Magnet-superconductor hybrid (MSH) structures represent one of the most promising platforms to realize, control and manipulate Majorana modes using scanning tunneling methods. By depositing either chains or islands of magnetic atoms on the surface of a conventional, elemental superconductor such as Pb or Re, topological superconducting phases can emerge. They feature either localised Majorana bound states at the chain ends or dispersing chiral Majorana modes at the island's boundary. Yet some of these experiments have not reached the spectral resolution to clearly distinguish between topological Majorana and trivial Shiba states due to very small superconducting gap sizes and experiments performed at sub-Kelvin temperatures. Here we consider superconducting substrates with unconventional spin-singlet pairing, including high-temperature d-wave and extended s-wave superconductors. We derive topological phase diagrams and compute edge states for cylinder and island geometries and discuss their properties. Several time-reversal invariant topological superconducting phases of the Zhang-Kane-Mele type are found and discussed. Moreover, we study one-dimensional MSH structures and show that parameters to realize topologically non-trivial magnetic chains embedded into a larger, two-dimensional substrate differ from the purely one-dimensional case. Quite generally we find that unconventional superconducting substrates work as well as the conventional s-wave substrates to realize topological phases. In particular, iron-based pnictide and chalcogenide superconductors are the most promising class of substrates for future high-temperature MSH systems.
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.
We present a pedagogical review of topological superconductivity and its consequences in spin-orbit coupled semiconductor/superconductor heterostructures. We start by reviewing the historical origins of the notions of Dirac and Majorana fermions in particle physics and discuss how lower dimensional versions of these emerge in one dimensional superconductors. Ultimately, we focus on Majorana zero-modes, which emerge at defects in the Majorana equation. We then review the definition of the topological invariant, and how it allows the prediction of such Majorana modes from the bulk bandstructure …
Topological superconductors are novel classes of quantum condensed phases, characterized by topologically nontrivial structures of Cooper pairing states. On the surfaces of samples and in vortex cores of topological superconductors, Majorana fermions, which are particles identified with their own anti-particles, appear as Bogoliubov quasiparticles. The existence and stability of Majorana fermions are ensured by bulk topological invariants constrained by the symmetries of the systems. Majorana fermions in topological superconductors obey a new type of quantum statistics referred to as non-Abeli…
We discuss electric transport through a point contact which bridges Majorana fermion modes appearing at edges of two helical superconductors. The contents focus on effects of interference and interaction unique to the Majorana fermions and role of spin-orbit interaction (SOI). Besides the Josephson current, the quasi-particle conductance depends sensitively on phase difference and relative helicity between the two superconductors. The interaction among the Majorana fermions causes the power-law temperature dependences of conductance for various tunneling channels. Especially, in the presence o…
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 …
Majorana corner modes appearing in two-dimensional second-order topological superconductors have great potential applications for fault-tolerant topological quantum computations. We demonstrate that in the presence of an in-plane magentic field two-dimensional ($s+p$)-wave superconductors host Majorana corner modes, whose location can be manipulated by the direction of the magnetic field. In addition, we discuss the effects of edge imperfections on the Majorana corner modes. We describe how different edge shapes and edge disorder affect the number and controllability of the Majorana corner mod…
Semiconducting nanowires in proximity to superconductors are promising experimental systems for Majorana fermions which may ultimately be used as building blocks for topological quantum computers. A serious challenge in the experimental realization of the Majorana fermion in these semiconductor-superconductor nanowire structures is tuning the semiconductor chemical potential in close proximity to the metallic superconductor. We show that, presently realizable structures in experiments with tunable chemical potential lead to Majorana resonances, which are interesting in their own right, but do …
Through a systematic symmetry and topology analysis we establish that three-dimensional chiral superconductors with strong spin-orbit coupling and odd-parity pairing generically host low-energy nodal quasiparticles that are spin-non-degenerate and realize Majorana fermions in three dimensions. By examining all types of chiral Cooper pairs with total angular momentum $J$ formed by Bloch electrons with angular momentum $j$ in crystals, we obtain a comprehensive classification of gapless Majorana quasiparticles in terms of energy-momentum relation and location on the Fermi surface. We show that t…
