In recent years,the study of Majorana signatures in quantum transport has become a central focus in condensed matter physics.Here,we present a rigorous and systematic derivation of the fermionic superoperator describi...In recent years,the study of Majorana signatures in quantum transport has become a central focus in condensed matter physics.Here,we present a rigorous and systematic derivation of the fermionic superoperator describing the open quantum dynamics of electron transport through Majorana zero modes,building on the techniques introduced in Phys.Rev.B 105,035121(2022).The numerical implementation of this superoperator is to construct its differential equivalence,the hierarchical equations of motion(HEOM).The HEOM approach describes the system-bath correlated dynamics.Furthermore,we also develop a functional derivative scheme that provides exact expressions for the transport observables in terms of the auxiliary density operators introduced in the HEOM formulation.The superoperator formalism establishes a solid theoretical foundation for analyzing key transport signatures that may uncover the unique characteristics of Majorana physics in mesoscopic systems.展开更多
We review the properties of Majorana fermions in particle physics and point out that Majorana modes in solid state systems are significantly different. The key reason is the concept of anti-particle in solid state sys...We review the properties of Majorana fermions in particle physics and point out that Majorana modes in solid state systems are significantly different. The key reason is the concept of anti-particle in solid state systems is different from its counterpart in particle physics. We define Majorana modes as the eigenstates of Majorana operators and find that they can exist both at edges and in the bulk. According to our definition, only one single Majorana mode can exist in a system no matter at edges or in the bulk. Kitaev＇s spinless p-wave superconductor is used to illustrate our results and the dynamical behavior of the Majorana modes.展开更多
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...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.展开更多
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 consi...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.展开更多
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 re...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.展开更多
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)...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.展开更多
We study the possibility to realize a Majorana zero mode that is robust and may be easily manipulated for braiding in quantum computing in the ground state of the Kitaev model in this work.To achieve this we first app...We study the possibility to realize a Majorana zero mode that is robust and may be easily manipulated for braiding in quantum computing in the ground state of the Kitaev model in this work.To achieve this we first apply a uniform[111]magnetic field to the gapless Kitaev model and turn the Kitaev model to an effective p+ip topological superconductor of spinons.We then study possible vortex binding in such system to a topologically trivial spot in the ground state.We consider two cases in the system:one is a vacancy and the other is a fully polarized spin.We show that in both cases,the system binds a vortex with the defect and a robust Majorana zero mode in the ground state at a weak uniform[111]magnetic field.The distribution and asymptotic behavior of these Majorana zero modes are studied.The Majorana zero modes in both cases decay exponentially in space,and are robust against local perturbations and other Majorana zero modes far away,which makes them promising candidates for braiding in topological quantum computing.展开更多
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...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.展开更多
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 ...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.展开更多
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 intera...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.展开更多
The iron-based superconductors(FeSCs)have recently emerged as a promising single-material Majorana platform by hosting isolated Majorana zero modes(MZMs)at relatively high temperatures.To further verify its Majorana n...The iron-based superconductors(FeSCs)have recently emerged as a promising single-material Majorana platform by hosting isolated Majorana zero modes(MZMs)at relatively high temperatures.To further verify its Majorana nature and move forward to build topological quantum qubits,it is highly desirable to achieve tunability for MZMs on homogeneous FeSCs.Here,with an in-situ strain device,we can controllably create MZMs on the homogeneous surface of stoichiometric superconductor LiFeAs by altering its chemical potential.The evolution of discrete energy modes inside a strained vortex is found to mimic exactly as the predicted topological vortex case,proving the Majorana nature of emerging zero modes of vortex.More importantly,our work provides a controllable method for MZM in a homogeneous FeSC,and such achievement of tunability of MZMs in the FeSC Majorana-material platform is an important step towards their application in topological quantum computation.展开更多
