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Lao J, Zhou T. Manipulating chiral Majorana mode with additional potential in superconductor-Chern insulator heterostructures. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2024; 36:405702. [PMID: 38955340 DOI: 10.1088/1361-648x/ad5e2c] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2024] [Accepted: 07/02/2024] [Indexed: 07/04/2024]
Abstract
We employed the self-consistent Bogoliubov-de Gennes equations to explore the states of chiral Majorana mode in quantum anomalous Hall insulators in proximity to a superconductor, leading to the development of an extensive topological phase diagram. Our investigation focused on how an additional potential affects the separation of chiral Majorana modes across different phase conditions. We substantiated our findings by examining the zero-energy Local Density of States spectrum and the probability distribution of the chiral Majorana modes. We established the universality of chiral Majorana mode separation by applying an additional potential. This finding serves as a vital resource for future endeavors aimed at controlling and detecting these particles, thereby contributing to the advancement of quantum computing and condensed matter physics.
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Affiliation(s)
- Junming Lao
- Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics, South China Normal University, Guangzhou 510006, People's Republic of China
| | - Tao Zhou
- Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics, South China Normal University, Guangzhou 510006, People's Republic of China
- Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research Institute for Physics, South China Normal University, Guangzhou 510006, People's Republic of China
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2
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Soori A. Majorana fermions in Kitaev chains side-coupled to normal metals. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2024; 36:395302. [PMID: 38906133 DOI: 10.1088/1361-648x/ad5ad3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/02/2024] [Accepted: 06/21/2024] [Indexed: 06/23/2024]
Abstract
Majorana fermions, exotic particles with potential applications in quantum computing, have garnered significant interest in condensed matter physics. The Kitaev model serves as a fundamental framework for investigating the emergence of Majorana fermions in one-dimensional systems. We explore the intriguing question of whether Majorana fermions can arise in a normal metal (NM) side-coupled to a Kitaev chain (KC) in the topologically trivial phase. Our findings reveal affirmative evidence, further demonstrating that the KC, when in the topological phase, can induce additional Majorana fermions in the neighboring NM region. Through extensive parameter analysis, we uncover the potential for zero, one, or two pairs of Majorana fermions in a KC side-coupled to an NM. Additionally, we investigate the impact of magnetic flux on the system and calculate the winding number -a topological invariant used to characterize topological phases.
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Affiliation(s)
- Abhiram Soori
- School of Physics, University of Hyderabad, Prof. C. R. Rao Road, GachibowliC. R. Rao Road, Gachibowli, Hyderabad 500046, India
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3
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Liu YB, Zhang WY, Yi TC, Li L, Liu M, You WL. Quantum criticality of generalized Aubry-André models with exact mobility edges using fidelity susceptibility. Phys Rev E 2024; 109:054123. [PMID: 38907436 DOI: 10.1103/physreve.109.054123] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2024] [Accepted: 04/23/2024] [Indexed: 06/24/2024]
Abstract
In this study, we explore the quantum critical phenomena in generalized Aubry-André models, with a particular focus on the scaling behavior at various filling states. Our approach involves using quantum fidelity susceptibility to precisely identify the mobility edges in these systems. Through a finite-size scaling analysis of the fidelity susceptibility, we are able to determine both the correlation-length critical exponent and the dynamical critical exponent at the critical point of the generalized Aubry-André model. Based on the Diophantine equation conjecture, we can determines the number of subsequences of the Fibonacci sequence and the corresponding scaling functions for a specific filling fraction, as well as the universality class. Our findings demonstrate the effectiveness of employing the generalized fidelity susceptibility for the analysis of unconventional quantum criticality and the associated universal information of quasiperiodic systems in cutting-edge quantum simulation experiments.
