Session S13: Majorana Bound States I

Chiral Majorana Modes

An emerging approach to quantum computing seeks to utilize topologically protected quantum states as Qubits to solve the error-correction problem, as the information encoded in such a "topological quantum computer" cannot be easily corrupted. A recent focus in condensed matter physics has been finding and fabricating such topological materials. In this talk, I will discuss two material systems that could host chiral Majorana modes and may have potential applications in topological quantum computing. The first system is the interface between a magnetically doped topological insulator and a superconductor, where we found experimental transport evidence for a chiral edge state of Majorana Fermions, which were proposed theoretically by Ettore Majorana in the 1930s by remained elusive. The second system is the ultra-thin bilayer film of bismuth and nickel, where we found experimental optical evidence for a superconducting state that breaks time-reversal symmetry, pointing to a 'd+id' superconducting state. Theories suggest that this state may have two chiral Majorana edge modes propagating around the sample edge, either clockwise or counterclockwise. These Majorana edge states, with further engineering and manipulation, could be useful for topological quantum computing. The works presented here were performed in collaboration with Qing Lin He, Lei Pan, Alexander L. Stern, Edward Burks, Xiaoyu Che, Gen Yin, Jing Wang, Biao Lian, Quan Zhou, Eun Sang Choi, Koichi Murata, Xufeng Kou, Tianxiao Nie, Qiming Shao, Yabin Fan, Shou-Cheng Zhang, Kai Liu, Kang L. Wang (Science, 357(6348), 294–299 (2017)) , and in collaboration with Xinxin Gong, Mehdi Kargarian, Alex Stern, Di Yue, Hexin Zhou, Xiaofeng Jin, Victor M. Galitski, Victor M. Yakovenko. (Science Advances, 3, 3, e1602579 (2017))   

Towards unveiling elusive Majorana fermions in gold nanowires via STS

One-dimensional topological superconductor has inspired extensive theoretical work and spurred several experimental thrusts to explore Majorana fermions (MFs). A particularly promising theory involves combining the surface state of gold to an s-wave superconductor subjected to an applied field [1]. This system promises robust MFs protected by an intense Rashba spin-orbit coupling much larger than in semiconductor systems. Spatially well separated MFs are expected in long nanowires of Au, for an unambiguous detection of MFs. We studied scanning tunneling spectroscopy (STS) of Au nanowires long compared to the MF coupling length. STS shows clean superconducting gap induced in the surface state of Au via proximity effect. Using spatially resolved STS at ~ 400mK with a magnetic field applied along the wire, we demonstrate a pair of zero-bias bound states emerging at the ends of nanowire. While the observed non-local features nicely agree with the earlier theoretical prediction, our temperature and energy resolution is not sufficient to distinguish between MF and mini-gap state that are expected to exist at the end of a multi-channel wire. [1] Potter, A. C. and Lee, P. A., Phys Rev Lett 105, 227003 (2010).   

2e-periodic Switching Current Modulation in Nanowire Single Cooper Pair Transistors

Over the past years, many studies have reported on signatures of Majorana modes via zero-bias peaks. However, to date there have been no experiments demonstrating their non-Abelian statistics. Recently, Aasen et al. proposed that the Majorana fusion channels, intimately related to their non-Abelian statistics, could be probed in a single-wire geometry [1]. An outstanding issue for the fusion experiment, however, is quasiparticle poisoning, because it can lead to decoherence and readout errors.

Therefore, we studied parity effects in gate-tuneable single Cooper pair transistors made from proximitized InAs nanowires. We measured the gate modulation of the switching currents as a function of temperature, island length, and the magnetic field applied parallel to the wire. We observed 2e-periodic signals for a wide range of gate settings. This implies there are few deep quasiparticle traps present in the island. We develop a thermodynamic model, which shows that our island does experience rapid parity fluctuations despite this clean 2e signal. When increasing the magnetic field, an oscillating even-odd pattern appears. We attribute this to the zero-energy crossing of an Andreev bound state.

