Zero-energy Modes Research Articles

Optical non-Hermitian para-Fermi oscillators

We present a proposal for the optical simulation of para-Fermi oscillators in arrays of coupled waveguides. We use a representation that arises as a deformation of the $su(2)$ algebra. This provides us with a set of chiral and a zero-energy-like normal modes. The latter is its own chiral pair and suggest the addition of controlled losses/gains following a pattern defined by parity. In these non-Hermitian para-Fermi oscillators, the analog of the zero-energy mode presents the largest effective loss/gain and it is possible to tune the system to show sequences of exceptional points and varying effective losses/gains. These arrays can be used for mode suppression or enhancement depending on the use of loss or gain, in that order. We compare our coupled mode theory predictions with finite element method simulations to good agreement.

Non-Hermitian topological Anderson insulators

Non-Hermitian systems can exhibit unique topological and localization properties. Here we elucidate the non-Hermitian effects on disordered topological systems by studying a non-Hermitian disordered Su-Schrieffer-Heeger model with nonreciprocal hoppings. We show that the non-Hermiticity can enhance the topological phase against disorders by increasing energy gaps. Moreover, we uncover a topological phase which emerges only under both moderate non-Hermiticity and disorders, and is characterized by localized insulating bulk states with a disorder-averaged winding number and zero-energy edge modes. Such topological phases induced by the combination of non-Hermiticity and disorders are dubbed non-Hermitian topological Anderson insulators. We also find that the system has non-monotonous localization behaviour and the topological transition is accompanied by an Anderson transition. These properties are general in other non-Hermitian models.

Bulk-boundary correspondence in non-Hermitian systems: stability analysis for generalized boundary conditions

The bulk-boundary correspondence (BBC), i.e. the direct relation between bulk topological invariants defined for infinite periodic systems and the occurrence of protected zero-energy surface states in finite samples, is a ubiquitous and widely observed phenomenon in topological matter. In non-Hermitian generalizations of topological systems, however, this fundamental correspondence has recently been found to be qualitatively altered, largely owing to the sensitivity of non-Hermitian eigenspectra to changing the boundary conditions. In this work, we report on two contributions towards comprehensively explaining this remarkable behavior unique to non-Hermitian systems with theory. First, we analytically solve paradigmatic non-Hermitian topological models for their zero-energy modes in the presence of generalized boundary conditions interpolating between open and periodic boundary conditions, thus explicitly following the breakdown of the conventional BBC. Second, addressing the aforementioned spectral fragility of non-Hermitian matrices, we investigate as to what extent the modified non-Hermitian BBC represents a robust and generically observable phenomenon.Graphical abstract

Zero-energy modes in serially coupled double quantum dots**Project supported by the National Natural Science Foundation of China (Grant Nos. 11774418 and 11374363).

We investigate symmetrically coupled double quantum dots via the hierarchical equations of motion method and propose a novel zero-energy mode (ZEM) at a temperature above the spin singlet–triplet transition temperature. Owing to the resonance of electron quasi-particle and hole quasi-particle, ZEM has a peak at ω = 0 in the spectral density function. We further examine the effect of the magnetic field on the ZEM, where an entanglement of spin and charge has been determined; therefore, the magnetic field can split the ZEM in the spectra.

The generation of non-ordinary state-based peridynamics by the weak form of the peridynamic method

A weak form of the peridynamic (PD) method derived from the classical Galerkin framework by substituting the traditional derivatives into the PD differential operators is proposed. The attractive features of the proposed weak form of PD method include the following: (1) a higher-order approximation than the non-ordinary state-based peridynamic (NOSB-PD) in the strain construction; (2) the NOSB-PD is demonstrated as a special case of the weak form of the PD method; (3) as an extension of the NOSB-PD, the zero-energy mode oscillations in the weak form of the PD can be significantly reduced by introducing higher-order PD derivatives. In addition, a series of numerical tests are conducted. The results show the following: (1) the three proposed stabilization items containing higher-order PD derivatives have a better accuracy and stability than the traditional items of the NOSB-PD. In particular, the stress point stabilization item is preferred since it has the highest accuracy and efficiency and does not introduce any additional parameters; (2) the weak form of PD method is very suitable in dealing with the crack propagation and bifurcation problems.

