Majorana Bound States Research Articles

Statistical Majorana Bound State Spectroscopy.

Tunnel spectroscopy data for the detection of Majorana bound states (MBS) is often criticized for its proneness to misinterpretation of genuine MBS with low-lying Andreev bound states. Here, we suggest a protocol removing this ambiguity by extending single shot measurements to sequences performed at varying system parameters. We demonstrate how such sampling, which we argue requires only moderate effort for current experimental platforms, resolves the statistics of Andreev side lobes, thus providing compelling evidence for the presence or absence of a Majorana center peak.

Josephson dc Current through T-Shaped Double-Quantum-Dots Hybridized to Majorana Nanowires

We study quantum interference effects on Josephson current in T-shaped double quantum dots (TDQDs) with one of them (the central dot) is sandwiched between the left and right topological superconductor nanowires hosting Majorana bound states (MBSs). We find that the current’s magnitude is suppressed by the inter-dot coupling that induces the quantum interference effect, with unchanged jump in the current at particular phase difference between the two nanowires from which the Josephson effect arises. The current remains as a sinusoidal function with respective to the phase difference in the presence of quantum interference effect, but with significant reduction. The central broad peak in the curve of the Josephson current versus the QDs’ levels are split in different ways depending on the configurations of the latter. We also find that the impacts of the non-z-axial direction magnetic field, bending angle between the two nanowires and the direct hybridization amplitude between the MBSs on the current all depend on the arrangement of the QDs’ energy levels.

Dynamic heat and charge transports through double-quantum-dot-interferometer modulated by Majorana bound states and time-oscillating Aharonov-Bohm flux

Dynamic properties of Majorana bound states (MBSs) coupled double-quantum-dot (DQD) interferometer threaded with ac magnetic flux are investigated, and the time-averaged thermal current formulas are derived. Photon-assisted local and nonlocal Andreev reflections contribute efficiently to the charge and heat transports. The modifications of source-drain electric, electric-thermal, thermal conductances (G, ξ, κ e ), Seebeck coefficient (S c ), and thermoelectric figure of merit (ZT) versus AB phase have been calculated numerically. These coefficients exhibit the shift of oscillation period from 2π to 4π distinctly due to attaching MBSs. The applied ac flux enhances the magnitudes of G, ξ, κ e obviously, and the detailed enhancing behaviors are relevant to the energy levels of DQD. The enhancements of S c and ZT are generated due to the coupling of MBSs, while the application of ac flux suppresses the resonant oscillations. The investigation provides a clue for detecting MBSs through measuring the photon-assisted S c and ZT versus AB phase oscillations.

Probing Majorana bound states through an inhomogeneous Andreev double dot interferometer

In modern experiments with hybrid superconducting (SC)/semiconducting nanowires the presence of zero-energy Andreev bound states (ABSs), characterized by a partial overlap of the Majorana wave functions, is a common problem that significantly complicates the detection of a genuine Majorana bound state (MBS). In this article, taking into account spatial inhomogeneity of experimentally investigated nanowire samples, we study interference transport features of a curved heterostructure in which two normal wires (or arms) are separated by a superconducting wire. Since Andreev reflection on the two N/S interfaces with smoothly changing electrostatic and SC pairing potentials results in the emergence of bound states, the low-energy interference transport is described in the framework of the model of two noninteracting Andreev levels or the Andreev double quantum dot. A set of limiting cases is analyzed allowing us to highlight the interference properties that are unique for the different types of ABSs, such as bulk ABS, inhomogeneous ABS and MBS. In particular, considerable attention is paid to the features of the Aharonov-Bohm (AB) effect. It is shown that the response of each state can be recognized analyzing both AB period and extrema positions of the conductance oscillations which take place without any fine tuning of the system parameters.

Robust and Fragile Majorana Bound States in Proximitized Topological Insulator Nanoribbons.

