1
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Majek P, Weymann I. Spin-selective transport in a correlated double quantum dot-Majorana wire system. Sci Rep 2024; 14:17762. [PMID: 39085311 PMCID: PMC11291930 DOI: 10.1038/s41598-024-66478-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2024] [Accepted: 07/01/2024] [Indexed: 08/02/2024] Open
Abstract
In this work we investigate the spin-dependent transport through a double quantum dot embedded in a ferromagnetic tunnel junction and side attached to a topological superconducting nanowire hosting Majorana zero-energy modes. We focus on the transport regime when the Majorana mode leaks into the double quantum dot competing with the two-stage Kondo effect and the ferromagnetic-contact-induced exchange field. In particular, we determine the system's spectral properties and analyze the temperature dependence of the spin-resolved linear conductance by means of the numerical renormalization group method. Our study reveals unique signatures of the interplay between the spin-resolved tunneling, the Kondo effect and the Majorana modes, which are visible in the transport characteristics. In particular, we uncover a competing character of the coupling to topological superconductor and that to ferromagnetic leads, which can be observed already for very low spin polarization of the electrodes. This is signaled by an almost complete quenching of the conductance in one of the spin channels which is revealed through perfect conductance spin polarization. Moreover, we show that the conductance spin polarization can change sign depending on the magnitude of spin imbalance in the leads and strength of interaction with topological wire. Thus, our work demonstrates that even minuscule spin polarization of tunneling processes can have large impact on the transport properties of the system.
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Affiliation(s)
- Piotr Majek
- Institute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 2, 61-614, Poznań, Poland.
| | - Ireneusz Weymann
- Institute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 2, 61-614, Poznań, Poland
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2
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Wrześniewski K, Weymann I. Cross-correlations between currents and tunnel magnetoresistance in interacting double quantum dot-Majorana wire system. Sci Rep 2024; 14:7815. [PMID: 38570697 PMCID: PMC10991420 DOI: 10.1038/s41598-024-58344-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2023] [Accepted: 03/27/2024] [Indexed: 04/05/2024] Open
Abstract
We theoretically investigate the spin and charge transport properties of a double quantum dot coupled to distinct edges of the nanowire hosting Majorana zero-energy modes. The focus is on the analysis of the currents flowing through the left and right junctions and their cross-correlations. We show that the system reveals very different transport properties depending on the detuning protocol of the quantum dot energy levels. For the symmetric detuning, the current dependencies reveal only two maxima associated with resonant tunneling, and currents in the left and right arms of the system reveal weak positive cross-correlations. On the other hand, for antisymmetric detuning, the flow of electrons into drains is maximized and strongly correlated in one bias voltage direction, while for the opposite bias direction a spin blockade is predicted. Furthermore, we observe a suppression of the current cross-correlations at a highly symmetric detuning point, indicating the involvement of the Majorana zero-energy modes in the transport processes. To gain insight into the role of the spin polarization of the Majorana edge states, we analyze the spin-dependent transport characteristics by considering the relationship between the spin canting angle, which describes the coupling of the Majorana modes to the spin of the quantum dots, and the magnetic configurations of the ferromagnetic drains. Moreover, we examine the non-local zero bias anomaly in the differential conductance, detailed analysis of which revealed a specific operational mode of the device that can facilitate the identification of the Majorana presence in the quantum dot-Majorana wire system. Finally, we also consider the transport properties in different magnetic configurations of the system and discuss the behavior of the associated tunnel magnetoresistance.
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Affiliation(s)
- Kacper Wrześniewski
- Faculty of Physics, Institute of Spintronics and Quantum Information, Adam Mickiewicz University, Uniwersytetu Poznańskiego 2, 61-614, Poznan, Poland.
| | - Ireneusz Weymann
- Faculty of Physics, Institute of Spintronics and Quantum Information, Adam Mickiewicz University, Uniwersytetu Poznańskiego 2, 61-614, Poznan, Poland
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3
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van Driel D, Wang G, Bordin A, van Loo N, Zatelli F, Mazur GP, Xu D, Gazibegovic S, Badawy G, Bakkers EPAM, Kouwenhoven LP, Dvir T. Spin-filtered measurements of Andreev bound states in semiconductor-superconductor nanowire devices. Nat Commun 2023; 14:6880. [PMID: 37898657 PMCID: PMC10613242 DOI: 10.1038/s41467-023-42026-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2023] [Accepted: 09/27/2023] [Indexed: 10/30/2023] Open
Abstract
Semiconductor nanowires coupled to superconductors can host Andreev bound states with distinct spin and parity, including a spin-zero state with an even number of electrons and a spin-1/2 state with odd-parity. Considering the difference in spin of the even and odd states, spin-filtered measurements can reveal the underlying ground state. To directly measure the spin of single-electron excitations, we probe an Andreev bound state using a spin-polarized quantum dot that acts as a bipolar spin filter, in combination with a non-polarized tunnel junction in a three-terminal circuit. We observe a spin-polarized excitation spectrum of the Andreev bound state, which can be fully spin-polarized, despite strong spin-orbit interaction in the InSb nanowires. Decoupling the hybrid from the normal lead causes a current blockade, by trapping the Andreev bound state in an excited state. Spin-polarized spectroscopy of hybrid nanowire devices, as demonstrated here, is proposed as an experimental tool to support the observation of topological superconductivity.
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Affiliation(s)
- David van Driel
- QuTech and Kavli Institute of NanoScience, Delft University of Technology, 2600, GA, Delft, The Netherlands
| | - Guanzhong Wang
- QuTech and Kavli Institute of NanoScience, Delft University of Technology, 2600, GA, Delft, The Netherlands
| | - Alberto Bordin
- QuTech and Kavli Institute of NanoScience, Delft University of Technology, 2600, GA, Delft, The Netherlands
| | - Nick van Loo
- QuTech and Kavli Institute of NanoScience, Delft University of Technology, 2600, GA, Delft, The Netherlands
| | - Francesco Zatelli
- QuTech and Kavli Institute of NanoScience, Delft University of Technology, 2600, GA, Delft, The Netherlands
| | - Grzegorz P Mazur
- QuTech and Kavli Institute of NanoScience, Delft University of Technology, 2600, GA, Delft, The Netherlands
| | - Di Xu
- QuTech and Kavli Institute of NanoScience, Delft University of Technology, 2600, GA, Delft, The Netherlands
| | - Sasa Gazibegovic
- Department of Applied Physics, Eindhoven University of Technology, 5600, MB, Eindhoven, The Netherlands
| | - Ghada Badawy
- Department of Applied Physics, Eindhoven University of Technology, 5600, MB, Eindhoven, The Netherlands
| | - Erik P A M Bakkers
- Department of Applied Physics, Eindhoven University of Technology, 5600, MB, Eindhoven, The Netherlands
| | - Leo P Kouwenhoven
- QuTech and Kavli Institute of NanoScience, Delft University of Technology, 2600, GA, Delft, The Netherlands
| | - Tom Dvir
- QuTech and Kavli Institute of NanoScience, Delft University of Technology, 2600, GA, Delft, The Netherlands.