References:
[1] D. Aasen et al. PRX 6, 031016 (2016)   

Signatures of Majorana Fermions in Hybrid Superconductor-Topological Insulator Josephson Junctions

We are studying the transport properties of lateral superconductor-topological insulator-superconductor Josephson junctions. In a magnetic field perpendicular to the TI surface, it is expected that localized zero-energy Majorana bound states will be stabilized in the junction at locations at which the phase difference is an odd-multiple of π, i.e. at the core of Josephson vortices. Our measurements of the critical current modulation patterns have shown lifting of odd nodes, which we interpret as evidence for a 4π-periodic sin(φ/2)-component in the Josephson current-phase relation (CPR), and motivated a model for the nucleation and manipulation of these Majorana states. In this talk, we summarize the results of a series of experiments designed to test in detail specific features of this model: (1) testing whether there is an even-odd pattern of node-lifting, (2) identifying irregular features in the diffraction pattern that could indicate the abrupt entry of Majorana states, (3) looking for changes in the noise distribution of critical currents that would indicate parity transitions of Majorana pairs, and (4) searching for non-sinusoidal components in direct measurements of the CPR using an asymmetric SQUID technique.   

Comparing Majorana ZBP with trivial Andreev bound states and super hard superconducting gap in InSb nanowire hybrid devices

Majorana bound states(MBS) has been predicted to emerge in 1D nanowire with induced superconductivity, spin-orbital coupling and external magnetic field. Tunneling spectroscopy can be used to probe the signal of MBS, namely Zero bias conductance peaks(ZBP) appearing at finite field with suitable chemical potential. However, trivial Andreev bound states due to quantum confinement can also give rise to ZBPs with very similar manner. We compare Majorana ZBPs with Andreev bound states in different aspects: ZBP phase diagram, onset magnetic field and field angle dependence. We will also show super hard induced superconducting gap in our system with further improved contact recipe.   

Replicas of Andreev spectra in InSb nanowires

A superconducting contact to a quantum dot can induce electron-hole correlations known as Andreev bound states (ABSs) inside the dot. We study transport through ABSs in InSb nanowires with superconducting contacts as both source and drain. Anomalous transport resonances are observed in these systems, namely one or more (up to 4) replicas of Andreev spectra at higher source-drain biases, as well as negative differential conductance shadows of the resonances inside the induced gap. We develop a model that captures these features by considering the effect of a soft-gap superconducting probe, and of additional ABSs induced in the dot, and in the nanowire segments adjacent to the dot. Our findings extend the framework for the interpretation of tunneling experiments in mesoscopic systems, and specifically of Majorana bound states which are studied with semiconductor nanowire probes similar to those used here.   

Effective g-factor in Majorana Wires

We use the effective g-factor of subgap states, g*, in hybrid InAs nanowires with an epitaxial Al shell to investigate how the superconducting density of states is distributed between the nanowire core and the shell. We find a step-like reduction of g* and improved hard gap with reduced carrier density in the nanowire, controlled by gate voltage. These observations are relevant for Majorana devices, which require tunable carrier density and g* exceeding the g-factor of the proximitizing superconductor. Additionally, we observe the closing and reopening of a gap in the subgap spectrum coincident with the appearance of a zero-bias conductance peak. Finally, more recent results from different device geometries are discussed.   

Long-range Topological Superconductors: Majorana & Non-local Dirac fermions

Long-range interactions can dramatically modify the phase diagram of condensed matter systems. Novel topological superconducting phases can be induced in both wire-shaped and planar structures by including long-range effects. In 1D, massless Majorana modes at the edge pair into a new topological quasiparticle: a massive Dirac fermion localized at both edges of the wire. This new topological phase has fractional quantum numbers as a consequence of the long-range couplings. Moreover, we remarkably observe a staircase of higher-order topological phase transitions. In 2D, we show how long-range interactions may greatly enhance the formation of chiral topological superconductivity. The inclusion of long-range effects is potentially applicable to recent experiments with magnetic impurities and islands in superconductors.
[1] Chiral Topological Superconductors Enhanced by Long-range Interactions, O. Viyuela, L. Fu, M.A. Martin-Delgado, arXiv: 1707.02326 (2017).
[2] Staircase to Higher-Order Topological Phase Transitions, P. Cats, A. Quelle, O. Viyuela, M.A. Martin-Delgado, C.Morais Smith, arXiv: 1710.05691 (2017).   

