Majorana Quasiparticles Research Articles

Minimal Zeeman field requirement for a topological transition in superconductors

Platforms for creating Majorana quasiparticles rely on superconductivity and breaking of time-reversal symmetry. By studying continuous deformations to known trivial states, we find that the relationship between superconducting pairing and time reversal breaking imposes rigorous bounds on the topology of the system. Applying these bounds to s-wave systems with a Zeeman field, we conclude that a topological phase transition requires that the Zeeman energy at least locally exceed the superconducting pairing by the energy gap of the full Hamiltonian. Our results are independent of the geometry and dimensionality of the system.

Majorana bound states in vortex lattices on iron-based superconductors

Majorana quasi-particles may arise as zero-energy bound states in vortices on the surface of a topological insulator that is proximitized by a conventional superconductor. Such a system finds its natural realization in the iron-based superconductor FeTe0.55Se0.45 that combines bulk s-wave pairing with spin helical Dirac surface states, and which thus comprises the ingredients for Majorana modes in absence of an additional proximitizing superconductor. In this work, we investigate the emergence of Majorana vortex modes and lattices in such materials depending on parameters like the magnetic field strength and vortex lattice disorder. A simple 2D square lattice model here allows us to capture the basic physics of the underlying materials system. To address the problem of disordered vortex lattice, which occurs in real systems, we adopt the technique of the singular gauge transformation which we modify such that it can be used in a system with periodic boundary conditions. This approach allows us to go to larger vortex lattices than otherwise accessible, and is successful in replicating several experimental observations of Majorana vortex bound states in the FeTe0.55Se0.45 platform. Finally it can be related to a simple disordered Majorana lattice model that should be useful for further investigations on the role of interactions, and toward topological quantum computation.

Observability of a sharp Majorana transition in a few-body model

We propose experimentally observable signatures of of topological Majorana quasiparticles in the few-body limit of the interacting cold-atom model of [Iemini et al. Phys. Rev. Lett. 118 200404 (2017)]. In this limit, the total on-site density and single-body correlations change smoothly with the model parameters, while the calculated mutual information of opposite ends of the lattice indicates a sharp transition of the system to a topological ground state. Furthermore, local density and parity measurements provide an experimentally viable path for observing the ground state Majorana quasiparticles in ultracold atoms. Our results lay out a promising future for utilizing few-body systems as a testing ground for Majorana physics.

Direct observation of the specific heat of Majorana quasiparticles in superfluid3He-B

The existence of Majorana quasiparticles was predicted in the edge state in topological insulators, especially in the p-wave superfluid medium {^3}He-B. Due to its purity and coherent quantum state, {^3}He-B is an ideal platform for searching for Majorana fermions in condensed matter systems. In the limit of extremely low temperatures, the density of Bogolyubov quasiparticles and the heat capacity of {^3}He-B decrease exponentially. In this article, we present the first observation of the deviation of its heat capacity from exponential dependence in the limit of record low cooling. We found an additional heat capacity that more than doubled the heat capacity of bulk {^3}He-B and changes as T^2. The additional heat capacity is in good agreement with the predicted heat capacity of 2D gas of Majorana. This observation is a direct proof of the existence of Majorana quasiparticles in {^3}He-B.

Braiding Majorana corner modes in a second-order topological superconductor

We propose the concept of a device based on a square-shaped sample of a two-dimensional second-order topological helical superconductor which hosts two zero-dimensional Majorana quasiparticles at the corners. The two zero-energy modes rely on particle-hole symmetry (PHS) and their spacial position can be shifted by rotating an in-plane magnetic field and tuning proximity-induced spin-singlet pairing. We consider an adiabatic cycle performed on the degenerate ground-state manifold and show that it realizes the braiding of the two modes whereby they accumulate a non-trivial statistical phase $\pi$ within one cycle. Alongside with the PHS-ensured operator algebra, the fractional statistics confirms the Majorana nature of the zero-energy excitations. A schematic design for a possible experimental implementation of such a device is presented, which could be a step towards realizing non-Abelian braiding.