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4
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Marra P, Nigro A. Majorana/Andreev crossover and the fate of the topological phase transition in inhomogeneous nanowires. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 34:124001. [PMID: 34929683 DOI: 10.1088/1361-648x/ac44d2] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2021] [Accepted: 12/20/2021] [Indexed: 06/14/2023]
Abstract
Majorana bound states (MBS) and Andreev bound states (ABS) in realistic Majorana nanowires setups have similar experimental signatures which make them hard to distinguishing one from the other. Here, we characterize the continuous Majorana/Andreev crossover interpolating between fully-separated, partially-separated, and fully-overlapping Majorana modes, in terms of global and local topological invariants, fermion parity, quasiparticle densities, Majorana pseudospin and spin polarizations, density overlaps and transition probabilities between opposite Majorana components. We found that in inhomogeneous wires, the transition between fully-overlapping trivial ABS and nontrivial MBS does not necessarily mandate the closing of the bulk gap of quasiparticle excitations, but a simple parity crossing of partially-separated Majorana modes (ps-MM) from trivial to nontrivial regimes. We demonstrate that fully-separated and fully-overlapping Majorana modes correspond to the two limiting cases at the opposite sides of a continuous crossover: the only distinction between the two can be obtained by estimating the degree of separations of the Majorana components. This result does not contradict the bulk-edge correspondence: indeed, the field inhomogeneities driving the Majorana/Andreev crossover have a length scale comparable with the nanowire length, and therefore correspond to a nonlocal perturbation which breaks the topological protection of the MBS.
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Affiliation(s)
- Pasquale Marra
- Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan
- Department of Physics, and Research and Education Center for Natural Sciences, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan
| | - Angela Nigro
- Dipartimento di Fisica 'E. R. Caianiello', Università degli Studi di Salerno, 84084 Fisciano (Salerno), Italy
- Consiglio Nazionale delle Ricerche CNR-SPIN, UOS Salerno, 84084 Fisciano (Salerno), Italy
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5
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Sedlmayr N, Bena C. Instability of Majorana states in Shiba chains due to leakage into a topological substrate. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 34:104004. [PMID: 34879363 DOI: 10.1088/1361-648x/ac413f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2021] [Accepted: 12/08/2021] [Indexed: 06/13/2023]
Abstract
We revisit the problem of Majorana states in chains of scalar impurities deposited on a superconductor with a mixed s-wave and p-wave pairing. We also study the formation of Majorana states for magnetic impurity chains. We find that the magnetic impurity chains exhibit well-localized Majorana states when the substrate is trivial, but these states hybridize and get dissolved in the bulk when the substrate is topological. Most surprisingly, and contrary to previous predictions, the scalar impurity chain does not support fully localized Majorana states except for very small and finely tuned parameter regimes, mostly for a non-topological substrate close to the topological transition. Our results indicate that a purely p-wave or a dominant p-wave substrate are not good candidates to support either magnetic or scalar impurity topological Shiba chains.
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Affiliation(s)
- Nicholas Sedlmayr
- Institute of Physics, Maria Curie-Skłodowska University, Plac Marii Skłodowskiej-Curie 1, PL-20031 Lublin, Poland
| | - Cristina Bena
- Université Paris Saclay, CNRS, CEA, Institut de Physique Théorique, 91191, Gif-sur-Yvette, France
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6
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Hassan SA, Wu BH, Xu XF, Wang CR, Cao JC. Bending effect on the Majorana bound states in planar Josephson junctions. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:385001. [PMID: 34171850 DOI: 10.1088/1361-648x/ac0ea7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/29/2021] [Accepted: 06/25/2021] [Indexed: 06/13/2023]
Abstract
We consider the bending effect on the formation of Majorana bound states (MBSs) in planar Josephson junctions where the normal stripe is tilted in a V shape. Our results show that the MBSs remain robust for moderate bending angles. Beyond some critical angles, the degradation of MBSs can be revealed by its eigenspectrum as well as the Majorana polarization (MP). Our results show that the parameter space of bending angle for robust MBSs can be significantly enlarged by tuning the superconducting phase difference across the Josephson junction. These findings suggest that the interplay of the junction geometry and the device parameters provides richer degree of freedom in designing topological superconducting devices for future applications. The MP analysis is an indispensable tool for characterizing the Majorana states.
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Affiliation(s)
- S A Hassan
- Department of Applied Physics, Donghua University, 2999 North Renmin Road, Shanghai, 201620, People's Republic of China
| | - B H Wu
- Department of Applied Physics, Donghua University, 2999 North Renmin Road, Shanghai, 201620, People's Republic of China
| | - X F Xu
- Department of Applied Physics, Donghua University, 2999 North Renmin Road, Shanghai, 201620, People's Republic of China
| | - C R Wang
- Department of Applied Physics, Donghua University, 2999 North Renmin Road, Shanghai, 201620, People's Republic of China
| | - J C Cao
- Key Laboratory of Terahertz Solid-State Technology, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, 865 Changning Road, Shanghai 200050, People's Republic of China
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7
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Assessing Bound States in a One-Dimensional Topological Superconductor: Majorana versus Tamm. Symmetry (Basel) 2021. [DOI: 10.3390/sym13061100] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Abstract
Majorana bound states in topological superconductors have attracted intense research activity in view of applications in topological quantum computation. However, they are not the only example of topological bound states that can occur in such systems. Here, we study a model in which both Majorana and Tamm bound states compete. We show both numerically and analytically that, surprisingly, the Tamm state remains partially localized even when the spectrum becomes gapless. Despite this fact, we demonstrate that the Majorana polarization shows a clear transition between the two regimes.
