Topological Zero Research Articles

Dual-band topological refractive properties in solid phononic crystals

The lossless transmission of topological edge states and the robustness of defect immunity make topological refractive materials have important application prospects in acoustic communication and integrated acoustic devices. However, most topological refractive materials only have a single operating frequency band, which obviously limits their application in multi-band and multi-function devices. Therefore, a phononic crystal with dual-frequency topological states is designed in this paper, and its dual-topological refractive properties are investigated and discussed. The existence of in-plane bulk elastic wave edge states is demonstrated by calculating the band structure of supercell comprising phononic crystals with different topological phases. Numerical simulations of the constructed acoustic topological waveguide are performed and negative refraction is observed at the low bandgap while zero refraction is observed at the high bandgap. Then, the topological zero refraction is investigated for different incidence angles. The designed dual-band acoustic topological insulator has potential applications in multi-band acoustic communication and acoustic information processing.

Dynamic Phase Enabled Topological Mode Steering in Composite Su‐Schrieffer–Heeger Waveguide Arrays

AbstractTopological boundary states localize at interfaces whenever the interface implies a change of the associated topological invariant encoded in the geometric phase. The generically present dynamic phase, however, which is energy and time‐dependent, is known to be non‐universal, and hence not to intertwine with any topological geometric phase. Using the example of topological zero modes in composite Su‐Schrieffer‐Heeger (c‐SSH) waveguide arrays with a central defect is reported on the selective excitation and transition of topological boundary mode based on dynamic phase‐steered interferences. This work thus provides a new knob for the control and manipulation of topological states in composite photonic devices, indicating promising applications where topological modes and their bandwidth can be jointly controlled by the dynamic phase, geometric phase, and wavelength in on‐chip topological devices.

Observation of monopole topological mode

Among the many far-reaching consequences of the potential existence of a magnetic monopole, it induces a topological zero mode in the Dirac equation, which was derived by Jackiw and Rebbi 48 years ago and has been elusive ever since. Here, we show that the monopole and multi-monopole solutions can be constructed in the band theory by gapping the three-dimensional Dirac points in hedgehog mass configurations. We then experimentally demonstrate such a monopole bound state in an optimized Dirac acoustic crystal structurally modulated in full solid angles. The monopole mode exhibits the optimal scaling behavior — whose modal spacing is inversely proportional to the cubic root of the modal volume. This work completes the kink-vortex-monopole zero-mode trilogy and paves the way for exploring higher-dimensional bulk-topological-defect correspondence.

Different topological phase transitions in the Su–Schrieffer–Heeger model under different disorder structures

Abstract We investigate the topological phase transition in the Su–Schrieffer–Heeger model with the long-range hopping and quasi-periodic modulation. By numerically calculating the real-space winding number, we obtain topological phase diagrams for different disordered structures. These diagrams suggest that topological phase transitions are different by selecting the specific disordered structure. When quasi-periodic modulation is applied to intracell hopping, the resulting disorder induces topological Anderson insulator (TAI) phase with high winding number (W = 2), but the topological states are destroyed as the disorder increases. Conversely, when intercell hoppings are modulated quasi-periodically, both TAI phase and the process of destruction and restoration of topological zero modes can be induced by disorder. These topological states remain robust even under strong disorder conditions. Our work demonstrates that disorder effects do not always disrupt topological states; rather, with a judicious selection of disordered structures, topological properties can be preserved.

Topological Photonic Alloy.

We present the new concept of photonic alloy as a nonperiodic topological material. By mixing nonmagnetized and magnetized rods in a nonperiodic 2D photonic crystal configuration, we realized photonic alloys in the microwave regime. Our experimental findings reveal that the photonic alloy sustains nonreciprocal chiral edge states even at very low concentration of magnetized rods. The nontrivial topology and the associated edge states of these nonperiodic systems can be characterized by the winding of the reflection phase. Our results indicate that the threshold concentrations for the investigated system within the first nontrivial band gap to exhibit topological behavior approach zero in the thermodynamic limit for substitutional alloys, while the threshold remains nonzero for interstitial alloys. At low concentration, the system exhibits an inhomogeneous structure characterized by isolated patches of nonpercolating magnetic domains that are spaced far apart within a topologically trivial photonic crystal. Surprisingly, the system manifests chiral edge states despite a local breakdown of time-reversal symmetry rather than a global one. Photonic alloys represent a new category of disordered topological materials, offering exciting opportunities for exploring topological materials with adjustable gaps.

