile of the circuit. Whereas the quantized bulk quadrupole moment of an electronic crystal do…
We propose an electric circuit array with topologically protected uni-directional voltage modes at its boundary. Instead of external bias fields or floquet engineering, we employ negative impedance converters with current inversion (INICs) to accomplish a non-reciprocal, time-reversal symmetry broken electronic network we call topolectrical Chern circuit (TCC). The TCC features an admittance bulk gap fully tunable via the resistors used in the INICs, along with a chiral voltage boundary mode reminiscent of the Berry flux monopole present in the admittance band structure. The active circuit ele…
We investigate the emergence of topological valley Hall and kink states in a two-dimensional topolectrical (TE) model as a result of broken chiral and reflection symmetries. The TE system consists of two segments hosting distinct topological states with opposite signs of the valley Hall index, and separated by a heterojunction. In the practical circuit, the valley Hall index can be flipped between the two segments by modulating the onsite potential on the sublattice nodes of the respective segments. The presence of resistive coupling, which introduces non-Hermiticity in the system, subsequentl…
Recent studies have shown that there are two types of topological quadrupolar semimetals. One is the quadrupolar bulk semimetal, which is a conventional semimetal with gapless nodes in the bulk spectrum. The other type is named the quadrupolar surface semimetal, possessing a gapped bulk but gapless nodes on the surface spectrum. There have been many experiments on the implementation of quadrupolar bulk semimetals. However, the quadrupolar surface semimetal has never been realized. Here we report on the experimental realization of the quadrupolar surface semimetal in the circuit system. To high…
It is well known that Anderson transition is a disorder-induced metal-insulator transition.Contrary to this conventional wisdom, some investigations have shown that disorders could destroy the phase coherence of localized modes in flatbands, making the localized states melt into extended states. This phenomenon is called the inverse Anderson transition. While, to date, the experimental observation of inverse Anderson transitions is still lacking. In this work, we report the implementation of inverse Anderson transitions based on Aharonov-Bohm topolectrical circuits. Different types of disorder…
A system is non-Hermitian when it exchanges energy with its environment and non-reciprocal when it behaves differently upon the interchange of input and response. Within the field of metamaterial research on synthetic topological matter, the skin effect describes the conspiracy of non-Hermiticity and non-reciprocity to yield extensive anomalous localization of all eigenmodes in a (quasi) one-dimensional geometry. Here, we introduce the reciprocal skin effect, which occurs in non-Hermitian but reciprocal systems in two or more dimensions: Eigenmodes with opposite longitudinal momentum exhibit o…
Recently, the theory of quantized dipole polarization has been extended to account for electric multipole moments, giving rise to the discovery of multipole topological insulators (TIs). Both two-dimensional (2D) quadrupole and three-dimensional (3D) octupole TIs with robust zero-dimensional (0D) corner states have been realized in various classical systems. However, due to the intrinsic 3D limitation, the higher dimensional multipole TIs, such as four-dimensional (4D) hexadecapole TIs, are supposed to be extremely hard to construct in real space, although some of their properties have been di…
Topological physics opens up a plethora of exciting phenomena allowing to engineer disorder-robust unidirectional flows of light. Recent advances in topological protection of electromagnetic waves suggest that even richer functionalities can be achieved by realizing topological states of quantum light. This area, however, remains largely uncharted due to the number of experimental challenges. Here, we take an alternative route and design a classical structure based on topolectrical circuits which serves as a simulator of a quantum-optical one-dimensional system featuring the topological state …