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8
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Udupa A, Banerjee A, Sengupta K, Sen D. One-dimensional spin-orbit coupled Dirac system with extended s-wave superconductivity: Majorana modes and Josephson effects. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:145301. [PMID: 33470985 DOI: 10.1088/1361-648x/abdd63] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/12/2020] [Accepted: 01/19/2021] [Indexed: 06/12/2023]
Abstract
Motivated by the spin-momentum locking of electrons at the boundaries of certain topological insulators, we study a one-dimensional system of spin-orbit coupled massless Dirac electrons withs-wave superconducting pairing. As a result of the spin-orbit coupling, our model has only two kinds of linearly dispersing modes, and we take these to be right-moving spin-up and left-moving spin-down. Both lattice and continuum models are studied. In the lattice model, we find that a single Majorana zero energy mode appears at each end of a finite system provided that thes-wave pairing has an extended form, with the nearest-neighbor pairing being larger than the on-site pairing. We confirm this both numerically and analytically by calculating the winding number. We find that the continuum model also has zero energy end modes. Next we study a lattice version of a model with both Schrödinger and Dirac-like terms and find that the model hosts a topological transition between topologically trivial and non-trivial phases depending on the relative strength of the Schrödinger and Dirac terms. We then study a continuum system consisting of twos-wave superconductors with different phases of the pairing, with aδ-function potential barrier lying at the junction of the two superconductors. Remarkably, we find that the system has asingleAndreev bound state (ABS) which is localized at the junction. When the pairing phase difference crosses a multiple of 2π, an ABS touches the top of the superconducting gap and disappears, and a different state appears from the bottom of the gap. We also study the AC Josephson effect in such a junction with a voltage bias that has both a constantV0and a term which oscillates with a frequencyω. We find that, in contrast to standard Josephson junctions, Shapiro plateaus appear when the Josephson frequencyωJ= 2eV0/ℏis a rational fraction ofω. We discuss experiments which can realize such junctions.
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Affiliation(s)
- Adithi Udupa
- Center for High Energy Physics, Indian Institute of Science, Bengaluru 560012, India
| | - Abhishek Banerjee
- Center for High Energy Physics, Indian Institute of Science, Bengaluru 560012, India
- Center for Quantum Devices and Microsoft Quantum Lab Copenhagen, Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
| | - K Sengupta
- School of Physical Sciences, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
| | - Diptiman Sen
- Center for High Energy Physics, Indian Institute of Science, Bengaluru 560012, India
- Department of Physics, Indian Institute of Science, Bengaluru 560012, India
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9
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Aligia AA, Pérez Daroca D, Arrachea L. Tomography of Zero-Energy End Modes in Topological Superconducting Wires. PHYSICAL REVIEW LETTERS 2020; 125:256801. [PMID: 33416361 DOI: 10.1103/physrevlett.125.256801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2020] [Accepted: 11/19/2020] [Indexed: 06/12/2023]
Abstract
We describe the Majorana zero modes in topological hybrid superconductor-semiconductor wires with spin-orbit coupling and magnetic field, in terms of generalized Bloch coordinates φ,θ,δ. When the spin-orbit coupling and the magnetic field are perpendicular, φ and δ are universal in an appropriate coordinate system. We show how to extract the angle θ from the behavior of the Josephson current-phase relation, which enables tomography of the Majorana modes. Simple analytical expressions describe accurately the numerical results.
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Affiliation(s)
- A A Aligia
- Centro Atómico Bariloche, Comisión Nacional de Energía Atómica, 8400 Bariloche, Argentina
- Instituto Balseiro, Comisión Nacional de Energía Atómica, 8400 Bariloche, Argentina
- Consejo Nacional de Investigaciones Científicas y Técnicas, 1025 CABA, Argentina
| | - D Pérez Daroca
- Consejo Nacional de Investigaciones Científicas y Técnicas, 1025 CABA, Argentina
- Gerencia de Investigación y Aplicaciones, Comisión Nacional de Energía Atómica, 1650 San Martín, Buenos Aires, Argentina
| | - Liliana Arrachea
- Consejo Nacional de Investigaciones Científicas y Técnicas, 1025 CABA, Argentina
- International Center for Advanced Studies, Escuela de Ciencia y Tecnología and ICIFI, Universidad Nacional de San Martín, 25 de Mayo y Francia, 1650 Buenos Aires, Argentina
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10
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Fu W, Yao N, Ke SS, Guo Y, Lü HF. Leakage of Majorana bound states in an inhomogeneous topological nanowire. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 32:435602. [PMID: 32604083 DOI: 10.1088/1361-648x/aba154] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2020] [Accepted: 06/30/2020] [Indexed: 06/11/2023]
Abstract
We present an exact solution of the continuum Bogolyubov-de-Gennes Hamiltonian for Majorana bound states (MBSs) generated in a superconductor-semiconductor hybrid topological nanowire. The full energy spectra that include the band states and in-gap states could be obtained. We show that for relatively short wire length, the zero energy mode could be induced even in the topological trivial regime, which also indicates oscillatory dependence on the chemical potential. With the increase of the Zeeman field, the MBSs are almost fully spin-polarized and do not localize at the wire ends gradually. We also extend our discussion to the property of Majorana modes in an inhomogeneous nanowire, in which a local gate voltage is applied to one end of the nanowire. It is found that the local potential barrier or well could modulate the Majorana energy splitting periodically. The leakage of MBSs to the potential region is exponentially suppressed for the barrier case. A potential well could induce near-zero-energy bound states and these states merge with MBSs, leading to the delocalization of MBSs. In the potential well region, both the spin-up and spin-down components of the trivial states account for a significant proportion, which can be detected experimentally.
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Affiliation(s)
- Wei Fu
- School of Physics and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, People's Republic of China
| | - Na Yao
- School of Physics and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, People's Republic of China
- College of Optoelectronics Technology, Chengdu University of Information Technology, Chengdu 610225, People's Republic of China
| | - Sha-Sha Ke
- School of Physics and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, People's Republic of China
| | - Yong Guo
- Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, People's Republic of China
| | - Hai-Feng Lü
- School of Physics and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, People's Republic of China
- Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, People's Republic of China
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11
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Schindler F, Bradlyn B, Fischer MH, Neupert T. Pairing Obstructions in Topological Superconductors. PHYSICAL REVIEW LETTERS 2020; 124:247001. [PMID: 32639809 DOI: 10.1103/physrevlett.124.247001] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2020] [Accepted: 05/11/2020] [Indexed: 06/11/2023]
Abstract
The modern understanding of topological insulators is based on Wannier obstructions in position space. Motivated by this insight, we study topological superconductors from a position-space perspective. For a one-dimensional superconductor, we show that the wave function of an individual Cooper pair decays exponentially with separation in the trivial phase and polynomially in the topological phase. For the position-space Majorana representation, we show that the topological phase is characterized by a nonzero Majorana polarization, which captures an irremovable and quantized separation of Majorana Wannier centers from the atomic positions. We apply our results to diagnose second-order topological superconducting phases in two dimensions. Our work establishes a vantage point for the generalization of topological quantum chemistry to superconductivity.
