Majorana zero modes induced by skyrmion lattice

One-dimensional s-wave superconductor with spin-orbit coupling is a platform for the realization of Majorana zero modes. The spin-exchange with the magnetic skyrmion lattice can induce spin-orbit coupling in a s-wave superconductor system and the effects are different from the constant spin–orbit coupling. The strength of the effective spin–orbit coupling as well as the rich topoloigcal phase diagram are directly connected to the radius of the skyrmion lattice R. We obtain the rich topological phase diagram of this system with different skyrmion lattice radii by numerically evaluating the spectrum of the system under the periodic boundary condition, and we also find the Majorana zero modes under the open boundary condition to verify the bulk–edge correspondence.

Exploring Majorana zero modes in iron-based superconductors

Majorana zero modes(MZMs) are Majorana-fermion-like quasiparticles existing in crystals or hybrid platforms with topologically non-trivial electronic structures. They obey non-Abelian braiding statistics and are considered promising to realize topological quantum computing. Discovery of MZM in the vortices of the iron-based superconductors(IBSs)has recently fueled the Majorana research in a way which not only removes the material barrier requiring construction of complicated hybrid artificial structures, but also enables observation of pure MZMs under higher temperatures. So far,MZMs have been observed in iron-based superconductors including FeTe_(0.55)Se_(0.45),(Li_(0.84)Fe_(0.16))OHFe Se, Ca KFe_(4)As_(4),and Li Fe As. In this topical review, we present an overview of the recent STM studies on the MZMs in IBSs. We start with the observation of MZMs in the vortices in FeTe_(0.55)Se_(0.45)and discuss the pros and cons of FeTe_(0.55)Se_(0.45) compared with other platforms. We then review the following up discovery of MZMs in vortices of Ca KFe_(4)As_(4), impurity-assisted vortices of Li Fe As, and quantum anomalous vortices in FeTe_(0.55)Se_(0.45), illustrating the pathway of the developments of MZM research in IBSs. Finally, we give perspective on future experimental works in this field.

Majorana zero modes,unconventional real-complex transition,and mobility edges in a one-dimensional non-Hermitian quasi-periodic lattice

A one-dimensional non-Hermitian quasiperiodic p-wave superconductor without PT-symmetry is studied.By analyzing the spectrum,we discovered that there still exists real-complex energy transition even if the inexistence of PT-symmetry breaking.By the inverse participation ratio,we constructed such a correspondence that pure real energies correspond to the extended states and complex energies correspond to the localized states,and this correspondence is precise and effective to detect the mobility edges.After investigating the topological properties,we arrived at a fact that the Majorana zero modes in this system are immune to the non-Hermiticity.

Topological phase diagrams and Majorana zero modes of the Kitaev ladder and tube

In this paper,we study two quasi-one-dimensional（1 D） Kitaev models with ladder-like and tube-like spatial structures,respectively.Our results provide the phase diagrams and explicit expressions of the Majorana zero modes.The topological phase diagrams are obtained by decomposing the topological invariants and the topological conditions for topologically nontrivial phases are given precisely.For systems which belongs to topological class BDI,we obtain the regions in the phase diagrams where the topological numbers show even-odd effect.For the Kitaev tube model a phase factor induced by the magnetic flux in the axial direction of the tube is introduced to alter the classification of the tube Hamiltonian from class BDI to D.The Kitaev tube of class D is characterized by the Z2 index when the number of chains is odd while 0,1,2 when the number of chains is even.The phase diagrams show periodic behaviors with respect to the magnetic flux.The bulk-boundary correspondence is demonstrated by the observations that the topological conditions for the bulk topological invariant to take nontrivial values are precisely those for the existence of the Majorana zero modes.

Cross correlation mediated by distant Majorana zero modes with no overlap

Existing studies via shot noise calculation conclude that the cross correlation between the currents in the two leads connected by a pair of Majorana zero modes(MZMs)vanishes when their coupling energy𝜖∈_(M)→0.Motivated by the intrinsic nature of nonlocality of the MZMs,we revisit this important problem and propose an experimental scheme to demonstrate the nonvanishing cross correlation even at the limit𝜖∈_(M)→0.The proposed scheme employs the Andreevprocess-associated branch circuit currents,which are theoretically obtained by applying a decomposition analysis for the total currents while are accessible directly in practical measurement.For different bias voltage setup,we find intriguing results of both negative and positive correlations and carry out simple physical understanding using a quantum jump technique.Importantly,combining together with the evidence of the zero-bias-peak of conductance,the nonlocal cross correlation predicted in this work can help to confirm the existence of the nonlocal MZMs.