Zero-Energy Modes, Fractional Fermion Numbers and The Index Theorem in a Vortex-Dirac Fermion System

Physics of topological materials has attracted much attention from both physicists and mathematicians recently. The index and the fermion number of Dirac fermions play an important role in topological insulators and topological superconductors. A zero-energy mode exists when Dirac fermions couple to objects with soliton-like structure such as kinks, vortices, monopoles, strings, and branes. We discuss a system of Dirac fermions interacting with a vortex and a kink. This kind of systems will be realized on the surface of topological insulators where Dirac fermions exist. The fermion number is fractionalized and this is related to the presence of fermion zero-energy excitation modes. A zero-energy mode can be regarded as a Majorana fermion mode when the chemical potential vanishes. Our discussion includes the case where there is a half-flux quantum vortex associated with a kink in a magnetic field in a bilayer superconductor. A normalizable wave function of fermion zero-energy mode does not exist in the core of the half-flux quantum vortex. The index of Dirac operator and the fermion number have additional contributions when a soliton scalar field has a singularity.

Quantum effects in graphitic materials: Colossal magnetoresistance, Andreev reflections, ferromagnetism, and granular superconductivity

Unlike the more common local conductance spectroscopy, nonlocal conductance can differentiate between nontopological zero-energy modes localized around inhomogeneities, and true Majorana edge modes in the topological phase. In particular, negative nonlocal conductance is dominated by the crossed Andreev reflection. Fundamentally, the effect reflects the system’s topology. In graphene, the Andreev reflection and the inter-band Klein tunneling couple electronlike and hole-like states through the action of either a superconducting pair potential or an electrostatic potential. We are here probing quantum phenomena in modified graphitic samples. Four-point contact transport measurements at cryogenic to room temperatures were conducted using a Quantum Design Physical Property Measurement System. The observed negative nonlocal differential conductance Gdiff probes the Andreev reflection at the walls of the superconducting grains coupled by Josephson effect through the semiconducting matrix. In addition, Gdiff shows the butterfly shape that is characteristic to resistive random-access memory devices. In a magnetic field, the Andreev reflection counters the effect of the otherwise lowered conduction. At low temperatures, the magnetoresistance shows irreversible yet strong giant oscillations that are known to be quantum in nature. In addition, we have found evidence for seemingly granular superconductivity. Thus, graphitic materials show potential for quantum electronics applications, including rectification and topological states.

Quantum phases of a one-dimensional Majorana-Bose-Hubbard model

Majorana zero modes (MZM-s) occurring at the edges of a 1D, p-wave, spinless superconductor, in absence of fluctuations of the phase of the superconducting order parameter, are quintessential examples of topologically-protected zero-energy modes occurring at the edges of 1D symmetry-protected topological phases. In this work, we numerically investigate the fate of the topological phase in the presence of phase-fluctuations using the density matrix renormalization group (DMRG) technique. To that end, we consider a one-dimensional array of MZM-s on mesoscopic superconducting islands at zero temperature. Cooper-pair and MZM-assisted single-electron tunneling, together with finite charging energy of the mesoscopic islands, give rise to a rich phase-diagram of this model. We show that the system can be in either a Mott-insulating phase, a Luttinger liquid (LL) phase of Cooper-pairs or a second gapless phase. In contrast to the LL of Cooper-pairs, this second phase is characterized by nonlocal string correlation functions which decay algebraically due to gapless charge-$e$ excitations. The three phases are separated from each other by phase-transitions of either Kosterlitz-Thouless or Ising type. Using a Jordan-Wigner transformation, we map the system to a generalized Bose-Hubbard model with two types of hopping and use DMRG to analyze the different phases and the phase-transitions.

Detecting non-Abelian statistics of topological states on a chain of superconducting circuits

In view of the fundamental importance and many promising potential applications, non-Abelian statistics of topologically protected states have attracted much attention recently. However, due to the operational difficulties in solid-state materials, experimental realization of non-Abelian statistics is lacking. The superconducting quantum circuit system is scalable and controllable, and thus is a promising platform for quantum simulation. Here we propose a scheme to demonstrate non-Abelian statistics of topologically protected zero-energy edge modes on a chain of superconducting circuits. Specifically, we can realize topological phase transition by varying the hopping strength and magnetic field in the chain, and the realized non-Abelian operation can be used in topological quantum computation. Considering the advantages of the superconducting quantum circuits, our protocol may shed light on quantum computation via topologically protected states.

