Zero-energy Modes Research Articles

Majorana mode leaking into a spin-charge entangled double quantum dot

The signatures of Majorana zero-energy mode leaking into a spin-charge entangled double quantum dot are investigated theoretically in the strong electron correlation regime. The considered setup consists of two capacitively coupled quantum dots attached to external contacts and side-attached to topological superconducting wire hosting Majorana quasiparticles. We show that the presence of Majorana mode gives rise to unique features in the local density of states in the $\text{SU}(4)$ Kondo regime. Moreover, it greatly modifies the gate voltage dependence of the linear conductance, leading to fractional values of the conductance. We also analyze the effect of a finite length of topological wire and demonstrate that nonzero overlap of Majorana modes at the ends of the wire is revealed in local extrema present in the local density of states of the dot coupled directly to the wire. The calculations are performed with the aid of the numerical renormalization group method.

Large Magnetoresistance and Nontrivial Berry Phase in Nb3Sb Crystals with A15 StructureSupported by the National Key R&D Program of China (Grant No. 2016YFA0300402), the National Natural Science Foundation of China (Grant Nos. 12074335, and 11974095), and the Fundamental Research Funds for the Central Universities.

Compounds with the A15 structure have attracted extensive attention due to their superconductivity and nontrivial topological band structures. We have successfully grown Nb3Sb single crystals with the A15 structure and systematically measured the longitudinal resistivity, Hall resistivity and quantum oscillations in magnetization. Similar to other topological trivial/nontrivial semimetals, Nb3Sb exhibits large magnetoresistance (MR) at low temperatures (717%, 2 K and 9 T), unsaturating quadratic field dependence of MR and up-turn behavior in ρxx (T) curves under magnetic field, which is considered to result from a perfect hole-electron compensation, as evidenced by the Hall resistivity measurements. The nonzero Berry phase obtained from the de-Hass van Alphen (dHvA) oscillations demonstrates that Nb3Sb is topologically nontrivial. These results indicate that Nb3Sb superconductor is also a semimetal with large MR and nontrivial Berry phase. This indicates that Nb3Sb may be another platform to search for the Majorana zero-energy mode.

Research progress of elastic topological materials

Topological insulators, originated in quantum systems, are recently generalized to electromagnetic, acoustic, and elastic wave fields due to interesting wave controlling provides, such as unidirectional and dissipationless energy transportation. This new kind of materials provide unprecedented possibilities for engineering wave flows. This paper will introduce the basic theory of topological insulators and summarize the research progress of topological insulators in elastic fields. Based on one-dimensional and two-dimensional discrete models, preliminary concepts of topological insulators, such as Dirac cone, band inversion, Berry curvatures, topological invariants, are introduced. The design and progress of valley Hall insulators, Chern insulators as well as spin Hall insulators are introduced afterward, followed by discussions on high-order topological insulators. The last part of this paper covers topological phenomena in static mechanics, including topological solitons and toplogical zero energy modes.

Characterization of Shapiro steps in the presence of a 4π-periodic Josephson current

The Majorana zero-energy modes (MZMs) residing at the boundary of topological superconductors have attracted a great deal of interest recently, as they provide a platform to explore fundamental physics such as non-Abelian statistics, as well as fault-tolerant quantum computation. Period doubling of Shapiro steps in a Josephson junction under microwave irradiation has been regarded as strong evidence for the emergence of the MZMs at the junction edges. However, questions remain as to how the Shapiro steps respond to the presence of a 4{\pi}-periodic Josephson current. In this study, we investigated the characteristic features of Shapiro steps with respect to the ratio ({\alpha}) of the 4{\pi}-periodic current to the topologically trivial 2{\pi}-periodic one, as well as the reduced microwave frequency ({\Omega}) and McCumber parameter ({\beta}) of the junction. Our analysis reproduced Shapiro steps similar to those observed experimentally for specific parameter sets of {\alpha},{\Omega} ({\lesssim 0.1}), and {\beta} ({gtrsim 1.0}). Full suppression of the first lobe of the n=1 step guarantees the presence of a 4{\pi}-periodic Josephson current.In addition, we discuss the range of {\Omega} and {\beta} needed for full suppression of the first lobe of the {n=1} step, even for small {\alpha} ({<0.1}). To observe "period-doubled Shapiro steps", even with a small {\alpha}, the junction should have a large {I_c}{R_N} product and sufficiently large junction capacitance.

