Session M54: Majorana and Parafermion Devices

Moving Majoranas across a quantum point contact in 2D topological insulators

Majorana fermions are zero-energy excitations of topological superconductors which obey non-Abelian exchange statistics and are basic building blocks for topological quantum computation. In order to observe and exploit their extraordinary properties, we need to be able to properly manipulate them, for instance, by braiding a couple of them in real space. We propose a setup based on the helical edges of two-dimensional topological insulators (2DTI) which allows for a high degree of tunability by only controlling a handful of superconducting phases. In particular, our setup allows to move the Majoranas along a single edge as well as to move them across two different edges coupled by a quantum point contact. The versatility of the system represents an essential step forward towards probing the non-Abelian exchange statistics of Majoranas.   

Transport properties of Majorana bound state networks in the Coulomb blockade regime

Topologically protected qubits based on nanostructures hosting Majorana bound states (MBSs) hold great promise for fault-tolerant quantum computing. We study the transport properties of nanowire networks hosting MBSs with a focus on the effects of the charging energy and the overlap between neighboring MBSs in short mesoscopic samples. In particular, we investigate structures hosting four MBSs such as T-junctions and Majorana boxes. Using a Markovian master equation, we discuss the leading transport processes mediated by the MBSs. Single-electron tunneling and a processes involving creation and annihilation of Cooper pairs dominate in the sequential tunneling limit. In the cotunneling regime, Andreev processes are suppressed due to the charging energy and transport is dominated by transitions via virtual intermediate states. Our results show that four-terminal measurements in the T-junction and Majorana box geometries can be useful tools for the characterization of the properties of MBSs with finite overlaps and charging energy.   

Josephson radiation from nonlinear dynamics of Majorana modes in topological Josephson junctions

Recently, in topological Josephson junctions, the electromagnetic emission with the quantized frequency of eV/h has been experimentally observed as a consequence of the 4π-periodic Josephson effect, showing a phase-sensitive signature of Majorana zero modes. However, experiments show that this nontrivial radiation vanishes above a critical voltage, sharply contradicting previous theoretical results of the standard resistively shunted junction model. We extend this model to include the Majorana dynamics and show a quantitative agreement with the experimental results. Furthermore, we predict a fragment emission line and a chaos regime which can be observed experimentally by altering the junction parameters. We reveal that all these unique features come from the nonlinear dynamics of Majorana zero modes. The fragmental emission line and its vanishment are well explained with a fixed-point portrait, while the chaotic behavior is understood as the result of the bifurcation of fixed points. Our work will inspire more works to examine the structure of the radiation spectrum of topological Josephson junctions, which is wildly presented in experimental devices.   

Signatures of topological ground state degeneracy in Majorana islands

We consider a mesoscopic superconducting island hosting multiple pairs of Majorana zero-energy modes. The Majorana island consists of multiple p-wave wires connected together by a trivial (s-wave) superconducting backbone and is characterized by an overall charging energy E_c; the wires are coupled to normal-metal leads via tunnel junctions. Using a combination of analytical and numerical techniques we calculate the average charge on the island as well as non-local conductance matrix as a function of a p-wave pairing gap Δ_P, charging energy E_c and dimensionless junction conductances g_i. We find that the presence of a topological ground-state degeneracy in the island dramatically enhances charge fluctuations and leads to the suppression of Coulomb blockade effects. Specifically, in contrast with conventional (s-wave) mesoscopic superconducting islands, we find that Coulomb blockade effects are suppressed in Majorana islands regardless of the ratio E_c/Δ_P or the magnitude of the conductances g_i. We also discuss our findings in relation to the so-called topological Kondo effect.   

Majorana Bound States at the Superconducting Vortex from Magnetic Skyrmions

We show topological superconductivity can arise at the interface of ferromagnetic skyrmion material in proximity to an s-wave superconductor in the presence of strong spin-orbit coupling. We study the Majorana zero modes localized at the interface of the heterostructure and propose a pathway towards a useful platform for the manipulation of Majorana fermions via controlling skyrmions.
  

Topological superconductivity and Majorana fermions in the heterostructure of EuS island and planar gold surface

In a recent work [1], possible signatures of a pair of Majorana zero-energy modes were found
within the platform forming by EuS island and gold wire. In this work, we show the gold wire can
be replaced by a large gold surface in this platform. Furthermore, we found the chemical potential
step between the EuS island covering region and bare gold surface region are essential for keeping
a sizable topological region in the large gold surface case. Our results suggest the feasibility of
depositing EuS wires on a large area of gold and simplify the fabrication process further.

