Majorana bound states are zero-energy excitations of topological superconductors which obey non-Abelian exchange statistics and are basic building blocks for topological quantum computation. In order to observe and exploit their extraordinary properties, we need to be able to properly manipulate them, for instance, by braiding a couple of them in real space. We propose a setup based on the helical edges of two-dimensional topological insulators (2DTI) which allows for a high degree of tunability by only controlling a handful of superconducting phases. In particular, our setup allows to move the Majoranas along a single edge as well as to move them across two different edges coupled by a quantum point contact. Robustness against non-optimal control of the phases is also discussed. This proposal constitutes an essential step forward towards realizing 2DTI-based architectures capable of performing braiding of Majoranas in a feasible way.
Majorana bound states exist inside a core of half-quantum vortex in spin-triplet superconductors. Despite intense efforts, they have not been discovered in any spin-triplet superconductor candidate materials. After the success of topological insulator, another route to achieve Majorana fermions has been suggested: heterostructures of s-wave superconductors and topological insulator or semimetal with strong spin-orbit coupling provide an effective spinless $p+ip$ pairing which supports Majorana bound states in a single vortex. This theoretical observation has stimulated both experimental and th…
Recent research on disorder effects in topological phases in quasicrystalline systems has received much attention. In this work, by numerically computing the (spin) Bott index and the thermal conductance, we reveal the effects of disorder on a class D chiral topological superconductor and a class DIII time-reversal-invariant topological superconductor in a two-dimensional Ammann-Beenker tiling quasicrystalline lattice. We demonstrate that both the topologically protected chiral and helical Majorana edge modes are robust against weak disorder in the quasicrystalline lattice. More fascinating is…
Helical Dirac-Majorana interferometer in a superconductor-topological insulator sandwich structure
在拓扑绝缘体与两个s波超导体构成的异质结构中，当两个s波超导体形成具有π相移的约瑟夫森结时，存在具有螺旋Majorana边缘模的时反不变性拓扑超导相.基于这样的异质结构，螺旋Dirac-Majorana干涉仪被提出来直接测量螺旋Majorana模式的独特输运特征。此外，我们设想如何在现有材料如Bi 2 Se 3或Bi 2 Te 3薄膜的基础上实现我们的提议。
The presence of a Majorana bound state in condensed matter systems is often associated to a zero bias peak in conductance measurements. Here, we analyze a system were this paradigm is violated. A Majorana bound state is always present at the interface between a quantum spin Hall system that is magnetically gapped and a quantum spin Hall system gapped by proximity induced s-wave superconductivity. However, the linear conductance could be either zero or nonzero and quantized depending on the energy and length scales of the barriers. The transition between the two values is reminiscent of the top…
A two-dimensional time-reversal symmetric topological superconductor is a fully gapped system possessing a helical Majorana mode on the edges. This helical Majorana edge mode (HMEM), which is a Kramer's pair of two chiral Majorana edge modes in the opposite propagating directions, is robust under time-reversal symmetry protection. We propose a feasible setup and accessible measurement to provide the preliminary step of the HMEM realization by studying superconducting antiferromagnetic quantum spin Hall insulators. Since this antiferromagnetic topological insulator hosts a helical electron edge…
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…
Helical spin texture and interference of Majorana bound states in one-dimensional topological superconductor
我们研究了在具有超导衬底和两个铁磁绝缘体（FI）的异质结构中根据2D量子自旋霍尔绝缘体（QSHI）的螺旋边缘态实现的一维（1D）马约拉纳束缚态（MBS）。利用Bogoliubov-de Gennes方法证明了在MBS波函数中存在螺旋自旋织构，其螺距与QSHI螺旋边缘态的费米动量成正比。此外，通过扫描隧道显微镜和能谱仪同时探测两个准粒子的能谱$\pm E \cos(ϕ/2)$，不仅可以绘制出两个准粒子的能谱图，而且还可以从实验上证明两个准粒子的激发态不会相互混合，这是由与准粒子激发态相关的宇称守恒所保护的.