Topological insulator (TI) nanoribbons with proximity-induced superconductivity are a promising platform for Majorana bound states (MBSs). In this work, we consider a detailed modeling approach for a TI nanoribbon in contact with a superconductor via its top surface, which induces a superconducting gap in its surface-state spectrum. The system displays a rich phase diagram with different numbers of end-localized MBSs as a function of chemical potential and magnetic flux piercing the cross section of the ribbon. These MBSs can be robust or fragile upon consideration of electrostatic disorder. We simulate a tunneling spectroscopy setup to probe the different topological phases of top-proximitized TI nanoribbons. Our simulation results indicate that a top-proximitized TI nanoribbon is ideally suited for realizing fully gapped topological superconductivity, in particular when the Fermi level is pinned near the Dirac point. In this regime, the setup yields a single pair of MBSs, well separated at opposite ends of the proximitized ribbon, which gives rise to a robust quantized zero-bias conductance peak.

Planar Josephson junctions in germanium: Effect of cubic spin-orbit interaction

Planar Josephson junctions comprising semiconductors with strong spin-orbit interaction (SOI) are promising platforms to host Majorana bound states (MBSs). Previous works on MBSs in planar Josephson junctions have focused on electron gases, where SOI is linear in momentum. In contrast, a two-dimensional hole gas in planar germanium (Ge) exhibits SOI that is cubic in momentum. Nevertheless, we show here that due to the particularly large SOI, Ge is a favorable material. Using a discretized model, we numerically simulate a Ge planar Josephson junction and demonstrate that also cubic SOI can lead to the emergence of MBSs. Interestingly, we find that the cubic SOI yields an asymmetric phase diagram as a function of the superconducting phase difference across the junction. We also find that trivial Andreev bound states can imitate the signatures of MBSs in a Ge planar Josephson junction, therefore making the experimental detection of MBSs difficult. We use experimentally realistic parameters to assess if the topological phase is accessible within experimental limitations. Our analysis shows that two-dimensional Ge is an auspicious candidate for topological phases.

Nonlocality of Majorana bound states modulated by an embedded quantum dot

We investigate the transport characteristics induced by the serially coupling of two pairs of Majorana bound states (MBSs) in two topological-superconducting nanowires. By considering the direct and indirect couplings of the MBSs via an embedded quantum dot (QD), we find that the coupling manner has an important influence on the nonlocality of MBSs. This is manifested as the change of conductance and Fano factors following the variation of the MBS-lead couplings, inter-MBS couplings, superconducting phase difference and the QD level, respectively. What is important is that the relationship of Fano factors can be altered by adjusting the coupling manner of the neighboring MBSs in the two nanowires. This indicates that the nonlocality of MBSs can be modulated by changing the inter-MBS couplings. • Nonlocality of Majorana bound states modulated by an embedded quantum dot is solved. • The influence of the direct and indirect couplings of the MBSs via an QD is considered. • The direct coupling of two pairs of MBSs is proposed. • The nonlocality of MBSs can be modulated by changing the inter-MBS couplings and the QD level. • Due to its nonlocality, braiding properties can be achieved, and thus quantum computing can be achieved.

Identifying Majorana bound states at quantum spin Hall edges using a metallic probe

We study the conductance afforded by a normal-metal probe which is directly contacting the helical edge modes of a quantum spin Hall insulator (QSHI). We show a $2e^2/h$ conductance peak at zero temperature in QSHI-based superconductor--ferromagnet hybrids due to the formation of a single Majorana bound state (MBS). In a corresponding Josephson junction hosting a pair of MBSs, a $4e^2/h$ conductance peak is found at zero temperature. The conductance quantization is robust to changes of the relevant system parameters and, remarkably, remains unaltered with increasing the distance between probe and MBSs. In the low temperature limit, the conductance peak is robust as the probe is placed within the localization length of MBSs. Our findings can therefore provide an effective way to detect the existence of MBSs in QSHI systems.

Creating and detecting poor man's Majorana bound states in interacting quantum dots

We propose and theoretically investigate an alternative way to create the poor man's Majorana bound states (MBSs) introduced in Phys. Rev. B 86, 134528 (2012). Our proposal is based on two quantum dots (QDs) with strong electron-electron interactions that couple via a central QD with proximity-induced superconductivity. In the presence of spin-orbit coupling and a magnetic field, gate control of all three QDs allows tuning the system into sweet spots with one MBS localized on each outer dot. We quantify the quality of these MBSs and show how it depends on the Zeeman energy and interaction strength. We also show how nonlocal transport spectroscopy can be used to identify sweet spots with high MBS quality. Our results provide a path for investigating MBS physics in a setting that is free of many of the doubts and uncertainties that plague other platforms.

