Majorana Zero-energy Modes Research Articles

Persistent current and zero-energy Majorana modes in a p -wave disordered superconducting ring

We discuss the emergence of zero-energy Majorana modes in a disordered finite-length $p$-wave one-dimensional superconducting ring, pierced by a magnetic flux $\mathrm{\ensuremath{\Phi}}$ tuned at an appropriate value $\mathrm{\ensuremath{\Phi}}={\mathrm{\ensuremath{\Phi}}}_{*}$. In the absence of fermion parity conservation, we evidence the emergence of the Majorana modes by looking at the discontinuities in the persistent current $I[\mathrm{\ensuremath{\Phi}}]$ at $\mathrm{\ensuremath{\Phi}}={\mathrm{\ensuremath{\Phi}}}_{*}$. By monitoring the discontinuities in $I[\mathrm{\ensuremath{\Phi}}]$, we map out the region in parameter space characterized by the emergence of Majorana modes in the disordered ring.

Transport properties of a hybrid Majorana wire-quantum dot system with ferromagnetic contacts

The transport properties of a quantum dot coupled to ferromagnetic contacts and attached to a topological superconductor wire hosting Majorana zero-energy modes at its ends are studied theoretically in the Kondo regime. By using the numerical renormalization group method, the temperature and dot level dependence of the spectral function, the conductance, and its spin polarization are studied for different coupling strengths to a topological superconductor. It is shown that the transport characteristics are generally determined by the interplay of three competing energy scales, resulting from Kondo correlations, a ferromagnetic-lead-induced exchange field, and topological-wire-induced splitting. These two splittings are found to have opposite signs. Moreover, they can compensate for each other in an appropriate parameter regime.

Impurity bound states in fully gapped d-wave superconductors with subdominant order parameters

Impurities in superconductors and their induced bound states are important both for engineering novel states such as Majorana zero-energy modes and for probing bulk properties of the superconducting state. The high-temperature cuprates offer a clear advantage in a much larger superconducting order parameter, but the nodal energy spectrum of a pure d-wave superconductor only allows virtual bound states. Fully gapped d-wave superconducting states have, however, been proposed in several cuprate systems thanks to subdominant order parameters producing d + is- or d + id′-wave superconducting states. Here we study both magnetic and potential impurities in these fully gapped d-wave superconductors. Using analytical T-matrix and complementary numerical tight-binding lattice calculations, we show that magnetic and potential impurities behave fundamentally different in d + is- and d + id′-wave superconductors. In a d + is-wave superconductor, there are no bound states for potential impurities, while a magnetic impurity produces one pair of bound states, with a zero-energy level crossing at a finite scattering strength. On the other hand, a d + id′-wave symmetry always gives rise to two pairs of bound states and only produce a reachable zero-energy level crossing if the normal state has a strong particle-hole asymmetry.

Detecting topological superconductivity using low-frequency doubled Shapiro steps

The fractional Josephson effect has been observed in many instances as a signature of a topological superconducting state containing zero-energy Majorana modes. We present a nontopological scenario which can produce a fractional Josephson effect generically in semiconductor-based Josephson junctions, namely, a resonant impurity bound state weakly coupled to a highly transparent channel. We show that the fractional ac Josephson effect can be generated by the Landau-Zener processes which flip the electron occupancy of the impurity bound state. The Josephson effect signature for Majorana modes become distinct from this nontopological scenario only at low frequency. We prove that a variant of the fractional ac Josephson effect, namely, the low-frequency doubled Shapiro steps, can provide a more reliable signature of the topological superconducting state.

Spin Seebeck effect in quantum dot side-coupled to topological superconductor

The spin-resolved thermoelectric transport properties of a quantum dot coupled to ferromagnetic leads and side-coupled to a topological superconductor wire hosting Majorana zero-energy modes are studied theoretically. The calculations are performed in the linear response regime by using the numerical renormalization group method. It is shown that transport characteristics are determined by the interplay of Kondo correlations, exchange field due to the presence of ferromagnets and the strength of coupling to Majorana wire. These different energy scales are revealed in the behavior of the Seebeck and spin Seebeck coefficients, which exhibit an enhancement for temperatures of the order of the coupling strength to topological wire. Moreover, it is demonstrated that additional sign changes of the thermopower can occur due to the presence of Majorana zero-energy modes. These findings may provide additional fingerprints of the presence of Majorana fermions.