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Affiliation(s)
| | | | | | - Liangsheng Li
- National Key Laboratory of Scattering and Radiation, Beijing 100854, China
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4
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Hu J, Yu F, Luo A, Pan XH, Zou J, Liu X, Xu G. Chiral Topological Superconductivity in Superconductor-Obstructed Atomic Insulator-Ferromagnetic Insulator Heterostructures. PHYSICAL REVIEW LETTERS 2024; 132:036601. [PMID: 38307042 DOI: 10.1103/physrevlett.132.036601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/15/2023] [Accepted: 12/08/2023] [Indexed: 02/04/2024]
Abstract
Implementing topological superconductivity (TSC) and Majorana states (MSs) is one of the most significant and challenging tasks in both fundamental physics and topological quantum computations. In this work, taking the obstructed atomic insulator (OAI) Nb_{3}Br_{8}, s-wave superconductor (SC) NbSe_{2}, and ferromagnetic insulator (FMI) CrI_{3} as an example, we propose a new setup to realize the 2D chiral TSC and MSs in the SC/OAI/FMI heterostructure, which could avoid the subband problem effectively and has the advantage of huge Rashba spin-orbit coupling. As a result, the TSC phase can be stabilized in a wide region of chemical potential and Zeeman splitting, and four distinct TSC phases with superconducting Chern number N=-1,-2,-3, 3 can be achieved. Moreover, a 2D Bogoliubov-de Gennes Hamiltonian based on the triangular lattice of obstructed Wannier charge centers, combined with the s-wave superconductivity paring and Zeeman splitting, is constructed to understand the whole topological phase diagram analytically. These results expand the application of OAIs and pave a new way to realize the TSC and MSs with unique advantages.
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Affiliation(s)
- Jingnan Hu
- Wuhan National High Magnetic Field Center and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Fei Yu
- Wuhan National High Magnetic Field Center and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Aiyun Luo
- Wuhan National High Magnetic Field Center and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Xiao-Hong Pan
- Wuhan National High Magnetic Field Center and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Jinyu Zou
- Wuhan National High Magnetic Field Center and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Xin Liu
- Wuhan National High Magnetic Field Center and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
- Institute for Quantum Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
- Wuhan Institute of Quantum Technology, Wuhan 430206, China
| | - Gang Xu
- Wuhan National High Magnetic Field Center and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
- Institute for Quantum Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
- Wuhan Institute of Quantum Technology, Wuhan 430206, China
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5
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Gonçalves M, Amorim B, Castro EV, Ribeiro P. Critical Phase Dualities in 1D Exactly Solvable Quasiperiodic Models. PHYSICAL REVIEW LETTERS 2023; 131:186303. [PMID: 37977623 DOI: 10.1103/physrevlett.131.186303] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2022] [Accepted: 09/25/2023] [Indexed: 11/19/2023]
Abstract
We propose a solvable class of 1D quasiperiodic tight-binding models encompassing extended, localized, and critical phases, separated by nontrivial mobility edges. Limiting cases include the Aubry-André model and the models of Sriram Ganeshan, J. H. Pixley, and S. Das Sarma [Phys. Rev. Lett. 114, 146601 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.146601] and J. Biddle and S. Das Sarma [Phys. Rev. Lett. 104, 070601 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.070601]. The analytical treatment follows from recognizing these models as a novel type of fixed points of the renormalization group procedure recently proposed in Phys. Rev. B 108, L100201 (2023)10.1103/PhysRevB.108.L100201 for characterizing phases of quasiperiodic structures. Beyond known limits, the proposed class of models extends previously encountered localized-delocalized duality transformations to points within multifractal critical phases. Besides an experimental confirmation of multifractal duality, realizing the proposed class of models in optical lattices allows stabilizing multifractal critical phases and nontrivial mobility edges in an undriven system without the need for the unbounded potentials required by previous proposals.
