Zero-energy Modes Research Articles

A neural network peridynamic method for modeling rubber-like materials

Peridynamic (PD) is a powerful tool for simulating the large deformation and failure process of many types of materials. However, its use in modeling rubber-like materials is limited due to the complex constitutive nature of the material, and low efficiency and numerical oscillation caused by PD. To address this issue, a neural network (NN) non-ordinary state-based peridynamics (NOSB PD) method is developed to model the large deformation and failure behavior of rubber-like materials. This method is free of the zero-energy modes, and can significantly improve the computational efficiency. Unlike the traditional NOSB PD method that formulates the force density vector based on the deformation gradient, this method uses a deep NN to map the bond related quantities to the force density vector. The accuracy and efficiency of the proposed method are demonstrated through a series of numerical examples. Additionally, this method can be applied to various hyperelastic materials for which analytical constitutive models exist.

Chiral symmetry breaking and topological charge of zigzag graphene nanoribbons

Interacting quasi-one-dimensional zigzag graphene nanoribbons display gapped edge excitations. Although the self-consistent Hartree–Fock fields break chiral symmetry, our work demonstrates that zigzag graphene nanoribbons maintain their status as short-range entangled symmetry-protected topological insulators. The relevant symmetry involves combined mirror and time-reversal operations. In undoped ribbons displaying edge ferromagnetism, the band gap edge states with a topological charge form on the zigzag edges. An analysis of the anomalous continuity equation elucidates that this topological charge is induced by the gap term. In low-doped zigzag ribbons, where the ground state exhibits edge spin density waves, this topological charge appears as a nearly zero-energy edge mode. Our system is outside the conventional classification for topological insulators.

Realization of Jackiw–Rebbi zero-energy modes at photonic crystal domain walls: Emergence of polarization-indiscriminate surface states

The Jackiw–Rebbi model is a relativistic quantum model credited with the theoretical predictions of zero-energy bound states and charge fractionalization prior to the discovery of topological insulators and the fractional quantum Hall effect. In this work, we demonstrate a photonic equivalent of the Jackiw–Rebbi model by resorting to photonic crystal band structure engineering. Specifically, our photonic realization employs two spatial inversion symmetric binary photonic crystals exhibiting complementary signs of differential effective mass parameter (δm) for their second bandgaps. Their concatenation manifests a step discontinuity in the spatial profile of the effective mass parameter, forming a domain wall at the photonic crystal interface. Upon analyzing the reflectance spectra of the concatenated photonic crystal structure, we find a midgap surface state localized at this domain wall. Furthermore, much in agreement with the Jackiw–Rebbi zero-energy solution, the materialized photonic surface state also exhibits a zero-energy character in a differential energy space corresponding to the δm parameter, which has been quantified experimentally. Crucially, the conceived zero-energy mode amounts to the observation of a peculiar surface state with polarization-indiscriminate dispersion that can help realize all-angle polarization neutral optics.

A nonlocal energy-informed neural network for peridynamic correspondence material models

This paper presents a nonlocal energy-informed neural network to address the inherent zero-energy mode instability in peridynamic correspondence material models. Starting from the principle of virtual work, the task of solving peridynamic equilibrium equation is reformulated as a variational minimum problem of total potential energy, serving as the natural loss function for neural network. By minimizing the potential energy of the peridynamic body, the proposed neural network effectively eliminates the zero-energy modes. Importantly, this approach preserves the original peridynamic formulation without introducing a supplementary force vector state or modifying the nonlocal deformation gradient tensor. Additionally, there is no need for an artificial damping coefficient in adaptive dynamic relaxation, and the high memory consumption in implicit schemes is avoided. To improve the prediction accuracy near boundaries and to impose boundary conditions consistently with the local continuum theory, variable horizon method is introduced. Comparative studies of the proposed neural network with analytical solutions, finite element analyses or other control methods are conducted to verify the feasibility and accuracy of the proposed neural network in suppressing zero-energy mode oscillations. Furthermore, the convergence of the proposed approach and the influence of different activation functions on the prediction accuracy and convergence rate are investigated.

