Topological Superconductor Wire Research Articles

Majorana Bound States Hallmarks in a Quantum Topological Interferometer Ring

AbstractIn this work, the conductance and current correlations properties of a quantum topological interferometer consisting of a quantum dot coupled to two Majorana bound states (MBSs) confined at both ends of a 1D topological superconductor ring nanowire is investigated. The ring in its topological nontrivial and trivial phases is analyzed to show that the tunneling conductance, shot noise, and Fano factor present unique characteristics to distinguish the hallmark of MBSs. The findings are reinforced by taking advantage of the full correspondence between the quantum topological interferometer and a quantum‐dot effectively coupled to a single Majorana state in a straight topological superconductor wire configuration. It is shown that besides the characteristic zero‐bias conductance and the already known shot noise features, the Fano factor provides significant information to distinguish the MBSs presence.

Spin Seebeck effect induced by a Majorana zero mode in a nanomagnet

Abstract In mesoscopic devices, spin-resolved thermoelectrics induced by Majorana zero modes (MZMs) has attracted much attention. In this work we study an exactly solvable model consisting of a quantum dot coupled to metallic leads and side-coupled to a local large spin and a topological superconductor wire hosting MZMs. Because of MZMs’ helical property, the spin-up component of the dot spin is coupled with the MZM We mainly focus on one of the thermoelectric quantities—the Seebeck coefficient. By using Green’s function method combined with Hubbard operators, we obtain an analytical result for the spectral function. Then it is shown that spin Seebeck effects are determined by the interplay of temperature, exchange coupling, and the strength of coupling to Majorana wire. It is found that the spin-up thermopower is immune to local impurities in the limit of low temperature or large coupling to topological superconductor. Then we show that the spin-resolved thermopowers exhibit angular-quantum-number-dependent zero points. Specifically, the spin-down thermopower exhibits 4 S + 1 zero points due to the presence of local large spin, and the spin-up thermopower exhibits 8 S + 3 zero points near the transition point (where the competition of hole and electron processes reaches equilibrium), due to the interplay of local large spin and MZM. The charge and spin Seebeck coefficients also show S -dependent and MZM-involved behaviors. The results show that the interplay of local large spin and Majorana zero mode generates some interesting results, which may provide additional information as the signature of Majorana zero modes.

Fraction conductivity induced by a Majorana zero mode in a nanomagnet

In mesoscopic devices, the spin-resolved transport physics induced by Majorana zero mode (MZM) has attracted much attention. Here we study a system consisting of a quantum dot coupled to metallic leads and side-coupled to a local large spin and a topological superconductor wire hosting MZMs. The interplay of MZM and impurity (local large spin) is investigated. By using Hubbard operator Green’s function method, we obtain an analytical result for retarded Green’s function with finite U. Then we find that in the case of on-resonance and zero Coulomb interaction, the 1/2 signature of MZM at zero energy survives, but the number of peaks in spectral function is dependent on the local large spin. When S = integer, the peak numbers are 2S+3; when S = half-integer, the numbers are 2S+2. Moreover, fraction conductivity arises in the charge conductance at zero energy for S = integer. When Coulomb correlation arises, the peak numbers and heights in conductance are affected, and the MZM-induced symmetry of conductance peaks around zero energy is also destroyed. Our work shows that the interplay of local large spin and MZM generates some interesting results, which may provide additional fingerprints for the detection of MZM.

Spin-resolved transport physics induced by a Majorana-fermion zero mode

By using the Hubbard operator Green’s function method, the spin-resolved transport properties of a quantum dot coupled to metallic leads and side-coupled to a topological superconductor wire hosting Majorana bound states (MBSs) are studied theoretically. Compared with the second quantization representation, the method can lead us to an analytical result for the retarded Green’s function with finite U. The spin-related current and conductance are discussed. In the case of zero Coulomb interaction and on-resonance, the MBS’s 1/2 signature is recovered, and furthermore, there exists a 1/2 negative differential spin conductance. In the case of infinite Coulomb interaction, the 1/2 signature does not survive due to the Coulomb correlation reducing the current and conductance. Also due to this correlation, the MBS-induced symmetry of conductance peaks around zero energy is destroyed. In addition to this, we find that there are two MBS-induced negative differential spin conductance peaks. Theoretically, our work is supplementary and contrastive to the mainstream second quantization method, and these spin-resolved results may be observed in future experiments.