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Affiliation(s)
- Miguel Gonçalves
- CeFEMA-LaPMET, Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal
- Centro de Física das Universidades do Minho e Porto, LaPMET, Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal
| | - Bruno Amorim
- Centro de Física das Universidades do Minho e Porto, LaPMET, University of Minho, Campus of Gualtar, 4710-057 Braga, Portugal
| | - Eduardo V Castro
- Centro de Física das Universidades do Minho e Porto, LaPMET, Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal
- Beijing Computational Science Research Center, Beijing 100193, China
| | - Pedro Ribeiro
- CeFEMA-LaPMET, Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal
- Beijing Computational Science Research Center, Beijing 100193, China
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6
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Shi YH, Liu Y, Zhang YR, Xiang Z, Huang K, Liu T, Wang YY, Zhang JC, Deng CL, Liang GH, Mei ZY, Li H, Li TM, Ma WG, Liu HT, Chen CT, Liu T, Tian Y, Song X, Zhao SP, Xu K, Zheng D, Nori F, Fan H. Quantum Simulation of Topological Zero Modes on a 41-Qubit Superconducting Processor. PHYSICAL REVIEW LETTERS 2023; 131:080401. [PMID: 37683167 DOI: 10.1103/physrevlett.131.080401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2023] [Revised: 06/29/2023] [Accepted: 07/25/2023] [Indexed: 09/10/2023]
Abstract
Quantum simulation of different exotic topological phases of quantum matter on a noisy intermediate-scale quantum (NISQ) processor is attracting growing interest. Here, we develop a one-dimensional 43-qubit superconducting quantum processor, named Chuang-tzu, to simulate and characterize emergent topological states. By engineering diagonal Aubry-André-Harper (AAH) models, we experimentally demonstrate the Hofstadter butterfly energy spectrum. Using Floquet engineering, we verify the existence of the topological zero modes in the commensurate off-diagonal AAH models, which have never been experimentally realized before. Remarkably, the qubit number over 40 in our quantum processor is large enough to capture the substantial topological features of a quantum system from its complex band structure, including Dirac points, the energy gap's closing, the difference between even and odd number of sites, and the distinction between edge and bulk states. Our results establish a versatile hybrid quantum simulation approach to exploring quantum topological systems in the NISQ era.
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Affiliation(s)
- Yun-Hao Shi
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
- Beijing Academy of Quantum Information Sciences, Beijing 100193, China
| | - Yu Liu
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Yu-Ran Zhang
- School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510640, China
- Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wako-shi, Saitama 351-0198, Japan
- Center for Quantum Computing, RIKEN, Wako-shi, Saitama 351-0198, Japan
| | - Zhongcheng Xiang
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Kaixuan Huang
- Beijing Academy of Quantum Information Sciences, Beijing 100193, China
| | - Tao Liu
- School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510640, China
| | - Yong-Yi Wang
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Jia-Chi Zhang
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Cheng-Lin Deng
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Gui-Han Liang
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Zheng-Yang Mei
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Hao Li
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Tian-Ming Li
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Wei-Guo Ma
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Hao-Tian Liu
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Chi-Tong Chen
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Tong Liu
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Ye Tian
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Xiaohui Song
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - S P Zhao
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
- Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
| | - Kai Xu
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
- Beijing Academy of Quantum Information Sciences, Beijing 100193, China
- Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
- CAS Center for Excellence in Topological Quantum Computation, UCAS, Beijing 100049, China
| | - Dongning Zheng
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
- Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
- CAS Center for Excellence in Topological Quantum Computation, UCAS, Beijing 100049, China
| | - Franco Nori
- Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wako-shi, Saitama 351-0198, Japan
- Center for Quantum Computing, RIKEN, Wako-shi, Saitama 351-0198, Japan
- Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
| | - Heng Fan
- Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
- Beijing Academy of Quantum Information Sciences, Beijing 100193, China
- Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
- CAS Center for Excellence in Topological Quantum Computation, UCAS, Beijing 100049, China
- Hefei National Laboratory, Hefei 230088, China
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7
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Mondal S, Sen D, Dutta A. Disconnected entanglement entropy as a marker of edge modes in a periodically driven Kitaev chain. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 35:085601. [PMID: 36541508 DOI: 10.1088/1361-648x/aca7f7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2022] [Accepted: 12/01/2022] [Indexed: 06/17/2023]
Abstract
We study the disconnected entanglement entropy (DEE) of a Kitaev chain in which the chemical potential is periodically modulated withδ-function pulses within the framework of Floquet theory. For this driving protocol, the DEE of a sufficiently large system with open boundary conditions turns out to be integer-quantized, with the integer being equal to the number of Majorana edge modes localized at each edge of the chain generated by the periodic driving, thereby establishing the DEE as a marker for detecting Floquet Majorana edge modes. Analyzing the DEE, we further show that these Majorana edge modes are robust against weak spatial disorder and temporal noise. Interestingly, we find that the DEE may, in some cases, also detect the anomalous edge modes which can be generated by periodic driving of the nearest-neighbor hopping, even though such modes have no topological significance and not robust against spatial disorder. We also probe the behavior of the DEE for a kicked Ising chain in the presence of an integrability breaking interaction which has been experimentally realized.