Effect of Interaction on the Majorana Zero Modes in the Kitaev Chain at Half Filling

The one-dimensional interacting Kitaev chain at half filling is studied. The symmetry of the Hamiltonian is examined by dual transformations, and various physical quantities as a function of the fermion-fermion interaction U are calculated systematically using the density matrix renormalization group method. A special value of interaction Up is revealed in the topological region of the phase diagram. We show that at Up the ground states are strictly two-fold degenerate even though the chain length is finite and the zero-energy peak due to the Majorana zero modes is maximally enhanced and exactly localized at the end sites. Here Up may be attractive or repulsive depending on other system parameters. We also give a qualitative understanding of the effect of interaction under the self-consistent mean field framework.

Majorana noise model and its influence on the power spectrum

Majorana quantum computation offers a potential approach to securely manipulating and storing quantum data in a topological manner that may effectively resist the decoherence induced by local noise. However, actual Majorana qubit setups are susceptible to noise. In this study, from a quantum dynamics perspective, we develop a noise model for Majorana qubits that accounts for quasi-particle poisoning and Majorana overlapping with fluctuation. Furthermore, we focus on Majorana parity readout methodologies, specifically those leveraging an ancillary quantum dot, and carry out an indepth exploration of continuous measurement techniques founded on the quantum jump model of a quantum point contact.Utilizing these methodologies, we proceed to analyze the influence of noise on the afore-mentioned noise model, employing numerical computation to evaluate the power spectrum and frequency curve. In the culmination of our study, we put forward a strategy to benchmark the presence and detailed properties of noise in Majorana qubits.

Topological phase transitions and Majorana zero modes induced by the periodic potential in an antiferromagnetic chain

We investigate the topological properties of an antiferromagnetic(AFM)chain with an on-site periodic potential,considering the intrinsic spin–orbit coupling and an external Zeeman field along with the nanowire.Our results indicate that Majorana zero modes(MZMs)can be observed by adjusting the strength of the periodic potential.We have calculated the energy spectrum,the wave-function and transport properties,and all these results support the existence of MZMs in the AFM chain.Additionally,multiple topological phase transitions occur as the strength of the periodic potential changes,and several regions support MZMs.

Recent progress on non-Abelian anyons:from Majorana zero modes to topological Dirac fermionic modes

Majorana zero modes(MZMs)have been intensively studied in recent decades theoretically and experimentally as the most promising candidate for non-Abelian anyons supporting topological quantum computation(TQC).In addition to the Majorana scheme,some non-Majorana quasiparticles obeying non-Abelian statistics,including topological Dirac fermionic modes,have also been proposed and therefore become new candidates for TQC.In this review,we overview the non-Abelian braiding properties as well as the corresponding braiding schemes for both the MZMs and the topological Dirac fermionic modes,emphasizing the recent progress on topological Dirac fermionic modes.A topological Dirac fermionic mode can be regarded as a pair of MZMs related by unitary symmetry,which can be realized in a number of platforms,including the one-dimensional topological insulator,higher-order topological insulator,and spin superconductor.This topological Dirac fermionic mode possesses several advantages compared with its Majorana cousin,such as superconductivity-free and larger gaps.Therefore,it provides a new avenue for investigating non-Abelian physics and possible TQC.

Majorana zero mode in the vortex of an artificial topological superconductor

Majorana fermion(MF),an exotic particle that is identical to its own antiparticle,was recently found in solid matter as a quasiparticle excitation,the Majorana zero mode(MZM),in the vortex of an artificial topological superconductor(TSC).This artificial TSC,first proposed by Fu and Kane in 2008,is a heterostructure made of a topological insulator Bi_2Te_3 and an s-wave superconductor NbSe_2.This paper will briefly review the experimental progresses based on the Bi_2Te_3/NbSe_2 heterostructure.All evidences are self-consistent and reveal that the MZM exists in the center of vortex.Those experimental results are also supported by theory.This finding is a milestone in the research of Majorana fermions in solid state physics and a starting point of MZM's application in topological quantum computation.

Hexagonal warping effect on Majorana zero modes at the ends of superconducting vortex lines in doped strong 3D topological insulators

In a superconducting topological insulator, a superconducting vortex line can trap a one-dimensional topological band with localized Majorana zero modes at the ends. Here, we study the effect of hexagonal warping and its corresponding symmetry-breaking effect on vortex phase transition. We perform both analytical calculations based on a semiclassical formula and numerical calculations based on full quantum mechanics using the Bogoliubov-de Gennes equation. We find that the hexagonal warping term extends the topological region of the vortex line as the chemical potential changes and leads to MZMs, even in the absence of topological surface states.