Orthogonality catastrophe and quantum speed limit for dynamical quantum phase transition

We investigate the orthogonality catastrophe and quantum speed limit in the Creutz model for dynamical quantum phase transitions. We show that exact zeros of the Loschmidt echo can exist in finite-size systems for specific discrete values. We highlight the role of the zero-energy mode when analyzing quench dynamics near the critical point. Additionally, we examine the behaviors of the time for the first exact zeros of the Loschmidt echo and the corresponding quantum speed limit time as the system size increases. While the bound is not tight, it can be attributed to the scaling properties of the band gap and energy variance with respect to system size. As such, we establish a link between the orthogonality catastrophe and quantum speed limit by referencing the full form of the Loschmidt echo. In addition, we reveal the potential that the quantum speed limit holds to detect static quantum phase transition point and a reduced amplitude of the noise induced behaviors of quantum speed limit.

A 3D pantographic metamaterial behaving as a mechanical shield: Experimental and numerical evidence

Multilayer pantographic metamaterials, in short, pantographic blocks, have shown peculiar mechanical behavior, especially when their constitutive hinges are revolving (i.e., perfect) joints. The pantographic block, which is the subject of the present paper, has been printed using a Powder Bed Fusion technology and its hinges may be modeled as perfect ones. In the reported in situ 3-point flexural test, the predictions obtained by second gradient models for its mechanical response are shown to be experimentally consistent thanks to measurements via Digital Volume Correlation. The deformation applied by the upper central support is almost entirely shielded by the pantographic block, namely, the specimen barely crosses through the reference bottom plane defined by the lower lateral supports, even when subjected to very large deformations. The mathematical model employed herein captures this observation in terms of a nonlinear ‘arching’ effect activated in the beams of the pantographic structure, provided elastic locking is introduced to prevent pantographic zero-energy modes.

A versatile implicit computational framework for continuum-kinematics-inspired peridynamics

Continuum-kinematics-inspired peridynamics (CPD) has been recently proposed as a novel reformulation of peridynamics that is characterized by one-, two- and three-neighbor interactions. CPD is geometrically exact and thermodynamically consistent and does not suffer from zero-energy modes, displacement oscillations or material interpenetration. In this manuscript, for the first time, we develop a computational framework furnished with automatic differentiation for the implementation of CPD. Thereby, otherwise tedious analytical differentiation is automatized by employing hyper-dual numbers (HDN). This differentiation method does not suffer from round-off errors, subtractive cancellation errors or truncation errors and is thereby highly stable with superb accuracy being insensitive to perturbation values. The computational framework provided here is compact and model-independent, thus once the framework is implemented, any other material model can be incorporated via modifying the potential energy solely. Finally, to illustrate the versatility of our proposed framework, various potential energies are considered and the corresponding material response is examined for different scenarios.

Edge state behavior in a Su–Schrieffer–Heeger like model with periodically modulated hopping

Su–Schrieffer–Heeger (SSH) model is one of the simplest models to show topological end/edge states and the existence of Majorana fermions. Here we consider a SSH like model both in one and two dimensions where a nearest neighbor hopping features spatially periodic modulations. In the 1D chain, we witness appearance of new in-gap end states apart from a pair of Majorana zero modes (MZMs) when the hopping periodicity go beyond two lattice spacings. The pair of MZMs, that appear in the topological regime, characterize the end modes each existing in either end of the chain. These, however, crossover to both-end end modes for small hopping modulation strength in a finite chain. Contrarily in a 2D SSH model with symmetric hopping that we consider, both non-zero and zero energy topological states appear in a finite square lattice even with a simple staggered hopping, though the zero energy modes disappear in a ribbon configuration. Apart from edge modes, the 2D system also features corner modes as well as modes with satellite peaks distributed non-randomly within the lattice. In both the dimensions, an increase in the periodicity of hopping modulation causes the zero energy Majorana modes to become available for either sign of the modulation. But interestingly with different periodicity for hopping modulations in the two directions, the zero energy modes in a 2D model become rarer and does not appear for all strength and sign of the modulation.