Zero energy modes in a superconductor with ferromagnetic adatom chains and quantum phase transitions

We study Majorana zero energy modes (MZEM) that occur in an s-wave superconducting surface, at the ends of a ferromagnetic (FM) chain of adatoms, in the presence of Rashba spin–orbit interaction (SOI) considering both non self-consistent and self-consistent superconducting order. We find that in the self-consistent solution, the average superconducting gap function over the adatom sites has a discontinuous drop with increasing exchange interaction at the same critical value where the topological phase transition occurs. We also study the MZEM for both treatments of superconducting order and find that the decay length is a linear function of the exchange coupling strength, chemical potential and superconducting order. For wider FM chains the MZEM occur at smaller exchange couplings and the slope of the decay length as a function of exchange coupling grows with chain width. Thus we suggest experimental detection of different delocalization of MZEM in chains of varying widths. We discuss similarities and differences between the MZEM for the two treatments of the superconducting order.

Transport through a quantum dot coupled to two Majorana bound states

We investigate electron transport inside a ring system composed of a quantum dot (QD) coupled to two Majorana bound states confined at the ends of a one-dimensional topological superconductor nanowire. By tuning the magnetic flux threading through the ring, the model system we consider can be switched into states with or without zero-energy modes when the nanowire is in its topological phase. We find that the Fano profile in the conductance spectrum due to the interference between bound and continuum states exhibits markedly different features for these two different situations, which consequently can be used to detect the Majorana zero-energy mode. Most interestingly, as a periodic function of magnetic flux, the conductance shows $2\pi$ periodicity when the two Majorana bound states are nonoverlapping (as in an infinitely long nanowire) but displays $4\pi$ periodicity when the overlapping becomes nonzero (as in a finite length nanowire). We map the model system into a QD--Kitaev ring in the Majorana fermion representation and affirm these different characteristics by checking the energy spectrum.

Majorana wave-function oscillations, fermion parity switches, and disorder in Kitaev chains

We study the decay and oscillations of Majorana fermion wavefunctions and ground state (GS) fermion parity in one-dimensional topological superconducting lattice systems. Using a Majorana transfer matrix method, we find that Majorana wavefunction properties are encoded in the associated Lyapunov exponent, which in turn is the sum of two independent components: a `superconducting component' which characterizes the gap induced decay, and the `normal component', which determines the oscillations and response to chemical potential configurations. The topological phase transition separating phases with and without Majorana end modes is seen to be a cancellation of these two components. We show that Majorana wavefunction oscillations are completely determined by an underlying non-superconducting tight-binding model and are solely responsible for GS fermion parity switches in finite-sized systems. These observations enable us to analytically chart out wavefunction oscillations, the resultant GS parity configuration as a function of parameter space in uniform wires, and special parity switch points where degenerate zero energy Majorana modes are restored in spite of finite size effects. For disordered wires, we find that band oscillations are completely washed out leading to a second localization length for the Majorana mode and the remnant oscillations are randomized as per Anderson localization physics in normal systems. Our transfer matrix method further allows us to i) reproduce known results on the scaling of mid-gap Majorana states and demonstrate the origin of its log-normal distribution, ii) identify contrasting behavior of disorder-dependent GS parity switches for the cases of even versus odd number of lattice sites, and iii) chart out the GS parity configuration and associated parity switch points as a function of disorder strength.

Synthesizing Majorana zero-energy modes in a periodically gated quantum wire

We explore a scheme for engineering a one-dimensional spinless p-wave superconductor hosting unpaired Majorana zero-energy modes, using an all-electric setup with a spin-orbit coupled quantum wire in proximity to an s-wave superconductor. The required crossing of the Fermi level by a single spin-split energy band is ensured by employing a periodically modulated Rashba interaction, which, assisted by electron-electron interactions and a uniform Dresselhaus interaction, opens a gap at two of the spin-orbit shifted Fermi points. While an implementation in a hybrid superconductor-semiconductor device requires improvements upon present-day capabilities, a variant of our scheme where spin-orbit-coupled cold fermions are effectively proximity-coupled to a BEC reservoir of Feshbach molecules may provide a ready-to-use platform.

Exact Solution of Quadratic Fermionic Hamiltonians for Arbitrary Boundary Conditions.

We present a procedure for exactly diagonalizing finite-range quadratic fermionic Hamiltonians with arbitrary boundary conditions in one of D dimensions, and periodic in the remaining D-1. The key is a Hamiltonian-dependent separation of the bulk from the boundary. By combining information from the two, we identify a matrix function that fully characterizes the solutions, and may be used to construct an efficiently computable indicator of bulk-boundary correspondence. As an illustration, we show how our approach correctly describes the zero-energy Majorana modes of a time-reversal-invariant s-wave two-band superconductor in a Josephson ring configuration, and predicts that a fractional 4π-periodic Josephson effect can only be observed in phases hosting an odd number of Majorana pairs per boundary.