Fermionizing Parafermions

Parafermion zero modes are Majorana-fermion generalizations that exhibit comparatively rich non-Abelian-anyon properties. We introduce exact mappings that connect parafermion chains, which can emerge in 2D fractionalized media, to strictly 1D fermionic systems. Most notably, Z₄ parafermions map onto physical electrons enjoying time-reversal symmetry. Phases for the former translate into a variety of interesting electronic states, including a time-reversal-invariant topological superconductor (TRITOPS) hosting Kramers pairs of edge Majorana zero modes. We further show that nontrivial parafermionic fusion rules survive in their electronic counterparts, and can be detected via a novel pumping cycle for the TRITOPS phase that yields an edge magnetization with quadrupled periodicity. We generalize these results to arbitrary Z_2N parafermion chains. Our work highlights new avenues for exploring 'beyond-Majorana' physics in experimentally realistic, weakly interacting 1D electronic platforms.   

Parafermionic Bound States in 1D Lattices of Spinful Fermions

Parafermions are emergent excitations, potentially relevant to topological quantum information, which generalize Majorana fermions. Many features of the latter are well captured by the seminal Kitaev model, a 1D lattice of spinless fermions with p-wave superconductivity. Here we present a non-trivial generalization of this model to the Ζ₄ parafermionic case. In particular, we identify and characterize a whole family of spinful fermionic Hamiltonians on a 1D lattice which support zero-energy parafermions, localized at the two ends of the system. Their properties are discussed as well as their robustness with respect to variations of parameters in the fermionic Hamiltonian. Our study provides a starting point to make a connection between fermionic lattice models and the low energy field theory description of 1D interacting systems featuring parafermionic bound states.   

Efficient real-space parameter optimization algorithm for Majorana nanowires

We present an efficient gradient-based method for the optimization of real-space parameter profiles in quasi-1D systems such as Majorana nanowires. Inspired by an analogy with the quantum optimal control algorithm GRAPE (Gradient Ascent Pulse Engineering), we can efficiently evaluate gradients of discretized real-space parameter profiles using analytical derivatives of recursive Green functions expressions. This new approach leads to a polynomial speedup over a simpler finite difference calculation. As an application, we optimize magnetic textures for the creation and stabilization of topological superconducting phases in nanowires without spin-orbit coupling. The optimization allows to go beyond intuitive and analytical results, leading to potentially new parameter regimes of experimental relevance in Majorana nanowires.   

A New Type of Many-Body Majorana Zero Modes in Fermionic Flux Ladder Model

One promising avenue to study one-dimensional (1D) topological phases is to realize them in synthetic materials such as cold atomic gases. Intriguingly, it is possible to realize Majorana boundary modes in a 1D number-conserving system consisting of two fermionic chains coupled only by pair-hopping processes. It is commonly believed that significant interchain single-particle tunneling necessarily destroys these Majorana modes, as it spoils the Z_2 fermion-parity symmetry that protects them. In this talk, we present a new mechanism to overcome this obstacle, by piercing a (synthetic) magnetic \pi-flux through each plaquette of the Fermi ladder. Using bosonization, we show that in this case there exists an exact leg-interchange symmetry that is robust to interchain hopping, and acts as fermion parity at long wavelengths. We utilize density matrix renormalization group and exact diagonalization to verify that the resulting model exhibits Majorana boundary modes up to large single-particle tunnelings, comparable to the intrachain hopping strength. Our work highlights the unusual impacts of different topologically trivial band structures on these interaction-driven topological phases, and identifies a distinct route to stabilizing Majorana boundary modes in 1D fermionic ladders.   

Conductance of a multi-Majorana island in the strong tunneling regime

We consider a mesoscopic superconducting island in a Coulomb blockade regime hosting more than two Majorana zero modes which are coupled to normal-metal leads. It has been predicted that such a system may exhibit a novel phenomenon dubbed as topological Kondo effect. We study transport properties of such an island in the strong tunneling limit and calculate the conductance matrix as a function of temperature and the gate voltage applied to the island. We construct a microscopic model explicitly incorporating the superconductor underlying the island, which we then study using the bosonization technique. This approach provides a more microscopic starting point than previous studies.