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Affiliation(s)
- Frank Schindler
- Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
| | - Barry Bradlyn
- Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA
| | - Mark H Fischer
- Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
| | - Titus Neupert
- Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
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12
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Wang HY, Zhuang L, Gao XL, Zhao XD, Liu WM. Robust Majorana edge modes with low frequency multiple time periodic driving. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 32:355404. [PMID: 32344387 DOI: 10.1088/1361-648x/ab8ddd] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/04/2020] [Accepted: 04/28/2020] [Indexed: 06/11/2023]
Abstract
Floquet Majorana edge modes capture the topological features of periodically driven p-wave superconductors. We present a Kitaev chain with multiple time periodic driving terms. Our results demonstrate how multiple driving will affect Floquet bands in frequency space, leading to more robust Floquet Majorana edge modes against driving frequencyωin comparison with the single driving scenario. Meanwhile, we have proposed how to predict Majorana edge modes via the Zak phase of Floquet bands. Besides, in contrast to the cases with single driving term, where the constant phase can be gauged out by properly choosing the initial time, we have shown the relative phase between multiple driving can not be gauged out and will play a dominant role in deciding topological phase transitions. For the sake of completeness, we also investigate the high frequency limit. Analytical results on effective Hamiltonian can be obtained via Magnus expansion and relative phase induced topological transitions can be shown explicitly.
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Affiliation(s)
- Huan-Yu Wang
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, People's Republic of China
| | - Lin Zhuang
- School of Physics, Sun Yat-Sen University, Guangzhou 510275, People's Republic of China
| | - Xian-Long Gao
- Department of Physics, Zhejiang Normal University, Jinhua 321004, People's Republic of China
| | - Xing-Dong Zhao
- College of Physics and Materials Science, Henan Normal University, Xinxiang 453007, People's Republic of China
| | - Wu-Ming Liu
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, People's Republic of China
- Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, People's Republic of China
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13
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Awoga OA, Cayao J, Black-Schaffer AM. Supercurrent Detection of Topologically Trivial Zero-Energy States in Nanowire Junctions. PHYSICAL REVIEW LETTERS 2019; 123:117001. [PMID: 31573272 DOI: 10.1103/physrevlett.123.117001] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2019] [Indexed: 06/10/2023]
Abstract
We report the emergence of zero-energy states in the trivial phase of a short nanowire junction with a strong spin-orbit coupling and magnetic field, formed by strong coupling between the nanowire and two superconductors. The zero-energy states appear in the junction when the superconductors induce a large energy shift in the nanowire, such that the junction naturally forms a quantum dot, a process that is highly tunable by the superconductor width. Most importantly, we demonstrate that the zero-energy states produce a π shift in the phase-biased supercurrent, which can be used as a simple tool for their unambiguous detection, ruling out any Majorana-like interpretation.
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Affiliation(s)
- Oladunjoye A Awoga
- Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden
| | - Jorge Cayao
- Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden
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14
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Jäck B, Xie Y, Li J, Jeon S, Bernevig BA, Yazdani A. Observation of a Majorana zero mode in a topologically protected edge channel. Science 2019; 364:1255-1259. [DOI: 10.1126/science.aax1444] [Citation(s) in RCA: 86] [Impact Index Per Article: 17.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2019] [Accepted: 06/04/2019] [Indexed: 12/23/2022]
Abstract
Superconducting proximity pairing in helical edge modes, such as those of topological insulators, is predicted to provide a unique platform for realizing Majorana zero modes (MZMs). We used scanning tunneling microscopy measurements to probe the influence of proximity-induced superconductivity and magnetism on the helical hinge states of bismuth(111) films grown on a superconducting niobium substrate and decorated with magnetic iron clusters. Consistent with model calculations, our measurements revealed the emergence of a localized MZM at the interface between the superconducting helical edge channel and the iron clusters, with a strong magnetization component along the edge. Our experiments also resolve the MZM’s spin signature, which distinguishes it from trivial in-gap states that may accidentally occur at zero energy in a superconductor.
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Affiliation(s)
- Berthold Jäck
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - Yonglong Xie
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - Jian Li
- Institute for Natural Sciences, Westlake Institute for Advanced Study, Hangzhou, Zhejiang, China
- School of Science, Westlake University, Hangzhou, China
| | - Sangjun Jeon
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - B. Andrei Bernevig
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - Ali Yazdani
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
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15
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Kobiałka A, Ptok A. Electrostatic formation of the Majorana quasiparticles in the quantum dot-nanoring structure. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2019; 31:185302. [PMID: 30703753 DOI: 10.1088/1361-648x/ab03bf] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
Zero-energy Majorana quasiparticles can be induced at the edges of low dimensional systems. Non-Abelian statistics of these states make them valid candidates for the realisation of topological quantum computer. From the practical point of view, it is crucial to obtain a system in which an on demand creation and manipulation of this type of bound states is feasible. In this article, we show such a possibility in a setup comprising a quantum nanoring in which we specify a quantum dot region via electrostatic means. The presence of quantum dot can lead to the emergence of Andreev and Majorana bound states in the investigated system. We study the differences between those two types of bound states and the possibility of their manipulation. Moreover, the exact calculation method for spectral function has been proposed, which can be used to study the influence of bound states on the band structure of the proposed system. Using this method, it can be shown that the Majorana bound states, induced at the edge of the system, present themselves as a dispersionless zero-energy flat-band.
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Affiliation(s)
- Aksel Kobiałka
- Institute of Physics, Maria Curie-Skłodowska University, Plac Marii Skłodowskiej-Curie 1, PL-20031 Lublin, Poland
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16
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Volpez Y, Loss D, Klinovaja J. Second-Order Topological Superconductivity in π-Junction Rashba Layers. PHYSICAL REVIEW LETTERS 2019; 122:126402. [PMID: 30978045 DOI: 10.1103/physrevlett.122.126402] [Citation(s) in RCA: 24] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2018] [Indexed: 06/09/2023]
Abstract
We consider a Josephson junction bilayer consisting of two tunnel-coupled two-dimensional electron gas layers with Rashba spin-orbit interaction, proximitized by a top and bottom s-wave superconductor with phase difference ϕ close to π. We show that, in the presence of a finite weak in-plane Zeeman field, the bilayer can be driven into a second order topological superconducting phase, hosting two Majorana corner states (MCSs). If ϕ=π, in a rectangular geometry, these zero-energy bound states are located at two opposite corners determined by the direction of the Zeeman field. If the phase difference ϕ deviates from π by a critical value, one of the two MCSs gets relocated to an adjacent corner. As the phase difference ϕ increases further, the system becomes trivially gapped. The obtained MCSs are robust against static and magnetic disorder. We propose two setups that could realize such a model: one is based on controlling ϕ by magnetic flux, the other involves an additional layer of randomly oriented magnetic impurities responsible for the phase shift of π in the proximity-induced superconducting pairing.
