Motivated by a recent experiment on superfluid 3He-A confined in narrow parallel plates using a rotating cryostat, we explore possible vortices stable under magnetic field applied to arbitrary angle relative to the plates in order to seek vortices which can accommodate the Majorana zero mode in the core. After proving that the singular vortex with the unit winding number provides the Majorana mode in the spinful situation, we establish the phase diagram in the plane; the rotation frequency $Ω$ vs system size R by finding possible order parameter textures within the Ginzburg-Landau framework. We also analyze the stability for a single and a pair of half-quantum vortices, which possesses the Majorana mode in its core. It is concluded from the above mention that the Majorana zero mode can be found in the present on-going experimental setting at ISSP, Univ. Tokyo.
We consider the one-dimensional (1D) topological superconductor that may form in a planar superconductor-metal-superconductor Josephson junction in which the metal is is subjected to spin orbit coupling and to an in-plane magnetic field. This 1D topological superconductor has been the subject of recent theoretical and experimental attention. We examine the effect of perpendicular magnetic field and a supercurrent driven across the junction on the position and structure of the Majorana zero modes that are associated with the topological superconductor. In particular, we show that under certain …
We present a simple model of Majorana fermions on a square lattice, and study zero-energy states due to Z$_2$ vortices. We show the relationship between the Chern number of the ground state and the number of the zero-energy states by numerical calculations for finite systems. We also discuss the relationship for the bulk system by observing the change of the spectral asymmetry. We finally present an effective continuum model with O(2) gauge potential which shows a topological change of the index at a half-flux.
It has been widely believed that half quantum vortices are indispensable to realize topological stable Majorana zero modes and non-Abelian anyons in spinful superconductors/superfluids. Contrary to this wisdom, we here demonstrate that integer quantum vortices in spinful superconductors can host topologically stable Majorana zero modes because of the mirror symmetry. The symmetry protected Majorana fermions may exhibit non-Abelian anyon braiding.
This is a short note presenting a repeated experiment of the half-quantized transport signature given by Majorana chiral modes living in a heterostructure formed by an s-wave superconductor and a quantum-anomalous-Hall insulator.
A concrete and experimentally feasible example for testing the putative Majorana zero energy state bound in a vortex is theoretically proposed for a parallel plate geometry of superfluid $^3$He-A phase. We examine the experimental setup in connection with ongoing rotating cryostat experiments. The theoretical analysis is based on the well-established Ginzburg--Landau functional, supplemented by microscopic calculations of the Bogoliubov--de Gennes equation, both of which allow the precise location of the parameter regions of the Majorana state to be found in realistic situations.
Half quantum vortices (HQVs) in superfluid $^3He-A$ have been speculated since 1976. Two years ago, Yamashita et al reported a very unusual NMR satellite in the rotating superfluid $^3He-A$ in a parallel plate geometry. Recently this satellite has been interpreted in terms of HQVs. Also, the strong flux pinnings in the triplet superconductors $UPt_3, Sr_2RuO_4$ and $U_{1-x}Th_xBe_{13}$ have been discovered by Mota and her colleagues which are interpreted in terms of HQVs. We shall first review the bound state spectra around a single vortex in unconventional superconductors, which provides the …
We show that the presence of Majorana zero modes (2D), and chiral-dispersing Majorana modes (3D), in the vortex cores of superconductors are neither sufficient nor necessary conditions for one to conclude the superconductivity is topologically non-trivial. We discuss the relevance of this result to the proximity-induced superconductivity, in the presence of magnetic field, on the surface of topological insulators.