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 Ma...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.展开更多
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 e...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.展开更多
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 addit...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.展开更多
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 numb...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.展开更多
We investigate the topological properties of a ladder model of the dimerized Kitaev superconductor chains.The topological class of the system is determined by the relative phase θ between the inter-and intra-chain su...We investigate the topological properties of a ladder model of the dimerized Kitaev superconductor chains.The topological class of the system is determined by the relative phase θ between the inter-and intra-chain superconducting pairing.One topological class is the class BDI characterized by the Z index,and the other is the class D characterized by the Z;index.For the two different topological classes,the topological phase diagrams of the system are presented by calculating two different topological numbers,i.e.,the Z index winding number W and the Z;index Majorana number M,respectively.In the case of θ=0,the topological class belongs to the class BDI,multiple topological phase transitions accompanying the variation of the number of Majorana zero modes are observed.In the case of θ = π/2 it belongs to the class D.Our results show that for the given value of dimerization,the topologically nontrivial and trivial phases alternate with the variation of chemical potential.展开更多
We study the topological properties of the one-dimensional non-Hermitian Kitaev model with complex either periodic or quasiperiodic potentials.We obtain the energy spectrum and the phase diagrams of the system by usin...We study the topological properties of the one-dimensional non-Hermitian Kitaev model with complex either periodic or quasiperiodic potentials.We obtain the energy spectrum and the phase diagrams of the system by using the transfer matrix method as well as the topological invariant.The phase transition points are given analytically.The Majorana zero modes in the topological nontrivial regimes are obtained.Focusing on the quasiperiodic potential,we obtain the phase transition from the topological superconducting phase to the Anderson localization,which is accompanied with the Anderson localization–delocalization transition in this non-Hermitian system.We also find that the topological regime can be reduced by increasing the non-Hermiticity.展开更多
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 ...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.展开更多
The Majorana fermions/quasi-particles obey non-Abelian statistics and play an essential role in topological quantum computing.Theoretically,they can be realized as a zero-energy excitation in the vortex core of topolo...The Majorana fermions/quasi-particles obey non-Abelian statistics and play an essential role in topological quantum computing.Theoretically,they can be realized as a zero-energy excitation in the vortex core of topological superconductors,called Majorana zero mode(MZM).However,the coexisting low-energy bound states in vortex cores often bring difficulties for identifying MZM.In this review,we summarized recent STM studies on the vortex states of electron-doped FeSe-based superconductors such as(Li,Fe)OHFeSe and single-layer FeSe/SrTiO_(3).Zero-bias conductance peaks(ZBCP)are observed in the unpinned vortex cores of(Li,Fe)OHFeSe.It displays nearly quantized conductance and distinct spatial distributions with respect to nonzero bound states,which indicates their nontrivial origin.Meanwhile,single-layer FeSe/SrTiO_(3)film only has conventional Caroli-de Gennes-Matricon(CdGM)bound states without zero-energy mode,reflecting the characteristics of s-wave superconductor.These results not only provide strong evidence of MZM,but also shed light on its origin in FeSe-based superconductors.展开更多