Dynamics of hybrid junctions of Majorana wires

We investigate the dynamics of hybrid junctions made of the topological superconductor (TS) and normal metal (N) wires. We consider an X-Y-Z configuration for the junctions where X, Y, Z = TS, N. We assume the wires X and Z being semi-infinite and in thermal equilibrium. We connect the wires X and Z through the short Y wire at some time, and numerically study time-evolution of the full device. For TS-N-TS device, we find a persistent, oscillating electrical current at both junctions even when there is no thermal or voltage bias, and the TS wires are identical. The amplitude and period of the oscillating current depend on the initial conditions of the middle N wire indicating the absence of thermalization. This zero-bias current vanishes at a long time for any of X and Z being an N wire or a TS wire near a topological phase transition. Using properties of different bound states within the superconducting gap, we develop a clear understanding of the oscillating currents.

Giant Shot Noise from Majorana Zero Modes in Topological Trijunctions.

The clear-cut experimental identification of Majorana bound states in transport measurements still poses experimental challenges. We here show that the zero-energy Majorana state formed at a junction of three topological superconductor wires is directly responsible for giant shot noise amplitudes, in particular at low voltages and for small contact transparency. The only intrinsic noise limitation comes from the current-induced dephasing rate due to multiple Andreev reflection processes.

Catalog of Andreev spectra and Josephson effects in structures with time-reversal-invariant topological superconductor wires

We study all the possible different two terminal configurations of Josephson junctions containing wires of time-reversal invariant topological superconductors (TRITOPS) and ordinary superconductors, including combinations with an interacting quantum dot between both wires in the junction. We introduce simple effective Hamiltonians which explain the different qualitative behaviors obtained. We analyze a wide range of phenomena, including occurrence and quenching of the so called $0-\pi$ transition, anomalous periodicity and jumps of the Josephson current as a function of the phase difference, and finite Josephson current in the absence of magnetic flux.

The Spin‐Dependent Coupling in the Hybrid Quantum Dot–Majorana Wire System

The transport properties of a quantum dot coupled to two metallic contacts and side‐coupled to a topological superconductor wire hosting Majorana zero energy modes (Majorana wire) have been studied. The spin‐dependent tunneling amplitude between quantum dot and Majorana wire has been assumed. The polarization of tunneling amplitudes causes the polarization of spin‐dependent zero‐bias conductance, but the total conductance of a quantum dot depends very weakly on the polarization of tunneling amplitudes. The complex behavior of the zero‐bias anomaly of conductance in the presence of an external magnetic field is shown. The zero‐bias conductance peaks at high external magnetic field can be the fingerprint of the presence of Majorana fermions.

Josephson effect in junctions of conventional and topological superconductors.

We present a theoretical analysis of the equilibrium Josephson current-phase relation in hybrid devices made of conventional s-wave spin-singlet superconductors (S) and topological superconductor (TS) wires featuring Majorana end states. Using Green’s function techniques, the topological superconductor is alternatively described by the low-energy continuum limit of a Kitaev chain or by a more microscopic spinful nanowire model. We show that for the simplest S–TS tunnel junction, only the s-wave pairing correlations in a spinful TS nanowire model can generate a Josephson effect. The critical current is much smaller in the topological regime and exhibits a kink-like dependence on the Zeeman field along the wire. When a correlated quantum dot (QD) in the magnetic regime is present in the junction region, however, the Josephson current becomes finite also in the deep topological phase as shown for the cotunneling regime and by a mean-field analysis. Remarkably, we find that the S–QD–TS setup can support φ0-junction behavior, where a finite supercurrent flows at vanishing phase difference. Finally, we also address a multi-terminal S–TS–S geometry, where the TS wire acts as tunable parity switch on the Andreev bound states in a superconducting atomic contact.