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Affiliation(s)
- Saikat Mondal
- Department of Physics, Indian Institute of Technology, Kanpur 208016, India
| | - Diptiman Sen
- Centre for High Energy Physics and Department of Physics, Indian Institute of Science, Bengaluru 560012, India
| | - Amit Dutta
- Department of Physics, Indian Institute of Technology, Kanpur 208016, India
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8
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Park JW, Do E, Shin JS, Song SK, Stetsovych O, Jelinek P, Yeom HW. Creation and annihilation of mobile fractional solitons in atomic chains. NATURE NANOTECHNOLOGY 2022; 17:244-249. [PMID: 34934195 PMCID: PMC8930762 DOI: 10.1038/s41565-021-01042-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2021] [Accepted: 11/02/2021] [Indexed: 06/04/2023]
Abstract
Localized modes in one-dimensional (1D) topological systems, such as Majonara modes in topological superconductors, are promising candidates for robust information processing. While theory predicts mobile integer and fractional topological solitons in 1D topological insulators, experiments so far have unveiled immobile, integer solitons only. Here we observe fractionalized phase defects moving along trimer silicon atomic chains formed along step edges of a vicinal silicon surface. By means of tunnelling microscopy, we identify local defects with phase shifts of 2π/3 and 4π/3 with their electronic states within the band gap and with their motions activated above 100 K. Theoretical calculations reveal the topological soliton origin of the phase defects with fractional charges of ±2e/3 and ±4e/3. Additionally, we create and annihilate individual solitons at desired locations by current pulses from the probe tip. Mobile and manipulable topological solitons may serve as robust, topologically protected information carriers in future information technology.
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Affiliation(s)
- Jae Whan Park
- Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS), Pohang, Korea
| | - Euihwan Do
- Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS), Pohang, Korea
- Department of Physics, Pohang University of Science and Technology, Pohang, Korea
| | - Jin Sung Shin
- Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS), Pohang, Korea
- Department of Physics, Pohang University of Science and Technology, Pohang, Korea
| | - Sun Kyu Song
- Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS), Pohang, Korea
| | | | - Pavel Jelinek
- Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic
| | - Han Woong Yeom
- Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS), Pohang, Korea.
- Department of Physics, Pohang University of Science and Technology, Pohang, Korea.
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9
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Topological Superconducting Transition Characterized by a Modified Real-Space-Pfaffian Method and Mobility Edges in a One-Dimensional Quasiperiodic Lattice. Symmetry (Basel) 2022. [DOI: 10.3390/sym14020371] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Abstract
A modified real-space-Pfaffian method is applied to characterize the topological superconducting transition of a one-dimensional p-wave superconductor with quasiperiodic potentials. We found that the Majorana zero-energy mode exists in the topological non-trivial phase, and its spatial distribution is localized at ends of the system, whereas in the topological trivial phase, there is no Majorana zero mode. Furthermore, we numerically found that due to the competition between the localized quasi-disorder and the extended p-wave pairing, there are mobility edges in the energy spectra. Our theoretical work enriches the research on the quasiperiodic p-wave superconducting models.
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10
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Kezilebieke S, Vaňo V, Huda MN, Aapro M, Ganguli SC, Liljeroth P, Lado JL. Moiré-Enabled Topological Superconductivity. NANO LETTERS 2022; 22:328-333. [PMID: 34978831 PMCID: PMC8759081 DOI: 10.1021/acs.nanolett.1c03856] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/06/2021] [Revised: 12/22/2021] [Indexed: 06/14/2023]
Abstract
The search for artificial topological superconductivity has been limited by the stringent conditions required for its emergence. As exemplified by the recent discoveries of various correlated electronic states in twisted van der Waals materials, moiré patterns can act as a powerful knob to create artificial electronic structures. Here, we demonstrate that a moiré pattern between a van der Waals superconductor and a monolayer ferromagnet creates a periodic potential modulation that enables the realization of a topological superconducting state that would not be accessible in the absence of the moiré. The magnetic moiré pattern gives rise to Yu-Shiba-Rusinov minibands and periodic modulation of the Majorana edge modes that we detect using low-temperature scanning tunneling microscopy (STM) and spectroscopy (STS). Moiré patterns and, more broadly, periodic potential modulations are powerful tools to overcome the conventional constraints for realizing and controlling topological superconductivity.