Model-free characterization of topological edge and corner states in mechanical networks

Topological materials can host edge and corner states that are protected from disorder and material imperfections. In particular, the topological edge states of mechanical structures present unmatched opportunities for achieving robust responses in wave guiding, sensing, computation, and filtering. However, determining whether a mechanical structure is topologically nontrivial and features topologically protected modes has hitherto relied on theoretical models. This strong requirement has limited the experimental and practical significance of topological mechanics to laboratory demonstrations. Here, we introduce and validate an experimental method to detect the topologically protected zero modes of mechanical structures without resorting to any modeling step. Our practical method is based on a simple electrostatic analogy: Topological zero modes are akin to electric charges. To detect them, we identify elementary mechanical molecules and measure their chiral polarization, a recently introduced marker of topology in chiral phases. Topological zero modes are then identified as singularities of the polarization field. Our method readily applies to any mechanical structure and effectively detects the edge and corner states of regular and higher-order topological insulators. Our findings extend the reach of chiral topological phases beyond designer materials and allow their direct experimental investigation.

Interaction- and phonon-induced topological phase transitions in double helical liquids.

Helical liquids, formed by time-reversal pairs of interacting electrons in topological edge channels, provide a platform for stabilizing topological superconductivity upon introducing local and nonlocal pairings through the proximity effect. Here, we investigate the effects of electron-electron interactions and phonons on the topological superconductivity in two parallel channels of such helical liquids. Interactions between electrons in different channels tend to reduce nonlocal pairing, suppressing the topological regime. Additionally, electron-phonon coupling breaks the self duality in the electronic subsystem and renormalizes the pairing strengths. Notably, while earlier perturbative calculations suggested that longitudinal phonons have no effect on helical liquids themselves to the leading order, our nonperturbative analysis shows that phonons can induce transitions between topological and trivial superconductivity, thereby weakening the stability of topological zero modes. Our findings highlight practical limitations in realizing topological zero modes in various systems hosting helical channels, including quantum spin Hall insulators, higher-order topological insulators, and their fractional counterparts recently observed in twisted bilayer systems.

Honing in on a topological zero-bias conductance peak

A popular signature of Majorana bound states in topological superconductors is the quantized zero-energy conductance peak. However, a similar zero energy conductance peak can also arise due to non-topological reasons. Here we show that these trivial and topological zero energy conductance peaks can be distinguished via the zero energy local density of states (LDOSs) and local magnetization density of states (LMDOSs). We find that the zero-energy LDOSs and the LMDOSs exhibit periodic oscillations for a trivial zero-bias conductance peak (ZBCP). In contrast, these oscillations disappear for the topological ZBCP because of perfect Andreev reflection at zero energy in topological superconductor junctions. Our results suggest that the zero-energy LDOSs and the LMDOSs can be used as an experimental probe to distinguish a trivial zero-energy conductance peak from a topological zero-energy conductance peak.

Symmetry‐Induced Selective Excitation of Topological States in Su–Schrieffer–Heeger Waveguide Arrays

The investigation of topological state transition in carefully designed photonic lattices is of high interest for fundamental research, as well as for applied studies such as manipulating light flow in on‐chip photonic systems. Herein, the topological phase transition between symmetric topological zero modes (TZM) and antisymmetric TZMs in Su–Schrieffer–Heeger mirror symmetric waveguides is reported. The transition of TZMs is realized by adjusting the coupling ratio between neighboring waveguide pairs, which is enabled by selective modulation of the refractive index in the waveguide gaps. Bidirectional topological transitions between symmetric and antisymmetric TZMs can be achieved with proposed switching strategy. Selective excitation of topological edge mode is demonstrated owing to the symmetry characteristics of the TZMs. The flexible manipulation of topological states is promising for on‐chip light flow control and may spark further investigations on symmetric/antisymmetric TZM transitions in other photonic topological frameworks.

Quantum Simulation of Topological Zero Modes on a 41-Qubit Superconducting Processor.

Quantum simulation of different exotic topological phases of quantum matter on a noisy intermediate-scale quantum (NISQ) processor is attracting growing interest. Here, we develop a one-dimensional 43-qubit superconducting quantum processor, named Chuang-tzu, to simulate and characterize emergent topological states. By engineering diagonal Aubry-André-Harper (AAH) models, we experimentally demonstrate the Hofstadter butterfly energy spectrum. Using Floquet engineering, we verify the existence of the topological zero modes in the commensurate off-diagonal AAH models, which have never been experimentally realized before. Remarkably, the qubit number over 40 in our quantum processor is large enough to capture the substantial topological features of a quantum system from its complex band structure, including Dirac points, the energy gap's closing, the difference between even and odd number of sites, and the distinction between edge and bulk states. Our results establish a versatile hybrid quantum simulation approach to exploring quantum topological systems in the NISQ era.