Planar and tunable quantum state transfer in a splicing Y-junction Su–Schrieffer–Heeger chain

We present a feasible scheme to implement a planar and tunable quantum state transfer (QST) via topologically protected zero-energy mode in a splicing Y-junction Su–Schrieffer–Heeger (SSH) chain. The introduction of the elaborate nearest-neighbor (NN) hopping enables one to generate a topological interface at the central site of the Y-junction. By modulating the NN hopping adiabatically in the chain, the quantum state initially prepared at the central site can be simultaneously transferred to the three endpoints of the Y-junction with the equal/unequal probabilities. The planar distribution of QST is expected to realize a quantum router, whose function is to make the quantum information on the central site (input port) appear equally/unequally at the three endpoints (output ports) with different directions. Moreover, the numerical simulations demonstrate that the scheme possesses the robustness on the fluctuations of the NN hopping and the on-site potential in the system. Furthermore, we show that the number of the output ports with different directions can be flexibly increased in an extended X-junction SSH chain, and the experimental feasibility for implementing special QST in a superconducting qubit-resonator system is briefly discussed. Our work extends the space distribution of QST from linear distribution to planar distribution and promotes the construction of large-scale quantum networks.

The simplified MITC4+ shell element and its performance in linear and nonlinear analysis

The recently developed improved MITC4+ shell element exhibits excellent convergence behavior in both bending- and membrane-dominated shell problems. Compared to the widely recognized MITC4 shell element, the membrane and in-plane shear locking were significantly alleviated. However, it is difficult to implement the assumed membrane strain field because of its complexity. In this study, a simplified MITC4+ shell element is developed. A simpler assumed membrane strain field that facilitates the extension to nonlinear formulations is proposed. To reduce the sensitivity of the element performance to mesh distortion, a geometry-dependent Gauss integration scheme is employed. The proposed element passes the patch, zero-energy mode, and isotropy tests. The performance of the simplified MITC4+ shell element is demonstrated using various linear and nonlinear benchmark problems.

Spin effects on transport and zero-bias anomaly in a hybrid Majorana wire-quantum dot system

We examine the impact of spin effects on the nonequilibrium transport properties of a nanowire hosting Majorana zero-energy modes at its ends, coupled to a quantum dot junction with ferromagnetic leads. Using the real-time diagrammatic technique, we determine the current, differential conductance and current cross-correlations in the nonlinear response regime. We also explore transport in different magnetic configurations of the system, which can be quantified by the tunnel magnetoresistance. We show that the presence of Majorana quasiparticles gives rise to unique features in all spin-resolved transport characteristics, in particular, to zero-bias anomaly, negative differential conductance, negative tunnel magnetoresistance, and it is also reflected in the current cross-correlations. Moreover, we study the dependence of the zero-bias anomaly on various system parameters and demonstrate its dependence on the magnetic configuration of the system as well as on the degree of spin polarization in the leads. A highly nontrivial behavior is also found for the tunnel magnetoresistance, which exhibits regions of enhanced or negative values—new features resulting from the coupling to Majorana wire.

An effective correspondence-based peridynamics-FEM coupling model for brittle fracture

Correspondence-based peridynamics (CBPD) is attractive and promising because of its outstanding ability to employ the traditional constitutive relations for material modeling. However, application of CBPD to brittle fracture problems is somewhat limited due to the numerical instability issue and lack of a rational bond failure criterion. To enable CBPD to accurately and stably predict brittle fracture problems, an improved energy-based bond failure criterion is firstly established in this work, in which the expression of bond potential is re-derived, and its simplified expression for isotropic linear elastic solids is clearly defined. The critical bond potential is determined by assuming it is directly proportional to the bond length, and the discretization error for the critical energy release rate in the discretized model is corrected with a correction factor. Besides the zero-energy mode control method, additional stabilization technique to reduce the influence of ill-posed or singular reference shape tensor on brittle fracture simulations is also explored to avoid the unexpected irregular damages during the fracture process. In addition, a direct coupling model of CBPD and FEM is established to improve the computational efficiency, and the ghost forces at the interface are verified to be negligible. Finally, several numerical examples involving typical quasi-static and dynamic brittle fracture problems are investigated, and fracture behaviors including the crack paths, load-displacement curves, and crack propagation speeds are all well predicted, which sufficiently demonstrates the accuracy and stability of the proposed method.

Gate-Defined Topological Josephson Junctions in Bernal Bilayer Graphene.

Recent experiments on Bernal bilayer graphene (BLG) deposited on monolayer WSe_{2} revealed robust, ultraclean superconductivity coexisting with sizable induced spin-orbit coupling. Here, we propose BLG/WSe_{2} as a platform to engineer gate-defined planar topological Josephson junctions, where the normal and superconducting regions descend from a common material. More precisely, we show that if superconductivity in BLG/WSe_{2} is gapped and emerges from a parent state with intervalley coherence, then Majorana zero-energy modes can form in the barrier region upon applying weak in-plane magnetic fields. Our results spotlight a potential pathway for "internally engineered" topological superconductivity that minimizes detrimental disorder and orbital-magnetic-field effects.