Tunable Dirac points and zero-energy modes in periodic curved graphene superlattices

We combined periodic ripples and electrostatic potentials to form curved graphene superlattices and studied the effects of space-dependent Fermi velocity induced from curvature on their electronic properties. With equal periods and symmetric potentials, the Dirac points do not move, but their locations shift under asymmetric potentials. This shift can be tuned by curvature and potentials. Tunable extra gaps in band structures can appear with unequal periods. The existence of new Dirac points is proposed, such that these new Dirac points can appear under smaller potentials with curvature, and their locations can be changed even under a fixed potential by adjusting the curvature. Our results suggest that curvature provides a new possible dimension to tune the electronic properties in graphene superlattices and a platform to more easier study physics near new Dirac points.

Bridging the gap between local and nonlocal numerical methods—A unified variational framework for non-ordinary state-based peridynamics

The paper aims to develop a unified variational framework to bridge the gap between the non-ordinary state-based peridynamics (NOSB-PD) and the classical continuum mechanics (CCM). First, a new force state vector is proposed by introducing the first Piola–Kirchhoff stress. This new force state vector enables the stress divergence of each material point to be expressed by averaging all the force state vectors in its support domain. The new force state vector also ensures the mathematical consistency between the strong form of PD and CCM when the horizon of a material point approaches to zero. Second, the displacement and traction boundaries in CCM are transformed into the non-local fictious boundary layers in PD, and a non-local Gauss’s formulation is presented by transforming the displacement and traction boundaries in CCM into the non-local fictious boundary layers in PD, and this formulation unifies the variational framework and boundary conditions of PD and CCM. Third, a fully implicit algorithm is developed to obtain the general nonlinear problems such as fracture and large deformation of solid materials. Further, a penalty method is employed to eliminate the zero-energy mode oscillation inherently observed in NOSB-PD, and the penalty force and penalty stiffness matrix are derived for the proposed implicit algorithm and numerical implementation. Numerical results demonstrate that the proposed method is accurate and can well capture the fracture and large deformation of solid materials. Results also indicate that the method can effectively prevent the zero-mode oscillations inherently observed in the original NOSB-PD, and thus ensures the computational stability.

Zero-energy modes in a super-chiral nanographene network of phenalenyl-tessellation molecules

We have derived a general rule for the appearance of zero-energy modes in super-chiral defective nanographene. This “super-zero-sum rule” defines the appearance of zero modes in a new class of materials, called polymerized phenalenyl-tessellation molecules (poly-PTMs). Through theoretical modeling of the electronic states in these molecular forms, we provide concrete solutions for achieving the quantum-spin systems needed in quantum-information devices. The two-dimensional graph of electronic π-orbitals in the poly-PTM possesses a number of localized zero modes equivalent to that of vacancies in PTMs. In addition to the modes confined to each PTM, another type of zero mode may appear according to the super-zero-sum rule supported by super-chirality. Since the magnetic interactions among quantum-spins in the zero modes are determined by how they appear (which is governed by the super-zero-sum rule), our rule is indispensable for designing quantum-information devices using zero modes in poly-aromatic hydrocarbons and defective graphene with vacancies.

Magneto-Optical Conductivity of Graphene: Signatures of a Uniform Y-Shaped Kekule Lattice Distortion

Recent experiments have revealed Y-shaped Kekulé lattice distortions in the honeycomb lattice on a graphene-copper superlattice[Nat. Phys. 12 (2016) 950]. In this paper the effects of such lattice distortions on the magneto-optical properties of graphene are investigated within Kubo linear-response formalism. It is found that the Y-shaped Kekulé lattice distortion lifts the valley degeneracy of all Landau levels except zero-energy mode, leading to six categories of absorption peaks in the optical absorption spectra with a distinct optical selection rule. Particular attention is paid to the absorption frequency, peak intensity of each category and its redistribution as the chemical potential is varied. Their main features are found to be different from those in undistorted graphene having their origin in the Kekulé lattice distortion. The optical response of the Y-shaped Kekulé-patterned graphene to left- and right-handed circular light is also discussed in details. The present results also suggest that magneto-optical measurements will allow to determine the Kekulé bond-texture amplitude.