Quantum memories with zero-energy Majorana modes and experimental constraints

In this work we address the problem of realizing a reliable quantum memory based on zero-energy Majorana modes in the presence of experimental constraints on the operations aimed at recovering the information. In particular, we characterize the best recovery operation acting only on the zero-energy Majorana modes and the memory fidelity that can be therewith achieved. In order to understand the effect of such restriction, we discuss two examples of noise models acting on the topological system and compare the amount of information that can be recovered by accessing either the whole system, or the zero-modes only, with particular attention to the scaling with the size of the system and the energy gap. We explicitly discuss the case of a thermal bosonic environment inducing a parity-preserving Markovian dynamics in which the introduced memory fidelity decays exponentially in time, independent from system size, thus showing the impossibility to retrieve the information by acting on the zero-modes only. We argue, however, that even in the presence of experimental limitations, the Hamiltonian gap is still beneficial to the storage of information.

Phase-tunable Majorana bound states in a topological N-SNS junction

We theoretically study the differential conductance of a one-dimensional normal-superconductor-normal-superconductor (N-SNS) junction with a phase bias applied between the two superconductors. We consider specifically a junction formed by a spin-orbit coupled semiconducting nanowire with regions of the nanowire having superconducting pairing induced by a bulk $s$-wave superconductor. When the nanowire is tuned into a topologically non-trivial phase by a Zeeman field, it hosts zero-energy Majorana modes at its ends as well as at the interface between the two superconductors. The phase-dependent splitting of the Majorana modes gives rise to features in the differential conductance that offer a clear distinction between the topologically trivial and non-trivial phases. We calculate the transport properties of the junction numerically and also present a simple analytical model that captures the main properties of the predicted tunneling spectroscopy.

Topological superconductivity in a multichannel Yu-Shiba-Rusinov chain

Chains of magnetic atoms placed on the surface of an s-wave superconductor with large spin-orbit coupling provide a promising platform for the realization of topological superconducting states characterized by the presence of Majorana zero-energy modes. In this work we study the properties of the one-dimensional chain of Yu-Shiba-Rusinov states induced by magnetic impurities using a realistic model for the magnetic atoms that include the presence of multiple scattering channels. These channels are mixed by the spin-orbit coupling and, via the hybridization of the Yu-Shiba-Rusinov states at different sites of the chain, result in a multi-band structure for the chain. We obtain the topological phase diagram for such band structure. We identify the parameter regimes for which the different bands lead to a topological phase and show that the inclusion of higher bands can greatly enlarge the phase space for the realization of topological states.

Majorana zero-energy modes and spin current evolution in mesoscopic superconducting loop systems with spin-orbit interaction

Majorana zero-energy modes and spin current evolution in mesoscopic superconducting loop systems with spin-orbit interaction

Statistical translation invariance protects a topological insulator from interactions

We investigate the effect of interactions on the stability of a disordered, two-dimensional topological insulator realized as an array of nanowires or chains of magnetic atoms on a superconducting substrate. The Majorana zero-energy modes present at the ends of the wires overlap, forming a dispersive edge mode with thermal conductance determined by the central charge $c$ of the low-energy effective field theory of the edge. We show numerically that, in the presence of disorder, the $c=1/2$ Majorana edge mode remains delocalized up to extremely strong attractive interactions, while repulsive interactions drive a transition to a $c=3/2$ edge phase localized by disorder. The absence of localization for strong attractive interactions is explained by a self-duality symmetry of the statistical ensemble of disorder configurations and of the edge interactions, originating from translation invariance on the length scale of the underlying mesoscopic array.

Observability of surface Andreev bound states in a topological insulator in proximity to an s-wave superconductor

To guide experimental work on the search for Majorana zero-energy modes, we calculate the superconducting pairing symmetry of a three-dimensional topological insulator in combination with an s-wave superconductor. We show how the pairing symmetry changes across different topological regimes. We demonstrate that a dominant p-wave pairing relation is not sufficient to realise a Majorana zero-energy mode useful for quantum computation. Our main result is the relation between odd-frequency pairing and Majorana zero energy modes by using Green functions techniques in three-dimensional topological insulators in the so-called Majorana regime. We discuss thereafter how the pairing relations in the different regimes can be observed in the tunneling conductance of an s-wave proximised three-dimensional topological insulator. We discuss the necessity to incorporate a ferromagnetic insulator to localise the zero-energy bound state to the interface as a Majorana mode.