Robust boundary states epitomize how deep physics can give rise to concrete experimental signatures with technological promise. Of late, much attention has focused on two distinct mechanisms for boundary robustness - topological protection, as well as the non-Hermitian skin effect. In this work, we report the first experimental realizations of hybrid higher-order skin-topological effect, in which the skin effect selectively acts only on the topological boundary modes, not the bulk modes. Our experiments, which are performed on specially designed non-reciprocal 2D and 3D topolectrical circuit l…
Knots are intricate structures that cannot be unambiguously distinguished with any single topological invariant. Momentum space knots, in particular, have been elusive due to their requisite finely tuned long-ranged hoppings. Even if constructed, probing their intricate linkages and topological "drumhead" surface states will be challenging due to the high precision needed. In this work, we overcome these practical and technical challenges with RLC circuits, transcending existing theoretical constructions which necessarily break reciprocity, by pairing nodal knots with their mirror image partne…
The study of the laws of nature has traditionally been pursued in the limit of isolated systems, where energy is conserved. This is not always a valid approximation, however, as the inclusion of features like gain and loss, or periodic driving, qualitatively amends these laws. A contemporary frontier of meta-material research is the challenge open systems pose to the established characterization of topological matter. There, one of the most relied upon principles is the bulk-boundary correspondence (BBC), which intimately relates the properties of the surface states to the topological classifi…
Topological corner states are exotic topological boundary states that are bounded to zero-dimensional geometry even the dimension of systems is large than one. As an elegant physical correspondence, their numbers are dictated by the bulk topological invariants. So far, all previous realizations of HOTIs are hallmarked by $\mathbb{Z}_2$ topological invariants and therefore have only one corner state at each corner. Here we report an experimental demonstration of $\mathbb{Z}$-class HOTI phases in electric circuits, hosting $N$ corner modes at each single corner structure. By measuring the impeda…
A non-Hermitian system is characterized by the violation of energy conservation. As a result of unbalanced gain or loss in the forward and backward directions due to non-reciprocal couplings, the eigenmodes of such systems exhibit extreme localization, also known as non-Hermitian skin effect (NHSE). This work explores unconventional scenarios where the interplay of multiple asymmetric couplings can cause the NHSE to vanish, with the admittance spectra taking identical dispersion under open boundary conditions (OBC) and periodic boundary conditions (PBC). This is unlike known non-Hermitian mode…
In this paper, we demonstrate the emergence and disappearance of topological zero modes (TZMs) in a coupled topolectrical (TE) circuit lattice. Specifically, we consider non-Hermitian TE chains in which TZMs do not occur in the individual uncoupled chains, but emerge when these chains are coupled by inter-chain capacitors. The coupled system hosts TZMs which show size-dependent behaviours and vanish beyond a certain critical size. In addition, the emergence or disappearance of the TZMs in the open boundary condition (OBC) spectra for a given size of the coupled system can be controlled by modu…
While chirality imbalances are forbidden in conventional lattice systems, non-Hermiticity can effectively avoid the chiral-doubling theorem to facilitate 1D chiral dynamics. Indeed, such systems support unbalanced unidirectional flows that can lead to the localization of an extensive number of states at the boundary, known as the non-Hermitian skin effect (NHSE). Recently, a generalized (rank-2) chirality describing a 2D robust gapless mode with dispersion $ω=k_{x}k_{y}$ has been introduced in crystalline systems. Here we demonstrate that rank-2 chirality imbalances can be established in a non…
We present a protected superconducting qubit based on an effective circuit element that only allows pairs of Cooper pairs to tunnel. These dynamics give rise to a nearly degenerate ground state manifold indexed by the parity of tunneled Cooper pairs. We show that, when the circuit element is shunted by a large capacitance, this manifold can be used as a logical qubit that we expect to be insensitive to multiple relaxation and dephasing mechanisms.