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Affiliation(s)
- Yanick Volpez
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
| | - Daniel Loss
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
| | - Jelena Klinovaja
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
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17
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Zhang K, Zeng J, Dong X, Cheng Q. Spin dependence of crossed Andreev reflection and electron tunneling induced by Majorana fermions. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:505302. [PMID: 30474619 DOI: 10.1088/1361-648x/aaedf6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
We investigate spin dependence of the nonlocal transport induced by Majorana fermions in a one-dimensional ferromagnet-ferromagnetic-insulator-superconductor-ferromagnetic-insulator-ferromagnet junction on the edge of a two-dimensional topological insulator. The results show that coupled Majorana fermions lead to the nonlocal transport processes including electron tunneling and crossed Andreev reflection, which can be tuned by adjusting the spin polarizations of the Majorana fermions. By manipulating the bands in the two ferromagnets, the nonlocal transport can be selected as either pure electron tunneling or pure crossed Andreev reflection, the transmission probability of which could be 100%. Furthermore, the pure electron tunneling and the pure crossed Andreev reflection are well controlled by the spin directions of the electron states in the two ferromagnets.
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Affiliation(s)
- Kunhua Zhang
- ICQD, Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China
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18
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Ptok A, Cichy A, Domański T. Quantum engineering of Majorana quasiparticles in one-dimensional optical lattices. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:355602. [PMID: 30051875 DOI: 10.1088/1361-648x/aad659] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
We propose a feasible way of engineering Majorana-type quasiparticles in ultracold fermionic gases on a one-dimensional (1D) optical lattice. For this purpose, imbalanced ultracold atoms interacting by the spin-orbit coupling should be hybridized with a three-dimensional Bose-Einstein condensate molecular cloud. We show that the Majorana-type excitations can be created or annihilated upon constraining the profile of a trapping potential and/or an internal scattering barier. This process is modeled within the Bogoliubov-de Gennes approach. Our study is relevant also to nanoscopic 1D superconductors, where both potentials can be imposed by electrostatic means.
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Affiliation(s)
- Andrzej Ptok
- Institute of Nuclear Physics, Polish Academy of Sciences, ul. E. Radzikowskiego 152, PL-31342 Kraków, Poland
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19
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Zazunov A, Iks A, Alvarado M, Levy Yeyati A, Egger R. Josephson effect in junctions of conventional and topological superconductors. BEILSTEIN JOURNAL OF NANOTECHNOLOGY 2018; 9:1659-1676. [PMID: 29977700 PMCID: PMC6009709 DOI: 10.3762/bjnano.9.158] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/30/2018] [Accepted: 04/27/2018] [Indexed: 06/08/2023]
Abstract
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.
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Affiliation(s)
- Alex Zazunov
- Institut für Theoretische Physik, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany
| | - Albert Iks
- Institut für Theoretische Physik, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany
| | - Miguel Alvarado
- Departamento de Física Teórica de la Materia Condensada C-V, Condensed Matter Physics Center (IFIMAC) and Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
| | - Alfredo Levy Yeyati
- Departamento de Física Teórica de la Materia Condensada C-V, Condensed Matter Physics Center (IFIMAC) and Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
| | - Reinhold Egger
- Institut für Theoretische Physik, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany
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20
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Głodzik S, Kobiałka A, Gorczyca-Goraj A, Ptok A, Górski G, Maśka MM, Domański T. Interplay between pairing and correlations in spin-polarized bound states. BEILSTEIN JOURNAL OF NANOTECHNOLOGY 2018; 9:1370-1380. [PMID: 29977671 PMCID: PMC6009651 DOI: 10.3762/bjnano.9.129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/15/2017] [Accepted: 04/09/2018] [Indexed: 06/08/2023]
Abstract
We investigate single and multiple defects embedded in a superconducting host, studying the interplay between the proximity-induced pairing and interactions. We explore the influence of the spin-orbit coupling on energies, polarization and spatial patterns of the bound (Yu-Shiba-Rusinov) states of magnetic impurities in a two-dimensional square lattice. We also address the peculiar bound states in the proximitized Rashba chain, resembling the Majorana quasiparticles, focusing on their magnetic polarization that has been recently reported by S. Jeon et al. (Science2017,358, 772). Finally, we study leakage of these polarized Majorana quasiparticles into side-attached nanoscopic regions and confront them with the subgap Kondo effect near to the singlet-doublet phase transition.
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Affiliation(s)
- Szczepan Głodzik
- Institute of Physics, M. Curie-Skłodowska University, 20-031 Lublin, Poland
| | - Aksel Kobiałka
- Institute of Physics, M. Curie-Skłodowska University, 20-031 Lublin, Poland
| | | | - Andrzej Ptok
- Institute of Nuclear Physics, Polish Academy of Sciences, 31-342 Kraków, Poland
| | - Grzegorz Górski
- Faculty of Mathematics and Natural Sciences, University of Rzeszów, 35-310 Rzeszów, Poland
| | - Maciej M Maśka
- Institute of Physics, University of Silesia, 41-500 Chorzów, Poland
| | - Tadeusz Domański
- Institute of Physics, M. Curie-Skłodowska University, 20-031 Lublin, Poland
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21
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Reeg C, Loss D, Klinovaja J. Proximity effect in a two-dimensional electron gas coupled to a thin superconducting layer. BEILSTEIN JOURNAL OF NANOTECHNOLOGY 2018; 9:1263-1271. [PMID: 29765804 PMCID: PMC5942388 DOI: 10.3762/bjnano.9.118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/27/2018] [Accepted: 03/23/2018] [Indexed: 06/08/2023]
Abstract
There have recently been several experiments studying induced superconductivity in semiconducting two-dimensional electron gases that are strongly coupled to thin superconducting layers, as well as probing possible topological phases supporting Majorana bound states in such setups. We show that a large band shift is induced in the semiconductor by the superconductor in this geometry, thus making it challenging to realize a topological phase. Additionally, we show that while increasing the thickness of the superconducting layer reduces the magnitude of the band shift, it also leads to a more significant renormalization of the semiconducting material parameters and does not reduce the challenge of tuning into a topological phase.