Overlap of parafermionic zero modes at a finite distance

Parafermion bound states (PBSs) are generalizations of Majorana bound states (MBSs) and have been predicted to exist as zero-energy eigenstates in proximitized fractional quantum Hall edge states. Similarly to MBSs, a finite distance between the PBS can split the ground state degeneracy. However, parafermionic modes have a richer exchange statistics than MBSs, so several interaction terms are allowed by the underlying $\mathbb{Z}_{2n}$ symmetry, rendering the effective Hamiltonian governing a pair of PBSs at a finite distance nontrivial. Here, we use a combination of analytical techniques (semiclassical instanton approximation) and numerical techniques (quantum Monte Carlo simulations) to determine the effective coupling Hamiltonian. For this purpose, we go beyond the dilute one-instanton gas approximation and show how finite-size effects can give rise to higher-order parafermion interactions. We find excellent agreement between the analytical results and Monte Carlo simulations. We estimate that these finite-size corrections should be observable in some of the recently proposed experiments to observe PBSs in strongly correlated systems.

Two-stage decoupling effect in electron transport through T-shaped quantum dots coupled via Majorana bound states

We investigate the electron transport through a multiple T-shaped quantum-dot structure in which the dots in the main channel are coupled to additional dots by Majorana bound states (MBSs) at the ends of one topological-superconducting nanowires. Calculation results show that at the zero-bias limit, the transport behavior displays the interesting two-stage decoupling effect, when the additional-dot levels shift around the Fermi energy. Firstly, if the levels of such dots are the same as the Fermi energy, their-coupled MBSs will be decoupled from the other MBSs. This leads to various zero-bias conductance results which are determined by the different dot-MBS coupling manners. Next when these levels depart away from the Fermi energy, the MBS pairs will be decoupled from the main channel simultaneously, and this system becomes equivalent to the structure with serially-coupled quantum dots. The findings in this work provide new understanding about the interplay between the quantum dot and MBSs in driving the electron transport through quantum-dot systems.

Signatures of the long-range phase transition in topological Josephson junctions

The extended Kitaev models spark a recent surge of interest due to its unprecedented topological phase with fractional invariants. In this extended model of topological superconductors, long-range pairing interactions can induce a pair of subgap massive edge modes (MEMs), posing novel challenges to the unambiguous detection of Majorana bound states (MBSs) sustained by the same model in the short-range limit. Here we investigate the effect of a power-law decayed long-range pairing on the dc and ac Josephson currents between two Majorana nanowires. By varying the magnetic field and long-range pairing, the nanowires can be driven into different phases hosting MEMs, MBSs, or trivial Andreev bound states (ABSs), which considerably modulate the Josephson current. For weakly linked Josephson junctions, we show that the MEMs emerging in the long-range phases can induce a zigzag profile in the dc current-phase relation featured by two sign reversals deviating from the point where the superconducting phase difference $\ensuremath{\phi}=\ensuremath{\pi}$. In contrast, the Josephson current indicates a sharp sign reversal at $\ensuremath{\phi}=\ensuremath{\pi}$ in the short-range phase that hosts MBSs, while a smooth sinelike dc Josephson current appears in the presence of ABSs. In the nonequilibrium junction biased by a voltage, we identify a $4\ensuremath{\pi}$ ac Josephson current in the short-range phases, indicating a MBSs-mediated tunneling process dominating the supercurrents. Differently, when the system enters MEMs-hosted long-range phases, the ordinary ac Josephson current with $2\ensuremath{\pi}$ periodicity can be restored. In addition, we find the ac Josephson current induced by MEMs is significantly suppressed under the strong long-range pairing interactions. These signatures can serve as probes to discriminate MBSs from MEMs and ABSs.

Quantum dot scanning tunneling microscopy for Majorana bound states in continuum

We propose a device composed of a quantum dot (QD) connected to a normal metal lead to detect Majorana bound states (MBSs), which are formed at the ends of a topological superconductor nanowire (TSNW) and coupled to the lead with spin-dependent hybridization strengths. The information of the MBSs leaked into the lead is inferred from the spectral function of the QD serving as the tip of a scanning tunneling microscope (STM). It is found that lead–MBSs interaction induces a bound state characterized by an infinitely high peak in the dot’s zero-energy spectral function. The overlap between the two modes of the MBSs turns this bound state into a resonant one, and thus the zero-energy peak is split into three with the height of the central one equaling that in the absence of lead–MBSs coupling. We also find that the MBSs have lower impacts on the additional peak in the dot’s spectral function induced by intradot Coulomb interaction.

