Majorana States Research Articles

Topological superconductivity and Majorana fermion

Topological superconductor is described by a full pairing gap in the bulk with nonzero topological number and gapless surface states consisting of Majorana fermion that is a hypothetical particle of its own. Here, being its own means that a Majorana fermion should be an equal superposition of an electron and a hole state. The emergence of Majorana fermions is the most prominent characteristic of topological superconductors. In particle physics, it is still unclear if there are some elementary particles that are Majorana fermions, but importantly, they are likely to exist as quasiparticle excitations in certain condensed matter systems. It has drawn much attention in the content of Majorana fermions recently in particular the condensed matter physics area, since these Majorana states are ideal platform for non-Abelian statistics studies and can be used to fabricate as topological qubit, thus having great potential application in fault-tolerant topological quantum computation. A general introduction to the remarkable properties of Majorana fermions in condensed matter systems will be introduced following the story from Dirac equation to Majorana equation. Then this review elaborates a variety of routes to topological superconductivity in the realm of condensed mater physics, from Majorana modes in zero dimension to one dimension, that is, Majorana bound state and Majorana edge state. These Majorana states are localized states and propagating states respectively, but still obey non-Abelian statistics and remain their topological properties. Specifically, one-dimensional tight-binding model of a p-wave superconductor and a magnetic Fe chain with superconducting pairing are representative platforms for hosting Majorana bound states at the wires ends. Some emergent topological material systems, such as topological insulator, quantum spin Hall insulator, and quantum anomalous Hall insulator are two-dimensional material systems that can not only accommodate Majorana bound states but also Majorana edge states. Different material systems, from one-dimension quantum wire to two-dimension material system, from hybridized system to intrinsic system, will also be discussed specifically. Relevant theoretical studies and experimental results that show possible signatures of topological superconductivity and Majorana states are summarized as well. Particularly, experimental signatures of these Majorana states, such as zero-bias conductance peaks in tunneling spectra due to Majorana bound states, quantization of conductance in magneto-electric transport measurements due to chiral Majorana edge states, and some unconventional superconductivities in intrinsic topological superconductors will be briefly reviewed. Insights about methods to perspectives to realize topological quantum computation is provided at the end, emphasizing the braiding using different Majorana states. The goal of this review is to provide a general introduction to the subject for either experimentalists or theorists who are new to this field, focusing on the aspects and current progresses which are most important for understanding these basic physics.

Effects of Majorana bound states on dissipation and charging in a quantum resistor-capacitor circuit

We investigate the effects of Majorana bound states on the ac response of a quantum resistor-capacitor circuit which is composed of a topological superconducting wire whose two ends are tunnel-coupled to a lead and a spinless quantum dot, respectively. The Majorana states formed at the two ends of the wire are found to suppress completely or enhance greatly the dissipation, depending on the strength of the overlap between two Majorana modes and/or the dot level. We compare the relaxation resistance and the quantum capacitance of the system with those of non-Majorana counterparts to find that the effects of the Majorana state on the ac response are genuine and cannot be reproduced in ordinary fermionic systems.

Yu-Shiba-Rusinov bound states induced by a spin flipper in the vicinity of a s-wave superconductor

We theoretically study the formation and characteristics of Yu-Shiba-Rusinov bound states within the superconducting gap using a BTK approach in presence of a spin flipper (high spin magnetic impurity). We focus on the zero energy in the conductance spectra and show how a peak is formed at E = 0 due to flipping of the magnetic impurity spin, but for no flip case a dip forms at E = 0 in the conductance spectra. This E = 0 conductance peak is almost quantized at 2e2/h values, however it arises due to non-topological reasons in contrast to the E = 0 peak formed due to Majorana states.

Engineering quantum spin liquids and many-body Majorana states with a driven superconducting box circuit

We design a driven superconducting box with four spins S=1/2 (qubits) such that coupled devices can give insight on the occurrence of quantum spin liquids and many-body Majorana states. Within one box or island, we introduce a generalized nuclear magnetic resonance algorithm to realize our models and study numerically the spin observables in time as well as the emergent gauge fields. We discuss the stability of the box towards various detuning effects and we include dissipation effects through a Lindblad master equation. Coupling boxes allows us to realize quantum spin liquid phases of Kitaev ${\cal Z}_2$ spin models in various geometries with applications in the toric code. Quantum phase transitions and Majorana physics might be detected by measuring local susceptibilities. We show how to produce a N\' eel state of fluxes by coupling boxes and we address the role of local impurity fluxes leading to random Ising models. We also present an implementation of the Sachdev-Ye-Kitaev Majorana model in coupled ladder systems.