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Affiliation(s)
- Christopher Reeg
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
| | - Daniel Loss
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
| | - Jelena Klinovaja
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
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22
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Polarization of the Majorana quasiparticles in the Rashba chain. Sci Rep 2017; 7:16193. [PMID: 29170399 PMCID: PMC5701041 DOI: 10.1038/s41598-017-16323-3] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/07/2017] [Accepted: 11/09/2017] [Indexed: 11/17/2022] Open
Abstract
We demonstrate that the selective equal–spin Andreev reflection (SESAR) spectroscopy can be used in STM experiments to distinguish the zero–energy Majorana quasiparticles from the ordinary fermionic states of the Rashba chain. Such technique, designed for probing the p–wave superconductivity, could be applied to the intersite pairing of equal–spin electrons in the chain of magnetic Fe atoms deposited on the superconducting Pb substrate. Our calculations of the effective pairing amplitude for individual spin components imply the magnetically polarized Andreev conductance, which can be used to ‘filter’ the Majorana quasiparticles from the ordinary in–gap states, although the pure spin current (i.e., perfect polarization) is impossible.
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23
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Jeon S, Xie Y, Li J, Wang Z, Bernevig BA, Yazdani A. Distinguishing a Majorana zero mode using spin-resolved measurements. Science 2017; 358:772-776. [DOI: 10.1126/science.aan3670] [Citation(s) in RCA: 153] [Impact Index Per Article: 21.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/03/2017] [Accepted: 09/29/2017] [Indexed: 12/13/2022]
Abstract
One-dimensional topological superconductors host Majorana zero modes (MZMs), the nonlocal property of which could be exploited for quantum computing applications. We use spin-polarized scanning tunneling microscopy to show that MZMs realized in self-assembled Fe chains on the surface of Pb have a spin polarization that exceeds that stemming from the magnetism of these chains. This feature, captured by our model calculations, is a direct consequence of the nonlocality of the Hilbert space of MZMs emerging from a topological band structure. Our study establishes spin-polarization measurements as a diagnostic tool to distinguish topological MZMs from trivial in-gap states of a superconductor.
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Affiliation(s)
- Sangjun Jeon
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - Yonglong Xie
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - Jian Li
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
- Institute for Natural Sciences, Westlake Institute for Advanced Study, Hangzhou, Zhejiang, China
- Westlake University, Hangzhou, Zhejiang, China
| | - Zhijun Wang
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - B. Andrei Bernevig
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
| | - Ali Yazdani
- Joseph Henry Laboratories and Department of Physics, Princeton University, Princeton, NJ 08544, USA
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24
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Ruby M, Heinrich BW, Peng Y, von Oppen F, Franke KJ. Exploring a Proximity-Coupled Co Chain on Pb(110) as a Possible Majorana Platform. NANO LETTERS 2017; 17:4473-4477. [PMID: 28640633 PMCID: PMC5515507 DOI: 10.1021/acs.nanolett.7b01728] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/24/2017] [Revised: 06/21/2017] [Indexed: 05/21/2023]
Abstract
Linear chains of magnetic atoms proximity coupled to an s-wave superconductor are predicted to host Majorana zero modes at the chain ends in the presence of strong spin-orbit coupling. Specifically, iron (Fe) chains on Pb(110) have been explored as a possible system to exhibit topological superconductivity and host Majorana zero modes [ Nadj-Perge , S. et al., Science 2014 , 346 , 602 - 607 ]. Here, we study chains of the transition metal cobalt (Co) on Pb(110) and check for topological signatures. Using spin-polarized scanning tunneling spectroscopy, we resolve ferromagnetic order in the d bands of the chains. Interestingly, also the subgap Yu-Shiba-Rusinov (YSR) bands carry a spin polarization as was predicted decades ago. Superconducting tips allow us to resolve further details of the YSR bands and in particular resonances at zero energy. We map the spatial distribution of the zero-energy signal and find it delocalized along the chain. Hence, despite the ferromagnetic coupling within the chains and the strong spin-orbit coupling in the superconductor, we do not find clear evidence of Majorana modes. Simple tight-binding calculations suggest that the spin-orbit-split bands may cross the Fermi level four times which suppresses the zero-energy modes.
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Affiliation(s)
- Michael Ruby
- Fachbereich
Physik, Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
| | - Benjamin W. Heinrich
- Fachbereich
Physik, Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
| | - Yang Peng
- Fachbereich
Physik, Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
| | - Felix von Oppen
- Fachbereich
Physik, Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
| | - Katharina J. Franke
- Fachbereich
Physik, Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
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25
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Barański J, Kobiałka A, Domański T. Spin-sensitive interference due to Majorana state on the interface between normal and superconducting leads. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:075603. [PMID: 28035092 DOI: 10.1088/1361-648x/aa5214] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
We investigate the subgap spectrum and transport properties of the quantum dot on the interface between the metallic and superconducting leads and additionally side-coupled to the edge of the topological superconducting (TS) chain, hosting the Majorana quasiparticle. Due to the chiral nature of the Majorana states only one spin component of the quantum dot electrons (say [Formula: see text]) is directly affected, however the proximity induced on-dot pairing transmits its influence on the opposite spin as well. We investigate the unique interferometric patterns driven by the Majorana quasiparticle that are different for each spin component. We also address the spin-sensitive interplay with the Kondo effect manifested at the same zero-energy and we come to the conclusion that quantum interferometry can unambiguously identify the Majorana quasiparticle.
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Affiliation(s)
- J Barański
- Institute of Physics, Polish Academy of Sciences, 02-668 Warsaw, Poland
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26
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Klinovaja J, Stano P, Loss D. Topological Floquet Phases in Driven Coupled Rashba Nanowires. PHYSICAL REVIEW LETTERS 2016; 116:176401. [PMID: 27176529 DOI: 10.1103/physrevlett.116.176401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2015] [Indexed: 06/05/2023]
Abstract
We consider periodically driven arrays of weakly coupled wires with conduction and valence bands of Rashba type and study the resulting Floquet states. This nonequilibrium system can be tuned into nontrivial phases such as topological insulators, Weyl semimetals, and dispersionless zero-energy edge mode regimes. In the presence of strong electron-electron interactions, we generalize these regimes to the fractional case, where elementary excitations have fractional charges e/m with m being an odd integer.