Topological crystalline insulators are known to support multiple Majorana zero modes(MZMs)at a single vortex,their hybridization is forbidden by a magnetic mirror symmetry MT.Due to the limited energy resolution of sc...Topological crystalline insulators are known to support multiple Majorana zero modes(MZMs)at a single vortex,their hybridization is forbidden by a magnetic mirror symmetry MT.Due to the limited energy resolution of scanning tunneling microscopes and the very small energy spacing of trivial bound states,it remains challenging to directly probe and demonstrate the existence of multiple MZMs.In this work,we propose to demonstrate the existence of MZMs by studying the hybridization of multiple MZMs in a symmetry breaking field.The different responses of trivial bound states and MZMs can be inferred from their spatial distribution in a vortex.However,the theoretical simulations are very demanding since it requires an extremely large system in real space.By utilizing the kernel polynomial method,we can efficiently simulate large lattices with over 108 orbitals to compute the local density of states which bridges the gap between theoretical studies based on minimal models and experimental measurements.We show that the spatial distribution of MZMs and trivial vortex bound states differs drastically in tilted magnetic fields.The zero-bias peak elongates when the magnetic field preserves MT,while it splits when MT is broken,giving rise to an anisotropic magnetic response.Since the bulk of SnTe are metallic,we also study the robustness of MZMs against the bulk states,and clarify when can the MZMs produce a pronounced anisotropic magnetic response.展开更多
基金supported by the National Natural Science Foundation of China(Nos.224B2305,22373091)the Innovation Program for Quantum Science and Technology(No.2021ZD0303301)。
文摘In recent years,the study of Majorana signatures in quantum transport has become a central focus in condensed matter physics.Here,we present a rigorous and systematic derivation of the fermionic superoperator describing the open quantum dynamics of electron transport through Majorana zero modes,building on the techniques introduced in Phys.Rev.B 105,035121(2022).The numerical implementation of this superoperator is to construct its differential equivalence,the hierarchical equations of motion(HEOM).The HEOM approach describes the system-bath correlated dynamics.Furthermore,we also develop a functional derivative scheme that provides exact expressions for the transport observables in terms of the auxiliary density operators introduced in the HEOM formulation.The superoperator formalism establishes a solid theoretical foundation for analyzing key transport signatures that may uncover the unique characteristics of Majorana physics in mesoscopic systems.
基金Q.Z. thanks Erhai Zhao for discussions on the Kitaev chain and the partial support by AFOSR FA9550-12-1- 0079 as a visiting scholar at George Mason University. This work was supported by the NationM Basic Research Program of China （Grants No. 2013CB921903） and the National Natural Science Foundation of China （Grants Nos. 11334001 and 11429402）.
文摘We review the properties of Majorana fermions in particle physics and point out that Majorana modes in solid state systems are significantly different. The key reason is the concept of anti-particle in solid state systems is different from its counterpart in particle physics. We define Majorana modes as the eigenstates of Majorana operators and find that they can exist both at edges and in the bulk. According to our definition, only one single Majorana mode can exist in a system no matter at edges or in the bulk. Kitaev＇s spinless p-wave superconductor is used to illustrate our results and the dynamical behavior of the Majorana modes.
基金the National Natural Science Foundation of China(Grant Nos.11835011 and 12174346).
文摘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.
基金supported by the Ministry of Science and Technology of China (Grant No. 2019YFA0308500)the Chinese Academy of Sciences (Grant Nos. XDB28000000 and YSBR-003)。
文摘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.
基金supported by the Hunan Provincial Natural Science Foundation of China(Nos.2020JJ4240,2018JJ2078)the Scientific Research Fund of Hunan Provincial Education Department(No.19A106,18C0699)the Postgraduate University-Level Research Programme of Jishou University(No.Jdy22043)。
文摘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.
基金the National Key Research and Development Program of China(Grant No.2017YFA0303304)the National Natural Science Foundation of China(Grant Nos.11675016,11974011,and 61905174).
文摘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.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11974166 and 11574134).
文摘We study the possibility to realize a Majorana zero mode that is robust and may be easily manipulated for braiding in quantum computing in the ground state of the Kitaev model in this work.To achieve this we first apply a uniform[111]magnetic field to the gapless Kitaev model and turn the Kitaev model to an effective p+ip topological superconductor of spinons.We then study possible vortex binding in such system to a topologically trivial spot in the ground state.We consider two cases in the system:one is a vacancy and the other is a fully polarized spin.We show that in both cases,the system binds a vortex with the defect and a robust Majorana zero mode in the ground state at a weak uniform[111]magnetic field.The distribution and asymptotic behavior of these Majorana zero modes are studied.The Majorana zero modes in both cases decay exponentially in space,and are robust against local perturbations and other Majorana zero modes far away,which makes them promising candidates for braiding in topological quantum computing.