Majorana fermions exist on the boundaries of two-dimensional topological superconductors (TSCs) as charge-neutral quasi-particles. The neutrality makes the detection of such states challenging from both experimental and theoretical points of view. Current methods largely rely on transport measurements in which Majorana fermions manifest themselves by inducing electron-pair tunneling at the lead-contacting point. Here we show that chiral Majorana fermions in TSCs generate {enhanced} local optical response. The features of local optical conductivity distinguish them not only from trivial superco…
We demonstrate that Majorana fermions exist in edges of systems and in a vortex core even for superconductors with nodal excitations such as the d-wave pairing state under a particular but realistic condition in the case with an anti-symmetric spin-orbit interaction and a nonzero magnetic field below the upper critical field. We clarify that the Majorana fermion state is topologically protected in spite of the presence of bulk gapless nodal excitations, because of the existence of a nontrivial topological number. Our finding drastically enlarges target systems where we can explore the Majorana…
We study two-dimensional noncentrosymmetric nodal superconductors under Zeeman field and clarify the field angle dependence of topological properties. It has been shown that the nodal excitation acquires an excitation gap due to the Zeeman field perpendicular to anti-symmetric spin-orbit coupling, and then gapful topological superconductivity is realized. We show that the system undergoes gapful-gapless transition against tilting of the field. The gapless phase remains to show a finite band gap and unusual Majorana edge states in between the bulk bands. The Majorana edge states naturally propa…
We theoretically study the effect of orbital-dependent exchange field in the formation of second order topological superconductors. We demonstrate that changing the orbital difference can induce topological transition and the Majorana corner modes therein can be manipulated. We further propose to detect the corner modes via a normal probe terminal. The conductance quantization is found to be robust to changes of the relevant system parameters.
We propose a novel platform for the creation and manipulation of Majorana zero modes consisting of a ferromagnetic metallic wire placed between conventional superconductors which are in proximity to ferromagnetic insulators. Our device relies on the interplay of applied supercurrents and exchange fields emerging from the ferromagnetic insulators. We assert that the proposed superconductor/ferromagnet heterostructures exhibit enhanced controllability, since topological superconductivity can be tuned apart from gate voltages applied on the ferromagnetic wire, also by manipulating the applied sup…
In this paper we find that confining a second-order topological superconductor with a harmonic potential leads to a proliferation of Majorana corner modes. As a consequence, this results in the formation of Majorana corner flat bands which have a fundamentally different origin from that of the conventional mechanism. This is due to the fact that they arise solely from the one-dimensional gapped boundary states of the hybrid system that become gapless without the bulk gap closing under the increase of the trapping potential magnitude. The Majorana corner states are found to be robust against th…
Braiding Majorana zero modes is essential for fault-tolerant topological quantum computing. Iron-based superconductors with nontrivial band topology have recently emerged as a surprisingly promising platform for creating distinct Majorana zero modes in magnetic vortices in a single material and at relatively high temperatures. The magnetic field-induced Abrikosov vortex lattice makes it difficult to braid a set of Majorana zero modes or to study the coupling of a Majorana doublet due to overlapping wave functions. Here we report the observation of the proposed quantum anomalous vortex with int…
Topologically nontrivial superconducting phases have been engineered in topological materials by the proximity effect in contact with conventional superconductors. In this paper, by using the method of the Kronig-Penney model, we study the superconducting proximity effect in the bulk electronic states of Weyl semimetals by considering a multilayer structure consisting of Weyl-semimetal and superconductor layers. Due to the proximity effect, two Weyl nodes are decoupled into four nodes of Majorana fermions resulting in Weyl-superconductor phases or three-dimensional extension of topological-sup…
A new method is introduced for probing Topological Superconductors. The integration of the superconding fermions generates a topological $\mathbf{Chern-Simons}$ sound action . Dislocations induce Majorana zero modes inside the sample, resulting in a new Hamiltonian which couple the Majorana modes, the electron field and the non-Abelian strain field sound. This Hamiltonian is used to compute the anomalous sound absorption. The Topological superconductor absorbs sound below the superconducting gap due to the transition between the quasi particles and the Majorana fermions. The sound waves offers…