Ising quantum criticality in Majorana nanowires

Finite-length one-dimensional topological superconductor wires host localized Majorana zero modes at their ends. In realistic models, these appear only after a topological quantum critical point is crossed by external tuning of parameters. Thus, there is a universal finite-size scaling, governed by the critical point, that dictates the evolution of the energy of the Majorana modes near the transition. We first describe this scaling, then apply it in detail to an explicit synthetic topological superconductor model. Our work not only connects Ising quantum criticality with realistic nanowires in the presence of spin-orbit coupling, Zeeman splitting and superconductivity, but also provides a viable experimental route for discerning the existence of the topological quantum critical point.

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.

Hanbury Brown and Twiss noise correlations in a topological superconductor beam splitter

We study Hanbury-Brown and Twiss current cross-correlations in a three-terminal junction where a central topological superconductor (TS) nanowire, bearing Majorana bound states at its ends, is connected to two normal leads. Relying on a non-perturbative Green function formalism, our calculations allow us to provide analytical expressions for the currents and their correlations at subgap voltages, while also giving exact numerical results valid for arbitrary external bias. We show that when the normal leads are biased at voltages $V_1$ and $V_2$ smaller than the gap, the sign of the current cross-correlations is given by $-\mbox{sgn}(V_1 \, V_2)$. In particular, this leads to positive cross-correlations for opposite voltages, a behavior in stark contrast with the one of a standard superconductor, which provides a direct evidence of the presence of the Majorana zero-mode at the edge of the TS. We further extend our results, varying the length of the TS (leading to an overlap of the Majorana bound states) as well as its chemical potential (driving it away from half-filling), generalizing the boundary TS Green function to those cases. In the case of opposite bias voltages, $\mbox{sgn}(V_1 \, V_2)=-1$, driving the TS wire through the topological transition leads to a sign change of the current cross-correlations, providing yet another signature of the physics of the Majorana bound state.

6π Josephson Effect in Majorana Box Devices.

We study Majorana devices featuring a competition between superconductivity and multichannel Kondo physics. Our proposal extends previous work on single-channel Kondo systems to a topologically nontrivial setting of a non-Fermi liquid type, where topological superconductor wires (with gap Δ) represent leads tunnel coupled to a Coulomb-blockaded Majorana box. On the box, a spin degree of freedom with Kondo temperature T_{K} is nonlocally defined in terms of Majorana states. For Δ≫T_{K}, the destruction of Kondo screening by superconductivity implies a 4π-periodic Josephson current-phase relation. Using a strong-coupling analysis in the opposite regime Δ≪T_{K}, we find a 6π-periodic Josephson relation for three leads, with critical current I_{c}≈eΔ^{2}/ℏT_{K}, corresponding to the transfer of fractionalized charges e^{*}=2e/3.

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.

Low-energy theory of transport in Majorana wire junctions

We formulate and apply a low-energy transport theory for hybrid quantum devices containing junctions of topological superconductor (TS) wires and conventional normal (N) or superconducting (S) leads. We model TS wires as spinless $p$-wave superconductors and derive their boundary Keldysh Green's function, capturing both the Majorana end state and continuum quasiparticle excitations in a unified manner. We also specify this Green's function for a finite-length TS wire. Junctions connecting different parts of the device are described by the standard tunneling Hamiltonian. Using this Hamiltonian approach, one also has the option to include many-body interactions in a systematic manner. For N-TS junctions, we provide the current-voltage ($I$-$V$) characteristics at arbitrary junction transparency and give exact results for the shot noise power and the excess current. For TS-TS junctions, analytical results for the thermal noise spectrum and for the $I$-$V$ curve in the high-transparency low-bias regime are presented. For S-TS junctions, we compute the entire $I$-$V$ curve and clarify the conditions for having a finite Josephson current.

Majorana fermion exchange in strictly one-dimensional structures

It is generally thought that the adiabatic exchange of two identical particles is impossible in one spatial dimension. Here we describe a simple protocol that permits the adiabatic exchange of two Majorana fermions in a one-dimensional topological superconductor wire. The exchange relies on the concept of “Majorana shuttle” whereby a π domain wall in the superconducting order parameter which hosts a pair of ancillary majoranas delivers one zero mode across the wire while the other one tunnels in the opposite direction. The method requires some tuning of parameters and does not, therefore, enjoy full topological protection. The resulting exchange statistics, however, remain non-Abelian for a wide range of parameters that characterize the exchange.