Long-range focusing of magnetic bound states in superconducting lanthanum

Quantum mechanical systems with long-range interactions between quasiparticles provide a promising platform for coherent quantum information technology. Superconductors are a natural choice for solid-state based quantum devices, while magnetic impurities inside superconductors give rise to quasiparticle excitations of broken Cooper pairs that provide characteristic information about the host superconductor. Here, we reveal that magnetic impurities embedded below a superconducting La(0001) surface interact via quasiparticles extending to very large distances, up to several tens of nanometers. Using low-temperature scanning probe techniques, we observe the corresponding anisotropic and giant oscillations in the LDOS. Theoretical calculations indicate that the quasi-two-dimensional surface states with their strongly anisotropic Fermi surface play a crucial role for the focusing and long-range extension of the magnetic bound states. The quasiparticle focusing mechanism should facilitate the design of versatile magnetic structures with tunable and directed magnetic interactions over large distances, thereby paving the way toward the design of low-dimensional magnet–superconductor hybrid systems exhibiting topologically non-trivial quantum states as possible elements of quantum computation schemes based on Majorana quasiparticles.

How to measure the Majorana polarization of a topological planar Josephson junction

We analyze the topological superconductivity and investigate the spectroscopic properties manifested by the zero-energy modes, induced in a metallic strip embedded into a Josephson-type junction. Focusing on the Majorana polarization of such quasiparticles, we propose feasible means for its empirical detection, using the spin-selective Andreev reflection method. Our study reveals a gradual development of a transverse gradient of the Majorana polarization across the metallic strip upon increasing its width. We also inspect the spatial profile and polarization of the Majorana quasiparticles in the presence of a strong electrostatic impurity. We show that, depending on its position, such a defect can lead to a substantial localization of the Majorana mode.

Majorana-Kondo interplay in T-shaped double quantum dots

The transport behavior of a double quantum dot side-attached to a topological superconducting wire hosting Majorana zero-energy modes is studied theoretically in the strong correlation regime. It is shown that Majorana modes can leak to the whole nanostructure, giving rise to a subtle interplay between the two-stage Kondo screening and the half-fermionic nature of Majorana quasiparticles. In particular, the coupling to the topological wire is found to reduce the effective exchange interaction between the two quantum dots in the absence of normal leads. Interestingly, it also results in an enhancement of the second-stage Kondo temperature when the normal leads are attached. Moreover, it is shown that the second stage of the Kondo effect can become significantly modified in one of the spin channels due to the interference with the Majorana zero-energy mode, yielding the low-temperature conductance equal to $G=G_0/4$, where $G_0 = 2e^2/h$, instead of $G=0$ in the absence of the topological superconducting wire. We also identify a nontrivial spin-charge ${\rm SU}(2)$ symmetry present in the system at a particular point in parameter space, despite lack of the spin nor charge conservation. Finally, we discuss the consequences of a finite overlap between two Majorana modes, as relevant for short Majorana wires.

Volkov-Pankratov states in topological superconductors

We study the in-gap states that appear at the boundaries of both 1D and 2D topological superconductors. While the massless Majorana quasiparticles are guaranteed to arise by the bulk-edge correspondence, we find that they could be accompanied by massive Volkov-Pankratov (VP) states which are present only when the interface is sufficiently smooth. These predictions can be tested in an s-wave superconductor with Rashba spin-orbit coupling placed on top of a magnetic domain wall. We calculate the spin-resolved local density of states of the VP states about the band inversion generated by a magnetic domain wall and find that they are oppositely spin-polarized on either side of the topological phase boundary. We also demonstrate that the spatial position, energy-level spacing, and spin polarization of the VP states can be modified by the introduction of in-plane electric fields.

Quantum Pairing Time Orders

AbstractThe concept of the time‐independent correlators for the even‐ and odd‐frequency pairing states that can be defined for both bosonic and fermionic quasiparticles is proposed. These correlators explicitly capture the existence of two distinct classes of pairing states and provide a direct probe of the hidden Berezinskii order. This concept is illustrated in the cases of pairings for Majorana fermions and quasiparticles in Dirac semimetals. It is shown that the time‐independent correlator is able to effectively capture the energy scale relevant for pairing.

Simulating the Majorana dynamics with ultracold atomic gases in a bilayer honeycomb lattice

The authors present theoretical results on the dynamical properties of General Majorana Quasiparticles and the unique Majorana Zitterbewegung. The results reveal the fidelity is a good observable to distinguish Majorana from Dirac or Weyl dynamics. Furthermore, a feasible method to detect the Majorana dynamics by using quench and quantum-state tomography has been provided in cold-atomic lattice system.

Majorana fermions go for a ride.