Explicit forms of zero modes in symmetric interacting Kitaev chain without and with dimerization

The fermionic and bosonic zero modes of the one-dimensional（1D） interacting Kitaev chain at the symmetric point are unveiled. The many-body structures of the Majorana zero modes in the topological region are given explicitly by carrying out a perturbation expansion up to infinite order. We also give the analytic expressions of the bosonic zero modes in the topologically trivial phase. Our results are generalized to the hybrid fermion system comprised of the interacting Kitaev model and the Su–Schrieffer–Heeger（SSH） model, in which we show that these two types of zero modes can coexist in a certain region of its phase diagram.

铁基超导体中的马约拉纳零能模及其阵列构筑

马约拉纳零能模服从非阿贝尔统计,其编织操作可用于构筑拓扑量子比特,是拓扑量子计算的基本单元,可从原理上解决量子计算中环境噪声带来的退相干问题.现有的马约拉纳零能模平台包括复合异质结构,如拓扑绝缘体/超导体、半导体纳米线/超导体或一维磁性原子链/超导体等,以及单一材料,如2M-WS_(2),4Hb-TaS_(2)和铁基超导体等.铁基超导体中的马约拉纳零能模具有材料平台简单、零能模纯净以及存活温度较高等一系列优势,引起了广泛关注.最近,大面积、有序和可调控的马约拉纳零能模晶格阵列在铁基超导体LiFeAs中被观测到,为未来的拓扑量子计算提供了一个理想平台.本综述首先回顾铁基超导体中马约拉纳零能模的实验观测,其中将重点介绍FeTe_(0.55)Se_(0.45),(Li_(0.84)Fe_(0.16))OHFeSe,CaKFe_(4)As_(4)和LiFeAs等材料体系.接着介绍给出铁基超导体中马约拉纳零能模关键性实验证据的一系列工作.然后进一步详细介绍近期LiFeAs中观测到有序和可调马约拉纳零能模晶格阵列的工作.最后给出总结和对未来马约拉纳领域研究的展望.

Topological phase boundary in a generalized Kitaev model

We study the effects of the next-nearest-neighbor hopping and nearest-neighbor interactions on topological phases in a one-dimensional generalized Kitaev model. In the noninteracting case, we define a topological number and calculate exactly the phase diagram of the system. With addition of the next-nearest-neighbor hopping, the change of phase boundary between the topological and trivial regions can be described by an effective shift of the chemical potential. In the interacting case, we obtain the entanglement spectrum, the degeneracies of which correspond to the topological edge modes, by using the infinite time-evolving block decimation method. The results show that the interactions change the phase boundary as adding an effective chemical potential which can be explained by the change of the average number of particles.

Majorana准粒子与超导体-半导体异质纳米线

Majorana准粒子是凝聚态物理版本的Majorana费米子.由于Majorana准粒子间的交换操作服从非阿贝尔统计,并基于此可构建更稳定的量子计算机,近年来在凝聚态物理界引起广泛关注.为帮助初学者快速理解Majorana准粒子的形成机理,本文回顾了在一维超导体-半导体异质纳米线系统中Majorana准粒子模型的提出和理论演化过程,介绍Kitaev链模型并分析了模型中各要素所起的作用.还介绍了典型Majorana器件的构成和测量方法,并结合最新的实验进展对探测到的零能电导峰进行了分析和述评.最后对超越一维系统的超导体-半导体异质系统的实验前景进行了展望.

电子型FeSe基高温超导体的磁通束缚态与Majorana零能模

作为凝聚态物理中一类新奇准粒子态,Majorana零能模(Majorana zero mode)由于可用来实现拓扑量子计算而成为当前的研究热点.理论预言,Majorana零能模可作为特殊的束缚态出现在一些拓扑超导体的磁通涡旋中.但实际超导体磁通中还可能存在其他低能束缚态或杂质态,这给Majorana零能模的辨别和具体应用带来了困难.目前实验上寻找合适的拓扑超导体系、分辨出清晰的Majorana零能模仍然是十分迫切的.本文主要介绍最近利用高能量分辨的扫描隧道显微镜,对电子掺杂铁硒类超导体(Li,Fe)OHFeSe和单层FeSe/SrTiO3磁通态进行的研究.实验上在前者的自由磁通中观测到清晰的零能模,并进一步测量到Majorana零能模的重要特征—量子化电导.而在后者磁通中只发现常规Caroli-de Gennes-Matricon(CdGM)束缚态,反映出s波对称性的特征.这系列实验既为Majorana零能模物性的进一步研究提供了合适平台,也为澄清铁基超导体中拓扑超导电性的来源提供了线索.