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Affiliation(s)
- Jelena Klinovaja
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
| | - Peter Stano
- RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan
- Institute of Physics, Slovak Academy of Sciences, 845 11 Bratislava, Slovakia
| | - Daniel Loss
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
- RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan
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27
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Thakurathi M, Deb O, Sen D. Majorana modes and transport across junctions of superconductors and normal metals. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:275702. [PMID: 26086479 DOI: 10.1088/0953-8984/27/27/275702] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We study Majorana modes and transport in one-dimensional systems with a p-wave superconductor (SC) and normal metal leads. For a system with an SC lying between two leads, it is known that there is a Majorana mode at the junction between the SC and each lead. If the p-wave pairing Δ changes sign or if a strong impurity is present at some point inside the SC, two additional Majorana modes appear near that point. We study the effect of all these modes on the sub-gap conductance between the leads and the SC. We derive an analytical expression as a function of Δ and the length L of the SC for the energy shifts of the Majorana modes at the junctions due to hybridization between them; the shifts oscillate and decay exponentially as L is increased. The energy shifts exactly match the location of the peaks in the conductance. Using bosonization and the renormalization group method, we study the effect of interactions between the electrons on Δ and the strengths of an impurity inside the SC or the barriers between the SC and the leads; this in turn affects the Majorana modes and the conductance. Finally, we propose a novel experimental realization of these systems, in particular of a system where Δ changes sign at one point inside the SC.
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Affiliation(s)
- Manisha Thakurathi
- Centre for High Energy Physics, Indian Institute of Science, Bengaluru 560 012, India
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28
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Haim A, Berg E, von Oppen F, Oreg Y. Signatures of majorana zero modes in spin-resolved current correlations. PHYSICAL REVIEW LETTERS 2015; 114:166406. [PMID: 25955064 DOI: 10.1103/physrevlett.114.166406] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/11/2014] [Indexed: 06/04/2023]
Abstract
We consider a normal lead coupled to a Majorana bound state. We show that the spin-resolved current correlations exhibit unique features which distinguish Majorana bound states from other low-energy resonances. In particular, the spin-up and spin-down currents from a Majorana bound state are anticorrelated at low bias voltages, and become uncorrelated at higher voltages. This behavior is independent of the exact form of coupling to the lead, and of the direction of the spin polarization. In contrast, an ordinary low-energy Andreev bound state gives rise to a positive correlation between the spin-up and spin-down currents, and this spin-resolved current-current correlation approaches a nonzero constant at high bias voltages. We discuss experimental setups in which this effect can be measured.
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Affiliation(s)
- Arbel Haim
- Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Erez Berg
- Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Felix von Oppen
- Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
| | - Yuval Oreg
- Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
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29
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He JJ, Ng TK, Lee PA, Law KT. Selective equal-spin Andreev reflections induced by Majorana fermions. PHYSICAL REVIEW LETTERS 2014; 112:037001. [PMID: 24484161 DOI: 10.1103/physrevlett.112.037001] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/10/2013] [Indexed: 06/03/2023]
Abstract
In this work, we find that Majorana fermions induce selective equal spin Andreev reflections (SESARs), in which incoming electrons with certain spin polarization in the lead are reflected as counterpropagating holes with the same spin. The spin polarization direction of the electrons of this Andreev reflected channel is selected by the Majorana fermions. Moreover, electrons with opposite spin polarization are always reflected as electrons with unchanged spin. As a result, the charge current in the lead is spin polarized. Therefore, a topological superconductor which supports Majorana fermions can be used as a novel device to create fully spin-polarized currents in paramagnetic leads. We point out that SESARs can also be used to detect Majorana fermions in topological superconductors.
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Affiliation(s)
- James J He
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
| | - T K Ng
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
| | - Patrick A Lee
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - K T Law
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
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30
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Korytár R, Schmitteckert P. Probing Majorana modes in the tunneling spectra of a resonant level. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2013; 25:475304. [PMID: 24172294 DOI: 10.1088/0953-8984/25/47/475304] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
Unambiguous identification of Majorana physics presents an outstanding problem whose solution could render topological quantum computing feasible. We develop a numerical approach to treat finite-size superconducting chains supporting Majorana modes, which is based on iterative application of a two-site Bogoliubov transformation. We demonstrate the applicability of the method by studying a resonant level attached to the superconductor subject to external perturbations. In the topological phase, we show that the spectrum of a single resonant level allows us to distinguish peaks coming from Majorana physics from the Kondo resonance.
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Affiliation(s)
- R Korytár
- Institute for Nanotechnology, Karlsruhe Institute of Technology (KIT), D-76021 Karlsruhe, Germany
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31
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Klinovaja J, Loss D. Topological edge states and fractional quantum Hall effect from umklapp scattering. PHYSICAL REVIEW LETTERS 2013; 111:196401. [PMID: 24266479 DOI: 10.1103/physrevlett.111.196401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/25/2013] [Indexed: 06/02/2023]
Abstract
We study anisotropic lattice strips in the presence of a magnetic field in the quantum Hall effect regime. At specific magnetic fields, causing resonant umklapp scattering, the system is gapped in the bulk and supports chiral edge states in close analogy to topological insulators. In electron gases with stripes, these gaps result in plateaus for the Hall conductivity exactly at the known fillings n/m (both positive integers and m odd) for the integer and fractional quantum Hall effect. For double strips, we find topological phase transitions with phases that support midgap edge states with flat dispersion. The topological effects predicted here could be tested directly in optical lattices.
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Affiliation(s)
- Jelena Klinovaja
- Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
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32
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Schnyder AP, Timm C, Brydon PMR. Edge currents as a signature of flatbands in topological superconductors. PHYSICAL REVIEW LETTERS 2013; 111:077001. [PMID: 23992077 DOI: 10.1103/physrevlett.111.077001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/21/2013] [Indexed: 06/02/2023]
Abstract
We study nondegenerate flatbands at the surfaces of noncentrosymmetric topological superconductors by exact diagonalization of Bogoliubov-de Gennes Hamiltonians. We show that these states are strongly spin polarized and acquire a chiral dispersion when placed in contact with a ferromagnetic insulator. This chiral mode carries a large edge current which displays a singular dependence on the exchange-field strength. The contribution of other edge states to the current is comparably weak. We hence propose that the observation of the edge current can serve as a test of the presence of nondegenerate flatbands.
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Stanescu TD, Tewari S. Majorana fermions in semiconductor nanowires: fundamentals, modeling, and experiment. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2013; 25:233201. [PMID: 23665894 DOI: 10.1088/0953-8984/25/23/233201] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
Abstract
After a recent series of rapid and exciting developments, the long search for the Majorana fermion-the elusive quantum entity at the border between particles and antiparticles-has produced the first positive experimental results, but is not over yet. Originally proposed by E Majorana in the context of particle physics, Majorana fermions have a condensed matter analogue in the zero-energy bound states emerging in topological superconductors. A promising route to engineering topological superconductors capable of hosting Majorana zero modes consists of proximity coupling semiconductor thin films or nanowires with strong spin-orbit interaction to conventional s-wave superconductors in the presence of an external Zeeman field. The Majorana zero mode is predicted to emerge above a certain critical Zeeman field as a zero-energy state localized near the order parameter defects, namely, vortices for thin films and wire ends for the nanowire. These Majorana bound states are expected to manifest non-Abelian quantum statistics, which makes them ideal building blocks for fault-tolerant topological quantum computation. This review provides an update on the current status of the search for Majorana fermions in semiconductor nanowires by focusing on the recent developments, in particular the period following the first reports of experimental signatures consistent with the realization of Majorana bound states in semiconductor nanowire-superconductor hybrid structures. We start with a discussion of the fundamental aspects of the subject, followed by considerations on the realistic modeling, which is a critical bridge between theoretical predictions based on idealized conditions and the real world, as probed experimentally. The last part is dedicated to a few intriguing issues that were brought to the fore by the recent encouraging experimental advances.