The chiral edge modes of a topological superconductor support two types of excitations: fermionic quasiparticles known as Majorana fermions and $π$-phase domain walls known as edge vortices. Edge vortices are injected pairwise into counter-propagating edge modes by a flux bias or voltage bias applied to a Josephson junction. An unpaired edge mode carries zero electrical current on average, but there are time-dependent current fluctuations. We calculate the shot noise power produced by a sequence of edge vortices and find that it increases logarithmically with their spacing - even if the spacin…
In recent years there has been an intensive search for Majorana fermion states in condensed matter systems. Predicted to be localized on cores of vortices in certain non-conventional superconductors, their presence is known to render the exchange statistics of bulk vortices non-Abelian. Here we study the equations governing the dynamics of phase solitons (fluxons) in a long Josephson junction in a topological superconductor. We show that the fluxon will bind a localized zero energy Majorana mode and will consequently behave as a non-Abelian anyon. The low mass of the fluxon, as well as its exp…
Majorana zero-modes bound to vortices in a topological superconductor have a non-Abelian exchange statistics expressed by a non-deterministic fusion rule: When two vortices merge they may or they may not produce an unpaired fermion with equal probability. Building on a recent proposal to inject edge vortices in a chiral mode by means of a Josephson junction, we show how the fusion rule manifests itself in an electrical measurement. A $2π$ phase shift at a pair of Josephson junctions creates a topological qubit in a state of even-even fermion parity, which is transformed by the chiral motion of…
We consider the exchange statistics of vortices, each of which traps an odd number ($N$) of Majorana fermions. We assume that the fermions in a vortex transform in the vector representation of the SO(N) group. Exchange of two vortices turns out to be non-Abelian, and the corresponding operator is further decomposed into two parts: a part that is essentially equivalent to the exchange operator of vortices having a single Majorana fermion in each vortex, and a part representing the Coxeter group. Similar decomposition was already found in the case with N=3, and the result shown here is a general…
We consider Majorana fermions tunneling among an array of vortices in a 2D chiral p-wave superconductor or equivalent material. The amplitude for Majorana fermions to tunnel between a pair of vortices is found to necessarily depend on the background superconducting phase profile; it is found to be proportional to the sine of half the difference between the phases at the two vortices. Using this result we study tight-binding models of Majorana fermions in vortices arranged in triangular or square lattices. In both cases we find that the aforementioned phase-tunneling relationship leads to the c…
In topological insulators doped with magnetic ions, spin-orbit coupling and ferromagnetism give rise to the quantum anomalous Hall effect. Here we show that in s-wave superconductors with strong spin-orbit coupling, magnetic impurity ions can generate topological vortices in the absence of external magnetic fields. Such vortices, dubbed quantum anomalous vortices, support robust Majorana zero-energy modes when superconductivity is induced in the topological surface states. We demonstrate that the zero-energy bound states observed in Fe(Te,Se) superconductors are possible realizations of the Ma…
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…
The possible stable singular vortex (SV) and half-quantum vortex (HQV) of the superfluid $^3$He-A phase confined in restricted geometries are investigated. The associated low-energy excitations are calculated in connection with the possible existence of Majorana zero modes obeying non-Abelian statistics. The energetics between those vortices is carefully examined using the standard Ginzburg-Landau (GL) functional with a strong-coupling correction. The Fermi liquid effect, which is not included in the GL functional, is considered approximately within the London approach. This allows us to deter…
We investigate how the vortex-vortex separation changes Majorana zero modes in the vicinity of the BCS-BEC (Bose-Einstein condensation) topological phase transition of p-wave resonant Fermi gases. By analytically and numerically solving the Bogoliubov-de Gennes equation for spinless p-wave superfluids with plural vortices, it is demonstrated that the quasiparticle tunneling between neighboring vortices gives rise to the quantum oscillation of the low-lying spectra on the scale of the Fermi wavelength in addition to the exponential splitting. This rapid oscillation, which appears in the weak co…
A nonuniform in-plane Zeeman field can induce spontaneous supercurrents of spin-orbit coupled electrons in superconducting two-dimensional systems and thin films. In this work it is shown that current vortices can be created at the ends of a long homogeneously magnetized strip of a ferromagnetic insulator, which is deposited on the surface of a three-dimensional topological insulator. The s-wave superconductivity on its surface is assumed to have an intrinsic origin, or to be induced by the proximity effect. It is shown that vortices with the odd vorticity can localize Majorana zero modes.The …
Recent experiments have observed bulk superconductivity in doped topological insulators. Here we ask whether vortex Majorana zero modes, previously predicted to occur when superconductivity is induced on the surface of topological insulators, survive even in these doped systems with metallic normal states. Assuming inversion symmetry, we find that Majorana zero modes indeed appear but only below a critical doping. The critical doping is associated with a topological phase transition of the vortex line, where it supports gapless excitations along its length. The critical point depends only on t…
We propose theoretically an effective scheme for braiding Majorana bound states by manipulating the point potential. The vortex pinning effect is carefully elucidated. This effect may be used to control the vortices and Majorana bound states in topological superconductors. The exchange of two vortices induced by moving the potentials is simulated numerically. The zero-energy state in the vortex core is robust with respect to the strength of the potential. The Majorana bound states in a pinned vortex are identified numerically.