(Invited) Bottom-up Grown Nanowire Quantum Devices

The main challenge for the development of a quantum computer is to solve decoherence. The promise of topological quantum computation is that local noise does not affect and destroy the quantum state, and therefore long coherence times are expected. In a topological system the quantum information is encoded in quasi particles, called Majorana Zero Modes (MZM). First signatures of MZM have been obtained in InSb nanowires [1]. Recently, different schemes for performing logical operations and uncovering the properties of Majorana states are proposed. For such a universal computational architecture the realization of a near-perfect nanowire network assembly is needed in which Majorana states are coherently coupled. Here, we demonstrate a generic process by which we can design any proposed braiding device by manipulating an InP substrate and thereby the nanowire growth position and orientation [2]. Our method leads to highly controlled growth of InSb nanowire networks with single crystalline wire-wire junctions. Additionally, nanowire “hashtag” structures are grown with a high yield and contacted. In these devices, the Aharonov–Bohm (AB) effect is observed, demonstrating phase coherent transport. These measurements reveal the high quality of these structures. Furthermore, this platform is used for in-situ epitaxial shadow growth of a superconductor on the nanowire facets, resulting in transport contacts. With this new generation devices we have observed quantized MZM [3]. Mourik et al., Science 2012, 336, 1003Gazibegovic et al. Nature 548 2017, 434Zhang et al. Nature 2018 , accepted

Parafermionic generalization of the topological Kondo effect

We propose and study a parafermionic generalization of the topological Kondo effect. The latter has been predicted to arise for a Coulomb-blockaded mesoscopic topological superconductor (Majorana box), where at least three normal leads are tunnel-coupled to different Majorana zero modes on the box. The Majorana states represent a quantum impurity spin that is partially screened due to cotunneling processes between leads, with a stable non-Fermi liquid ground state. Our theory studies a generalization where (i) Majorana states are replaced by topologically protected parafermionic zero modes, (ii) charging effects again define a spin-like quantum impurity on the resulting parafermion box, and (iii) normal leads are substituted by fractional edge states. In this multi-terminal problem, different fractional edge leads couple only via the parafermion box. We show that although the linear conductance tensor exhibits similar behavior as in the Majorana case, both at weak and strong coupling, our parafermionic generalization is actually not a Kondo problem but defines a rich new class of quantum impurity problems. At the strong-coupling fixed point, a current injected through a reference lead will be isotropically partitioned into outgoing currents in all other leads, together with a universal negative current scattered into the reference lead. The device can thus be operated as current extractor, where the current partitioning is noiseless at the fixed point. We describe a fractional quantum Hall setup proximitized by superconductors and ferromagnets, which could allow for an experimental realization in the near future.

Майорановские состояния вблизи примеси в p-волновой сверхпроводящей нанопроволоке

Майорановские состояния вблизи примеси в p-волновой сверхпроводящей нанопроволоке

A general algorithm for computing bound states in infinite tight-binding systems

We propose a robust and efficient algorithm for computing bound states of infinite tight-binding systems that are made up of a finite scattering region connected to semi-infinite leads. Our method uses wave matching in close analogy to the approaches used to obtain propagating states and scattering matrices. We show that our algorithm is robust in presence of slowly decaying bound states where a diagonalization of a finite system would fail. It also allows to calculate the bound states that can be present in the middle of a continuous spectrum. We apply our technique to quantum billiards and the following topological materials: Majorana states in 1D superconducting nanowires, edge states in the 2D quantum spin Hall phase, and Fermi arcs in 3D Weyl semimetals.

Local density of states in two-dimensional topological superconductors under a magnetic field: Signature of an exterior Majorana bound state

