Majorana Fermions In Topological Superconductors Research Articles

Room-temperature continuous-wave topological Dirac-vortex microcavity lasers on silicon

Robust laser sources are a fundamental building block for contemporary information technologies. Originating from condensed-matter physics, the concept of topology has recently entered the realm of optics, offering fundamentally new design principles for lasers with enhanced robustness. In analogy to the well-known Majorana fermions in topological superconductors, Dirac-vortex states have recently been investigated in passive photonic systems and are now considered as a promising candidate for robust lasers. Here, we experimentally realize the topological Dirac-vortex microcavity lasers in InAs/InGaAs quantum-dot materials monolithically grown on a silicon substrate. We observe room-temperature continuous-wave linearly polarized vertical laser emission at a telecom wavelength. We confirm that the wavelength of the Dirac-vortex laser is topologically robust against variations in the cavity size, and its free spectral range defies the universal inverse scaling law with the cavity size. These lasers will play an important role in CMOS-compatible photonic and optoelectronic systems on a chip.

Ternary logic design in topological quantum computing

A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of topological quantum phases of matter. These phases have quasiparticles excitations called anyons. The anyons are charge-flux composites and show exotic fractional statistics. When the order of exchange matters, then the anyons are called non-abelian anyons. Majorana fermions in topological superconductors and quasiparticles in some quantum Hall states are non-abelian anyons. Such topological phases of matter have a ground state degeneracy. The fusion of two or more non-abelian anyons can result in a superposition of several anyons. The topological quantum gates are implemented by braiding and fusion of the non-abelian anyons. The fault-tolerance is achieved through the topological degrees of freedom of anyons. Such degrees of freedom are non-local, hence inaccessible to the local perturbations. In this paper, the Hilbert space for a topological qubit is discussed. The Ising and Fibonacci anyonic models for binary gates are briefly given. Ternary logic gates are more compact than their binary counterparts and naturally arise in a type of anyonic model called the metaplectic anyons. The mathematical model, for the fusion and braiding matrices of metaplectic anyons, is the quantum deformation of the recoupling theory. We proposed that the existing quantum ternary arithmetic gates can be realized by braiding and topological charge measurement of the metaplectic anyons.

Experimental signature of topological superconductivity and Majorana zero modes on β-Bi2Pd thin films

The search for Majorana fermions in topological superconductors is one of paramount research targets in physics today. Using a cryogenic scanning tunneling microscopy, we here report the signature of topologically nontrivial superconductivity on a single material of β-Bi2Pd films grown by molecular beam epitaxy. The superconducting gap associated with spinless odd-parity pairing opens on the surface and appears much larger than the bulk one due to the Dirac-fermion enhanced parity mixing of surface pair potential. Zero bias conductance peaks, probably from Majorana zero modes supported by such superconducting states, are identified at magnetic vortices. The superconductivity exhibits resistance to nonmagnetic defects, characteristic of time-reversal-invariant topological superconductors. Our study reveals β-Bi2Pd as a prime platform to generate, manipulate and braid Majorana zero modes for quantum computation.

Majorana Fermions and Topology in Superconductors

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-Abelian statistics, which is distinct from bose and fermi statistics, and can be utilized for application to topological quantum computation. Also, Majorana fermions give rise to various exotic phenomena such as "fractionalization", non-local correlation, and "teleportation". A pedagogical review of these subjects is presented. We also discuss interaction effects on topological classification of superconductors, and the basic properties of Weyl superconductors.

Verifying non-Abelian statistics by numerical braiding Majorana fermions

Recently, Majorana fermions have attracted intensive attention because of their possible non-Abelian statistics and potential applications in topological quantum computation. This paper describes an approach to verify the non-Abelian statistics of Majorana fermions in topological superconductors. From the relationship between the braiding operator of Majorana fermions and that of Bogoliubov–de Gennes states, we determine that Majorana fermions in one-dimensional and two-dimensional topological superconductors both obey non-Abelian statistics.

Charged-impurity-induced Majorana fermions in topological superconductors

We study numerically Majorana fermions (MFs) induced by a charged impurity in topological superconductors. It is revealed from the relevant Bogoliubov-de Gennes equations that (i) for quasi-one dimensional systems, a pair of MFs are bounded at the two sides of one charge impurity and well separated; and (ii) for a two dimensional square lattice, the charged-impurity-induced MFs are similar to the known pair of vortex-induced MFs, in which one MF is bounded by the impurity while the other appears at the boundary. Moreover, the corresponding local density of states is explored, demonstrating that the presence of MF states may be tested experimentally.