Curved spaces play a fundamental role in many areas of modern physics, from cosmological length scales to subatomic structures related to quantum information and quantum gravity. In tabletop experiments, negatively curved spaces can be simulated with hyperbolic lattices. Here we introduce and experimentally realize hyperbolic matter as a paradigm for topological states through topolectrical circuit networks relying on a complex-phase circuit element. The experiment is based on hyperbolic band theory that we confirm here in an unprecedented numerical survey of finite hyperbolic lattices. We imp…
A recent theoretical proposal suggests that a simple circuit utilizing two superinductors may produce a qubit with ground state degeneracy [P. Brooks et al., Phys. Rev. A 87, 052306 (2013)]. We perform a full circuit analysis along with exact diagonalization of the circuit Hamiltonian to elucidate the nature of the spectrum and low-lying wave functions of this $0-π$ device. We show that the ground state degeneracy is robust to disorder in charge, flux and critical current as well as insensitive to modest variations in the circuit parameters. Our treatment is non-perturbative, provides access t…
Long-range interactions (LRIs) are ubiquitous in nature. Higher-order topological (HOT) insulator represents a new phase of matter. A critical issue is how LRI dictates the HOT phases. In this work, we discover four topologically distinct phases, i.e., HOT phase, the bound state in the continuum, Dirac semimetal (DSM) phase, and trivial insulator phase in a breathing kagome circuit with tunable LRIs. We find an emerging type-III corner state in the HOT phase, which splits from the edge state continuum and originates from the strong couplings between nodes at different edges. We experimentally …
The two degenerate ground states of the anisotropic Heisenberg (XY) spin model of a chain of qubits (pseudo-spins) can encode quantum information, but their degree of protection against local perturbations is known to be only partial. We examine the properties of the system in the presence of non-local spin-spin interactions, possibly emerging from the quantum electrodynamics of the device. We find a phase distinct from the XY phase admitting two ground states which are highly protected against all local field perturbations, persisting across a range of parameters. In the context of the XY cha…
Precise control of elastic waves in modes and coherences is of great use in reinforcing nowadays elastic energy harvesting/storage, nondestructive testing, wave-mater interaction, high sensitivity sensing and information processing, etc. All these implementations are expected to have elastic transmission with lower transmission losses and higher degree of freedom in transmission path. Inspired by topological states of quantum matters, especially quantum spin Hall effects (QSHEs) providing passive solutions of unique disorder-immune surface states protected by underlying nontrivial topological …
We investigate theoretically the dynamical behavior of a qubit obtained with the two ground eigenstates of an ultrastrong coupling circuit-QED system consisting of a finite number of Josephson fluxonium atoms inductively coupled to a transmission line resonator. We show an universal set of quantum gates by using multiple transmission line resonators (each resonator represents a single qubit). We discuss the intrinsic 'anisotropic' nature of noise sources for fluxonium artificial atoms. Through a master equation treatment with colored noise and manylevel dynamics, we prove that, for a general c…
We have studied the low-energy excitations in a minimalistic protected Josephson circuit which contains two basic elements (rhombi) characterized by the $π$ periodicity of the Josephson energy. Novel design of these elements, which reduces their sensitivity to the offset charge fluctuations, has been employed. We have observed that the life time $T_{1}$ of the first excited state of this quantum circuit in the protected regime is increased up to $70μs$, a factor of $\sim100$ longer than that in the unprotected state. The quality factor $ω_{01}T_{1}$ of this qubit exceeds $10^{6}$ . Our results…
The quest for new realizations of higher-order topological system has garnered much recent attention. In this work, we propose a paradigmatic brick lattice model where corner modes requires protection by nonsymmorphic symmetry in addition to two commuting mirror symmetries. Unlike the well-known square corner mode lattice, it has an odd number of occupied bands, which necessitates a different definition for the $\mathbb Z_2\times \mathbb Z_2$ topological invariant. By studying both the quadrupolar polarization and effective edge model, our study culminates in a phase diagram containing two dis…
According to the mathematical classification of topological band structures, there exist a number of fascinating topological states in dimensions larger than three with exotic boundary phenomena and interesting topological responses. While these topological states are not accessible in condensed matter systems, recent works have shown that synthetic systems, such as photonic crystals or electric circuits, can realize higher-dimensional band structures. Here, we argue that the 4D spinless topological insulator, due to its symmetry properties, is particularly well suited to be implemented in the…
