Exploring Majorana zero modes in iron-based superconductors

Majorana zero modes (MZMs) are Majorana-fermion-like quasiparticles existing in crystals or hybrid platforms with topologically non-trivial electronic structures. They obey non-Abelian braiding statistics and are considered promising to realize topological quantum computing. Discovery of MZM in the vortices of the iron-based superconductors (IBSs) has recently fueled the Majorana research in a way which not only removes the material barrier requiring construction of complicated hybrid artificial structures, but also enables observation of pure MZMs under higher temperatures. So far, MZMs have been observed in iron-based superconductors including FeTe_0.55Se_0.45, (Li_0.84Fe_0.16)OHFeSe, CaKFe₄As₄, and LiFeAs. In this topical review, we present an overview of the recent STM studies on the MZMs in IBSs. We start with the observation of MZMs in the vortices in FeTe_0.55Se_0.45 and discuss the pros and cons of FeTe_0.55Se_0.45 compared with other platforms. We then review the following up discovery of MZMs in vortices of CaKFe₄As₄, impurity-assisted vortices of LiFeAs, and quantum anomalous vortices in FeTe_0.55Se_0.45, illustrating the pathway of the developments of MZM research in IBSs. Finally, we give perspective on future experimental works in this field.     

Fidelity in topological superconductors with end Majorana fermions

Fidelity and fidelity susceptibility are introduced to investigate the topological superconductors with end Majorana fermions. A general formalism is established to calculate the fidelity and fidelity susceptibility by solving Bogoliubov–de Gennes equations. Both clean and disordered systems are studied within this formalism, and the results show that the fidelity susceptibility serves as a valid indicator for the topological quantum phase transition which signals the appearance of Majorana fermions. Our study provides a useful tool to investigate the topological quantum phase transition in superconductors, which is helpful to find topological phases in various systems.     

Realistic time-reversal invariant topological insulators with neutral atoms

We lay out an experiment to realize time-reversal invariant topological insulators in alkali atomic gases. We introduce an original method to synthesize a gauge field in the near field of an atom chip, which effectively mimics the effects of spin-orbit coupling and produces quantum spin-Hall states. We also propose a feasible scheme to engineer sharp boundaries where the hallmark edge states are localized. Our multiband system has a large parameter space exhibiting a variety of quantum phase transitions between topological and normal insulating phases. Because of their remarkable versatility, cold-atom systems are ideally suited to realize topological states of matter and drive the development of topological quantum computing.     

A brief review on Majorana bound states in topological superconductors

Topological superconductivity has drawn much attention recently, and most interests are focused on the Majorana bound states existing at the edges of one-dimensional topological superconductors. These Majorana bound states are ideal platform for studying non-Abelian statistics. Meanwhile, they are proposed to be useful in quantum computation. In this review, we introduce the basic concepts and models in this area. We begin from the Kitaev model, which is the most concise model for one-dimensional topological superconductivity. Then, we discuss how to realize this model with spin-orbit coupling in realistic materials. Finally, we show some simple methods to detect the Majorana bound states and study their novel properties with the help of adjacent quantum dots.     

Magnetic flux control of chiral Majorana edge modes in topological superconductor

We study the transport of chiral Majorana edge modes (CMEMs) in a hybrid quantum anomalous Hall insulator-topological superconductor (QAHI-TSC) system in which the TSC region contains a Josephson junction and a cavity. The Josephson junction undergoes a topological transition when the magnetic flux through the cavity passes through half-integer multiples of magnetic flux quantum. For the trivial phase, the CMEMs transmit along the QAHI-TSC interface as without magnetic flux. However, for the nontrivial phase, a zero-energy Majorana state appears in the cavity, leading to that a CMEM can resonantly tunnel through the Majorana state to a different CMEM. These findings may provide a feasible scheme to control the transport of CMEMs by using the magnetic flux and the transport pattern can be customized by setting the size of the TSC.     

Majorana modes at the ends of superconductor vortices in doped topological insulators

Recent experiments have observed bulk superconductivity in doped topological insulators. Here we ask whether vortex Majorana zero modes, previously predicted to occur when s-wave superconductivity is induced on the surface of topological insulators, survive in these doped systems with metallic normal states. Assuming inversion symmetry, we find that they do but only below a critical doping. The critical doping is tied to a topological phase transition of the vortex line, at which it supports gapless excitations along its length. The critical point depends only on the vortex orientation and a suitably defined SU(2) Berry phase of the normal state Fermi surface. By calculating this phase for available band structures we determine that superconducting p-doped Bi(2)Te(3), among others, supports vortex end Majorana modes. Surprisingly, superconductors derived from topologically trivial band structures can support Majorana modes too.     

Recent progress on non-Abelian anyons: from Majorana zero modes to topological Dirac fermionic modes

Majorana zero modes(MZMs) have been intensively studied in recent decades theoretically and experimentally as the most promising candidate for non-Abelian anyons supporting topological quantum computation(TQC). In addition to the Majorana scheme, some non-Majorana quasiparticles obeying non-Abelian statistics, including topological Dirac fermionic modes, have also been proposed and therefore become new candidates for TQC. In this review, we overview the non-Abelian braiding properties as well as...     

