We propose a systematic magnetic-flux-free approach to detect, manipulate and braid Majorana fermions in a semiconductor nanowire-based topological Josephson junction by utilizing the Majorana spin degree of freedom. We find an intrinsic $π$-phase difference between spin-triplet pairings enforced by the Majorana zeros modes (MZMs) at the two ends of a one-dimensional spinful topological superconductor. This $π$-phase is identified to be a spin-dependent superconducting phase, referred to as the spin-phase, which we show to be tunable by controlling spin-orbit coupling strength via electric gates. This electric controllable spin-phase not only affects the coupling energy between MZMs but also leads to a fractional Josephson effect in the absence of any applied magnetic flux, which enables the efficient topological qubit readout. We thus propose an all-electrically controlled superconductor-semiconductor hybrid circuit to manipulate MZMs and to detect their non-Abelian braiding statistics properties. Our work on spin properties of topological Josephson effects potentially opens up a new thrust for spintronic applications with Majorana-based semiconductor quantum circuits.
We explore the tunneling transport properties of a quantum dot embedded in an optical microcavity and coupled to a semiconductor-superconductor one-dimensional nanowire (Majorana nanowire) hosting Majorana zero modes (MZMs) at their edges. Conductance profiles reveal that strong light-matter coupling can be employed to distinguish between the cases of highly nonlocal MZMs, overlapped MZMs and MZMs with less degree of nonlocal feature. Moreover, we show that it is possible to access the degree of Majorana nonlocality (topological quality factor) by changing the dot spectrum through photon-induc…
One-dimensional topological superconductors host Majorana zero modes (MZMs), the non-local property of which could be exploited for quantum computing applications. Spin- polarized scanning tunneling microscopy measurements show that MZMs realized in self- assembled Fe chains on the surface of Pb have a spin polarization that exceeds that due to the magnetism of these chains. This feature, captured by our model calculations, is a direct consequence of the nonlocality of the Hilbert space of MZMs emerging from a topological band structure. Our study establishes spin polarization measurements as …
We propose a scheme to braid Majorana zero modes (MZMs) through steering the spin degree of freedom, without moving, measuring, or more generically fusing the modes. For a spinful Majorana system, we show that braiding two MZMs is achieved by adiabatically reversing the Majorana spins, which topologically corresponds to twisting two associated worldribbons, the extention of worldlines that track the braiding history of MZMs. We demonstrate the feasibility of applying the current scheme to the superconductor/2D-topological-insulator/ferromagnetic-insulator (SC/2DTI/FI) hybrid system which is cu…
A pair of Majorana zero modes (MZMs) constitutes a nonlocal qubit whose entropy is $\log 2$. Upon strongly coupling one of the constituent MZMs to a reservoir with a continuous density of states, a universal entropy change of $\frac{1}{2}\log 2$ is expected to be observed across an intermediate temperature plateau. We adapt the entropy-measurement scheme that was the basis of a recent experiment [Hartman et. al., Nat. Phys. 14, 1083 (2018)] to the case of a proximitized topological system hosting MZMs, and propose a method to measure this $\frac{1}{2}\log 2$ entropy change --- an unambiguous s…
In this work, we demonstrate that making a cut (a narrow vacuum regime) in the bulk of a quantum anomalous Hall insulator (QAHI) creates a topologically protected single helical channel with counter-propagating electron modes, and inducing superconductivity on the helical channel through proximity effect will create Majorana zero energy modes (MZMs) at the ends of the cut. In this geometry, there is no need for the proximity gap to overcome the bulk insulating gap of the QAHI to create MZMs as in the two-dimensional QAHI/superconductor (QAHI/SC) heterostructures. Therefore, the topological reg…
We study a vortex chain in a thin film of a topological insulator with proximity-induced superconductivity---a promising platform to realize Majorana zero modes (MZMs)---by modeling it as a two-leg Majorana ladder. While each pair of MZMs hybridizes through vortex tunneling, we hereby show that MZMs can be stabilized on the ends of the ladder with the presence of tilted external magnetic field and four-Majorana interaction. Furthermore, a fruitful phase diagram is obtained by controlling the direction of magnetic field and the thickness of the sample. We reveal many-body Majorana states and in…
By tuning the magnetic flux, the two ends of a 1D topological superconductor weakly coupled to a normal metal as a ring-shaped junction can host split Majorana zero modes (MZMs). When this ring geometry becomes Coulomb blockaded, and the two leads come into contact with the two wire ends, the current moves through the superconductor or the normal metal as an interferometer. The two-terminal interference conductance can be experimentally measured as a function of gate voltage and magnetic flux through the ring. However, a $4π$ periodicity in the conductance-phase relation (often considered the …