Evidence for propagating Majorana quasiparticles is found in a topological superconductor

Majorana quasi-particles and superconductor-semiconductor hybrid nanowires

Majorana fermions are known for being their own anti-particles. As the condensed matter version of Majorana fermions, Majorana quasiparticles have drawn extensive interests for being an ideal candidate for building a fault-tolerant quantum computer, due to their non-abelian statistics. This paper provides an introduction for beginners to the rapidly growing research field of Majorana quasiparticles focusing on one dimensional semiconductor nanowire-superconductor hybrid system. We aim to help readers to quickly understand Majorana quasiparticles and its formation mechanism and the latest experimental results. We first review the theoretical model of the Majorana quasiparticles with its historical background. We then discuss the Kitaev chain and analyze its key elements. We also introduce typical Majorana devices and their corresponding measurement methods. Furthermore, we discuss the observation of robust signatures of Majorana zero modes in recent experiments, with particular attention to tunneling conductance measurements. Finally, we give prospects on future experiments for advancing one dimensional semiconductor nanowire-superconductor hybrid system.

Emergent vortex Majorana zero mode in iron-based superconductors

During the recent years, the iron-based superconductors with a topological band structure have attracted intensive attention from the science community as a new and promising platform for emerging Majorana zero modes in their vortex core. These topological iron-based superconductors possess all of the desirable properties, <i>i.e.</i> single material, high-<i>T</i><sub>c</sub> superconductivity, strong electron-electron correlation and topological band structure, thus successfully avoiding the difficulties suffered by previous Majorana platforms, such as intrinsic topological superconductors and multiple types of proximitized heterostructures. So far, one has observed pristine vortex Majorana zero modes in several different compounds of iron-based superconductors. The systematic studies performed on those systems show that the vortex Majorana zero modes are quite evident experimentally and very clear theoretically, leading to a bright future in applications. The vortex cores of iron-based superconductors can become one of the major candidates for exploring topological quantum computing in the future. In this review article, we will focus on Fe(Te, Se) single crystal, to introduce the original ideas and research progress of the new emerging “iron home” for Majorana zero modes. Having elabrated the basic band structures and the experimental facts of the observed vortex zero modes in Fe(Te, Se), we will systematically summarize the main observations and fundamental physics of vortex Majorana zero modes in Fe(Te, Se). First of all, with the help of the observed behavior of Majorana wavefunction and quasiparticle poisioning, we will analyze the emerging mechanism of vortex Majorana zero modes in Fe(Te, Se). Then we will elaborate the measurements on Majorana symmetry and topological nature of vortex Majorana zero modes, assisted by several existing Majorana theories. After that, we will switch our view angle from quantum physics to quantum engineering, and comprehensively analyze the fate of vortex Majorana zero modes in a real material under a real environment, which may benefit the potential engineering applications in the future. This review article follows the physical properties of vortex Majorana zero modes, and emphasizes the link between theories and experiments. Our goal is to bridge the gap between the classical Majorana theories and the new emerging Majorana platform in iron-based superconductors, and help the readers to understand the experimental observations of the newly discovered “iron home” for Majoranas.

Exploration of Majorana bound states in topological superconductors

Majorana bound states are considered useful for realizing topological quantum computation since they obey the non-Abelian quantum statistics. Recent experiments have provided evidences for their existence in some superconducting systems, triggering significant interests from scientists in the field of condensed matter physics and related materials science. In this article, we briefly review the basic concepts and recent developments in the study of Majorana bound states. We first discuss about the origin of the nontrivial topology in superconducting systems within the Bogoliubov-de Gennes mean-field scheme. Then we show the construction of Majorana quasiparticle excitations from an electronic state, and the realization of non-Abelian statistics based on position exchanges of the Majorana bound states hosted in superconductivity vortices. Afterwards we talk about specific one-dimensional and two-dimensional topological superconductors, and propose possible experimental methods for detecting Majorana bound states and operating the Majorana qubits. In particular, a quantum device for Majorana braiding without moving vortices is introduced. Finally, perspectives of the study on Majorana bound states are provided.