Enhanced topological superconductivity in an asymmetrical planar Josephson junction

As a platform for holding Majorana zero models(MZMs),the two-dimensional planar topological Josephson junction that can be used as carriers for topological quantum computing faces some challenges.One is a combination of mirror and time-reversal symmetries may make the system hold multiple pairs of MZMs.The other is that a soft gap dominated by a large momentum occurs in a clean system.To solve these problems,asymmetric junction can be introduced.Breaking this symmetry changes the symmetry class from class BDI to class D,and only a single pair of MZMs can be left at the boundary of the system.We numerically study four cases that create an asymmetric system and find out different superconducting pairing potential,different coupling coefficients between two-dimensional electron gases(2DEGs)and two superconducting bulks,different widths of two superconducting bulks make the gap of the system decrease at the optimal value,but make the gap at the minimum value increases.And the zigzag-shape quasi-one-dimensional junction eliminates the large momentum parallel to the junction and enhances the gap at the large momentum.However,the zigzag-shape junction cannot increase the gap at the region of multiple pairs of MZMs in a symmetric system.We show that by combining zigzag-shape junction with different coupling coefficients,the system can maintain a large gap(≈0.2△)in a wide region of the parameter space.

拓扑半金属-超导体异质结的约瑟夫森效应

拓扑半金属是一类受对称性保护的无能隙量子材料.因其相对论性能带色散关系,拓扑半金属中涌现出丰富的量子态和量子效应,例如费米弧表面态和手征反常.近年来,因在拓扑量子计算的潜在应用,拓扑与超导的耦合体系受到广泛关注.本文从两方面回顾拓扑半金属-超导体异质结体系近年来的实验进展:1)超导电流对拓扑量子态的模式过滤;2)拓扑超导和Majorana零能模的探测与调控.对于前者,利用约瑟夫森电流对电磁场的响应,拓扑半金属中费米弧表面态的弹道输运被揭示,高阶拓扑半金属相被证实,有限动量配对及超导二极管效应被实现.对于后者,通过交流约瑟夫森效应,狄拉克半金属中4π周期的拓扑超导态被发现,纯电学栅压调控的拓扑相变被实现.本文最后展望了拓扑半金属-超导体异质结体系的发展前景和在Majorana零能模编织和拓扑量子计算上的潜在应用.

马约拉纳零能模的输运探测

理论预言拓扑超导体能够承载服从非阿贝尔统计的伊辛任意子—马约拉纳零能模,因而可用于实现容错的拓扑量子计算,是凝聚态领域最受关注的前沿课题之一.本文重点回顾了电学输运手段在研究马约拉纳零能模中的应用.在简要介绍拓扑超导、马约拉纳零能模和非阿贝尔统计等基本概念的基础上,对当前实现拓扑超导的多种方案进行了总结.重点介绍了利用低温输运手段探测马约拉纳零能模的实验方法,涵盖了超导/纳米线中广泛使用的电子隧穿谱、库仑阻塞谱和非局域电导探测,以及约瑟夫森器件中使用的(逆)交流约瑟夫森效应探测和电流(能量)相位关系的探测.同时,对利用上述测量手段得到的实验结果可能存在的平庸解释进行了必要的补充和说明.最后对马约拉纳零能模的输运探测进行了总结与展望.

选区外延生长的PbTe-超导杂化纳米线:一个可能实现拓扑量子计算的新体系

半导体-超导体杂化纳米线是用于研究马约拉纳零能模和拓扑量子计算的主要平台之一,而基于Ⅲ-Ⅴ族半导体InAs和InSb的纳米线则是当前此方向研究的主流材料体系.尽管经过多年制备技术的改进和优化,样品中过多的缺陷和杂质仍是阻碍此方向进一步发展的核心问题.近年来,一个新的马约拉纳纳米线候选体系——Ⅳ-Ⅵ族半导体PbTe-超导杂化纳米线吸引了很大关注并获得了快速的研究进展.PbTe的介电常数巨大,且具有晶格匹配的衬底,这些优势使其有潜力突破纳米线样品质量提升的瓶颈,成为马约拉纳零能模的研究和拓扑量子计算实现的理想平台.本文将简单介绍最近几年在PbTe纳米线和PbTe-超导杂化纳米线器件的选区分子束外延生长、输运性质研究方面取得的重要进展,并对这种新的马约拉纳纳米线候选体系的优势、问题及基于其实现拓扑量子计算的前景进行讨论.