In the superconducting regime of FeTe$_{(1-x)}$Se$_x$, there exist two types of vortices which are distinct by the presence or absence of zero energy states in their core. To understand their origin, we examine the interplay of Zeeman coupling and superconducting pairings in three-dimensional metals with band inversion. Weak Zeeman fields are found to suppress the intra-orbital spin-singlet pairing, known to localize the states at the ends of the vortices on the surface. On the other hand, an orbital-triplet pairing is shown to be stable against Zeeman interactions, but leads to delocalized ze…
Abrikosov vortices in Fe-based superconductors are a promising platform for hosting Majorana zero modes. Their adiabatic exchange is a key ingredient for Majorana-based quantum computing. However, the adiabatic braiding process can not be realized in state-of-the-art experiments. We propose to replace the infinitely slow, long-path braiding by only slightly moving vortices in a special geometry without actually physically exchanging the Majoranas, like a Majorana carousel. Although the resulting finite-time gate is not topologically protected, it is robust against variations in material specif…
By analytically solving the Bogoliubov-de Gennes equations we study the fermion bound states at the center of the core of a vortex in a two-dimensional superconductor. We consider three kinds of 2D superconducting models: (a) a standard type II superconductor in the mixed state with low density of vortex lines, (b) a superconductor with strong spin-orbit coupling locking the spin parallel to the momentum and (c) a superconductor with strong spin-orbit coupling locking the spin perpendicular to the momentum. The 2D superconducting states are induced via proximity effect between an $s$-wave supe…
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…
As part of the intense effort towards identifying platforms in which Majorana bound states can be realized and manipulated to perform qubit operations, we propose a topological Josephson junction architecture that achieves these capabilities and which can be experimentally implemented. The platform uses conventional superconducting electrodes deposited on a topological insulator film to form networks of proximity-coupled lateral Josephson junctions. Magnetic fields threading the network of junction barriers create Josephson vortices that host Majorana bound states localized in the junction whe…
We propose a mechanism for opening a full bulk energy gap and inducing vortex Majorana zero modes (MZMs) in nodal superconductors (SCs). We show that this becomes possible by coupling the nodal SC of interest to a magnetic texture crystal. The latter consists of superpositions of magnetic textures which repeat periodically in space according to suitable wave vectors that enable spin-flip scattering between all pairs of nodes of the SC, and thus open a full gap in its bulk energy spectrum. In this event, MZMs can be trapped in spin or shift vortices introduced in the magnetic texture crystal. O…
We investigate the effect of the Majorana Fermions which are formed at the boundary of a p-wave superconductor. When the Majorana overlapping energy is finite we construct the scattering matrix $\mathbf{S}$ by maping the Majorana zero mode to Fermions for which coherent states are defined and a path integral is obtained . The path integral is used to compute the scattering matrix in terms of the electrons in the leads . This method is suitable for computing the conductivity. We investigate a chiral Majorana Hamiltonian and show that in the absence of vortices the conductivity vanish. We comput…
Motivated by recent experiments on FeTe$_{1-x}$Se$_{x}$, we construct an explicit minimal model of an iron-based superconductor with band inversion at the $Z$ point and non-topological bulk $s_{\pm}$ pairing. While there has been considerable interest in Majorana zero modes localized at vortices in such systems, we find that our model - without any vortices - intrinsically supports 1D helical Majorana modes localized at the hinges between (001) and (100) or (010) surfaces, suggesting that this is a viable platform for observing "higher-order" topological superconductivity. We provide a general…
We consider topological superconductors and topological insulator/superconductor structures in the presence of multiple static vortices that host Majorana modes and focus on the Majorana tunneling processes between vortices. It is shown that these tunnelings generally lift the degeneracy of the many-body ground state in a non-universal way, sensitive to microscopic details at the smallest length-scales determined by the underlying physical problem. We also discuss an explicit realization of the Jackiw-Rossi zero-mode in a topological insulator/superconductor structure with zero chemical potent…
We consider Majorana zero modes in a Josephson junction on top of a topological insulator in transverse magnetic field. Majorana zero modes reside at periodically located nodes of Josephson vortices. We find that hybridization of these modes is prohibited by symmetries of the problem at vanishing chemical potential, which ensures better protection of zero modes and yields methods to control the tunnel coupling between Majorana modes for quantum information processing applications.