Dimensional Transmutation from Non-Hermiticity.

Dimensionality plays a fundamental role in the classification of novel phases and their responses. In generic lattices of 2D and beyond, however, we found that non-Hermitian couplings do not merely distort the Brillouin zone (BZ), but can in fact alter its effective dimensionality. This is due to the fundamental noncommutativity of multidimensional non-Hermitian pumping, which obstructs the usual formation of a generalized complex BZ. As such, basis states are forced to assume "entangled" profiles that are orthogonal in a lower dimensional effective BZ, completely divorced from any vestige of lattice Bloch states unlike conventional skin states. Characterizing this reduced dimensionality is an emergent winding number intimately related to the homotopy of noncontractible spectral paths. We illustrate this dimensional transmutation through a 2D model whose topological zero modes are protected by a 1D, not 2D, topological invariant. Our findings can be readily demonstrated via the bulk properties of nonreciprocally coupled platforms such as circuit arrays, and provokes us to rethink the fundamental role of geometric obstruction in the dimensional classification of topological states.

Floquet Gauge Anomaly Inflow and Arbitrary Fractional Charge in Periodically Driven Topological-Normal Insulator Heterostructures.

Usually, when coupling in a background gauge field, topological zero modes would yield an anomalous current at the interface, culminating in the zero-mode anomaly inflow, which is ultimately conserved by extra contributions from the topological bulk. However, the anomaly inflow mechanism for guiding Floquet steady states is rarely explored in periodically driven systems. Here we synthesize a driven topological-normal insulator heterostructure and propose a Floquet gauge anomaly inflow, associated with the occurrence of arbitrary fractional charge. Through our photonic modeling, we experimentally observed a Floquet gauge anomaly as the system was driven into anomalous topological phases. Prospectively, we believe our findings could pave a novel avenue on exploring Floquet gauge anomalies in driven systems of condensed matter, photonics, and ultracold atoms.

Non-Hermitian Topolectrical Circuit Sensor with High Sensitivity.

Electronic sensors play important roles in various applications, such as industry and environmental monitoring, biomedical sample ingredient analysis, wireless networks and so on. However, the sensitivity and robustness of current schemes are often limited by the low quality-factors of resonators and fabrication disorders. Hence, exploring new mechanisms of the electronic sensor with a high-level sensitivity and a strong robustness is of great significance. Here, a new way to design electronic sensors with superior performances based on exotic properties of non-Hermitian topological physics is proposed. Owing to the extreme boundary-sensitivity of non-Hermitian topological zero modes, the frequency shift induced by boundary perturbations can show an exponential growth trend with respect to the size of non-Hermitian topolectrical circuit sensors. Moreover, such an exponential growth sensitivity is also robust against disorders of circuit elements. Using designed non-Hermitian topolectrical circuit sensors, the ultrasensitive identification of the distance, rotation angle, and liquid level is further experimentally verified with the designed capacitive devices. The proposed non-Hermitian topolectrical circuit sensors can possess a wide range of applications in ultrasensitive environmental monitoring and show an exciting prospect for next-generation sensing technologies.

Crossed products, the weak ideal property, and topological dimension zero

Let α be an action of a finite group G on a C*-algebra A and assume that A has a composition series consisting of α-invariant ideals in which every quotient of successive ideals is α-simple (this happens, e.g., if A has finitely many α-invariant ideals). We prove that in this case, the crossed product C⁎(G,A,α) has the weak ideal property ⇔ the fixed point algebra Aα has the weak ideal property ⇔ A has the weak ideal property. We show that crossed products of many C*-algebras with the ideal property by finite groups have the weak ideal property. We prove that if G is a second countable locally compact group with a closed normal subgroup H such that the group G/H is finite and abelian, then a crossed product of a separable C*-algebra by G has the weak ideal property (respectively, topological dimension zero) ⇔ the crossed product given by the restriction of the action to H has the weak ideal property (respectively, topological dimension zero). We show that, many times, if the reduced group C*-algebra of a locally compact group G has the weak ideal property, or if it has topological dimension zero, then the same is true of the reduced group C*-algebra of many subquotients of G.