In this work, we show that a quasi-one-dimensional $d_{x^2-y^2}$-wave superconductor with Rashba spin-orbit coupling is a DIII class, time-reversal invariant, topological superconductor (TS) which supports a Majorana Kramers Doublet (MKD) at each end of the TS. A MKD is a pair of Majorana end states (MESs) protected by time-reversal symmetry (TRS). An external magnetic field breaks TRS and drives the system from DIII to D class in which case a single MES appears at each end of the TS. We show that a MKD induces resonant Andreev reflection with zero bias conductance peak of $4e^2/h$. Experiment…
Disorder is known to suppress the gap of a topological superconducting state that would support non-Abelian Majorana zero modes. In this paper, we study using the self-consistent Born approximation the robustness of the Majorana modes to disorder within a suitably extended Eilenberger theory, in which the spatial dependence of the localized Majorana wave functions is included. We find that the Majorana mode becomes delocalized with increasing disorder strength as the topological superconducting gap is suppressed. However, surprisingly, the zero bias peak seems to survive even for disorder stre…
Odd-frequency Cooper pairs are gathering attention for the convenience of investigating the edge state of topological superconductors including Majorana fermions. Although a spinless p-wave superconductor has only one Majorana fermion in a topological phase, the system with magnetic fields can reach the topological phases with multiple Majorana fermions. To distinguish these multiple Majorana fermion phases, we correlate the energy spectrum with the odd-frequency pair amplitude as increasing the system size. The system size dependence tells us three pieces of information: the parity of the num…
We explore Majorana zero modes bound to a vortex line in a three dimensional topological superconductor model, focusing our attention on the validity of the index theorem previously derived. We first solve the Bogoliubov-de Gennes equation at the zero energy to obtain the analytical index. We next calculate the topological index given by the order parameters. It turns out that they indeed coincide and that index theorem, which has been derived on the implicit assumption that a defect is point-like, is also valid for a line defect.
Majorana bound states (MBS) are well-established in the clean limit in chains of ferromagnetically aligned impurities deposited on conventional superconductors with finite spin-orbit coupling. Here we show that these MBS are very robust against disorder. By performing self-consistent calculations we find that the MBS are protected as long as the surrounding superconductor show no large signs of inhomogeneity. We find that longer chains offer more stability against disorder for the MBS, albeit the minigap decreases, as do increasing strengths of spin-orbit coupling and superconductivity.
Motivated by the recent experiment indicating that superconductivity in the doped topological insulator Cu$_x$Bi$_2$Se$_3$ has an odd-parity pairing symmetry with rotational symmetry breaking, we study the general class of odd-parity superconductors with two-component order parameters in trigonal and hexagonal crystal systems. In the presence of strong spin-orbit interaction, we find two possible superconducting phases below $T_c$, a time-reversal-breaking (i.e., chiral) phase and an anisotropic (i.e., nematic) phase, and determine their relative energetics from the gap function in momentum sp…
Due to the charge neutral and localized nature of surface Majorana modes, detection schemes usually rely on local spectroscopy or interference through the Josephson effect. Here, we theoretically study the magnetic response of a two-dimensional cone of Majorana fermions localized at the surface of class DIII Topological Superconductors. For a field parallel to the surface the Zeeman term vanishes and the orbital term induces a Doppler shift of the Andreev levels resulting in a tilting of the surface Majorana cone. For fields larger than a critical threshold field $H^*$ the system undergoes a t…
A number of ferromagnetic superconductors have been recently discovered which are believed to be in the so-called "equal spin pairing" (ESP) state. In the ESP state the Cooper pairs condense forming order parameters $Δ_{\uparrow\uparrow}, Δ_{\downarrow\downarrow}$ which are decoupled in the spin-sector. We show that these three-dimensional systems should generically support topologically protected surface Majorana arcs and bulk Weyl fermions as gapless excitations. Similar protected low-energy exotic quasiparticles should also appear in the recently discovered non-centrosymmteric superconducto…
We show that corner Majorana zero modes in a two-dimensional $p+id$ topological superconductor can be controlled by the manipulation of the parent $p-$wave superconducting order. Assuming that the $p$-wave superconducting order is in either a chiral or helical phase, we find that when a $d_{x^2-y^2}$ wave superconducting order is induced, the system exhibits quite different behavior depending on the nature of the parent $p$-wave phase. In particular, we find that while in the helical phase, a localized Majorana mode appears at each of the four corners, in the chiral phase, it is localized only…