Stabilized state-based peridynamics for elasticity emanating from constrained Lagrangian

State-based peridynamics for elasticity paves the way to bridge the local theory with the inherent capabilities of general elasticity with non-local effects. However, the original non-ordinary state-based peridynamics with correspondence materiel models suffers from high-frequency oscillations due to zero-energy mode instability. To address this issue, we start from the fundamental Lagrangian with the introduction of a constraint potential energy that minimizes the numerical error and suppresses the zero-energy mode oscillations. A stabilized non-ordinary state-based peridynamics is then derived from this constrained Lagrangian that forms an integral–differential–algebraic system. In specifics, the equation of motion for particles is described by integral-differential equations while the constraint suppressing the zero-energy mode instability is expressed as algebraic equations (zero-error constraint). An energy–momentum conserving implicit time-stepping scheme for this integral–differential–algebraic system is then derived. End-to-end 2D and 3D simulations, including scalar wave propagation, crack path, and large deformation problems, are presented to show that the proposed method is promising for these applications in terms of accurately capturing the crack path and large deformation, efficiently eliminating the numeral oscillations, conserving the total energy for long duration simulations, and preserving the zero-error constraint.

Wiener–Hopf factorization approach to a bulk-boundary correspondence and stability conditions for topological zero-energy modes

Both the physics and applications of fermionic symmetry-protected topological phases rely heavily on a principle known as bulk-boundary correspondence, which predicts the emergence of protected boundary-localized energy excitations (boundary states) if the bulk is topologically non-trivial. Current theoretical approaches formulate a bulk-boundary correspondence as an equality between a bulk and a boundary topological invariant, where the latter is a property of boundary states. However, such an equality does not offer insight into the stability or the sensitivity of the boundary states to external perturbations. To solve this problem, we adopt a technique known as the Wiener–Hopf factorization of matrix functions. Using this technique, we first provide an elementary proof of the equality of the bulk and the boundary invariants for one-dimensional systems with arbitrary boundary conditions in all Altland–Zirnbauer symmetry classes. This equality also applies to quasi-one-dimensional systems (e.g., junctions) formed by bulks belonging to the same symmetry class. We then show that only topologically non-trivial Hamiltonians can host stable zero-energy edge modes, where stability refers to continuous deformation of zero-energy excitations with external perturbations that preserve the symmetries of the class. By leveraging the Wiener–Hopf factorization, we establish bounds on the sensitivity of such stable zero-energy modes to external perturbations. Our results show that the Wiener–Hopf factorization is a natural tool to investigate bulk-boundary correspondence in quasi-one-dimensional fermionic symmetry-protected topological phases. Our results on the stability and sensitivity of zero-energy modes are especially valuable for applications, including Majorana-based topological quantum computing.

Photonic topological insulators induced by non-Hermitian disorders in a coupled-cavity array

Recent studies of disorder or non-Hermiticity induced topological insulators inject new ingredients for engineering topological matter. Here, we consider the effect of purely non-Hermitian disorders, a combination of these two ingredients, in a 1D coupled-cavity array with disordered gain and loss. Topological photonic states can be induced by increasing gain-loss disorder strength with topological invariants carried by localized states in the complex bulk spectra. The system showcases rich phase diagrams and distinct topological states from Hermitian disorders. The non-Hermitian critical behavior is characterized by the biorthogonal localization length of zero-energy edge modes, which diverges at the critical transition point and establishes the bulk-edge correspondence. Furthermore, we show that the bulk topology may be experimentally accessed by measuring the biorthogonal chiral displacement, which can be extracted from a proper Ramsey interferometer that works in both clean and disordered regions. The proposed coupled-cavity photonic setup relies on techniques that have been experimentally demonstrated and, thus, provides a feasible route toward exploring such non-Hermitian disorder driven topological insulators.