Dirac fermions and superconductivity in the two-dimensional transition metals MOH(M=Zr,Hf)

Discovery of the new two-dimensional (2D) Dirac semimetals incorporating both superconductivity and the topological band structure has provided a novel platform for realizing the intriguing applications of Dirac fermions and Majorana quasiparticles, ranging from high-speed quantum devices at the nanoscale to topological quantum computations. In this work, utilizing first-principles calculations and symmetry analysis, we introduce MOH (M= Zr, Hf) as a new topological superconductor with Dirac points close to a Fermi level which are connected with nearly flat edge states as a striking feature of topological semimetals. Our calculations show that ZrOH as a 2D topological semimetal can exhibit superconductivity and is a novel platform for studying the interplay between superconductivity and Dirac states in low-dimensional materials.

Nonequilibrium Majorana dynamics by quenching a magnetic field in Kitaev spin liquids

The honeycomb Kitaev spin model provides a quantum spin liquid in the ground state, where the spin excitations are fractionalized into itinerant and localized Majorana fermions; the former spectrum has a broad continuum ranging up to a high energy, while the latter has a sharp peak at a low energy. Despite tremendous efforts, it remains elusive to clearly identify these distinct Majorana excitations in experiments. Here we show their manifestation in the time evolution after quenching the magnetic field, by using the time-dependent Majorana mean-field theory for both the ferromagnetic and antiferromagnetic Kitaev models. We find that the transient spin dynamics from the quantum spin liquid states is qualitatively different from the conventional spin precessions by the quench from the high-field forced-ferromagnetic state. We obtain peculiar time evolutions with distinct time scales, i.e., short-time decay of high-energy components associated with the itinerant Majorana excitations, and long-lived excitations at a low energy by the localized ones. These peculiar behaviors are caused by the energy transfer between the two Majorana quasiparticles after the field quench. Moreover, we find that the Majorana semimetal with the point nodes in equilibrium turns into a Majorana metal with the transient "Fermi surfaces" by the energy transfer. In particular, for the quench from the intermediate-field quantum spin liquid in the antiferromagnetic Kitaev model, the Fermi surfaces change their topology in the time evolution, which is regarded as a dynamical version of the Majorana "Lifshitz transition". Our results unveil that the real-time dynamics provides another route to not only the identification of the fractional Majorana excitations in candidate materials of Kitaev magnets but also unprecedented quantum phases that cannot be stabilized as the equilibrium states.

Delocalisation of Majorana quasiparticles in plaquette\u2013nanowire hybrid system

Interplay between superconductivity, spin-orbit coupling and magnetic field can lead to realisation of the topologically non–trivial states which in finite one dimensional nanowires are manifested by emergence of a pair of zero-energy Majorana bound states. On the other hand, in two dimensional systems the chiral edge states can appear. We investigate novel properties of the bound states in a system of mixed dimensionality, composed of one-dimensional nanowire connected with two-dimensional plaquette. We study this system, assuming either its part or the entire structure to be in topologically non–trivial superconducting state. Our results show delocalisation of the Majorana modes, upon leaking from the nanowire to the plaquette with some tendency towards its corners.

Synthetic spin-orbit interaction for Majorana devices.

The interplay of superconductivity with non-trivial spin textures is promising for the engineering of non-Abelian Majorana quasiparticles. Spin-orbit coupling is crucial for the topological protection of Majorana modes as it forbids other trivial excitations at low energy but is typically intrinsic to the material1-7. Here, we show that coupling to a magnetic texture can induce both a strong spin-orbit coupling of 1.1 meV and a Zeeman effect in a carbon nanotube. Both of these features are revealed through oscillations of superconductivity-induced subgap states under a change in the magnetic texture. Furthermore, we find a robust zero-energy state-the hallmark of devices hosting localized Majorana modes-at zero magnetic field. Our findings are generalizable to any low-dimensional conductor, and future work could include microwave spectroscopy and braiding operations, which are at the heart of modern schemes for topological quantum computation.

Topological phase diagram of a three-terminal Josephson junction: From the conventional to the Majorana regime

We study the evolution of averaged transconductances in three-terminal Josephson junctions when the superconducting leads are led throughout a topological phase transition from an $s$-wave to a $p$-wave (Majorana) phase by an in-plane magnetic field ${B}_{x}$. We provide a complete description of this transition as a function of ${B}_{x}$ and a magnetic flux $\mathrm{\ensuremath{\Phi}}$ threading the junction. For that we use a spinful model within a formalism that allows us to treat on an equal footing the contribution to the transconductance from both the Andreev subgap levels and the continuum spectrum. We unveil a fractionalization in the quantization of the transconductance due to the presence of Majorana quasiparticles, reflecting the effective pumping of half a Cooper pair charge in the $p$-wave regime.