The recent realization of pristine Majorana zero modes (MZMs) in vortices of iron-based superconductors (FeSCs) provides a promising platform for long-sought-after fault-tolerant quantum computation. A large topological gap between the MZMs and the lowest excitations enabled detailed characterization of vortex MZMs in those materials. Despite those achievements, a practical implementation of topological quantum computation based on MZM braiding remains elusive in this new Majorana platform. Among the most pressing issues are the lack of controllable tuning methods for vortex MZMs and inhomogen…
The prospect of using semiconductor quantum dots as an experimental tool to distinguish Majorana zero modes (MZMs) from other zero-energy excitations such as Kondo resonances has brought up the fundamental question of whether topological superconductivity and the Kondo effect can coexist in these systems. Here, we study the Kondo effect in a quantum dot coupled to a metallic contact and to a pair of MZMs. We consider a situation in which the MBS are spin polarized in opposite directions. By using numerical renormalization-group calculations and scaling analysis of the renormalization group equ…
We investigate the one-dimensional $p$-wave superconducting model with periodically modulated hopping and show that under time-reversal symmetry, the number of the Majorana zero modes (MZMs) strongly depends on the modulation period. If the modulation period is odd, there can be at most one MZM. However if the period is even, the number of the MZMs can be zero, one and two. In addition, the MZMs will disappear as the chemical potential varies. We derive the condition for the existence of the MZMs and show that the topological properties in this model are dramatically different from the one wit…
It is difficult to unambiguously confirm the existence of Majorana zero modes (MZMs) due to the absence of smoking-gun signatures in charge transport measurements. Recent studies suggest that the spin degree of freedom of MZMs may provide an alternative detection method. We study the spin properties of the superconducting state in Majorana nanowires and the associated unconventional Josephson effect with realistic experimental parameters taken from [Zhang {\it et al.}, Nature 556, 74 (2018)]. For a superconducting thin film with in-plane polarized spin-triplet pairing, an out-of-plane electric…
Motivated by the recent experimental observation of the topological Anderson insulator in disordered atomic wires based on the Su-Schrieffer-Heeger (SSH) model, we study disorder effects on a dimerized Kitaev superconductor chain which is regarded as the superconductor version of the SSH model. By computing the real-space winding number and the zero-bias differential conductance, we analyze the topological phase transitions occurring in a dimerized Kitaev superconductor chain with disorder. It is found that disorder can induce a topologically nontrivial superconductor phase hosting Majorana ze…
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 qubit, 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 inducing a topological phase transition. The evolution of discrete energy modes inside a strained vortex…
Quantum Advanced Research Center (QuARC), Consejo Superior de Investigaciones Científicas (CSIC)（量子先进研究中心，西班牙国家科学研究委员会）；Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superior de Investigaciones Científicas (CSIC)（马德里材料科学研究所，西班牙国家科学研究委员会）
We investigate the effect of correlated disorder on Majorana zero modes (MZMs) bound to magnetic vortices in two-dimensional topological superconductors. By starting from a lattice model of interacting fermions with a $p_x \pm i p_y$ superconducting ground state in the disorder-free limit, we use perturbation theory to describe the enhancement of the Majorana localization length at weak disorder and a self-consistent numerical solution to understand the breakdown of the MZMs at strong disorder. We find that correlated disorder has a much stronger effect on the MZMs than uncorrelated disorder a…
Symmetry-protected topological superconductors (TSCs) can host multiple Majorana zero modes (MZMs) at their edges or vortex cores, while whether the Majorana braiding in such systems is non-Abelian in general remains an open question. Here we uncover in theory the unitary symmetry-protected non-Abelian statisitcs of MZMs and propose the experimental realization. We show that braiding two vortices with each hosting $N$ unitary symmetry-protected MZMs generically reduces to $N$ independent sectors, with each sector braiding two different Majorana modes. This renders the unitary symmetry-protecte…
Majorana zero modes (MZMs) realize a representation of non-abelian braid groups that enable topological quantum computation, wherein the storage and manipulation of information occur in decoherence-free degrees of freedom. This paper is intended to build off existing texts in the topic [1, 2] to serve as an introduction to MZMs at the undergraduate level, providing an overview of many important concepts in condensed matter physics, such as the Bogoliubov-de Gennes formalism and topological phases. We provide an overview of the motivation behind finding MZMs and discuss how the nature of these …
Topological stability is an important property for topological materials. However, the non-Hermitian effects may change this situation. Here, we investigate the robustness of edge states in the non-Hermitian Kitaev chain with imbalanced tunneling term and superconducting pairing term. By defining the similarity of Majorana zero modes (MZMs) and magnetic factor, the coalescing phase diagram of the MZMs and corresponding spin polarization phase diagram are provided. Because of the non-Hermitian coalescence effect and non-Hermitian suppression effect induced by the breakdown of sublattice symmetr…