Braiding flux-tubes in topological quantum and classical lattice models from class-D

We use magnetic flux-tubes to stabilize zero-energy modes in a lattice realization of a 2-dimensional superconductor from class D of classification table of topological condensed matter systems. The zero modes are exchanged by slowly displacing the flux-tubes and an application of the adiabatic theorem demonstrates the geometric nature of the resulting unitary time-evolution operators. Furthermore, an explicit numerical evaluation reveals that the evolutions are in fact topological, hence supplying a representation of the braid group, which turns out to be non-abelian. This physical representation is further formalized using single-strand planar diagrams. Lastly, we discuss how these predictions can be implemented with and observed in classical meta-materials and how the standard Majorana representation of the braid group can be generated by measuring derived physical observables.

Dimerization-induced topological superconductivity in a Rashba nanowire

We analyze influence of a dimerization on the topological phases of the Rashba nanowire proximitized to a superconducting substrate. We find that periodic alternations of the hopping integral and spin-orbit coupling can lead to band inversion, inducing transition to topologically nontrivial superconducting phase that hosts the Majorana zero-energy modes. This "dimerization-induced topological superconductivity" completely repels the topological phase of the uniform nanowire, whenever they happen to overlap. We provide an analytical justification for this puzzling behavior based on the symmetry/parity considerations and discuss feasible spectroscopic methods for its empirical observation. We also test stability of the topological superconducting phases against electrostatic disorder.

Assessment of MITC plate elements based on CUF with respect to distorted meshes

This paper discusses the robustness of plate elements based on Mixed Interpolation of Tensorial Components (MITC) technique and the variable kinematics approach of Carrera Unified Formulation (CUF) with respect to the problem of distorted meshes. MITC was originally proposed for Reissner-Mindlin type plates to develop shear locking free plate elements. In the present framework, refined plate elements are obtained by referring to high-order Equivalent Single Layer as well as Layer-Wise models expressed in CUF for the analysis of multilayered anisotropic structures. Four-node and nine-node elements are considered and some applications are developed for both isotropic and multilayered composite plates. Results related to the MITC approach are compared to the isoparametric elements, including selectively reduced quadrature schemes, for both, the static and free-vibration analysis. They show that CUF-MITC elements maintain their effectiveness also in the case of distorted meshes, for all the materials studied and kinematic models considered: the obtained elements are robust as free from shear locking and spurious zero-energy modes.

Robust Zero Energy Modes on Superconducting Bismuth Islands Deposited on Fe(Te,Se).

Topological superconductivity is one of the frontier research directions in condensed matter physics. One of the unique elementary excitations in topological superconducting state is the Majorana Fermion (mode) which is its own antiparticle and obeys the non-Abelian statistics and is thus useful for constructing the fault-tolerant quantum computation. The evidence for Majorana Fermions (mode) in condensed matter is now quickly accumulated. We deposit Bi islands on the iron-based superconductors Fe(Te,Se) and find the easily achievable zero energy modes on the tunneling spectra on some Bi islands. The zero energy mode is robust and appears everywhere on the island. Temperature and magnetic field dependence of the zero energy mode are also investigated. We attribute these zero energy modes to the Majorana modes due to the proximity effect-induced topological superconductivity on the Bi islands with strong spin-orbit coupling effect.

Topological phases in non-Hermitian Aubry-André-Harper models

Topological phases have recently witnessed a rapid progress in non-Hermitian systems. Here we study a one-dimensional non-Hermitian Aubry-Andr\'e-Harper model with imaginary periodic or quasiperiodic modulations. We demonstrate that the non-Hermitian off-diagonal AAH models can host zero-energy modes at the edges. In contrast to the Hermitian case, the zero-energy mode can be localized only at one edge. Such a topological phase corresponds to the existence of a quarter winding number defined by eigenenergy in momentum space. We further find the coexistence of a zero-energy mode located only at one edge and topological nonzero energy edge modes characterized by a generalized Bott index. In the incommensurate case, a topological non-Hermitian quasicrystal is predicted where all bulk states and two topological edge states are localized at one edge. Such topological edge modes are protected by the generalized Bott index. Finally, we propose an experimental scheme to realize these non-Hermitian models in electric circuits. Our findings add a new direction for exploring topological properties in Aubry-Andr\'e-Harper models.