Prevalence of trivial zero-energy subgap states in nonuniform helical spin chains on the surface of superconductors

Helical spin chains, consisting of magnetic (ad-)atoms, on the surface of bulk superconductors are predicted to host Majorana bound states (MBSs) at the ends of the chain. Here, we investigate the prevalence of trivial zero-energy bound states in these helical spin chain systems. First, we show that the Hamiltonian of a helical spin chain on a superconductor can be mapped to an effective Hamiltonian reminiscent of a semiconductor nanowire with strong Rashba spin-orbit coupling. In particular, we show that a varying rotation rate between neighbouring magnetic moments maps to smooth non-uniform potentials in the effective nanowire Hamiltonian. Previously it has been found that trivial zero-energy states are abundant in nanowire systems with smooth potentials. Therefore, we perform an extensive search for zero-energy bound states in helical spin chain systems with varying rotation rates. Although bound states with near zero-energy do exist for certain dimensionalities and rotation profiles, we find that zero-energy bound states are far less prevalent than in semiconductor nanowire systems with equivalent non-uniformities. In particular, utilising varying rotation rates, we do not find zero-energy bound states in the most experimentally relevant setup consisting of a one-dimensional helical spin chain on the surface of a three-dimensional superconductor, even for profiles that produce near zero-energy states in equivalent one- and two- dimensional systems. Although our findings do not rule them out, the much reduced prevalence of zero-energy bound states in long non-uniform helical spin chains compared with equivalent semi-conductor nanowires, as well as the ability to measure states locally via STM, should reduce the experimental barrier to identifying MBSs in such systems.

Transport through the T-shaped double-quantum-dot structure with side-coupled Majorana bound states

We perform theoretical investigation of the transport properties in the T-shaped double-quantum-dot structure, by considering Majorana bound states (MBSs) to be coupled to the dot in the main channel. Calculation results show that in the linear transport regime, the side-coupled dot covers the influence of the MBS on the transport behaviors, independent of the Coulomb interaction in the QDs. In the absence of superconducting pairing potential in the QDs, tuning the side-coupled dot level to the Fermi energy can erase the contribution of MBS to the conductance, due to the occurrence of antiresonance. Otherwise, the linear conductance exhibits two distinct results determined by the inter-MBS coupling manners. In the case of Majorana zero mode, the linear conductance value keeps equal to e 2 2 h when the side-coupled dot departs from the Fermi energy. Alternatively in the case of Majorana nonzero mode, the linear conductance is always analogous to the MBS-absent case. On the other hand, when the superconducting pairing potential is incorporated, the linear conductance is also independent of the QD-MBS coupling. Especially in the case of Majorana nonzero mode, the MBSs are decoupled from the QDs. These findings provide new information for understanding the interplay between the MBSs and the quantum dots. • We study transport in T-shaped double quantum dots, by considering MBSs to be coupled to the main dot. • Tuning the side-coupled dot level to the Fermi level can erase the contribution of MBS to the conductance. • For Majorana zero mode, the linear conductance is equal to e 2 /2h when the side-coupled dot departs from the Fermi level. • For Majorana nonzero mode, the linear conductance is always analogous to the MBS-absent case. • If superconducting pairing potential exists in QDs, MBSs are decoupled from QDs for Majorana nonzero mode.

Probing electron-hole weights of an Andreev bound state by transient currents

Andreev bound states (ABSs) are localized quantum states that contain both electron and hole components. They ubiquitously reside in inhomogeneous superconducting systems. Following theoretical analysis, we propose to probe the electron-hole weights of an ABS via a local tunneling measurement that detects the transient current under a steplike pulse bias. With our protocol, the ABS energy level can also be obtained from peaks of the Fourier spectrum of the transient current. Our protocol can be applied to detect robust zero-energy Majorana bound states (MBSs), which have equal electron-hole weights, in candidate platforms where local tunneling spectroscopy measurement is possible. In the one-dimensional Majorana nanowire model, we numerically calculate the electron-hole weights for different types of low-energy bound states, including ABSs, quasi-MBSs, and MBSs.