We study quasiparticle states on a surface of a topological insulator (TI) with proximity-induced superconductivity under an external magnetic field. An applied magnetic field creates two Majorana bound states: a vortex Majorana state localized inside a vortex core and an exterior Majorana state localized along a circle centered at the vortex core. We calculate the spin-resolved local density of states (LDOS) and demonstrate that the shrinking of the radius of the exterior Majorana state, predicted in Ref. [R. S. Akzyanov et al., Phys. Rev. B 94, 125428 (2016)], under a strong magnetic field can be seen in LDOS without smeared out by non-zero-energy states. The spin-resolved LDOS further reveals that the spin of the exterior Majorana state is strongly polarized. Accordingly, the induced odd-frequency spin-triplet pairs are found to be spin-polarized as well. In order to detect the exterior Majorana states, however, the Fermi energy should be closed to the Dirac point to avoid contributions from continuum levels. We also study a different two-dimensional topological-superconducting system where a two-dimensional electron gas with the spin-orbit coupling is sandwiched between an s-wave superconductor and a ferromagnetic insulator. We show that the radius of an exterior Majorana state can be tuned by an applied magnetic field. However, on the contrary to the results at a TI surface, neither the exterior Majorana state nor the induced odd-frequency spin-triplet pairs are spin-polarized. We conclude that the spin-polarization of the Majorana state is attributed to the spin-polarized Landau level which is characteristic for systems with the Dirac-like dispersion.

Circular dichroism of chiral Majorana states.

Background: Majorana states in condensed matter devices may be of a localized nature, such as in hybrid semiconductor/superconductor nanowires, or chirally propagating along the edges such as in hybrid 2D quantum-anomalous Hall/superconductor structures.Results: We calculate the circular dichroism due to chiral Majorana states in a hybrid structure made of a quantum-anomalous Hall insulator and a superconductor. The optical absorption of chiral Majorana states is characterized by equally spaced absorption peaks of both positive and negative dichroism. In the limit of a very long structure (a 2D ribbon) peaks of a single sign are favored.Conclusion: Circular-dichroism spectroscopy of chiral Majorana states is suggested as a relevant probe for these peculiar states of topological matter.

Chiral Majorana fermion modes regulated by a scanning tunneling microscope tip

The Majorana fermion can be described by a real wave function with only two phases (0 and {\pi}) which provide a controllable degree of freedom. We propose a strategy to regulate the phase of the chiral Majorana state by coupling with a scanning tunneling microscope tip in a system consisting of quantum anomalous Hall insulator coupled with a superconductor. With the change of the chemical potential, the chiral Majorana state can be tuned alternately between 0 and {\pi}, in which, correspondingly, the perfect normal tunneling and perfect crossed Andreev reflection appear. The perfect crossed Andreev reflection, by which a Cooper pair can be split into two electrons going into different terminals completely, leads to a pumping current and distinct quantized resistances. These findings may provide a signature of Majorana fermions and pave a feasible avenue to regulate the phase of Majorana state.

Observation of topological superconductivity on the surface of an iron-based superconductor.

Topological superconductors are predicted to host exotic Majorana states that obey non-Abelian statistics and can be used to implement a topological quantum computer. Most of the proposed topological superconductors are realized in difficult-to-fabricate heterostructures at very low temperatures. By using high-resolution spin-resolved and angle-resolved photoelectron spectroscopy, we find that the iron-based superconductor FeTe1-x Se x (x = 0.45; superconducting transition temperature Tc = 14.5 kelvin) hosts Dirac-cone-type spin-helical surface states at the Fermi level; the surface states exhibit an s-wave superconducting gap below Tc Our study shows that the surface states of FeTe0.55Se0.45 are topologically superconducting, providing a simple and possibly high-temperature platform for realizing Majorana states.

Majorana States in Presence of Electron Interactions: Spinful Fractional Josephson Junction with a Quantum Dot

Majorana States in Presence of Electron Interactions: Spinful Fractional Josephson Junction with a Quantum Dot

Spin Dependent Conductance of a Quantum Dot Side attached to Topological Superconductors as a Probe of Majorana Fermion States

Spin-polarized transport through a quantum dot side attached to a topological superconductor and coupled to a pair of normal leads is discussed in Coulomb and Kondo regimes. For discussion of Coulomb range equation of motion method with extended Hubbard I approximation is used and Kondo regime is analyzed by Kotliar-Ruckenstein slave boson approach. Apart from the occurrence of zero bias anomaly the presence of Majorana states reflects also in splitting of Coulomb lines. In the region of Coulomb borders the spin dependent negative differential conductance is observed. Due to the low energy scale of Kondo effect this probe allows for detection of Majorana states even for extremely weak coupling with topological wire. In this range no signatures of Majorana states appear in Coulomb blockade dominated transport.

