Session B7: Focus Session: Majorana Fermions

Engineering Majorana fermions in atomic chains with collinear magnetic order.

We propose new mechanisms for engineering Majorana fermions (MFs) based on atomic chains with collinear magnetic order, on top of a conventional superconductor. For antiferromagnetic ordering, we show [1] that a weak Zeeman field and a supercurrent convert the preexisting topologically-unprotected Shiba states into MFs, without the requirement of Rashba spin-orbit coupling (SOC). Remarkably, the electronic spin-polarization of the arising edge MF wavefunctions depends solely on the parity of the number of magnetic moments, which can serve as a unique signature of the MFs. Instead, if Rashba SOC is present, both ferro- and anti-ferromagnetic orders can lead to topological phases which can harbor one or even two MFs per edge [2]. We demonstrate how to experimentally access the diverse MF phases by varying the adatom spacing, the SOC strength, or the magnetic moment of the adatoms in consideration.The two magnetic orderings lead to qualitatively and quantitatively distinct topological features that are reflected in the spatial profile of the MF wavefunctions. Our findings connect to the recent observations of MFs in atomic chains [3] and can open alternative routes for confirming the emergence of MFs. [1] A. Heimes, P. Kotetes, and G. Sch\"{o}n,PRB 90, 060507(R) (2014). [2] A. Heimes, D. Mendler, and P. Kotetes, arXiv:1410.6367. [3] S. Nadj-Perge et al., Science 346, 602 (2014).   

Majorana Fermions in Quantum Wires with Helical Magnetic Textures

Magnetic textures can lead to the formation of Majorana states in a quantum wire located nearby an s-wave superconductor [1-3]. We investigate theoretically the formation of Majorana fermions in quasi-one-dimensional quantum wires in the presence of different kinds of helical-like magnetic textures. Our calculations reveal that finite-size effects as well as non-trivial helical magnetic configurations can lead to the formation of multiple Majorana states. Under some specific conditions it is possible to localize some of the Majorana modes at certain points within the wire. The positions of these points can be controlled by properly tuning the system parameters. The conditions governing the transitions from the trivial to the topological state are found and the different Majorana modes are characterized according to their local spin polarization. The possibility of experimentally realizing the proposed models in actual physical systems is also discussed.\\[4pt] [1] M. Kjaergaard, K. W\"olms, and K. Flensberg, Phys. Rev. B 85, 020503(R) (2012).\\[0pt] [2] J. Klinovaja, P. Stano, and A. Yazdani, and D. Loss, Phys. Rev. Lett. 111, 186805 (2013).\\[0pt] [3] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Science 346, 602 (2014).   

Interplay of Majorana and Kondo modes in an interacting quantum dot coupled to a topological quantum wire

We investigate the low-temperature conductance of an interacting quantum dot (QD) coupled to a topological quantum wire. Our realistic model includes the Rashba spin-orbit interaction, proximity $s$-wave superconductivity, an applied magnetic field, and an Anderson-type interacting QD. Using recursive Green's function techniques we find a QD conductance of $0.5e^2/h$, associated with the emergent Majorana end mode in the wire ``leaking'' into the QD. This signature is robust, appearing in the presence of Zeeman fields and even when the QD is deep in the Coulomb blockade. We further study the Kondo regime using the numerical renormalization group. Our results indicate a strong interplay between Majorana \& Kondo physics. The Kondo effect can be quenched by Zeeman fields, revealing a persistent $0.5e^2/h$ conductance coming from the Majorana mode leaking into the QD. These properties can be used for the experimental identification of Majorana-Kondo physics in these systems. Our results here corroborate and extend those of [${\it Phys. Rev. B}$ $\textbf{89}$, 165314 (2013)] by showing that the Majorana resonance pinned to the Fermi level arising in the QD is robust and survives even in the presence of Coulomb effects within the QD.   