Custodial symmetries are common in the Standard Model of particle physics. They arise when quantum corrections to a parameter are proportional to the parameter itself. Here, we show that a custodial symmetry of the chiral type is also present in a classical Su-Schrieffer-Heeger (SSH) electrical circuit with memory (memcircuit). In the absence of memory, the SSH circuit supports a symmetry-protected topological edge state. Memory induces nonlinearities that break chiral symmetry explicitly and spreads the state across the circuit. However, the resulting state is still protected against perturba…
There have been considerable efforts devoted to the study of topological phases in certain non-Hermitian systems that possess real eigenfrequencies in the presence of gain and loss. However, it is challenging to experimentally realize such non-Hermitian topological insulators in either quantum or photonic systems, due to the difficulties in introducing controlled gain and loss. On the other hand, the wide choices of active circuit components provide us with unprecedented convenience and flexibility in engineering non-Hermitian topological insulators in electrical circuits. Here, we report expe…
Encoding a qubit in logical quantum states with wavefunctions characterized by disjoint support and robust energies can offer simultaneous protection against relaxation and pure dephasing. Using a circuit-quantum-electrodynamics architecture, we experimentally realize a superconducting $0-π$ qubit, which hosts protected states suitable for quantum-information processing. Multi-tone spectroscopy measurements reveal the energy level structure of the system, which can be precisely described by a simple two-mode Hamiltonian. We find that the parity symmetry of the qubit results in charge-insensiti…
Topological phases of matter are conventionally characterized by the bulk-boundary correspondence in Hermitian systems: The topological invariant of the bulk in $d$ dimensions corresponds to the number of $(d-1)$-dimensional boundary states. By extension, higher-order topological insulators reveal a bulk-edge-corner correspondence, such that $n$-th order topological phases feature $(d-n)$-dimensional boundary states. The advent of non-Hermitian topological systems sheds new light on the emergence of the non-Hermitian skin effect (NHSE) with an extensive number of boundary modes under open boun…
Recently, higher-order topological phases that do not obey the usual bulk-edge correspondence principle have been introduced in electronic insulators and brought into classical systems, featuring with in-gap corner/hinge states. So far, second-order topological insulators have been realized in mechanical metamaterials, microwave circuit, topolectrical circuit and acoustic metamaterials. Here, using near-field scanning measurements, we show the direct observation of corner states in second-order topological photonic crystal (PC) slabs consisting of periodic dielectric rods on a perfect electric…
Topologically gapless edge states, characterized by topological invariants and Berry's phases of bulk energy bands, provide amazing techniques to robustly control the reflectionless propagation of electrons, photons and phonons. Recently, a new family of topological phases, dictated by the bulk polarization, has been observed, leading to the discovery of the higher-order topological insulators (HOTIs). So far, the HOTIs are only demonstrated in discrete mechanical and electromagnetic systems and electrical circuits with the quantized quadrupole polarization. Here, we realize the higher-order t…
Quadrupole topological insulators are a new class of topological insulators with quantized quadrupole moments, which support protected gapless corner states. The experimental demonstrations of quadrupole-topological insulators were reported in a series of artificial materials, such as photonic crystals, acoustic crystals, and electrical circuits. In all these cases, the underlying structures have discrete translational symmetry and thus are periodic. Here we experimentally realize two-dimensional aperiodic-quasicrystalline quadrupole-topological insulators by constructing them in electrical ci…
Exploring the properties and applications of topological quantum states is essential to better understand topological matter. Here, we theoretically study a quasi-one-dimensional topological atom array. In the low-energy regime, the atom array is equivalent to a topological superatom. Driving the superatom in a cavity, we study the interaction between light and topological quantum states. We find that the edge states exhibit topology-protected quantum coherence, which can be characterized from the photon transmission. This quantum coherence helps us to find a superradiance-subradiance transiti…
We show the emergence of topological Bogoliubov bosonic excitations in the relatively strong coupling limit of an LC (inductance-capacitance) one-dimensional quantum circuit. This dimerized chain model reveals a ${\cal Z}_2$ local symmetry as a result of the counter-rotating wave (pairing) terms. The topology is protected by the sub-lattice symmetry, represented by an anti-unitary transformation. We present a methos to measure the winding of the topological Zak phase across the Brillouin zone by a reflection measurement of (microwave) light. Our method probes bulk quantities and can be impleme…