Helical Majorana fermions and flat edge states in the heterostructures of iridates and high- TC cuprates

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 theoretical communities to search for Majorana fermions, and recently a localized Majorana state at the end of one-dimensional wire has been reported. Here we study the two-dimensional interface of Weyl semimetal and d-wave superconductor which promotes a pair of helical Majorana fermions propagating along the edge of the interface. We suggest that Iridium oxide layer IrO$_2$ classified as two-dimensional Weyl semimetal in close proximity to high temperature Cuprates would be a best example to explore these helical Majorana modes.

Quasiclassical Theory of Spin Dynamics in Superfluid $$^3$$ 3 He: Kinetic Equations in the Bulk and Spin Response of Surface Majorana States

We develop a theory based on the formalism of quasiclassical Green's functions to study the spin dynamics in superfluid $^3$He. First, we derive kinetic equations for the spin-dependent distribution function in the bulk superfluid reproducing the results obtained earlier without quasiclassical approximation. Then we consider a spin dynamics near the surface of fully gapped $^3$He-B phase taking into account spin relaxation due to the transitions in the spectrum of localized fermionic states. The lifetime of longitudinal and transverse spin waves is calculate taking into account the Fermi-liquid corrections which lead to the crucial modification of fermionic spectrum and spin responses.

Dynamic current susceptibility as a probe of Majorana bound states in nanowire-based Josephson junctions

We theoretically study a Josephson junction based on a semiconducting nanowire subject to a time-dependent flux bias. We establish a general density matrix approach for the dynamical response of the Majorana junction and calculate the resulting flux-dependent susceptibility using both microscopic and effective low-energy descriptions for the nanowire. We find that the diagonal component of the susceptibility, associated with the dynamics of the Majorana states populations, dominates over the standard Kubo contribution for a wide range of experimentally relevant parameters. The diagonal term, thus far unexplored in the context of Majorana physics, allows to probe accurately the presence of Majorana bound states in the junction.

Nonlocality and dynamic response of Majorana states in fermionic superfluids

We suggest a microscopic model describing the nonlocal ac response of a pair of Majorana states in fermionic superfluids beyond the tunneling approximation. The time-dependent perturbations of quasiparticle transport are shown to excite finite period beating of the wavefunction between the distant Majorana states. We propose an experimental test to measure the characteristic time scales of quasiparticle transport through the pair of Majorana states defining, thus, quantitative characteristics of nonlocality known to be a generic feature of Majorana particles.

Mixed-pairing superconductivity in 5d Mott insulators with antisymmetric exchange: Application to Sr2IrO4

We study the symmetry of the potential superconducting order parameter in $5d$ Mott insulators with an eye toward hole-doped ${\mathrm{Sr}}_{2}{\mathrm{IrO}}_{4}$. Using a mean-field method, a mixed singlet-triplet superconductivity, $d+p$, is observed due to the antisymmetric exchange originating from a quasi-spin-orbit coupling. Our calculation on ribbon geometry shows the possible existence of the topologically protected edge states, because of the nodal structure of the superconducting gap. These edge modes are spin polarized and emerge as zero-energy flat bands, supporting a symmetry-protected Majorana state, verified by evaluation of the winding number and ${\mathbb{Z}}_{2}$ topological invariant. At the end, a possible experimental approach for observation of these edge states and determination of the superconducting gap symmetry is discussed based on the quasiparticle interference technique.

Majorana states in prismatic core-shell nanowires

We consider core-shell nanowires with conductive shell and insulating core and with polygonal cross section. We investigate the implications of this geometry on Majorana states expected in the presence of proximity-induced superconductivity and an external magnetic field. A typical prismatic nanowire has a hexagonal profile, but square and triangular shapes can also be obtained. The low-energy states are localized at the corners of the cross section, i.e., along the prism edges, and are separated by a gap from higher energy states localized on the sides. The corner localization depends on the details of the shell geometry, i.e., thickness, diameter, and sharpness of the corners. We study systematically the low-energy spectrum of prismatic shells using numerical methods and derive the topological phase diagram as a function of magnetic field and chemical potential for triangular, square, and hexagonal geometries. A strong corner localization enhances the stability of Majorana modes to various perturbations, including the orbital effect of the magnetic field, whereas a weaker localization favorizes orbital effects and reduces the critical magnetic field. The prismatic geometry allows the Majorana zero-energy modes to be accompanied by low-energy states, which we call pseudo Majorana, and which converge to real Majoranas in the limit of small shell thickness. We include the Rashba spin-orbit coupling in a phenomenological manner, assuming a radial electric field across the shell.