Topological superconductivity and Majorana fermions in chains of magnetic atoms on the surface of a superconductor

Chain of magnetic atoms on the surface of a BCS superconductor is a versatile platform for the realization of one-dimensional superconductors with Majorana bound states that lends itself to high-resolution scanning tunneling microscopy studies [1,2]. In this talk, I will describe experimental efforts to realize this platform using self-assembled chains of Fe atoms on the surface of Pb (110) and to directly visualize Majorana quasi-particle bound states at their edges [2]. Using spin-polarized STM studies, we show that Fe chains are ferromagnetic while tunneling into Pb's substrate demonstrates signatures of strong spin-orbit interaction at its surface. Comparison of experimental measurements of structure and normal state electronic structure with DFT calculations suggest that these are triple zigzag chains with an odd number of band-crossings at the Fermi level. The onset of superconductivity in the Pb strongly modifies the low energy density of states of the Fe chains and induces a zero energy state at their ends. I will describe how these observations are consistent with the formation of a topological superconducting phase with Majorana edge states.\\[4pt] [1] S. Nadj-Perge, I.K. Drozdov, B. A. Bernevig, and A. Yazdani, Physical Review B 88, 020407 (2013).\\[0pt] [2] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, A. Yazdani, Science 346, 6209 (2014).   

Topological phase transitions of interacting Majorana fermions in an array of vortices

We study a vortex array in a p-wave superconducting thin film that produces ladder-like lattices with interacting Majorana fermions. We construct a model Hamiltonian with parameters which are tunable via the deformation of the vortex array. We explore topological phase transitions of the system and compute phase diagrams using the density-matrix-renormalization-group method. We further investigate transport properties for experimental detection. Our results have potential application on building devices to engineer strongly correlated Majorana fermions.   

Majorana Fermions in Chiral Topological Ferromagnetic Nanowires

Motivated by a recent experiment in which zero-bias peaks have been observed in STM experiments performed on chains of magnetic atoms on a superconductor, we show that a multichannel ferromagnetic wire deposited on a spin-orbit coupled superconducting substrate can realize a non-trivial chiral topological superconducting state with Majorana bound states localized at the wire ends. The non-trivial topological state occurs for generic parameters requiring no fine tuning, at least for very large exchange spin splitting in the wire. We theoretically obtain the signatures which appear in the presence of an arbitrary number of Majorana modes in multi-wire systems incorporating the role of finite temperature, finite potential barrier at the STM tip, and finite wire length. These signatures are presented in terms of spatial profiles of STM differential conductance which clearly reveal zero energy Majorana end modes and the prediction of a multiple Majorana based fractional Josephson effect.   

Majorana bound states without topological superconductivity

Recent experimental efforts towards the detection of Majorana bound states have focused on creating the conditions for topological superconductivity. Here we demonstrate an alternative route, which achieves fully localised zero-energy Majorana bound states when a topologically trivial superconductor is opened to a normal region. The emergence of Majorana states is a consequence of non-hermitian degeneracies of the resulting open quantum system, while arbitrarily large Majorana lifetimes follow from high junction transparency and helicity of the normal side. At these degeneracies, also known as ``exceptional points,'' both the eigenvalues and the eigenstates coalesce, and acquire Majorana properties (zero-energy, self-conjugation, $4\pi$-periodic braiding...) despite the trivial band topology. Exceptional Majoranas are thus the open-system counterparts of conventional Majorana bound states, to which they are continuously connected, and exhibit all their phenomenology while not requiring topological superconductivity.   