Noncollinear magnetism opens exciting possibilities to generate topological superconductivity. Here, we focus on helical and cycloidal magnetic textures in magnet-superconductor hybrid structures in a background magnetic field. We demonstrate that this system can enter a topological phase which can be understood as a set of parallel topological wires. We explore and confirm this idea in depth with three different approaches: a continuum model, a tight-binding model based on the magnetic unit cell, and exact diagonalization on a finite two-dimensional lattice. The key signature of this topologi…
Chiral and helical Majorana edge modes are two archetypal gapless excitations of two-dimensional topological superconductors. They belong to superconductors from two different Altland-Zirnbauer symmetry classes characterized by $\mathbb{Z}$ and $\mathbb{Z}_2$ topological invariant respectively. It seems improbable to tune a pair of co-propagating chiral edge modes to counter-propagate without symmetry breaking. Here we show that such a direct topological transition is in fact possible, provided the system possesses an additional symmetry $\mathcal{O}$ which changes the bulk topological invaria…
Recent experimental efforts towards the detection of Majorana bound states have focused on creating the conditions for topological superconductivity. Here we demonstrate an alternative route, which achieves fully localised zero-energy Majorana bound states when a topologically trivial superconductor is strongly coupled to a helical normal region. Such a junction can be experimentally realised by e.g. proximitizing a finite section of a nanowire with spin-orbit coupling, and combining electrostatic depletion and a Zeeman field to drive the non-proximitized (normal) portion into a helical phase.…
Superconducting proximity pairing in helical edge modes, such as those of topological insulators (TI), is predicted to provide a unique platform for realizing Majorana zero modes (MZMs). We use scanning tunneling microscopy measurements to probe the influence of proximity induced superconductivity and magnetism on the helical hinge states of Bi(111) films, grown on a superconducting Nb substrate and decorated with magnetic Fe clusters. Consistent with model calculations, our measurements reveal the emergence of a localized MZM at the interface between the superconducting helical edge channel a…
We establish quasi-two-dimensional thin films of iron-based superconductors (FeSCs) as a new high-temperature platform for hosting intrinsic time-reversal-invariant helical topological superconductivity (TSC). Based on the combination of Dirac surface state and bulk extended $s$-wave pairing, our theory should be directly applicable to a large class of experimentally established FeSCs, opening a new TSC paradigm. In particular, an applied electric field serves as a "topological switch" for helical Majorana edge modes in FeSC thin films, allowing for an experimentally feasible design of gate-co…
We propose a novel realization for a topologically superconducting phase hosting Majorana zero-modes on the basis of quantum spin Hall systems. Remarkably, our proposal is completely free of ferromagnets. Instead, we confine helical edge states around a narrow defect line of finite length in a two-dimensional topological insulator. We demonstrate the formation of a new topological regime, hosting protected Majorana modes in the presence of s-wave superconductivity and Zeeman coupling. Interestingly, when the system is weakly tunnel-coupled to helical edge state reservoirs, a particular transpo…
Spin-helical states, which arise in quasi-one-dimensional (1D) channels with spin-orbital (SO) coupling, underpin efforts to realize topologically-protected quantum bits based on Majorana modes in semiconductor nanowires. Detecting helical states is challenging due to non-idealities present in real devices. Here we show by means of tight-binding calculations that by using ferromagnetic contacts it is possible to detect helical modes with high sensitivity even in the presence of realistic device effects, such as quantum interference. This is possible because of the spin-selective transmission p…
We study a general Majorana junction, where N helical nanowires are connected to a common s-wave superconductor proximity-inducing Majorana bound states in the wires. The normal part of each wire (j=1,...,N) acts as connected lead, where electrons can tunnel into the respective Majorana state γ_{A,j}. The Majorana states at the other end, γ_{B,j}, are coupled to each other by an arbitrary tunnel matrix. We examine the conditions for even-odd parity effects in the tunnel conductance for various junction topologies.