Spin dynamics of the antiferromagnetic Heisenberg model on a kagome bilayer

We study the spin dynamics of classical Heisenberg antiferromagnet with nearest neighbor interactions on a quasi-two-dimensional kagome bilayer. This geometrically frustrated lattice consists of two kagome layers connected by a triangular-lattice linking layer. By combining Monte Carlo with precessional spin dynamics simulations, we compute the dynamical structure factor of the classical spin liquid in kagome bilayer and investigate the thermal and dilution effects. While the low frequency and long wavelength dynamics of the cooperative paramagnetic phase is dominated by spin diffusion, weak magnon excitations persist at higher energies, giving rise the half moon pattern in the dynamical structure factor. In the presence of spin vacancies, the dynamical properties of the diluted system can be understood within the two population picture. The spin diffusion of the "correlated" spin clusters is mainly driven by the zero-energy weather-van modes, giving rise to an autocorrelation function that decays exponentially with time. On the other hand, the diffusive dynamics of the quasi-free "orphan" spins leads to a distinctive longer time power-law tail in the autocorrelation function. We discuss the implications of our work for the glassy behaviors observed in the archetypal frustrated magnet SrCr$_{9p}$Ga$_{12-9p}$O$_{19}$ (SCGO).

Increasing the number of topological nodal lines in semimetals via uniaxial pressure

The application of pressure has been demonstrated to induce intriguing phase transitions in topological nodal-line semimetals. In this work we analyze how uniaxial pressure affects the topological character of BaSn_2, a Dirac nodal-line semimetal in the absence of spin-orbit coupling. Using calculations based on the density functional theory and a model tight-binding Hamiltonian, we find the emergence of a second nodal line for pressures higher than 4 GPa. We examine the topological features of both phases demonstrating that a nontrivial character is present in both of them. Thus, providing evidence of a topological-to-topological phase transition in which the number of topological nodal lines increases. The orbital overlap increase between Ba d_{xz} and d_{yz} orbitals and Sn p_z orbitals and the preservation of crystal symmetries are found to be responsible for the advent of this transition. Furthermore, we pave the way to experimentally test this kind of transition by obtaining a topological relation between the zero-energy modes that arise in each phase when a magnetic field is applied.

Oligomodal metamaterials with multifunctional mechanics

Mechanical metamaterials are artificial composites that exhibit a wide range of advanced functionalities such as negative Poisson's ratio, shape shifting, topological protection, multistability, extreme strength-to-density ratio, and enhanced energy dissipation. In particular, flexible metamaterials often harness zero-energy deformation modes. To date, such flexible metamaterials have a single property, for example, a single shape change, or are pluripotent, that is, they can have many different responses, but typically require complex actuation protocols. Here, we introduce a class of oligomodal metamaterials that encode a few distinct properties that can be selectively controlled under uniaxial compression. To demonstrate this concept, we introduce a combinatorial design space containing various families of metamaterials. These families include monomodal (i.e., with a single zero-energy deformation mode); oligomodal (i.e., with a constant number of zero-energy deformation modes); and plurimodal (i.e., with many zero-energy deformation modes), whose number increases with system size. We then confirm the multifunctional nature of oligomodal metamaterials using both boundary textures and viscoelasticity. In particular, we realize a metamaterial that has a negative (positive) Poisson's ratio for low (high) compression rate over a finite range of strains. The ability of our oligomodal metamaterials to host multiple mechanical responses within a single structure paves the way toward multifunctional materials and devices.

A fluid–structure interaction model for free-surface flows and flexible structures using smoothed particle hydrodynamics on a GPU

This paper presents the development of a numerical model for violent hydrodynamics of free-surface flows interacting with flexible structures using the meshless smoothed particle hydrodynamics (SPH) method accelerated with a graphics processing unit (GPU). The present work implements a unified SPH framework to solve both the fluid and structural dynamics within the open-source SPH code DualSPHysics. Well-known deficiencies with SPH-based structural modelling (linear inconsistency, tensile instability and rank deficiency) are addressed by adopting a Total Lagrangian formulation with kernel correction and zero-energy mode suppression. The fluid–structure coupling is handled via manipulation of the existing boundary condition within DualSPHysics, which does not require any geometrical knowledge of the interface. This, together with the unified SPH framework, permits straightforward implementation within DualSPHysics and retains its parallel scalability on hardware acceleration platforms, in particular on GPUs. An extensive validation and profiling of the proposed model is carried out for a range of challenging 2-D and 3-D test cases incorporating violent free-surface flows, nonlinear structural dynamics and complex fluid–structure interactions, and its performance is characterised and quantified. The analysis shows good agreement between the proposed model and analytical/benchmark solutions from the literature. Furthermore, the GPU profiling shows speedups on a single GPU of up to 40, compared to an optimised 12-core (24 threads) central processing unit (CPU) version. These findings indicate the proposed model is a viable and efficient approach to simulating violent free-surface fluid–structure interactions.