In thermodynamics a macroscopic state of a system results from a number of its microscopic states. This number is given by the exponent of the system's entropy $\exp(S)$. In non-interacting systems with discrete energy spectra, such as large scale quantum dots, $S$ as a function of the temperature has usually a plateau shape with integer values of $\exp(S)$ on these plateaus. Plateaus with non-integer values of $\exp(S)$ are fundamentally forbidden and would be thermodynamically infeasible. Here we investigate the entropy of a non-interacting quantum dot coupled via tunneling to normal metals …
${}^{3}P_{2}$ superfluids realized in neutron stars are the largest topological quantum matters in our Universe. We establish the existence and stability of non-Abelian half-quantum vortices (HQVs) in ${}^{3}P_{2}$ superfluids with strong magnetic fields. Using a self-consistent microscopic approach, we find that a singly quantized vortex is energetically destabilized into a pair of two non-Abelian HQVs owing to the strongly spin-orbit-coupled pairing. We find a topologically protected Majorana fermion on each HQV, thereby providing two-fold non-Abelian anyons characterized by both Majorana fe…
Recently, a quantum anomalous Hall insulator (QAHI)/superconductor heterostructure has been realized and shows half-quantized conductance plateaus in two-terminal conductance measurements [Q. L. He \textit{et al.}, Science {\bf357}, 294 (2017)]. The half-quantized conductance plateaus are considered as a solid evidence of chiral Majorana edge modes. However, there is a strong debate over the origin of the half-quantized conductance plateaus. In this work, we propose a Josephson junction based on the QAHI/superconductor heterostructure to identify the existence of chiral Majorana edge modes. We…
We propose a scheme for the use of magnetic force microscopy to manipulate Majorana zero modes emergent in vortex cores of topological superconductors in the Fe(Se,Te) family. We calculate the pinning forces necessary to drag two vortices together and the resulting change in current and charge density of the composite fermion. A possible algorithm for measuring and altering Majorana pair parity is demonstrated.
We show that periodic lattice of Abrikosov vortices in chiral $p$ wave superconductor in general supports fermionic states with zero energy provided the intervortex distance is smaller than the critical one. The zero modes appear at the intersection with the Fermi level of electronic magnetic Bloch bands formed by the overlapping vortex core states. The Bloch bands are robust against lattice disorder induced by fluctuations of vortex positions and can transmit the energy flow across the lattice. The hallmark of zero modes on Bloch bands in electronic heat conductivity is discussed.
We study a Kitaev model on a square lattice, which describes topologically trivial superconductor when gap opens, while supports topological gapless phase when gap closes. The degeneracy points are characterized by two vortices in momentum space, with opposite winding numbers, which are not removable unless meet together. We show rigorously that the topological gapless phase always hosts a partial Majorana flat band edge modes in a ribbon geometry, although such a single band model has zero Chern number as a topologically trivial superconductor. The flat band disappears when the gapless phase …
Majorana zero-modes in a superconductor are midgap states localized in the core of a vortex or bound to the end of a nanowire. They are anyons with non-Abelian braiding statistics, but when they are immobile one cannot demonstrate this by exchanging them in real space and indirect methods are needed. As a real-space alternative, we propose to use the chiral motion along the boundary of the superconductor to braid a mobile vortex in the edge channel with an immobile vortex in the bulk. The measurement scheme is fully electrical and deterministic: edge vortices ($π$-phase domain walls) are creat…
The thermal fluctations of vortices in a superconductor can be usefully mapped onto the quantum fluctations of a collection of bosons at T=0 moving in 2 dimensions. When the superconductor is a thin platelet with the magnetic field parallel to its surface, the interacting quantum bosons are effectively moving in 1 dimension, allowing for powerful Luttinger liquid methods to be applied. Here we consider how this 1 dimensional limit is approached, studying the interaction of vortices with the platelet surfaces and each other. Using realistic parameters and vortex interactions for an underdoped Y…
Proposed approaches to topological quantum computation based on Majorana bound states may enable new paths to fault-tolerant quantum computing. Several recent experiments have suggested that the vortex cores of topological superconductors, such as iron-based superconductors, may host Majorana bound states at zero energy. To facilitate quantum computation with these zero-energy vortex bound states, a precise and fast manipulation of individual vortices is crucial. However, handling individual vortices remains a challenge, and a theoretical framework for describing individually controlled vortex…
FeSeTe has recently emerged as a leading candidate material for the two-dimensional topological superconductivity (TSC). Two reasons for the excitement are the high $T_c$ of the system and the fact that the Majorana zero modes (MZMs) inside the vortex cores live on the exposed surface rather than at the interface of a heterostructure as in the proximitized topological insulators. However, the recent scanning tunneling spectroscopy data have shown that, contrary to the theoretical expectation, the MZM does not exist inside every vortex core. Hence there are ``full'' vortices with MZMs and ``emp…
Higher-Order Topological Superconductivity and Electrically Tunable Majorana Corner Modes in Monolayer MnXPb$_2$ (X=Se, Te)-Pb Heterostructure
Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；Anhui Provincial Key Laboratory of Low-Energy Quantum Materials and Devices, High Magnetic Field Laboratory, HFIPS, Chinese Academy of Sciences（中国科学院合肥物质科学研究院强磁场科学中心安徽低能量子材料与器件重点实验室）；Science Island Branch of Graduate School, University of Science and Technology of China（中国科学技术大学研究生院合肥科学岛分院）；National Key Laboratory of Computational Physics（计算物理学国家重点实验室）；School of Physics and Physical Engineering, Qufu Normal University（曲阜师范大学物理工程学院）
We consider a two-dimensional px+i py superconductor in the presence of multiple vortices, which support zero-energy Majorana fermion states in their cores. Intervortex tunnelings of the Majorana fermions lift the topological state degeneracy. Using the Bogoliubov-de Gennes equation, we calculate splitting of the zero-energy modes due to these tunneling events. We also discuss superconducting fluctuations and, in particular, their effect on the energy splitting.
