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Leumer N, Marganska M, Muralidharan B, Grifoni M. Exact eigenvectors and eigenvalues of the finite Kitaev chain and its topological properties. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 32:445502. [PMID: 32320961 DOI: 10.1088/1361-648x/ab8bf9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/24/2020] [Accepted: 04/22/2020] [Indexed: 06/11/2023]
Abstract
We present a comprehensive, analytical treatment of the finite Kitaev chain for arbitrary chemical potential and chain length. By means of an exact analytical diagonalization in the real space, we derive the momentum quantization conditions and present exact analytical formulas for the resulting energy spectrum and eigenstate wave functions, encompassing boundary and bulk states. In accordance with an analysis based on the winding number topological invariant, and as expected from the bulk-edge correspondence, the boundary states are topological in nature. They can have zero, exponentially small or even finite energy. Further, for a fixed value of the chemical potential, their properties are ruled by the ratio of the decay length to the chain length. A numerical analysis confirms the robustness of the topological states against disorder.
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Affiliation(s)
- Nico Leumer
- Institute for Theoretical Physics, University of Regensburg, 93053 Regensburg, Germany
| | - Magdalena Marganska
- Institute for Theoretical Physics, University of Regensburg, 93053 Regensburg, Germany
| | - Bhaskaran Muralidharan
- Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India
| | - Milena Grifoni
- Institute for Theoretical Physics, University of Regensburg, 93053 Regensburg, Germany
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He QL, Pan L, Stern AL, Burks EC, Che X, Yin G, Wang J, Lian B, Zhou Q, Choi ES, Murata K, Kou X, Chen Z, Nie T, Shao Q, Fan Y, Zhang SC, Liu K, Xia J, Wang KL. RETRACTED: Chiral Majorana fermion modes in a quantum anomalous Hall insulator-superconductor structure. Science 2017; 357:294-299. [PMID: 28729508 DOI: 10.1126/science.aag2792] [Citation(s) in RCA: 130] [Impact Index Per Article: 18.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/02/2016] [Revised: 02/26/2017] [Accepted: 06/09/2017] [Indexed: 01/14/2023]
Abstract
Majorana fermion is a hypothetical particle that is its own antiparticle. We report transport measurements that suggest the existence of one-dimensional chiral Majorana fermion modes in the hybrid system of a quantum anomalous Hall insulator thin film coupled with a superconductor. As the external magnetic field is swept, half-integer quantized conductance plateaus are observed at the locations of magnetization reversals, giving a distinct signature of the Majorana fermion modes. This transport signature is reproducible over many magnetic field sweeps and appears at different temperatures. This finding may open up an avenue to control Majorana fermions for implementing robust topological quantum computing.
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Affiliation(s)
- Qing Lin He
- Department of Electrical and Computer Engineering, Department of Physics, and Department of Materials Science and Engineering, University of California, Los Angeles, CA 90095, USA.
| | - Lei Pan
- Department of Electrical and Computer Engineering, Department of Physics, and Department of Materials Science and Engineering, University of California, Los Angeles, CA 90095, USA
| | - Alexander L Stern
- Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA
| | - Edward C Burks
- Physics Department, University of California, Davis, CA 95616, USA
| | - Xiaoyu Che
- Department of Electrical and Computer Engineering, Department of Physics, and Department of Materials Science and Engineering, University of California, Los Angeles, CA 90095, USA
| | - Gen Yin
- Department of Electrical and Computer Engineering, Department of Physics, and Department of Materials Science and Engineering, University of California, Los Angeles, CA 90095, USA
| | - Jing Wang
- State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200433, China.,Department of Physics, Stanford University, Stanford, CA 94305, USA
| | - Biao Lian
- Department of Physics, Stanford University, Stanford, CA 94305, USA
| | - Quan Zhou
- Department of Physics, Stanford University, Stanford, CA 94305, USA
| | - Eun Sang Choi
- National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310-3706, USA
| | - Koichi Murata
- Department of Electrical and Computer Engineering, Department of Physics, and Department of Materials Science and Engineering, University of California, Los Angeles, CA 90095, USA
| | - Xufeng Kou
- Department of Electrical and Computer Engineering, Department of Physics, and Department of Materials Science and Engineering, University of California, Los Angeles, CA 90095, USA. .,School of Information Science and Technology, ShanghaiTech University, Shanghai 200031, China
| | - Zhijie Chen
- Physics Department, University of California, Davis, CA 95616, USA
| | - Tianxiao Nie
- Department of Electrical and Computer Engineering, Department of Physics, and Department of Materials Science and Engineering, University of California, Los Angeles, CA 90095, USA
| | - Qiming Shao
- Department of Electrical and Computer Engineering, Department of Physics, and Department of Materials Science and Engineering, University of California, Los Angeles, CA 90095, USA
| | - Yabin Fan
- Department of Electrical and Computer Engineering, Department of Physics, and Department of Materials Science and Engineering, University of California, Los Angeles, CA 90095, USA
| | - Shou-Cheng Zhang
- Department of Physics, Stanford University, Stanford, CA 94305, USA.