Quasi-stationary solutions in 1D non-Hermitian systems

Eigenstates exhibit localization at an open edge in a non-Hermitian lattice due to non-Hermitian skin effect. We here explore another interesting feature of non-Hermitian skin effect and predict quasi-stationary solutions, which are approximately time-independent. We show that the transition from such states to eigenstates is dramatically non-perturbative. We consider a non-Hermitian variant of the Su-Schrieffer-Heeger (SSH) model and predict non-topological but robust quasi-stationary zero energy modes.

Topolectric circuits: Theory and construction

We highlight a general theory to engineer arbitrary Hermitian tight-binding lattice models in electrical LC circuits, where the lattice sites are replaced by the electrical nodes, connected to its neighbors and to the ground by capacitors and inductors. In particular, by supplementing each node with $n$ subnodes, where the phases of the current and voltage are the $n$ distinct roots of \emph{unity}, one can in principle realize arbitrary hopping amplitude between the sites or nodes via the \emph{shift capacitor coupling} between them. This general principle is then implemented to construct a plethora of topological models in electrical circuits, \emph{topolectric circuits}, where the robust zero-energy topological boundary modes manifest through a large boundary impedance, when the circuit is tuned to the resonance frequency. The simplicity of our circuit constructions is based on the fact that the existence of the boundary modes relies only on the Clifford algebra of the corresponding Hermitian matrices entering the Hamiltonian and not on their particular representation. This in turn enables us to implement a wide class of topological models through rather simple topolectric circuits with nodes consisting of only two subnodes. We anchor these outcomes from the numerical computation of the on-resonance impedance in circuit realizations of first-order ($m=1$), such as Chern and quantum spin Hall insulators, and second- ($m=2$) and third- ($m=3$) order topological insulators in different dimensions, featuring sharp localization on boundaries of codimensionality $d_c=m$. Finally, we subscribe to the \emph{stacked topolectric circuit} construction to engineer three-dimensional Weyl, nodal-loop, quadrupolar Dirac and Weyl semimetals, respectively displaying surface and hinge localized impedance.

A gauge-fixing procedure for spherical fluid membranes and application to computations

A distinguishing feature of lipid (bilayer) membranes is their in-plane fluidity caused by free-flowing lipid molecules on the membrane surface. In continuum models for lipid membranes (e.g., the Helfrich–Canham model), fluidity manifests as invariance of the free energy to change in parametrization of the reference surface; a property termed reparametrization invariance. Two different parametric equations of the surface, related through a reparametrization, have identical equilibrium and stability properties. They can therefore be considered equivalent representations of the same surface. Since there are infinitely many ways to parametrize a surface, there are infinitely many equivalent representations for the surface. This highly redundant representation for a surface poses significant challenges to computations. For example, in computational studies using finite element analysis, extreme mesh distortion and spurious zero-energy modes are reported (Feng and Klug, 2006; Ma and Klug, 2008). In this work, by viewing reparametrization invariance as a form of gauge symmetry, we propose a gauge-fixing procedure for the case of topologically spherical membranes. We show that this procedure breaks gauge symmetry and tames the extreme redundancy of the system. We also demonstrate that this procedure is suitable for efficient numerical computations. We obtain accurate equilibrium configurations for the Helfrich–Canham model while circumventing computational issues noted above.

Observation of Distinct Spatial Distributions of the Zero and Nonzero Energy Vortex Modes in (Li_{0.84}Fe_{0.16})OHFeSe.

The energy and spatial distributions of vortex bound state in superconductors carry important information about superconducting pairing and the electronic structure. Although discrete vortex states, and sometimes a zero energy mode, had been observed in several iron-based superconductors, their spatial properties are rarely explored. In this study, we used low-temperature scanning tunneling microscopy to measure the vortex state of (Li,Fe)OHFeSe with high spatial resolution. We found that the nonzero energy states display clear spatial oscillations with a period corresponding to bulk Fermi wavelength; while in contrast, the zero energy mode does not show such oscillation, which suggests its distinct electronic origin. Furthermore, the oscillations of positive and negative energy states near E_{F} are found to be clearly out of phase. Based on a two-band model calculation, we show that our observation is more consistent with an s_{++} wave pairing in the bulk of (Li, Fe)OHFeSe, and superconducting topological states on the surface.