[1] S. Manna, P. Wei, Y. Xie, K. T. Law, P. A. Lee, and
J. Moodera, in press.   

Transmission Amplitude through a Coulomb blockaded Majorana Wire

We study electronic transport through a Coulomb blockaded superconducting Rashba wire in the co-tunneling regime between conductance resonances. Embedding such a wire into one arm of an electron interferometer allows to study the amplitude for coherent transmission. Varying an external Zeeman field allows to tune the wire into a topological regime, where localized Majorana zero modes are formed at both ends of the wire. In this topological phase, the nonlocal transport through Majorana zero modes is the dominant mechanism and gives rise to a maximum in the transmission amplitude as a function of Zeeman field, whose height is proportional to the wire length. On the other hand, for tunneling through a generic extended state or through a pair of Andreev bound states, the transmission amplitude is independent of wire length. Hence, the Zeeman field and length dependence of the transmission amplitude are unique signatures for the presence of Majorana zero modes.   

Parafermion braiding in fractional quantum Hall edge states with finite chemical potential

Parafermions are non-Abelian anyons which generalize Majorana fermions and hold great promise for topological quantum computation. We study the braiding of Z_2n parafermions which have been predicted to emerge as bound states in fractional quantum Hall systems at filling factor ν=1/n (n odd). Using a combination of bosonization and refermionization, we calculate the energy splitting as a function of distance and chemical potential for a pair of parafermions separated by a gapped region. Braiding of parafermions in quantum Hall edge states can be implemented by repeated fusion and nucleation of parafermion pairs. We simulate the conventional braiding protocol of parafermions numerically, taking into account the finite separation and finite chemical potential. We show that a nonzero chemical potential poses challenges for the adiabaticity of the braiding process because it leads to accidental crossings in the spectrum. To remedy this, we propose an improved braiding protocol which avoids those degeneracies.   

Entanglement in topological spin Josephson junctions

We study the spin transport through 1D quantum Ising-XY-Ising junctions that emulates topological Superconducting-Normal-Superconducting junctions via Jordan-Wigner transformation. We calculate, both numerically and analytically, the spectrum of Andreev bound states and the resulting Z₂ fractional spin Josephson effect from Majorana Fermions. Deep in the topological regime, we identify an effective time-reversal symmetry that leads to Z₄ fractional spin Josephson effect in the presence of interactions within junctions. Interestingly, in the lattice model, a hidden lattice time-reversal symmetry is revealed to protect Z₄ fractional spin Josephson effect in odd chain sites that persists in the absence of interactions. We also evaluate the resulting spin texture in the presence of the spin currents and highlight the effects of Majorana bound states on the entanglement of neighboring spin within junctions quantified by the concurrence. We propose to use a microwave cavity setup (cQED) for detecting the aforementioned Josephson effects by dispersive readout methods. Our results are relevant for a plethora of spin systems, such as trapped ions, coupled quantum dots, or magnetic impurities on surfaces.

[1] Pei-Xin Shen, Silas Hoffman, and Mircea Trif (to be submitted).   

Effect of Magnetic Field on Multi-terminal Josephson Junctions

Junctions with three or more superconducting terminals gained broad interest as they provide means to study physics and topology in higher dimensions and to braid Majorana fermions for fault-tolerant quantum computation. We study effect of perpendicular magnetic field on Andreev energy levels and critical currents in a 3-terminal Josephson junction with conventional s-wave superconducting leads and a normal 2DEG scattering region. In a 3-terminal junction, currents through two terminals determine the DC Josephson effect which occurs when the two currents are limited by the Critical Current Contour (CCC). We study the Fraunhofer diffraction patterns that manifest itself as oscillations in the diameter and area of the CCC. We show that the oscillations remain in 3-terminal devices but the additional terminal reduces the amplitude of the oscillations. We also show that magnetic field mixes with the superconducting phases in the leads and deforms the ground state energy landscape. We argue that a peculiar modulation of CCC with magnetic flux is the signature of coherent Josephson effect in multi-terminal Josephson junctions.   