The topological properties of a multiband topological superconductor in two dimensions are studied, when the latter is obtained by introducing electron pairing in an otherwise topological insulator. The type of pairing, doping and Fermi surface topology play an essential role. Considering the Andreev reflection problem, we use a previously developed quantum waveguide theory for multiorbital systems and find that, when the Fermi surface has several pockets, this theory retrieves the correct number of Majorana fermion states as predicted by the topological index. By varying band structure parame…
As a half-metal is spin-polarized at its Fermi level by definition, it was conventionally thought to have little proximity effect to an s-wave superconductor. Here we show that, with interface spin-orbit coupling, $p_x +ip_y$ superconductivity without spin degeneracy is induced on the half-metal, and we give an estimate of its bulk energy gap. Therefore a single-band half-metal can give us a topological superconductor with a single chiral Majorana edge state. Our band calculation shows that two atomic layers of VTe or CrO$_2$ is a single-band half-metal for a wide range ($\sim$0.1eV) of Fermi …
Noncentrosymmetric superconductors with strong spin-orbit coupling and the B phase of ${}^3$He are possible realizations of topological superconductors with time-reversal symmetry. The nontrivial topology of these time- reversal invariant superconductors manifests itself at the material surface in the form of helical Majorana modes. In this paper, using extensive numerical simulations, we investigate the stability and properties of these Majorana states under strong surface disorder, which influences both bulk and surface states. To characterize the effects of strong disorder, we compute the l…
Nodal topological superconductors characterized by $p_x$-wave pairing symmetry host flat-band Majorana bound states causing drastic anomalies in low-energy electromagnetic responses. Nevertheless, the study of flat-band Majorana bound states has been at a standstill owing to a serious lack of candidate materials for $p_x$-wave superconductors. In this paper, by expanding a scheme of planar topological Josephson junctions, we propose a promising device realizing an effective $p_x$-wave superconductor. Specifically, we consider a three-dimensional Josephson junction consisting of a thin-film sem…
Recent experiments reported the puzzling zero energy modes associated with ferromagnetic impurities in some iron-based superconductors with topological band structures. Here, we show that the sufficiently strong exchange coupling between a ferromagnetic impurity and substrate can trigger a quantum phase transition, beyond which, the phase of the topological surface superconducting order parameter around the impurity acquires a sign-change. In such a case, we prove that a Kramers degenerate pair of Majorana modes can be induced at the boundary separating the two sign-change regimes and trapped …
We study Majorana edge states and their topological properties of one-dimensional(1D) and two-dimensional(2D) Rashba semiconductor deposited on iron-based superconductor under the applied Zeeman field for various directions. Using the recursive Green's function method, we calculate the local density of states(LDOS) both for $s_{\pm}$ and $s_{++}$-wave pairings. We elucidate that it shows anisotropic response to the applied Zeeman field specific to Majorana edge states. This anisotropy can be understood by the winding number, which shows whether the present system is topological or not. The res…
We investigate non-Abelian statistics of Majorana Kramers pairs (MKPs) in a network system of one-dimensional time-reversal invariant topological superconductors by using numerical simulations of braiding dynamics, and examine the tolerance against various perturbations which may cause decoherence of MKPs. We, first, consider effects of a magnetic field which breaks time-reversal symmetry. In contrast to a naive expectation, the non-Abelian braiding of MKPs is robust against applied magnetic fields provided that the initial and final states of a braiding process are invariant under the combina…
We show that a topological phase supporting Majorana fermions can form in a 2D electron gas (2DEG) adjacent to an interdigitated superconductor-ferromagnet structure. An advantage of this setup is that the 2DEG can induce the required Zeeman splitting and superconductivity from a single interface, allowing one to utilize a wide class of 2DEGs including the surface states of bulk InAs. We demonstrate that the interdigitated device supports a robust topological phase when the finger spacing lambda is smaller than half of the Fermi wavelength lambda_F. In this regime, the electrons effectively se…
In this paper we study an s-wave topological SC with a square vortex-lattice. We proposed a topological Majorana lattice model to describe this topological state. We found that the Majorana lattice model is really a "topological SC" on the parent topological SC. Such correspondence between the Majorana lattice model and the topological SC with vortex-lattice is another holographic feature of the topological state. In addition, the numerical calculations confirm our theoretical prediction.