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Affiliation(s)
- Shawulienu Kezilebieke
- Department
of Applied Physics, Aalto University, 00076 Aalto, Finland
- Department
of Physics, Department of Chemistry, and Nanoscience Center, University of Jyväskylä, FI-40014 Jyväskylä, Finland
| | - Viliam Vaňo
- Department
of Applied Physics, Aalto University, 00076 Aalto, Finland
| | - Md N. Huda
- Department
of Applied Physics, Aalto University, 00076 Aalto, Finland
| | - Markus Aapro
- Department
of Applied Physics, Aalto University, 00076 Aalto, Finland
| | - Somesh C. Ganguli
- Department
of Applied Physics, Aalto University, 00076 Aalto, Finland
| | - Peter Liljeroth
- Department
of Applied Physics, Aalto University, 00076 Aalto, Finland
| | - Jose L. Lado
- Department
of Applied Physics, Aalto University, 00076 Aalto, Finland
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11
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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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12
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Du CX, Xu N, Du L, Zhang Y, Wu JH. Topological edge states controlled by next-nearest-neighbor coupling and Peierls phase in a P T-symmetric trimerized lattice. OPTICS EXPRESS 2021; 29:37722-37732. [PMID: 34808839 DOI: 10.1364/oe.438779] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2021] [Accepted: 10/23/2021] [Indexed: 06/13/2023]
Abstract
We study the topological features in a trimerized lattice of parity-time symmetry with comparable nearest-neighbor (NN) and next-nearest-neighbor (NNN) couplings as well as a Peierls phase. Eigen energies of four edge states in two bandgaps, of topological origin verified by the quantized total Zak phase, are surprisingly independent of the NNN coupling and the Peierls phase. Topological regions with respect to the intercell NN coupling, as the intracell NN coupling is fixed, can be extended with reinforced localization strengths for one pair of edge states but reduced with weakened localization strengths for the other pair of edge states, by increasing the NNN coupling. The partial overlapping between extended and reduced topological regions promises then a two-step phase transition of 'zero - two - four' edge states, viable to be periodically modulated by the Peierls phase.
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13
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Bandyopadhyay S, Bhattacharjee S, Sen D. Driven quantum many-body systems and out-of-equilibrium topology. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:393001. [PMID: 34225268 DOI: 10.1088/1361-648x/ac1151] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2021] [Accepted: 07/05/2021] [Indexed: 06/13/2023]
Abstract
In this review we present some of the work done in India in the area of driven and out-of-equilibrium systems with topological phases. After presenting some well-known examples of topological systems in one and two dimensions, we discuss the effects of periodic driving in some of them. We discuss the unitary as well as the non-unitary dynamical preparation of topologically non-trivial states in one and two dimensional systems. We then discuss the effects of Majorana end modes on transport through a Kitaev chain and a junction of three Kitaev chains. Following this, transport through the surface states of a three-dimensional topological insulator has also been reviewed. The effects of hybridization between the top and bottom surfaces in such systems and the application of electromagnetic radiation on a strip-like region on the top surface are described. Two unusual topological systems are mentioned briefly, namely, a spin system on a kagome lattice and a Josephson junction of three superconducting wires. We have also included a pedagogical discussion on topology and topological invariants in the appendices, where the connection between topological properties and the intrinsic geometry of many-body quantum states is also elucidated.