| | - Kai Liu
- Physics Department, University of California, Davis, CA 95616, USA
| | - Jing Xia
- Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA
| | - Kang L Wang
- Department of Electrical and Computer Engineering, Department of Physics, and Department of Materials Science and Engineering, University of California, Los Angeles, CA 90095, USA. .,King Abdulaziz City for Science and Technology (KACST), Center of Excellence in Green Nanotechnology, Riyadh, Saudi Arabia
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Liu XP, Zhou Y, Wang YF, Gong CD. Multifarious topological quantum phase transitions in two-dimensional topological superconductors. Sci Rep 2016; 6:28471. [PMID: 27329219 PMCID: PMC4916467 DOI: 10.1038/srep28471] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/29/2016] [Accepted: 06/03/2016] [Indexed: 11/29/2022] Open
Abstract
We study the two-dimensional topological superconductors of spinless fermions in a checkerboard-lattice Chern-insulator model. With the short-range p-wave superconducting pairing, multifarious topological quantum phase transitions have been found and several phases with high Chern numbers have been observed. We have established a rich phase diagram for these topological superconducting states. A finite-size checkerboard-lattice cylinder with a harmonic trap potential has been further investigated. Based upon the self-consistent numerical calculations of the Bogoliubov-de Gennes equations, various phase transitions have also been identified at different regions of the system. Multiple pairs of Majorana fermions are found to be well-separated and localized at the phase boundaries between the phases characterized by different Chern numbers.
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Affiliation(s)
- Xiao-Ping Liu
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China
| | - Yuan Zhou
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China.,Condensed Matter Physics and Material Science Department, Brookhaven National Loboratory, Upton, New York 11973, USA
| | - Yi-Fei Wang
- Center for Statistical and Theoretical Condensed Matter Physics and Department of Physics, Zhejiang Normal University, Jinhua 321004, China
| | - Chang-De Gong
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China.,Center for Statistical and Theoretical Condensed Matter Physics and Department of Physics, Zhejiang Normal University, Jinhua 321004, China
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Wu WH, Zhu KD. Sensitive detection of Majorana fermions based on a hybrid spin-microcantilever via enhanced spin resonance spectrum. NANOTECHNOLOGY 2015; 26:195501. [PMID: 25895653 DOI: 10.1088/0957-4484/26/19/195501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Motivated by recent experimental progress towards the detection and manipulation of Majorana fermions in ferromagnetic atomic chains on a superconductor, we present a novel proposal based on a single-crystal diamond (SCD) microcantilever with a single nitrogen-vacancy (NV) center spin embedded in ultrapure diamond substrate to probe Majorana fermions in an all-optical domain. With this scheme, a possible distinct Majorana signature is investigated via the electron spin resonance spectrum. In the proposal, the SCD microcantilever behaves as a phonon cavity and is robust for detecting of Majorana fermions, while the NV center spin can be considered as a sensitive probe. Further, the vibration of the microcantilever will enhance the coupling effect, which makes the Majorana fermions more sensitive to detection and the well-established optical NV spin readout technology will certainly promote the detection. This proposed method may provide a potential supplement for the detection of Majorana fermions.
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Affiliation(s)
- Wen-Hao Wu
- Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Department of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, People's Republic of China
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