基金Project supported by the National Natural Science Foundation of China(Grant No.11274379)the Fundamental Research Funds for the Central Universitiesthe Research Funds of Renmin University of China(Grant No.14XNLQ07)
文摘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.
基金Project supported by the National Key R&D Program of China (Grant Nos.2021YFA1400900,2021YFA0718300,and 2021YFA1400243)the National Natural Science Foundation of China (Grant No.61835013)。
文摘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.
基金Supported by the National Natural Science Foundation of China under Grant No 11274379the Research Funds of Renmin University of China under Grant No 14XNLQ07
文摘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.
基金supported by grants from the National Natural Science Foundation of China(11888101,11234014,61888102,11920101005,11921004)the Ministry of Science and Technology of China(2016YFA0202300,2018YFA0305700,2017YFA0302900,2019YFA0308500)+3 种基金the Chinese Academy of Sciences(XDB28000000,XDB07000000,112111KYSB20160061)supported by JSPS Grants-in-Aid for Scientific Research(JP19H01818,JP19H05826,JP21H04439)MEXT as the“Program for Promoting Researches on the Supercomputer Fugaku”(JPMXP1020200104)supported by the Synergic Extreme Condition User Facility,Beijing,China。
文摘The iron-based superconductors(FeSCs)have recently emerged as a promising single-material Majorana platform by hosting isolated Majorana zero modes(MZMs)at relatively high temperatures.To further verify its Majorana nature and move forward to build topological quantum qubits,it is highly desirable to achieve tunability for MZMs on homogeneous FeSCs.Here,with an in-situ strain device,we can controllably create MZMs on the homogeneous surface of stoichiometric superconductor LiFeAs by altering its chemical potential.The evolution of discrete energy modes inside a strained vortex is found to mimic exactly as the predicted topological vortex case,proving the Majorana nature of emerging zero modes of vortex.More importantly,our work provides a controllable method for MZM in a homogeneous FeSC,and such achievement of tunability of MZMs in the FeSC Majorana-material platform is an important step towards their application in topological quantum computation.
基金supported by the Innovation Program for Quantum Science and Technology (Grant No.2021ZD0302400)the National Natural Science Foundation of China (Grants No.11974198)the Natural Science Foundation of Shandong Province of China (Grant No.ZR2021MA091)。
文摘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.
基金supported by the National Natural Science Foundation of China(Grant No.11274379)the Research Funds of Renmin University of China(Grant No.14XNLQ07)
文摘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.
基金financially supported by the Innovation Program for Quantum Science and Technology(Grant No.2021ZD0302400)the National Natural Science Foundation of China(Grant No.11974271)+2 种基金the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDB28000000)the National Basic Research Program of China(Grant No.2015CB921102)the China Postdoctoral Science Foundation(Grant No.2021M690233)。
文摘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.
基金Project supported by the National Basic Research Program of China(Grant No.2012CB921704)
文摘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.
基金supported by the National Natural Science Foundation of China(Grant No.11274102)the Program for New Century Excellent Talents in University of Ministry of Education of China(Grant No.NCET-11-0960)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20134208110001)
文摘We investigate the topological properties of a ladder model of the dimerized Kitaev superconductor chains.The topological class of the system is determined by the relative phase θ between the inter-and intra-chain superconducting pairing.One topological class is the class BDI characterized by the Z index,and the other is the class D characterized by the Z;index.For the two different topological classes,the topological phase diagrams of the system are presented by calculating two different topological numbers,i.e.,the Z index winding number W and the Z;index Majorana number M,respectively.In the case of θ=0,the topological class belongs to the class BDI,multiple topological phase transitions accompanying the variation of the number of Majorana zero modes are observed.In the case of θ = π/2 it belongs to the class D.Our results show that for the given value of dimerization,the topologically nontrivial and trivial phases alternate with the variation of chemical potential.