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Affiliation(s)
- Souvik Bandyopadhyay
- Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India
| | - Sourav Bhattacharjee
- Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India
| | - Diptiman Sen
- Center for High Energy Physics and Department of Physics, Indian Institute of Science, Bengaluru 560012, India
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Wang Y, Zhang L, Niu S, Yu D, Liu XJ. Realization and Detection of Nonergodic Critical Phases in an Optical Raman Lattice. PHYSICAL REVIEW LETTERS 2020; 125:073204. [PMID: 32857567 DOI: 10.1103/physrevlett.125.073204] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2020] [Accepted: 07/21/2020] [Indexed: 06/11/2023]
Abstract
The critical phases, being delocalized but nonergodic, are fundamental phases different from both the many-body localization and ergodic extended quantum phases, and have so far not been realized in experiment. Here we propose an incommensurate topological insulating model of AIII symmetry class to realize such critical phases through an optical Raman lattice scheme, which possesses a one-dimensional (1D) spin-orbit coupling and an incommensurate Zeeman potential. We show the existence of both noninteracting and many-body critical phases, which can coexist with the topological phase, and show that the critical-localization transition coincides with the topological phase boundary in noninteracting regime. The dynamical detection of the critical phases is proposed and studied in detail based on the available experimental techniques. Finally, we demonstrate how the proposed critical phases can be achieved within the current ultracold atom experiments. This work paves the way to observe the novel critical phases.
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Affiliation(s)
- Yucheng Wang
- Shenzhen Institute for Quantum Science and Engineering, and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
| | - Long Zhang
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
| | - Sen Niu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
| | - Dapeng Yu
- Shenzhen Institute for Quantum Science and Engineering, and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
| | - Xiong-Jun Liu
- Shenzhen Institute for Quantum Science and Engineering, and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
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15
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Habibi A, Ghadimi R, Jafari SA. Topological phase diagram of the disordered 2XY model in presence of generalized Dzyaloshinskii-Moriya interaction. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 32:015604. [PMID: 31470436 DOI: 10.1088/1361-648x/ab401c] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
The topological index of a system determines its edge physics. However, in situations such as strong disorder where due to level repulsion the spectral gap closes, the topological indices are not well-defined. In this paper, we show that the localization length of zero modes determined by the transfer matrix method reveals much more information than the topological index. The localization length can provide not only information about the topological index of the Hamiltonian itself, but it can also provide information about the topological indices of the 'relative' Hamiltonians. As a case study, we study a generalized XY model (2XY model) further augmented by a generalized Dziyaloshinskii-Moriya-like (DM) interaction parameterized by [Formula: see text] that after fermionization breaks the time-reversal invariance. The parent Hamiltonian at [Formula: see text] which belongs to the BDI class is indexed by an integer winding number while the [Formula: see text] daughter Hamiltonian which belongs to class D is specified by a Z 2 index [Formula: see text]. We show that the localization length, in addition to determining Z 2, can count the number of Majorana zero modes leftover at the boundary of the daughter Hamiltonian-which are not protected by the winding number anymore. Therefore the localization length outperforms the standard topological indices in two respects: (i) it is much faster and more accurate to calculate and (ii) it can count the winding number of the parent Hamiltonian by looking into the edges of the daughter Hamiltonian.
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Affiliation(s)
- Alireza Habibi
- Department of Physics, Sharif University of Technology, Tehran 11155-9161, Iran. Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg
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16
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Abstract
In recent years, tools from quantum information theory have become indispensable in characterizing many-body systems. In this work, we employ measures of entanglement to study the interplay between disorder and the topological phase in 1D systems of the Kitaev type, which can host Majorana end modes at their edges. We find that the entanglement entropy may actually increase as a result of disorder, and identify the origin of this behavior in the appearance of an infinite-disorder critical point. We also employ the entanglement spectrum to accurately determine the phase diagram of the system, and find that disorder may enhance the topological phase, and lead to the appearance of Majorana zero modes in systems whose clean version is trivial.