Index theorem for the zero modes of Majorana fermion vortices in chiral p-wave superconductors

We show that the Majorana fermion zero modes in the cores of odd winding number vortices of a 2D (p(x)+ip(y))-paired superconductor is due to an index theorem. This theorem is analogous to that proven by Jackiw and Rebbi for the existence of localized Dirac fermion zero modes on the mass domain walls of a 1D Dirac theory. The important difference is that, in our case, the theorem is proven for a two component fermion field theory where the first and second components are related by parity reversal and Hermitian conjugation.     

Majorana modes and topological superfluids for ultracold fermionic atoms in anisotropic square optical lattices

Motivated by the recent experimental realization of two-dimensional spin-orbit couplingthrough optical Raman lattice scheme, we study attractive interacting ultracold gases withspin-orbit interaction in anisotropic square optical lattices, and find that richs-wavetopological superfluids can be realized, including Z₂ topological superfluids beyondthe characterization of “tenfold way” in addition to chiral topological superfluids. Thetopological defects-superfluid vortex and edge dislocations-may host Majorana modes insome topological superfluids, which are helpful for realizing topological quantumcomputation and Majorana fermionic quantum computation. In addition, we also discuss theBerezinsky-Kosterlitz-Thouless phase transitions for different topologicalsuperfluids.     

Strongly anisotropic s-wave gaps in exotic superconductors

The exotic superconductors, defined as those that follow the phenomenological trend reported by Uemura (T _c approximately proportional to λ_L ^?2), at present constitute the most broad and general class of superconductors which can reasonably be considered 'similar to the high-T _c cuprates'. It is therefore of much interest to determine the forms of their pairing gap functions. We examine evidence for the gap forms in non-cuprate exotics and demonstrate some general features. The cubic materials often have highly anisotropic gaps. The planar materials tend to have an even stronger gap anisotropy, to the extent that they often have gap nodes, but nevertheless they usually have an s-wave-like gap symmetry. There is good evidence for the latter even in the controversial cases of planar organics and nickel borocarbides. Exceptions to these generalizations are also pointed out and discussed.     

Existence of Majorana fermions and topological order in nodal superconductors with spin-orbit interactions in external magnetic fields

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 antisymmetric 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 fermion state.     

Edge currents in superconductors with a broken time-reversal symmetry

We analyze edge currents and edge bands at the surface of a time-reversal symmetry breaking dx2-y2 + id(xy) superconductor. We show that the currents have large Friedel oscillations with two interfering frequencies: square root of 2kF from subgap states, and 2kF from the continuum. The results are based independently on a self-consistent slave-boson mean-field theory for the t-J model on a triangular lattice, and on a T-matrix scattering theory calculation. The shape of the edge-state band, as well as the particular frequency square root of 2kF of the Friedel oscillations, are attributes unique for the dx2-y2 + id(xy) case, and may be used as a fingerprint for its identification. Extensions to different time-reversal symmetry breaking superconductors can be achieved within the same approach.     

Holographic correspondence in topological superconductors

We analytically derive a compatible family of effective field theories that uniquely describe topological superconductors in 3D, their 2D boundary and their 1D defect lines. We start by deriving the topological field theory of a 3D topological superconductor in class DIII, which is consistent with its symmetries. Then we identify the effective theory of a 2D topological superconductor in class D living on the gapped boundary of the 3D system. By employing the holographic correspondence we derive the effective chiral conformal field theory that describes the gapless modes living on the defect lines or effective boundary of the class D topological superconductor. We demonstrate that the chiral central charge is given in terms of the 3D winding number of the bulk which by its turn is equal to the Chern number of its gapped boundary.     

Spin and Majorana polarization in topological superconducting wires

We study a one-dimensional wire with strong Rashba and Dresselhaus spin-orbit coupling (SOC), which supports Majorana fermions when subject to a Zeeman magnetic field and in the proximity of a superconductor. Using both analytical and numerical techniques we calculate the electronic spin texture of the Majorana end states. We find that the spin polarization of these states depends on the relative magnitude of the Rashba and Dresselhaus SOC components. Moreover, we define and calculate a local "Majorana polarization" and "Majorana density" and argue that they can be used as order parameters to characterize the topological transition between the trivial system and the system exhibiting Majorana bound modes. We find that the local Majorana polarization is correlated to the transverse spin polarization, and we propose to test the presence of Majorana fermions in a 1D system by a spin-polarized density of states measurement.     

Delocalization of quasiparticles in quasi-two-dimensional disordered superconductors with s-wave pairing

We investigate localization behavior of quasiparticles in disordered multi-plane superconductors with s-wave pairing. By introducing disorder with random site energies, the spatial fluctuations of Bogoliubov-de Gennes pairing potential are self-consistently determined. The size dependence of rescaled localization length for a long bar is calculated by using the transfer-matrix method. From the finite-size scaling analysis we show that there exists a critical point of the disorder strength W_c which separates the extended and localized quasiparticle states in such quasi-two-dimensional systems. The associated critical behavior is studied and the relationship of the results to the number of planes is discussed.