We present a new platform for creating Majorana bound states from 2D gapless superconducting state in spin-helical systems under the in-plane Zeeman field. Topological 1D channels are formed by quantum confinement of quasiparticles via Andreev reflection from the surrounding fully gapped superconducting region. Our proposal can be realized using narrow strips of magnetic insulators on top of proximitized 3D topological insulators. This setup has key advantages that include: small Zeeman fields, no required fine-tuning of chemical potential, removal of the low-energy detrimental states, and lar…
Majorana bound states (MBS) differ from the regular zero energy Andreev bound states in their nonlocal properties, since two MBS form a single fermion. We design strategies for detection of this nonlocality by using the phenomenon of Coulomb-mediated Majorana coupling in a simplest setting which still retains falsifiability. Focusing on the implementation of MBS based on the quantum spin Hall effect, we also design a way to probe Majoranas without the need to open a magnetic gap in the helical edge states. In the setup that we analyse, long range MBS coupling manifests in the $h/e$ magnetic fl…
The surface of a higher order topological insulator (HOTI) comprises a two-dimensional topological insulator (TI) with broken inversion symmetry, whose mass is determined by the microscopic details of the surface such as surface potentials and termination. It hosts a helical mode pinned to selected hinges where the surface gap changes its sign. We study the effect of perturbations that break time-reversal and particle-conservation on this helical mode, such as a Zeeman field and a proximate superconductor. We find that in contrast to the helical modes of inversion symmetric TIs, which are gapp…
Cascade of topological phase transitions and revival of topological zero modes in imperfect double helical liquids
Institute of Physics, Academia Sinica（中央研究院物理研究所）；Department of Electronics and Electrical Engineering, National Yang Ming Chiao Tung University（国立阳明交通大学电子工程学系）；Physics Division, National Center for Theoretical Sciences（国家理论科学中心物理组）
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…
Spin-momentum locking in a semiconductor device with strong spin-orbit coupling (SOC) is a fundamental goal of nanoscale spintronics and an important prerequisite for the formation of Majorana bound states. Such a helical state is predicted in one-dimensional (1D) nanowires subject to strong Rashba SOC and spin-mixing, its hallmark being a characteristic reentrant behaviour in the conductance. Here, we report the first direct experimental observations of the reentrant conductance feature, which reveals the formation of a helical liquid, in the lowest 1D subband of an InAs nanowire. Surprisingl…
The helical state is a fundamental prerequisite for many spintronics applications and Majorana zero mode engineering in nanoscopic semiconductors. Its existence in quasi-one-dimensional nanowires was predicted to be detectable as a characteristic reentrant behavior in the conductance, which in a typical two-terminal architecture may be difficult to distinguish from other possible phenomena such as Fabry-Perot oscillations. Here we present an alternative method of helical gap detection free of the mentioned ambiguity, and based on the nonlocal conductance measurements in a three-terminal juncti…
In contrast to elementary Majorana particles, emergent Majorana fermions (MFs) in condensed-matter systems may have electromagnetic multipoles. We developed a general theory of magnetic multipoles for surface helical MFs on time-reversal-invariant superconductors. The results show that the multipole response is governed by crystal symmetry, and that a one-to-one correspondence exists between the symmetry of Cooper pairs and the representation of magnetic multipoles under crystal symmetry. The latter property provides a way to identify nonconventional pairing symmetry via the magnetic response …
We study how the shape of a periodic magnetic field affects the presence of Majorana bound states (MBS) in a nanowire-superconductor system. Motivated by the field configurations that can be produced by an array of nanomagnets, we consider spiral fields with an elliptic cross-section and fields with two sinusoidal components. We show that MBS are robust to imperfect helical magnetic fields. In particular, if the amplitude of one component is tuned to the value determined by the superconducting order parameter in the wire, the MBS can exist even if the second component has a much smaller amplit…
One-dimensional helical liquids can appear at boundaries of certain condensed matter systems. Two prime examples are the edge of a quantum spin Hall insulator and the hinge of a three-dimensional second-order topological insulator. For these materials, the presence of a helical state at the boundary serves as a signature of their nontrivial electronic bulk topology. Additionally, these boundary states are of interest themselves, as a novel class of strongly correlated low-dimensional systems with interesting potential applications. Here, we review existing results on such helical liquids in se…
Electronic systems harboring one dimensional helical modes, where the spin and momentum of the electron are locked, have lately become an important field of its own. When coupled to a conventional superconductor, such systems are expected to manifest topological superconductivity, a unique phase that gives rise to exotic Majorana zero modes. Even more interesting are fractional helical states which have not been observed before and which open the route for the realization of the generalized para fermions quasiparticles. Possessing non abelian exchange statistics, these quasiparticles may serve…
A hybrid setup consisting of a superconductivity-proximitized quantum spin Hall (QSH) insulator and a quantum anomalous Hall (QAH) insulator is proposed for chiral injection of electrons into the Majorana bound state (MBS). An unexplored region of the phase space involving the exchange field induced boost of the helical edge state is then proposed for the detection of the MBS. 2-D transport simulations of our proposed setup is compared with the corresponding setup in the absence of the QAH region, when moderate disorder and a small but finite bulk out-of-plane magnetic field and a Rashba field…