Majorana Corner Modes and Flat-Band Majorana Edge Modes in Superconductor/Topological-Insulator/Superconductor Junctions

Recently, superconductors with higher-order topology have stimulated extensive attention and research interest. Higher-order topological superconductors exhibit unconventional bulk-boundary correspondence, thus allow exotic lower-dimensional boundary modes, such as Majorana corner and hinge modes. However, higher-order topological superconductivity has yet to be found in naturally occurring materials. We investigate higher-order topology in a two-dimensional Josephson junction comprised of two s-wave superconductors separated by a topological insulator thin film. We find that zero-energy Majorana corner modes, a boundary fingerprint of higher-order topological superconductivity, can be achieved by applying magnetic field. When an in-plane Zeeman field is applied to the system, two corner modes appear in the superconducting junction. Furthermore, we also discover a two-dimensional nodal superconducting phase which supports flat-band Majorana edge modes connecting the bulk nodes. Importantly, we demonstrate that zero-energy Majorana corner modes are stable when increasing the thickness of topological insulator thin film.

Boson star normal modes

Boson stars are gravitationally bound objects that arise in ultralight dark matter models and form in the centers of galactic halos or axion miniclusters. We systematically study the excitations of a boson star, taking into account the mixing between positive and negative frequencies introduced by gravity. We show that the spectrum contains zero-energy modes in the monopole and dipole sectors resulting from spontaneous symmetry breaking by the boson star background. We analyze the general properties of the eigenmodes and derive their orthogonality and completeness conditions which have non-standard form due to the positive-negative frequency mixing. The eigenvalue problem is solved numerically for the first few energy levels in different multipole sectors and the results are compared to the solutions of the Schrödinger equation in fixed boson star gravitational potential. The two solutions differ significantly for the lowest modes, but get close for higher levels. We further confirm the normal mode spectrum in 3D wave simulations where we inject perturbations with different multipoles. As an application of the normal mode solutions, we compute the matrix element entering the evaporation rate of a boson star immersed in a hot axion gas. The computation combines the use of exact wavefunctions for the low-lying bound states and of the Schrödinger approximation for the high-energy excitations.

Refined Simulation of Reinforced Concrete Beam Based on a Hybrid Peridynamic Method

Reinforced concrete (RC) structures under earthquake excitation may fail and cause significant casualties and economic losses, highlighting the importance of studying their seismic failure mechanisms. Considering that the commonly used finite element method and discrete element method have inherent limitations, a more efficient meshless method, known as peridynamics (PD), has been proposed and applied in various areas. PD has two types, namely, bond-based peridynamics (BPD) and state-based peridynamics (SPD). BPD is limited by its fixed Poisson’s ratio, while SPD suffers from the zero-energy mode issue. A hybrid peridynamics (HPD) method is introduced in this paper to overcome these limitations, as it establishes bonds between each PD point and other PD points within its horizon and sums up all bond forces on the PD point to calculate the total force. The proposed HPD method is then applied to simulate three RC beams with different shear span-to-depth ratios. The simulation results, including the shear force–deflection of the beams, shear force–strain of stirrups, crack formation and propagation, and diagonal crack width, are compared against experimental data. The proposed HPD method is demonstrated as being capable of simulating RC structures’ behaviors in an accurate and stable manner.

A meshfree model of hard-magnetic soft materials

Hard-magnetic soft materials comprising a soft matrix embedded with hard-magnetic particles can exhibit large deformations under external magnetic fields, which have attracted much interest due to their non-contact activation, flexible programmability, and rapid response in various applications such as biomedical devices, soft robotics and flexible electronics. Precise predictions of large deformations of hard-magnetic soft materials would be a key for relevant applications. Given that the rational designs in the programmable shape morphing of ferromagnetic structures requires high precision, here we develop a meshfree model based on the radial point interpolation method to quantitatively predict large deformations of hard-magnetic soft materials. We apply the stabilized conforming nodal integration instead of direct nodal integration to eliminate the influence of zero-energy mode, and the central difference scheme is implemented for the resolution procedure. We explore the bending of a hard-magnetic beam and snap buckling of a bistable hard-magnetic beam under external magnetic stimuli. Uniform and non-uniform discretizations are compared to show the insensitivity of the radial point interpolation method to nodal mesh. To demonstrate the versatile programmability of our model in the design of magnetic robotics, we further simulate complex locomotion (crawling, walking and rolling) of hard-magnetic soft robots under external magnetic excitation. The results could be used to guide rational designs of ferromagnetic structures and soft continuum robots.