Disorder-induced phase transitions in a quasi 1-D Majorana wire

In a strictly 1D spinless p-wave superconductor, disorder is known to induce a phase transition between a topologically nontrivial phase and a trivial insulating phase when the mean free path $l$ becomes of the order of the superconducting coherence length $\xi$. We show that, in constrast, a multichannel spinless p-wave superconductor goes through a series of phase transitions alternating between topologically trivial and nontrivial phases upon increasing the disorder strength. The number of phase transitions equals the channel number $N$ and each phase transition is accompanied by a Dyson singularity in the density of states $\nu(\varepsilon) \propto \varepsilon^{-1}|\ln\varepsilon|^{-3} $. The observed behavior is the result of an effective chiral symmetry allowing us to analytically investigate the phase boundaries and the density of states. The latter displays a power-law singularity $\nu(\varepsilon) \propto \varepsilon^{|\alpha|-1}$ for small energies $\varepsilon$ away from the critical points. Using the concept of ``superuniversality,'' we relate the exponent $\alpha$ to the wire's transport properties at zero energy and, hence, to the mean free path and the superconducting coherence length.   

Parity crossings of Shiba states and Majorana-like zero bias anomalies in hybrid superconductor-normal nanowire systems with quantum dot behavior

Although recent experiments with semiconducting nanowires are partially consistent with the the existence of Majorana bound states (MBS) at normal-superconductor junctions, other mechanisms cannot be completely ruled out. In this talk, I will focus on short nanowires with quantum dot behavior and discuss the magnetic field dependence of Shiba states [1] as well as novel Kondo features induced by the weak coupling to a normal lead [2]. Our results are based on the phase diagram of an Anderson impurity in contact with superconducting and normal-state leads. This phase diagram is obtained by means of the numerical renormalization group technique and is valid for arbitrary ratio of the superconducting gap to the Kondo temperature. Overall, we find a very rich behavior of spectral functions with zero-bias anomalies which can emerge irrespective of whether the ground state is a doublet or a singlet. This phenomenology originates from crossings of the ground state fermionic parity and mimics that of MBS. [1] E. J. H. Lee, X. Jiang, M. Houzet, R. Aguado, C. M. Lieber, and S. De Franceschi, Nature Nanotechnology, 9, 79 (2014). [2]R. Zitko, J. S. Lim, R. Lopez and R. Aguado, arXiv:1405.6084 (2014).   

Helical Majorana surface states of strongly disordered topological superconductors with time-reversal symmetry

Noncentrosymmetric superconductors with strong spin-orbit coupling and the B phase of ${}^3$He are possible realizations of topological superconductors with time-reversal symmetry. The nontrivial topology manifests itself at the material's surface in terms of linearly dispersive helical Majorana modes protected by symmetry from disorder weaker than the superconducting gap. Using extensive numerical simulations, we investigate the stability and properties of these Majorana states under strong surface disorder, which influences both bulk and surface states. A critical crossover from weak to strong disorder is observed in both two and three dimensions, through which an extended state exactly at zero energy always persists. The localization properties of the ingap states are studied through the distribution of the local density of states and level repulsion statistics.   

Floquet Majorana Modes in Graphene Quantum Dots

We propose a possible way to realize Floquet Majorana fermions in graphene quantum dots connected by a superconducting island. The effective crossed Andreev reflection and hopping amplitudes between the dots are calculated as a function of system parameters. It is shown that the spin degeneracy is broken when the dots are driven out of phase. This all-electric, highly tunable device could be a realistic platform for uncovering dynamically generated Majorana fermions in graphene system.   

Exploring signatures of Majorana fermions in Fe chains on Pb using a dilution refrigerator scanning tunneling microscope

Recently, it was shown that Majorana fermions can be realized by depositing chains of magnetic atoms on a superconductor if the spins are non-collinear or if they are ferromagnetic and subject to large spin-orbit coupling. Scanning tunneling microscopy (STM) studies of Fe chains on a Pb(110) surface revealed zero-energy states localized to the ends, and a spin-polarized tip was used to show ferromagnetism in the chains as well as spin-orbit coupling on the Pb surface, providing strong evidence for Majorana fermions. These measurements were performed at 1.4 K, and the width of the zero-energy mode was limited by thermal broadening. In this talk, I will present subsequent measurements in which we extend these results to temperatures below 250 mK using a dilution refrigerator STM. The low temperature allows us to obtain significantly higher energy resolution, and I will discuss our results in the context of Majorana fermions.