Probing Majorana bound states via a pn junction containing a quantum dot

We propose an alternative route to transport experiments for detecting Majorana bound states (MBSs) by combining topological superconductivity with quantum optics in a superconducting $pn$ junction containing a quantum dot (QD). We consider a topological superconductor (TSC) hosting two Majorana bound states at its boundary ($n$ side). Within an effective low-energy model, the MBSs are coherently tunnel-coupled to a spin-split electron level on the QD which is placed close to one of the MBSs. Holes on the QD are tunnel-coupled to a normal conducting reservoir ($p$ side). Via electron-hole recombination, photons in the optical range are emitted, which have direct information on the MBS-properties through the recombined electrons. Using a master equation approach, we calculate the polarization-resolved photon emission intensities (PEIs). In the weak coupling regime between MBSs and QD, we find an analytical expression for the PEI which allows to clearly distinguish the cases of well separated MBSs at zero energy from overlapping MBSs. For separated MBSs, the Majorana spinor-polarization is given by the relative widths of the two PEI peaks associated with the two spin states on the QD. For overlapping MBSs, a coupling to the distant (nonlocal) MBS causes a shift of the emission peaks. Additionally, we show that quasiparticle poisoning (QP) influences the PEI drastically and changes its shot noise from super-Poissonian to sub-Poissonian. In the strong coupling regime, more resonances emerge in the PEI due to spin-mixing effects. Finally, we comment on how our proposal could be implemented using a Majorana nanowire.

Reshaping the Jaynes-Cummings ladder with Majorana bound states

We study the optical properties of a hybrid device composed by a quantum dot (QD) resonantly coupled to a photonic mode of an optical microcavity and a Majorana nanowire: a topological superconducting segment hosting Majorana bound states (MBSs) at the opposite ends. In the regime of strong light-matter coupling, it is demonstrated that the leakage of the Majorana mode into the QD opens new optical transitions between polaritonic states formed due to hybridisation of material excitation with cavity photons, which leads to the reshaping of the Jaynes-Cummings ladder and can lead to the formation of a robust single-peak at cavity eigenfrequency in the emission spectrum. Moreover, weak satellite peaks in the low and high frequency regions are revealed for the distinct cases of highly isolated MBSs, overlapped MBSs and MBSs not well localized at the nanowire ends.

Braiding Operations for a Topological Josephson Junction Array using SFQ Current Pulses

Majorana bound states (MBSs) can occur in topological Josephson junction (JJ) arrays with the presence of a fractional vortex with 2π-phase winding. It has been theoretically shown that the braiding operation can be achieved by using current pulses, compatible with single flux quantum (SFQ) technology. Therefore, we designed an SFQ logic circuit for the braiding operations of the topological JJ array using single flux quanta in a conventional JJ array. Braiding operations could be realized by an SFQ logic circuit with double flux quantum driver. To obtain positive and negative current pulses using the SFQ logic circuit, we used SFQ logic circuits with ac biasing for simulation. Finally, we confirmed the CNOT operation using an array-type SFQ circuit by means of simulation and estimated the CNOT operation speed.

Majorana bound states with chiral magnetic textures

The aim of this Tutorial is to give a pedagogical introduction into realizations of Majorana fermions, usually termed as Majorana bound states (MBSs), in condensed matter systems with magnetic textures. We begin by considering the Kitaev chain model of “spinless” fermions and show how two “half” fermions can appear at chain ends due to interactions. By considering this model and its two-dimensional generalization, we emphasize intricate relation between topological superconductivity and possible realizations of MBS. We further discuss how “spinless” fermions can be realized in more physical systems, e.g., by employing the spin-momentum locking. Next, we demonstrate how magnetic textures can be used to induce synthetic or fictitious spin–orbit interactions, and, thus, stabilize MBS. We describe a general approach that works for arbitrary textures and apply it to skyrmions. We show how MBS can be stabilized by elongated skyrmions, certain higher order skyrmions, and chains of skyrmions. We also discuss how braiding operations can be performed with MBS stabilized on magnetic skyrmions. This Tutorial is aimed at students at the graduate level.