Generic quantized zero-bias conductance peaks in superconductor-semiconductor hybrid structures

We show theoretically that quantized zero-bias conductance peaks should be ubiquitous in superconductor-semiconductor hybrids by employing a random matrix model with continuous tuning parameters. We demonstrate that NS junction conductance spectra can be generically obtained in this model replicating all features seen in recent experimental results. The theoretical quantized conductance peaks, which explicitly do not arise from spatially isolated Majorana zero modes, are easily found by preparing a contour plot of conductance over several independent tuning parameters, mimicking the effect of Zeeman splitting and voltages on gates near the junction. This suggests that even stable, apparently quantized, conductance peaks need not correspond to isolated Majorana modes; rather the a priori expectation should be that such quantized peaks generically occupy a significant fraction of the high-dimensional tuning parameter space that characterizes the NS tunneling experiments.   

Repulsive interactions enhance the coupling of superconductors to fractional quantum Hall edges

We study helical gapless modes arising on the edges of Abelian fractional quantum Hall liquids proximity coupled to a superconductor. This setup can be utilized to create non-Abelian parafermion zero modes if the coupling to the superconductor opens a gap in the helical modes. However, when the coupling to the superconductor is weak it is ineffective and does not open a gap due to the competition with the repulsive interactions stabilizing the fractional quantum Hall liquid. We therefore investigate the possibility for obtaining a gapped phase at strong coupling to the superconductor. To this end, we use an effective wire construction model for the quantum Hall liquid and employ renormalization group methods to obtain the phase diagram of the system. Surprisingly, at strong coupling we find a gapped phase which is stabilized by strong repulsive interactions in the bulk of the quantum Hall fluids. To investigate the possibility for obtaining a gap in the intermediate coupling regime, we identify a duality transformation that maps between the weak coupling and strong coupling regimes. We find conditions on the existence of a gap in the intermediate coupling regime by investigating self-dual points of the model.   

Survival of the fractional Josephson effect in time reversal invariant topological superconductors

A one-dimensional time reversal invariant topological superconductor (TRITOPS) hosts a Kramers pair of Majorana zero modes at each end. Previous work has established that TRITOPS phases do not enable rigid non-Abelian braiding, even when time-reversal symmetry is preserved throughout: local operators can mix the Majorana Kramers pair, yielding non-universal contributions to the non-Abelian Berry phase. Nonetheless, TRITOPS phases have been predicted to display other topological signatures, including a 4π periodic fractional Josephson effect. This talk will revisit the fractional Josephson effect and explore to what extent the anomalous periodicity remains robust in the presence of local mixing terms that destroy rigidity of non-Abelian braiding.   

Presence versus absence of end-to-end nonlocal conductance correlations in Majorana nanowires: Majorana bound states versus Andreev bound states

By calculating the differential tunneling conductance spectra from the two ends of a Majorana nanowire with a quantum dot embedded at one end, we establish that a careful examination of the nonlocal correlations of the zero bias conductance peaks, as measured separately from the two ends of the wire, can distinguish between topological Majorana bound states and trivial Andreev bound states. In particular, there will (not) be identical correlated zero bias peaks from both ends for Majorana bound states (Andreev bound states), and thus the presence (absense) of correlated zero bias conductance from the two wire ends could imply the presence (absence) of topological Majorana zero modes in the system. We present detailed results for the calculated conductance, energy spectra, and wavefunctions for different chemical potentials at the same magnetic field values to motivate end-to-end conductance correlation measurements in Majorana nanowires.   

Dissipative response of a topological Josephson junction at the critical point

A semiconducting nanowire proximitized by a superconductor is emerging as a building block of a topological qubit. Magnetic field applied to a proximitized nanowire drives it through the critical point into a topological state. We investigate signatures of this quantum phase transition in the dissipative response of a nanowire Josephson junction. The gap between the ground and excited states vanishes at the transition point. We find, that the low-frequency dissipation may remain weak despite the gap closing; this is markedly different from the conventional transition between the superconducting and normal states. In the absence of phase bias across the junction, the dissipative component of the admittance scales as σ(ω) ∼ ω². At a finite phase bias φ across the junction, σ(ω) ∼ φ² is frequency-independent in the interval ω ≤ Δ (here Δ is the value of the proximity-induced gap at zero field). We establish the complete scaling functions for the admittance in the vicinity of the transition. In the presence of a finite gap (i.e., upon detuning from the critical point) the scaling function for σ(ω) at a finite φ agrees with conventional Mattis-Bardeen formula. However, it is modified substantially and characterized by a much stronger frequency dependence if φ = 0.