Disorder-induced subgap states and Majorana zero-energy edge modes in two-dimensional topological insulator-superconductor hybrid structures

Contrary to the widespread belief that Majorana zero-energy modes, existing as bound edge states in 2D topological insulator (TI)-superconductor (SC) hybrid structures, are unaffected by non-magnetic static disorder by virtue of Anderson's theorem, we show that such a protection against disorder does not exist in realistic multi-channel TI/SC/ferromagnetic insulator (FI) sandwich structures of experimental relevance since the time-reversal symmetry is explicitly broken locally at the SC/FI interface where the end Majorana mode (MM) resides. We find that although the MM itself and the \emph{bulk} topological superconducting phase inside the TI are indeed universally protected against disorder, disorder-induced subgap states are generically introduced at the TI edge due to the presence of the FI/SC interface as long as multiple edge channels are occupied. We discuss the implications of the finding for the detection and manipulation of the edge MM in realistic TI/SC/FI experimental systems of current interest.

Majorana vortex-bound states in three-dimensional nodal noncentrosymmetric superconductors

Noncentrosymmetric superconductors (NCSs), characterized by antisymmetric spin-orbit coupling and a mixture of spin-singlet and spin-triplet pairing components, are promising candidate materials for topological superconductivity. An important hallmark of topological superconductors is the existence of protected zero-energy states at surfaces or in vortex cores. Here we investigate Majorana vortex-bound states in three-dimensional nodal and fully gapped NCSs by combining analytical solutions of Bogoliubov--de Gennes (BdG) equations in the continuum with exact diagonalization of BdG Hamiltonians. We show that depending on the crystal point-group symmetries and the topological properties of the bulk Bogoliubov-quasiparticle wave functions, different types of zero-energy Majorana modes can appear inside the vortex core. We find that for nodal NCSs with tetragonal point group ${C}_{4v}$ the vortex states are dispersionless along the vortex line, forming one-dimensional Majorana flat bands, while for NCSs with ${D}_{4}$ point-group symmetry the vortex modes are helical Majorana states with a linear dispersion along the vortex line. NCSs with monoclinic point group ${C}_{2}$, on the other hand, do not exhibit any zero-energy vortex-bound states. We show that in the case of the ${C}_{4v}$ $({D}_{4})$ point group the stability of these Majorana zero modes is guaranteed by a combination of reflection $(\ensuremath{\pi}$ rotation), time-reversal, and particle-hole symmetry. Considering continuous deformations of the quasiparticle spectrum in the presence of vortices, we show that the flat-band vortex-bound states of ${C}_{4v}$ point-group NCSs can be adiabatically connected to the dispersionless vortex-bound states of time-reversal symmetric Weyl superconductors. Experimental implications of our results for thermal transport and tunneling measurements are discussed.

Interplay between Kondo and Majorana Interactions in Quantum Dots

We study the properties of a quantum dot coupled to a one-dimensional topological superconductor and a normal lead and discuss the interplay between Kondo and Majorana-induced couplings in quantum dot. The latter appears due to the presence of Majorana zero-energy modes localized at the ends of the one-dimensional superconductor. We investigate the phase diagram of the system as a function of Kondo and Majorana interactions using a renormalization-group analysis, a slave-boson mean-field theory and numerical simulations using the density-matrix renormalization group method. We show that, in addition to the well-known Kondo fixed point, the system may flow to a new fixed point controlled by the Majorana-induced coupling which is characterized by non-trivial correlations between a localized spin on the dot and the fermion parity of the topological superconductor and normal lead. We compute several measurable quantities such as differential tunneling conductance and impurity spin susceptibility which highlight some peculiar features characteristic to the Majorana fixed point.

Robustness of quantum memories based on Majorana zero modes

We analyze the rate at which quantum information encoded in zero-energy Majorana modes is lost in the presence of perturbations. We show that information can survive for times that scale exponentially with the size of the chain both in the presence of quenching and time-dependent quadratic dephasing perturbations, even when the latter have spectral components above the system's energy gap. The origin of the robust storage, namely the fact that a sudden quench affects in the same way both parity sectors of the original spectrum, is discussed, together with the memory performance at finite temperatures and in the presence of particle exchange with a bath.