基金the National Key R&D Program of China(Grant Nos.2016YFA0300600 and 2016YFA0302104)the National Natural Science Foundation of China(Grant Nos.12074410,12047502,11934015,11947301,and 11774397)+1 种基金the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDB33000000)the fellowship of China Postdoctoral Science Foundation(Grant No.2020M680724)。
文摘We study the topological properties of the one-dimensional non-Hermitian Kitaev model with complex either periodic or quasiperiodic potentials.We obtain the energy spectrum and the phase diagrams of the system by using the transfer matrix method as well as the topological invariant.The phase transition points are given analytically.The Majorana zero modes in the topological nontrivial regimes are obtained.Focusing on the quasiperiodic potential,we obtain the phase transition from the topological superconducting phase to the Anderson localization,which is accompanied with the Anderson localization–delocalization transition in this non-Hermitian system.We also find that the topological regime can be reduced by increasing the non-Hermiticity.
基金Project supported by the National Natural Science Foundation of China(Grant No.11974271)。
文摘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.
基金supported by the Innovation Program for Quantum Science and Technology(Grant no.2021ZD0302800)National Natural Science Foundation of China(Grants Nos.92065202,11888101,11790312,11961160717,12225403,12104094)+2 种基金Shanghai Municipal Science and Technology Major Project(Grant No.2019SHZDZX01)Science and Technology Commission of Shanghai Municipality,China(Grant Nos.19JC1412702,21TQ1400100)the China Postdoctoral Science Foundation(Grant Nos.BX20200097,2020M681137).
文摘The Majorana fermions/quasi-particles obey non-Abelian statistics and play an essential role in topological quantum computing.Theoretically,they can be realized as a zero-energy excitation in the vortex core of topological superconductors,called Majorana zero mode(MZM).However,the coexisting low-energy bound states in vortex cores often bring difficulties for identifying MZM.In this review,we summarized recent STM studies on the vortex states of electron-doped FeSe-based superconductors such as(Li,Fe)OHFeSe and single-layer FeSe/SrTiO_(3).Zero-bias conductance peaks(ZBCP)are observed in the unpinned vortex cores of(Li,Fe)OHFeSe.It displays nearly quantized conductance and distinct spatial distributions with respect to nonzero bound states,which indicates their nontrivial origin.Meanwhile,single-layer FeSe/SrTiO_(3)film only has conventional Caroli-de Gennes-Matricon(CdGM)bound states without zero-energy mode,reflecting the characteristics of s-wave superconductor.These results not only provide strong evidence of MZM,but also shed light on its origin in FeSe-based superconductors.
基金support from the National Key R&D Program of China(Grants No.2021YFA1401500)the Hong Kong Research Grants Council(Project No.16306220).
文摘Topological crystalline insulators are known to support multiple Majorana zero modes(MZMs)at a single vortex,their hybridization is forbidden by a magnetic mirror symmetry MT.Due to the limited energy resolution of scanning tunneling microscopes and the very small energy spacing of trivial bound states,it remains challenging to directly probe and demonstrate the existence of multiple MZMs.In this work,we propose to demonstrate the existence of MZMs by studying the hybridization of multiple MZMs in a symmetry breaking field.The different responses of trivial bound states and MZMs can be inferred from their spatial distribution in a vortex.However,the theoretical simulations are very demanding since it requires an extremely large system in real space.By utilizing the kernel polynomial method,we can efficiently simulate large lattices with over 108 orbitals to compute the local density of states which bridges the gap between theoretical studies based on minimal models and experimental measurements.We show that the spatial distribution of MZMs and trivial vortex bound states differs drastically in tilted magnetic fields.The zero-bias peak elongates when the magnetic field preserves MT,while it splits when MT is broken,giving rise to an anisotropic magnetic response.Since the bulk of SnTe are metallic,we also study the robustness of MZMs against the bulk states,and clarify when can the MZMs produce a pronounced anisotropic magnetic response.