Selective Equal-Spin Andreev Reflections Induced by Majorana Fermions

In this work, we find that Majorana fermions induce selective equal spin Andreev reflections (SESARs), in which incoming electrons with certain spin polarization in the lead are reflected as counterpropagating holes with the same spin. The spin polarization direction of the electrons of this Andreev reflected channel is selected by the Majorana fermions. Moreover, electrons with opposite spin polarization are always reflected as electrons with unchanged spin. As a result, the charge current in the lead is spin polarized. Therefore, a topological superconductor which supports Majorana fermions can be used as a novel device to create fully spin-polarized currents in paramagnetic leads. We point out that SESARs can also be used to detect Majorana fermions in topological superconductors.

Majorana fermions in topological superconductors with spin-orbit interaction

When a topological order is realized in a certain class of superconductors, it supports Majorana fermions on the edge or in the vortex core. Vortices with Majorana fermion modes are neither fermions nor bosons but non-Abelian anyons, obeying the non-Abelian statistics for which the exchange operations of particles are not commutative. Because of this remarkable feature, Majorana fermions plays an important role for the realization of fault-tolerant topological quantum computation.Here I would like to report our recent work on Majorana fermions and non-Abelian anyons in topological superconductors with spin-orbit interaction [1, 2, 3, 4, 5, 6]. Until recently, only spin-triplet superconducting states, such as such ν = 5/2 fractional quantum Hall states or Sr2RuO4, had been known to exhibit Majorana fermions [7, 8, 9, 10, 11]. However, now it is revealed that spin-singlet superconductors also support Majorana fermions if we take into account the spin-orbit interaction. While a possible realization of non-Abelian anyon in an s-wave superconducting state was first discussed in the context of cosmology [1], by using a zero mode solution in a vortex by Jackiw and Rossi [12] (or its non-gauged version [13]), now there are many examples of such systems in condensed matter physics. In particular, a scheme using the spin-orbit interaction and Zeeman one, which was proposed in [2, 3, 4], is powerful to realize Majorana fermions in various systems. It is shown that Majorana fermions are realized in full-gapped s-wave superfluids/superconductors [3, 4, 14, 15]. Also, Majorana fermions is shown to possible even in nodal superconductors such as high-Tc cuprates [5].We also show that a peculiar Majorana fermion is obtained in a certain non-centrosymmetric superconductor [6, 16, 17]. In sharp contrast to ordinary Majorana fermions, the novel Majorana fermion has a flat dispersion and preserves the time-reversal invariance.

Introduction to topological superconductivity and Majorana fermions

This short review paper provides a pedagogical introduction to the rapidly growing research field of Majorana fermions in topological superconductors. We first discuss in some detail the simplest ‘toy model’ in which Majoranas appear, namely a one-dimensional tight-binding representation of a p-wave superconductor, introduced more than 10 years ago by Kitaev. We then give a general introduction to the remarkable properties of Majorana fermions in condensed matter systems, such as their intrinsically non-local nature and exotic exchange statistics, and explain why these quasiparticles are suspected to be especially well suited for low-decoherence quantum information processing. We also discuss the experimentally promising (and perhaps already successfully realized) possibility of creating topological superconductors using semiconductors with strong spin–orbit coupling, proximity-coupled to standard s-wave superconductors and exposed to a magnetic field. The goal is to provide an introduction to the subject for experimentalists or theorists who are new to the field, focusing on the aspects which are most important for understanding the basic physics. The text should be accessible for readers with a basic understanding of quantum mechanics and second quantization, and does not require knowledge of quantum field theory or topological states of matter.

Nonadiabatic effects in the braiding of non-Abelian anyons in topological superconductors

Qubits in topological quantum computation are built from non-Abelian anyons. Adiabatic braiding of anyons is exploited as topologically protected logical gate operations. Thus, the adiabaticity upon which the notion of quantum statistics is defined, plays a fundamental role in defining the non-Abelian anyons. We study the non-adiabatic effects in braidings of Ising-type anyons, namely Majorana fermions in topological superconductors, using the formalism of time-dependent Bogoliubov-de Gennes equations. Using this formalism, we consider non-adiabatic corrections to non-Abelian statistics from: (1) tunneling splitting of anyons imposing an additional dynamical phase to the transformation of ground states; (2) transitions to excited states that are potentially destructive to non-Abelian statistics since the non-local fermion occupation can be spoiled by such processes. However, if the bound states are localized and being braided together with the anyons, non-Abelian statistics can be recovered once the definition of Majorana operators is appropriately generalized taking into account the fermion parity in these states. On the other hand, if the excited states are extended over the whole system and form a continuum, the notion of local fermion parity no longer holds. We then quantitatively characterize the errors introduced in this situation.