Conductance spectroscopy of Majorana zero modes in superconductor-magnetic insulator nanowire hybrid systems

There has been recent interest in superconductor-magnetic insulator hybrid Rashba nanowire setups for potentially hosting Majorana zero modes at smaller external Zeeman fields. Using the non-equilibrium Green’s function technique, we develop a quantum transport model that accounts for the interplay between the quasiparticle dynamics in the superconductor-magnetic insulator bilayer structure and the transport processes through the Rashba nanowire. We provide an analysis of three-terminal setups to probe the local and non-local conductance in clean and disordered nanowires. We uncover the gap closing and reopening followed by the emergence of near-zero energy states, which can be attributed to topological zero modes in the clean limit. In the presence of a disordered potential, trivial Andreev bound states may form with signatures reminiscent of topological zero modes. Our results provide transport-based analysis of regimes that support the formation of Majorana modes in these hybrid systems while investigating the effect of disorder on devices.

Nonlocal pseudospin dynamics in a quantum Ising chain

The existence of topological zero modes in nontrivial phase of quantum Ising chain results in not only the Kramers-like degeneracy spectrum, but also dynamic response for non-Hermitian perturbation in the ordered phase (2021 Phys. Rev. Lett. 126 116 401). In this work, we investigate the possible response of the degeneracy spectrum for Hermitian perturbations. We provide a single-particle description of the model in the ordered phase, associating with an internal degree of freedom characterized as a pseudospin. The effective magnetic field, arising from both local and nonlocal perturbations in terms of string operators, acts on the pseudospin. We show that the action of string operator can be realized via a quench under the local perturbations. As an application, any ground states and excited states for the Hamiltonian with perturbation can be selected to identify the quantum phase, by adding the other perturbations to trigger a quench and measuring the Loschmidt echo.

System size dependent topological zero modes in coupled topolectrical chains

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 modulating the signs of its inverse decay length. Analytically, trivial and non-trivial phases of the coupled system can be distinguished by the differing ranks of their corresponding Laplacian matrix. The TE circuit framework enables the physical detection of the TZMs via electrical impedance measurements. Our work establishes the conditions for inducing TZMs and modulating their behavior in coupled TE chains.

Two-dimensional microtwist modeling of topological polarization in hinged Kagome lattices and its experimental validation

Kagome lattices have recently attracted a great attention because of the unique mechanical properties including their topological polarization and localized zero modes at certain edges, which challenge the standard effective continuum theories. The previous study of these systems has been predominantly focused on the ideal Kagome lattice with the spring–mass models. In this study, we stretch this paradigm by exploring the hinged Kagome lattices towards practical application to understand the topological polarization under the framework of the microtwist continuum. The hinges are modeled by ligaments capable of supporting stretching, shear and bending deformations. The microtwist elasticity is then formulated thanks to leading order two-scale asymptotics and its constitutive and balance equations are derived. Performance of the proposed theory is validated by the exact solution for predicting dispersion relations and periodic zero modes. We further demonstrate the effectiveness of this theory through numerical simulations as well as experimental testing. Finally, nonuniform deformation under complex loadings and parity asymmetric surface waves in microtwist media are explored. Our study provides a great potential of using the microtwist medium to design, control and program hinge-based metamaterials.

The Topological Dimension of Radial Julia Sets

We prove that the meandering set for $$f_a(z)=e^z+a$$ is homeomorphic to the space of irrational numbers whenever a belongs to the Fatou set of $$f_a$$ . This extends recent results by Vasiliki Evdoridou and Lasse Rempe. It implies that the radial Julia set of $$f_a$$ has topological dimension zero for all attracting and parabolic parameters, including all $$a\in (-\infty ,-1]$$ . Similar results are obtained for Fatou’s function $$f(z)=z+1+e^{-z}$$ .

Frustrated magnetic interactions in a cyclacene crystal

We study the emergence of magnetism and its interplay with structural properties in a two-dimensional molecular crystal of cyclacenes, using density functional theory (DFT). Isolated cyclacenes with an even number of fused benzenes host two unpaired electrons in two topological protected zero modes, at the top and bottom carbon rings that form the molecule. We show that, in the gas phase, electron repulsion promotes an open-shell singlet with strong intramolecular antiferromagnetic exchange. We consider a closed packing triangular lattice crystal phase and we find a strong dependence of the band structure and magnetic interactions on the rotation angle of the cyclacenes with respect to the crystal lattice vectors. The orientational ground state maximizes the intermolecular hybridization, yet local moments survive. Intermolecular exchange is computed to be antiferromagnetic, and DFT predicts a broken symmetry ${120}^{\ensuremath{\circ}}$ spin phase reflecting the frustration of the intermolecular spin coupling. Thus, the cyclacene crystal realizes a bilayer of two antiferromagnetically coupled $S=1/2$ triangular lattices. Our results provide a bottom-up route towards carbon based strongly correlated platforms in two dimensions.