Spin chains proximitized with superconducting condensates have emerged as one of the most promising platforms for the realization of Majorana modes. Here, we craft diluted spin chains atom-by-atom following seminal theoretical proposal suggesting indirect coupling mechanisms as a viable route to trigger topological superconductivity. Starting from single adatoms hosting deep Shiba states, we use the highly anisotropic Fermi surface of the substrate to create spin chains characterized by different magnetic configurations along distinct crystallographic directions. By scrutinizing a large set of…
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.
Topological superconductors are exotic gapped phases of matter hosting Majorana mid-gap states on their boundary. In conventional topological superconductors, Majorana in-gap states appear in the form of either localized zero-dimensional modes or propagating spin-1/2 fermions with a quasi-relativistic dispersion relation. Here we show that unconventional propagating Majorana states can emerge on the surface of three-dimensional topological superconductors protected by rotational symmetry. The unconventional Majorana surface states fall into three different categories: a spin-$S$ Majorana fermi…
Semiconducting nanowires in proximity to superconductors are promising experimental systems for Majorana fermions, which may ultimately be used as building blocks for topological quantum computers. A serious challenge in the experimental realization of the Majorana fermions is the supression of topological superconductivity by disorder. We show that Majorana fermions protected by a robust topological gap can occur at the ends of a chain of quantum dots connected by s-wave superconductors. In the appropriate parameter regime, we establish that the quantum dot/superconductor system is equivalent…
We show that Majorana fermions associated with Majorana flat bands emerge as zero energy modes on the [110] edge of single layer or multilayer nodal superconductors with $d_{x^2-y^2}$-wave pairing and Rashba spin-orbit coupling. Moreover, as long as the global inversion symmetry of the single layer or multilayer superconductor is broken, and in the presence of an in-plane magnetic field parallel to the [110] edge, the Majorana fermions together with usual fermionic Andreev bound states induce a triple-peak structure in tunnelling spectroscopy experiments. Importantly, we show that the zero bia…
We analyze the prospects for stabilizing Majorana zero modes in semiconductor nanowires that are proximity-coupled to higher-temperature superconductors. We begin with the case of iron pnictides which, though they are s-wave superconductors, are believed to have superconducting gaps that change sign. We then consider the case of cuprate superconudctors. We show that a nanowire on a step-like surface, especially in an orthorhombic material such as YBCO, can support Majorana zero modes at an elevated temperature.
In the heterostructure composed of a topological insulator sandwiched by two s-wave superconductors, the time reversal invariant topological superconducting phase, possessing helical Majorana edge modes, is found to exist when the two s-wave superconductors form a Josephson junction with a pi phase shift. Based on such a heterostructure, a helical Dirac-Majorana interferometer is proposed to directly measure a unique transport signature of the helical Majorana modes. Furthermore, we envision how our proposal can be realized on the basis of existing materials such as Bi2Se3 or Bi2Te3 thin films…
Recently, Majorana Fermions (MFs) have attracted intensive attention due to their exotic statistics and possible applications in topological quantum computation (TQC). They are proposed to exist in various two-dimensional (2D) topological systems, such as px+ipy topological superconductor and nanowire-superconducting hybridization system. In this paper, two types of Majorana Fermions with different polygon sign rules are pointed out. A "smoking gun" numerical evidence to identify MF's classification is presented through looking for the signature of a first order topological quantum phase trans…
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.