Application of hourglass control to Eulerian smoothed particle hydrodynamics

Being a truly meshless method, smoothed particle hydrodynamics (SPH) raises expectations to naturally handle solid mechanics problems of large deformations. However, in a simple formulation it severely suffers from two instabilities, namely tensile instability and zero-energy modes, which hinders SPH from being an popular numerical tool in that area. Although Lagrangian SPH completely removes tensile instability, it is not yet able to prevent zero-energy modes. Furthermore, kernel updates are required to properly handle very large deformations which again triggers tensile instability. Additionally, Lagrangian SPH cannot naturally deal with contact problems. Pursuing an alternative route, this paper aims at stabilizing Eulerian SPH in order to accurately deal with large deformations while preserving the fundamental properties of SPH to easily handle contact problems as well as fluid–structure interaction in a straightforward monolithic manner. For this purpose, an hourglass control scheme already employed to prevent zero-energy modes in Lagrangian SPH framework is used. The advantage of the present scheme is that the stabilization method can be easily implemented in any Eulerian SPH code by making only few changes to the code. The proposed scheme is employed to simulate several cases of elasticity, plasticity, fracture and fluid–structure interaction in order to assess its accuracy and effectiveness. The obtained results are compared with analytical solutions and finite element results where very good agreement is found.

Modeling Deficiencies in the Eight-Node Mindlin Plate Finite Element Physically Explained

AbstractModeling errors that prevent the eight-node Mindlin plate finite element to behave correctly, namely shear locking and spurious zero-energy modes, can be explained using strain gradient not...

Defect-position-dependent -symmetry breaking in coupled Su–Schrieffer–Heeger chains

We investigate the spontaneous parity-time (-) symmetry-breaking behavior under three different cases in three coupled Su–Schrieffer–Heeger chains whose Hermiticity is broken by the presence of two conjugated imaginary potentials (defect terms). We show that the defect terms located at different positions in the system have significant impacts on the energy spectra in topologically trivial and nontrivial regimes. Meanwhile, we find that the topologically nontrivial zero-energy edge modes exist in symmetric region and the system undergoes several -symmetry-breaking transitions when the defect terms are located at the middle positions. Interestingly, when the defect terms are located at each position, the symmetry of the system is always spontaneously broken no matter how the loss rate changes. Under the same parameter regimes, furthermore, the energy spectra of the system for the cases that the defect terms are located at the two end positions and at the middle positions are more robust than to the case that the defect terms are located at each position.

An elementary rigorous proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model

We generalize the Su-Schrieffer-Heeger (SSH) model with the inclusion of arbitrary long-range hopping amplitudes, providing a simple framework to investigate arbitrary adiabatic deformations that preserve the chiral symmetry upon the bulk energy bands with any arbitrary winding numbers. Using only elementary techniques of solving linear difference equations and applying Cauchy's integral formula, we obtain a mathematically rigorous and physically transparent proof of the bulk-boundary correspondence for the generalized SSH model. The multiplicity of robust zero-energy edge modes is shown to be identical to the winding number. On the other hand, nonzero-energy edge modes, if any, are shown to be unstable under adiabatic deformations and not related to the topological invariant. Furthermore, under deformations of small spatial disorder, the zero-energy edge modes remain robust.

Multilayered triangular and quadrilateral flat shell elements based on the Refined Zigzag Theory

The paper presents a class of C0-continuous, flat shell elements based on the Refined Zigzag Theory (RZT) for the analysis of multilayered and curved composite and sandwich structures. The use of the interdependent interpolation strategy allows eliminating the shear-locking phenomenon and introducing the drilling rotation necessary to complete the set of classical nodal degrees of freedom (three displacements and three rotations). Additional kinematic variables are present in RZT, the zigzag rotations around the in-plane axes that measure the normal distortion typical of multilayered structures. An additional “drilling” zigzag rotation is therefore included among the nodal degrees of freedom in order to properly model curved and built-up structures. A stabilization procedure is adopted to suppress spurious zero-energy modes. A three-node triangular and a four-node quadrilateral flat shell element are formulated with 9 degrees of freedom per node. Example problems involving flat and curved multilayered structures are presented and discussed in order to assess the accuracy and convergence properties of the presented elements. Both static response predictions and free vibrations analyses are considered and the comparison is made with analytic RZT solutions, high-fidelity 3D finite element models and FSDT-based flat shell elements.

Photonic zero-energy modes in a metal-based Lieb lattice

We design a photonic tight-binding system using the dispersive background, and observe the noncompact photonic zero-energy modes for both monopole and dipolar states in a finite Lieb lattice with flat truncations. In such a photonic Lieb system, the compact localization of s, p, and d flat bands is also checked. We show that this photonic zero-energy mode is provided by one dispersive band for singular touching, which has the same frequency with the flat band states. Specially, the zero-energy mode can be completely excited by merely one point source at the flat band frequency, covering all the minority sites and forming a noncompact state. This work may provide a deep understanding about the photonic zero-energy mode for higher order states in the Lieb or other flat band models.