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Affiliation(s)
- T D Stanescu
- Department of Physics, West Virginia University, Morgantown, WV 26506, USA.
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Ren CD, Yang YH, Wang J. Supercurrent modulated by magnetization in a composite topological superconductor junction. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2013; 25:205702. [PMID: 23628709 DOI: 10.1088/0953-8984/25/20/205702] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
We report a theoretical study of the supercurrent flowing in a composite topological superconductor (TS) junction, where a magnetized quantum dot is considered in between the two TS leads. Based on the tight-binding model and the Keldysh Green's function method, we found that the magnetization (m) in the dot can significantly modify the current-phase relationship, and a nonzero supercurrent flows at zero superconducting phase gradient when m has a component perpendicular to both the external magnetic field h and the pseudomagnetic field Bsoc from the spin-orbit coupling. The induced zero-phase supercurrent can be modulated substantially by both the magnitude and the direction of m, and takes its maximum value at the critical magnetic field of the topological phase transition. Our findings might provide an experimentally simple and accessible scheme for the detection of Majorana fermions.
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Affiliation(s)
- C D Ren
- Department of Physics, Southeast University, Nanjing 210096, People's Republic of China
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DeGottardi W, Sen D, Vishveshwara S. Majorana fermions in superconducting 1D systems having periodic, quasiperiodic, and disordered potentials. PHYSICAL REVIEW LETTERS 2013; 110:146404. [PMID: 25167015 DOI: 10.1103/physrevlett.110.146404] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2012] [Indexed: 05/22/2023]
Abstract
We present a unified study of the effect of periodic, quasiperiodic, and disordered potentials on topological phases that are characterized by Majorana end modes in one-dimensional p-wave superconducting systems. We define a topological invariant derived from the equations of motion for Majorana modes and, as our first application, employ it to characterize the phase diagram for simple periodic structures. Our general result is a relation between the topological invariant and the normal state localization length. This link allows us to leverage the considerable literature on localization physics and obtain the topological phase diagrams and their salient features for quasiperiodic and disordered systems for the entire region of parameter space.
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Affiliation(s)
- Wade DeGottardi
- Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, USA
| | - Diptiman Sen
- Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India
| | - Smitha Vishveshwara
- Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, USA
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Perfetto E. Dynamical formation and manipulation of Majorana fermions in driven quantum wires in contact with a superconductor. PHYSICAL REVIEW LETTERS 2013; 110:087001. [PMID: 23473188 DOI: 10.1103/physrevlett.110.087001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2012] [Indexed: 06/01/2023]
Abstract
Controlling the dynamics of Majorana fermions (MF) subject to time-varying driving fields is of fundamental importance for the practical realization of topological quantum computing. In this work we study how it is possible to dynamically generate and maintain the topological phase in cold-atom nanowires after the temporal variation of the Hamiltonian parameters. Remarkably we show that for a sudden quench the system can never relax toward a state exhibiting fully developed MF, independently of the initial and final Hamiltonians. Only for sufficiently slow protocols the system behaves adiabatically, and the topological phase can be reached. Finally we address the crucial question of how "adiabatic" a protocol must be in order to manipulate the MF inside the topological phase without deteriorating their Majorana character.
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Affiliation(s)
- E Perfetto
- Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Rome, Italy
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Alicea J. New directions in the pursuit of Majorana fermions in solid state systems. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2012; 75:076501. [PMID: 22790778 DOI: 10.1088/0034-4885/75/7/076501] [Citation(s) in RCA: 479] [Impact Index Per Article: 39.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
Abstract
The 1937 theoretical discovery of Majorana fermions-whose defining property is that they are their own anti-particles-has since impacted diverse problems ranging from neutrino physics and dark matter searches to the fractional quantum Hall effect and superconductivity. Despite this long history the unambiguous observation of Majorana fermions nevertheless remains an outstanding goal. This review paper highlights recent advances in the condensed matter search for Majorana that have led many in the field to believe that this quest may soon bear fruit. We begin by introducing in some detail exotic 'topological' one- and two-dimensional superconductors that support Majorana fermions at their boundaries and at vortices. We then turn to one of the key insights that arose during the past few years; namely, that it is possible to 'engineer' such exotic superconductors in the laboratory by forming appropriate heterostructures with ordinary s-wave superconductors. Numerous proposals of this type are discussed, based on diverse materials such as topological insulators, conventional semiconductors, ferromagnetic metals and many others. The all-important question of how one experimentally detects Majorana fermions in these setups is then addressed. We focus on three classes of measurements that provide smoking-gun Majorana signatures: tunneling, Josephson effects and interferometry. Finally, we discuss the most remarkable properties of condensed matter Majorana fermions-the non-Abelian exchange statistics that they generate and their associated potential for quantum computation.
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Affiliation(s)
- Jason Alicea
- Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA
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Gangadharaiah S, Trifunovic L, Loss D. Localized end states in density modulated quantum wires and rings. PHYSICAL REVIEW LETTERS 2012; 108:136803. [PMID: 22540720 DOI: 10.1103/physrevlett.108.136803] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2011] [Indexed: 05/31/2023]
Abstract
We study finite quantum wires and rings in the presence of a charge-density wave gap induced by a periodic modulation of the chemical potential. We show that the Tamm-Shockley bound states emerging at the ends of the wire are stable against weak disorder and interactions, for discrete open chains and for continuum systems. The low-energy physics can be mapped onto the Jackiw-Rebbi equations describing massive Dirac fermions and bound end states. We treat interactions via the continuum model and show that they increase the charge gap and further localize the end states. The electrons placed in the two localized states on the opposite ends of the wire can interact via exchange interactions and this setup can be used as a double quantum dot hosting spin qubits. The existence of these states could be experimentally detected through the presence of an unusual 4π Aharonov-Bohm periodicity in the spectrum and persistent current as a function of the external flux.
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Affiliation(s)
- Suhas Gangadharaiah
- Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
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