The search for the Majorana fermions in condensed matter physics has attracted much attention, partially because they may be used for the fault-tolerant quantum computation. It has been predicted that the Majorana zero mode may exist in the vortex core of topological superconductors. Recently, many iron-based superconductors are claimed to exhibit a topologically nontrivial surface state, including Fe(Te,Se). Some previous experiments through scanning tunneling microscopy (STM) have found zero-bias conductance peaks (ZBCP) within the vortex cores of Fe(Te,Se). However, our early experimental r…
We study the effect of thermal fluctuations on the topological stability of chiral p-wave superconductors. We consider two models of superconductors: spinless and spinful with a focus on topological properties and Majorana zero-energy modes. We show that proliferation of vortex-antivortex pairs above the Kosterlitz-Thouless temperature T_KT drives the transition from a thermal Quantum Hall insulator to a thermal metal/insulator, and dramatically modifies the ground-state degeneracy splitting. Therefore, in order to utilize 2D chiral p-wave superconductors for topological quantum computing, the…
We have found the precise stability region of the half quantum vortex (HQV) for superfluid $^3$He A phase confined in parallel plates with a narrow gap under rotation. Standard Ginzburg-Landau free energy, which is well established, is solved to locate the stability region spanned by temperature $T$ and rotation speed ($/Omega$). This $/Omega$-$T$ stability region is wide enough to check it experimentally in available experimental setup. The detailed order parameter structure of HQV characterized by A$_1$ core is given to facilitate the physical reasons of its stability over other vortices or …
Many clever routes to Majorana fermions have been discovered by exploiting the interplay between superconductivity and band topology in metals and insulators. However, realizations in semimetals remain less explored. We ask, ``under what conditions do superconductor vortices in time-reversal symmetric Weyl semimetals -- three-dimensional semimetals with only time-reversal symmetry -- trap Majorana fermions on the surface?'' If each constant-$k_{z}$ plane, where $z$ is the vortex axis, contains equal numbers of Weyl nodes of each chirality, we predict a generically gapped vortex and derive a to…
Excitation spectrum of a half-quantum vortex in a p-wave superconductor contains a zero-energy Majorana fermion. This results in a degeneracy of the ground state of the system of several vortices. From the properties of the solutions to Bogoliubov-de-Gennes equations in the vortex core we derive the non-abelian statistics of vortices identical to that for the Moore-Read (Pfaffian) quantum Hall state.
The Majorana zero mode (MZM), which manifests as an exotic neutral excitation in superconductors, is the building block of topological quantum computing. It has recently been found in the vortices of several iron-based superconductors as a zero-bias conductance peak (ZBCP) in tunneling spectroscopy. In particular, a clean and robust MZM has been observed in the cores of free vortices in (Li0.84Fe0.16)OHFeSe. Here using scanning tunneling spectroscopy (STS), we demonstrate that Majorana-induced resonant Andreev reflection occurs between the STM tip and this zero-bias bound state, and consequent…
We characterize topological features of Josephson junctions formed by coupled mesoscopic chiral $p$-wave superconducting islands. Through analytic and numerical studies of the low-lying BdG (Bogoliubov-de Gennes) spectrum, we identify localized MBS (Majorana bound states) nucleated in Josephson vortices by the application of a perpendicular magnetic field. Additionally, we demonstrate the existence of an extended MBS that is delocalized around the outer perimeter of the coupled islands, which has measurable consequences on the Josephson supercurrent and phase dynamics of the junction. In parti…
We study the conditions for the existence of unpaired Majorana modes at the ends of vortex lines or the side edges of a layered topological superconductor. We show that the problem is mapped to that of a general Majorana chain and extend Kitaev's condition for the existence of its nontrivial phase by providing an additional condition when a supercurrent flows in the chain. Unpaired Majorana bound states may exist in a vortex line that threads the layers if the spin-orbit coupling has certain in-layer components but, interestingly, only if a nonzero supercurrent is maintained along the vortex. …