Self-sustained eversion or inversion of a thermally responsive torus.

When a cylindrical rod is placed on a flat and hot surface with a constant temperature, it can reach a steady state after certain time. In the steady state, though the temperature field inside the rod is inhomogeneous, it does not change with time. The inhomogeneous temperature change in the rod may induce inhomogeneous thermal expansion in it. Recent experiments have determined that if the rod is slightly curved, the inhomogeneous thermal expansion in the rod can drive its continuous and self-sustained rolling on a hot surface. It has been further shown that if the rod is bent to a closed torus and placed on a hot surface, the torus everts or inverts continuously due to the cross-coupling between the thermal field and the cyclic rotation. Such cyclic eversion or inversion of a torus can be regarded as a zero-elastic-energy mode because both the elastic energy and the shape of the torus remain unchanged during the rotation. In this article, we develop a coupled mechanics theory to model the continuous self-sustained eversion or inversion of a viscoelastic torus on a hot surface. We hope our modeling will inspire more novel designs of elastic motors being capable of zero-energy mode motion and help to quantitatively predict their performance.

Signatures of interfacial topological chiral modes via RKKY exchange interaction in Dirac and Weyl systems

We theoretically investigate the features of Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction between two magnetic impurities, mediated by the interfacial bound states inside a domain wall (DW). The latter separates the two regions with oppositely signed inversion symmetry broken terms in graphene and Weyl semimetal. The DW is modelled by a smooth quantum well which hosts a number of discrete bound states including a pair of gapless, metallic zero-energy modes with opposite chiralities. We find clear signatures of these interfacial chiral bound states in spin response (RKKY exchange interaction) which is robust to the deformation of the quantum well.

Magnetization dynamics in a Majorana-wire–quantum-dot setup

We theoretically study the quench dynamics of the local magnetization in a hybrid Majorana-wire--quantum-dot system coupled to external leads. In order to thoroughly understand the origin of the dot magnetization dynamics, we consider either normal metal or ferromagnetic electrodes. In the first case, the magnetization arises exclusively from the proximity to the topological superconductor hosting Majorana zero-energy modes and the associated development of an induced exchange field. We predict a nonmonotonic dependence of the dot's magnetization in the odd-occupation regime and show that the dynamics is governed by the magnitude of the coupling to Majorana wire. However, when the system is coupled to ferromagnetic leads, the ferromagnet and Majorana contributions to the effective exchange field are competing with each other and reveal a nontrivial dynamical behavior. As a result, the time-dependent magnetization can undergo multiple sign changes preceding the relaxation to a new thermal value. We also identify the transport regime, where fine tuning of the coupling to Majorana wire within a narrow range allows one to manipulate the magnetic state of the system. The effect of spin polarization of the leads and influence of the finite overlap between the Majorana edge modes are also examined. Moreover, we analyze the quench in the energy of the quantum dot orbital level and demonstrate that the rather straightforward charge dynamics can disguise nontrivial time evolution of the magnetization. Finally, we compare predicted dynamics with results obtained for quantum dot coupled to spin-polarized fermionic bound state instead of Majorana zero-energy mode.

Bond-associated non-ordinary state-based peridynamic model for multiple spalling simulation of concrete

The non-ordinary state-based peridynamic (NOSB PD) model has the capability of incorporating existing constitutive relationships in the classical continuum mechanics. In the present work, we first develop an NOSB PD model corresponding to the Johnson–Holmquist II (JH-2) constitutive damage model, which can describe the severe damage of concrete under intense impact compression. Besides, the numerical oscillation problem of the NOSB PD caused by zero-energy mode is analyzed and hence a bond-associated non-ordinary state-based peridynamic (BA-NOSB PD) model is adopted to remove the oscillation. Then, the elastic deformation of a three-dimensional bar is analyzed to verify the capability of BA-NOSB PD in eliminating the numerical oscillation. Furthermore, concrete spalling caused by the interaction of incident compression wave and reflected tension wave is simulated. The dynamic tensile fracture process of concrete multiple spalling is accurately reproduced for several examples according to the spalling number and spalling thickness analysis, illustrating the approach can well simulate and analyze the concrete spalling discontinuities.