We analyze charging-energy-protected Majorana-based qubits, focusing on the residual dephasing that is present when the distance between Majorana zero modes (MZMs) is insufficient for full topological protection. We argue that the leading source of dephasing is $1/f$ charge noise. This noise affects the qubit as a result of the hybridization energy and charge distribution associated with weakly-overlapping MZMs, which we calculate using a charge-conserving formalism. We estimate the coherence time to be hundreds of nanoseconds for Majorana-based qubits whose MZM separation is $L\sim 5ξ$ (with …
Magnet-superconductor hybrid (MSH) systems are a key platform for custom-designed topological superconductors. Ideally, the ends of a one-dimensional MSH structure will host Majorana zero-modes (MZMs), the fundamental unit of topological quantum computing. However, some of the experiments with ferromagnetic chains show a more complicated picture. Due to tiny gap sizes and hence long coherence lengths MZMs might hybridize and lose their topological protection. Recent experiments on a niobium surface have shown that both ferromagnetic and antiferromagnetic chains may be engineered, with the magn…
Phase-Controlled Majorana Zero Modes in Altermagnetic Topological-Insulator Josephson Junctions
Huazhong University of Science and Technology（华中科技大学）；Tsung-Dao Lee Institute（李政道研究所）；School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；High Magnetic Field Laboratory, HFIPS, Chinese Academy of Sciences（中国科学院合肥物质科学研究院高磁场实验室）；College of Physics and Optoelectronic Engineering, Department of Physics, Jinan University（暨南大学物理与光电工程学院）；Hefei National Laboratory（合肥国家实验室）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）
We numerically investigate non-Abelian braiding dynamics of vortices in two-dimensional topological superconductors, such as $s$-wave superconductors with Rashba spin-orbit coupling. Majorana zero modes (MZMs) hosted by the vortices constitute a topological qubit, which offers a fundamental building block of topological quantum computation. As the MZMs are protected by $\mathbb{Z}_2$ invariant, however, the Majorana qubit and quantum gate operations may be sensitive to intrinsic decoherence caused by quasiparticle interference. Numerically simulating the time-dependent Bogoliubov-de Gennes equ…
Majorana zero modes (MZMs) obey non-Abelian statistics and are considered building blocks for constructing topological qubits. Iron-based superconductors with topological band structures have emerged as promising hosting materials, since isolated candidate MZMs in the quantum limit have been observed inside the topological vortex cores. However, these materials suffer from issues related to alloying-induced disorder, uncontrolled vortex lattices and a low yield of topological vortices. Here, we report the formation of an ordered and tunable MZM lattice in naturally-strained stoichiometric LiFe…
Unveiling In-Gap States and Majorana Zero Modes in Superconductor-Topological Insulator Bilayer model
Materials Science and Technology Division, Oak Ridge National Laboratory（橡树岭国家实验室材料科学与技术分部）；Institute of Physics, Polish Academy of Sciences（波兰科学院物理研究所）；Microsoft Quantum, Station Q, University of California, Santa Barbara（微软量子、Station Q、加州大学圣塔芭芭拉分校）；Department of Physics and Astronomy, Purdue University（普渡大学物理与天文系）
Using a cryogenic scanning tunneling microscopy, we report the observation of topologically nontrivial superconductivity on a single material of β-Bi2Pd films grown by molecular beam epitaxy. The superconducting gap associated with spinless odd-parity pairing opens on the surface and appears much larger than the bulk one due to the Dirac-fermion enhanced parity mixing of surface pair potential. Majorana zero modes (MZMs), supported by such superconducting states, are identified at magnetic vortices. The superconductivity and MZMs exhibit resistance to nonmagnetic defects, characteristic of tim…
In a superconducting topological insulator, superconducting vortex line may trap 1-dimensional topological band with Majorana zero modes (MZMs) localized at its ends. In this work, we study the effect of hexagonal warping to the vortex phase transition. We carry out both analytical calculations based on a semiclassical formula and numerical calculations based on full quantum mechanics of Bogliubov de Gennes equation. We find that the hexagonal warping term enlarges the topological region of the vortex line as the chemical potential changes, and leads to the MZMs even in the absence of topologi…
It is well known that non-Abelian Majorana zero modes (MZM) harbor at vortex cores in a $p_{x}+\text{i}p_{y}$ topological superconductor, which can be realized in a 2D spin-orbit coupled system with a single Fermi surface and by proximity coupling to an $s$-wave superconductor. Here we show that existence of non-Abelian MZMs is unrelated to the bulk topology of a 2D superconductor, and propose that such exotic modes can be resulted in much broader range of superconductors, being topological or trivial. For a generic 2D system with multiple Fermi surfaces and gapped out by superconducting pairi…
The issue on the effect of interactions in topological states concerns not only interacting topological phases but also novel symmetry-breaking phases and phase transitions. Here we study the interaction effect on Majorana zero modes (MZMs) bound to a square vortex lattice in two-dimensional (2D) topological superconductors. Under the neutrality condition, where single-body hybridization between MZMs is prohibited by an emergent symmetry, a minimal square-lattice model for MZMs can be faithfully mapped to a quantum spin model, which has no sign problem in the world-line quantum Monte Carlo sim…