We study theoretically the role of topological invariants to protect the Majorana fermions in a model of two-dimensional chiral superconductors which belong to class D of topological periodic table. A rich phase diagram is revealed. Each phase is characterized by the topological invariants for 2d (Z) and 1d (Z2), which lead to the Majorana fermion at the edge of the dislocation and the core of the vortex. Interference of the Majorana fermions originated from the different topological invariants is studied. The stability of the Majorana fermion with respect to the interlayer coupling, i.e., in …
One of the goals of modern spectroscopy is to invent techniques which detect neutral excitations that have been theoretically proposed. For superconductors, two point transport measurements detect the Andreev crossed reflection which confirms the existence of the Majorana fermions. Similar information can be obtained from a measurement using two piezoelectric transducers. One transducer measures the stress tensor response from the strain field generated by the second transducer. The ratio between the stress response and strain velocity determines the dissipative response. We will show that the…
Majorana zero mode is an exotic quasi-particle excitation with non-Abelian statistics in topological superconductor systems, and can serve as the cornerstone for topological quantum computation, a new type of fault-tolerant quantum computation architecture. This review paper highlights recent progress in realizing Majorana modes in iron-based high-temperature superconductors. We begin with the discussion on topological aspect of electronic band structures in iron-based superconductor compounds. Then we focus on several concrete proposals for Majorana modes, including the Majorana zero modes in…
We found parity-violating Majorana fermions in a chiral superconductor Sr2RuO4. The current-voltage curves show an anomalous behavior: The induced voltage is an even function of the bias current. The magnetic field dependent results suggest the excitation of the Majorana fermions along the closed chiral edge current of the single domain under bias current. We also discuss the relationship between a change of the chirality and spontaneous magnetization of the single domain Sr2RuO4.
Second-order topological superconductors host Majorana corner and hingemodes in contrast to conventional edge and surface modes in two and three dimensions. However, the realization of such second-order corner modes usually demands unconventional superconducting pairing or complicated junctions or layered structures. Here we show that Majorana corner modes could be realized using a 2D quantum spin Hall insulator in proximity contact with an $s$-wave superconductor and subject to an in-plane Zeeman field. Beyond a critical value, the in-plane Zeeman field induces opposite effective Dirac masses…
As platforms of Majorana modes, topological insulator (quantum anomalous Hall insulator)/superconductor (SC) heterostructures have attracted tremendous attention over the past decade. Here we substitute the topological insulator by its higher-order counterparts. Concretely, we consider second-order topological insulators (SOTIs) without time-reversal symmetry and investigate SOTI/SC heterostructures in both two and three dimensions. Remarkably, we find that such novel heterostructures provide natural realizations of second-order topological superconductors (SOTSCs) which host Majorana corner m…
Realizations of Majorana fermions in solid state materials have attracted great interests recently in connection to topological order and quantum information processing. We propose a novel way to create Majorana fermions in superconductors. We show that an incipient non-collinear magnetic order turns a spin-singlet superconductor with nodes into a topological superconductor with a stable Majorana bound state in the vortex core; at a topologically-stable magnetic point defect; and on the edge. We argue that such an exotic non-Abelian phase can be realized in extended t-J models on the triangula…
A counting formula for computing the number of (Majorana) zero modes bound to topological point defects is evaluated in a gradient expansion for systems with charge-conjugation symmetry. This semi-classical counting of zero modes is applied to some examples that include graphene and a chiral p-wave superconductor in two-dimensional space. In all cases, we explicitly relate the counting of zero modes to Chern numbers.
We study a realistic Floquet topological superconductor, a periodically driven nanowire proximitized to an equilibrium s-wave superconductor. Due to both strong energy and density fluctuations caused from the superconducting proximity effect, the Floquet Majorana wire becomes dissipative. We show that the Floquet band structure is still preserved in this dissipative system. In particular, we find that both the Floquet Majorana zero and pi modes can no longer be simply described by the Floquet topological band theory. We also propose an effective model to simplify the calculation of the lifetim…
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.
Coupled Majorana zero modes with nonzero energies are generally detrimental to the non-Abelian statistics due to the additional dynamic phase. Nevertheless, we show that a well-connected lead can introduce a local non-Hermitian dissipation term to shift the energies of the both coupled Majorana modes to zero, and surprisingly turn the coupled Majorana mode far from the lead into a dark Majorana mode with exponentially small dissipation. This dark Majorana mode can conquer the drawback of the partially overlapped Majorana zero modes and possess the properties of true Majorana zero mode such as …
The study of non-Abelian Majorana zero modes advances our understanding of the fundamental physics in quantum matter, and pushes the potential applications of such exotic states to topological quantum computation. It has been shown that in two-dimensional (2D) and 1D chiral superconductors, the isolated Majorana fermions obey non-Abelian statistics. However, Majorana modes in a $Z_2$ time-reversal invariant (TRI) topological superconductor come in pairs due to Kramers' theorem. Therefore, braiding operations in TRI superconductors always exchange two pairs of Majoranas. In this work, we show i…