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17
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Majorana Fermions in One-Dimensional Structures at LaAlO3/SrTiO3 Oxide Interfaces. CONDENSED MATTER 2018. [DOI: 10.3390/condmat3040037] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
We study one-dimensional structures that may be formed at the LaAlO 3 /SrTiO 3 oxide interface by suitable top gating. These structures are modeled via a single-band model with Rashba spin-orbit coupling, superconductivity and a magnetic field along the one-dimensional chain. We first discuss the conditions for the occurrence of a topological superconducting phase and the related formation of Majorana fermions at the chain endpoints, highlighting a close similarity between this model and the Kitaev model, which also reflects in a similar condition the formation of a topological phase. Solving the model in real space, we also study the spatial extension of the wave function of the Majorana fermions and how this increases with approaching the limit condition for the topological state. Using a scattering matrix formalism, we investigate the stability of the Majorana fermions in the presence of disorder and discuss the evolution of the topological phase with increasing disorder.
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18
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Cai X. Phase diagram of the incommensurate off-diagonal Aubry-André model with p-wave pairing. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:345601. [PMID: 30015625 DOI: 10.1088/1361-648x/aad3ef] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Using exact numerical methods, we study the interplay between p-wave superconductivity and off-diagonal modulation in the incommensurate off-diagonal Aubry-André model with p-wave pairing. When chemical potential is zero, the modulation not only leads to an Anderson-like localization with all bulk states changing into critical ones, but also induces a topological phase transition accompanying the disappearing of exponentially localized zero-energy edge Majorana fermions. These two transitions happen at different places and true Anderson localization is absent. With a finite chemical potential, the quasi-periodicity of the modulation causes band splitting and drives the system into a band insulator phase. It has a fixed incommensurate particle density which is determined by the modulation frequency. Phase diagrams are identified, and can be tested with existing proposals for experimental realizations of the effective off-diagonal disorder and the p-wave pairing.
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Affiliation(s)
- Xiaoming Cai
- State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, People's Republic of China
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19
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Crowley PJD, Chandran A, Laumann CR. Quasiperiodic Quantum Ising Transitions in 1D. PHYSICAL REVIEW LETTERS 2018; 120:175702. [PMID: 29756831 DOI: 10.1103/physrevlett.120.175702] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2018] [Indexed: 06/08/2023]
Abstract
Unlike random potentials, quasiperiodic modulation can induce localization-delocalization transitions in one dimension. In this Letter, we analyze the implications of this for symmetry breaking in the quasiperiodically modulated quantum Ising chain. Although weak modulation is irrelevant, strong modulation induces new ferromagnetic and paramagnetic phases which are fully localized and gapless. The quasiperiodic potential and localized excitations lead to quantum criticality that is intermediate to that of the clean and randomly disordered models with exponents of ν=1^{+} (exact) and z≈1.9, Δ_{σ}≈0.16, and Δ_{γ}≈0.63 (up to logarithmic corrections). Technically, the clean Ising transition is destabilized by logarithmic wandering of the local reduced couplings. We conjecture that the wandering coefficient w controls the universality class of the quasiperiodic transition and show its stability to smooth perturbations that preserve the quasiperiodic structure of the model.
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Affiliation(s)
- P J D Crowley
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - A Chandran
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - C R Laumann
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
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20
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Nag T, Rajak A. Entanglement entropy of a three-spin-interacting spin chain with a time-reversal-breaking impurity at one boundary. Phys Rev E 2018; 97:042108. [PMID: 29758713 DOI: 10.1103/physreve.97.042108] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2017] [Indexed: 06/08/2023]
Abstract
We investigate the effect of a time-reversal-breaking impurity term (of strength λ_{d}) on both the equilibrium and nonequilibrium critical properties of entanglement entropy (EE) in a three-spin-interacting transverse Ising model, which can be mapped to a p-wave superconducting chain with next-nearest-neighbor hopping and interaction. Importantly, we find that the logarithmic scaling of the EE with block size remains unaffected by the application of the impurity term, although, the coefficient (i.e., central charge) varies logarithmically with the impurity strength for a lower range of λ_{d} and eventually saturates with an exponential damping factor [∼exp(-λ_{d})] for the phase boundaries shared with the phase containing two Majorana edge modes. On the other hand, it receives a linear correction in term of λ_{d} for an another phase boundary. Finally, we focus to study the effect of the impurity in the time evolution of the EE for the critical quenching case where the impurity term is applied only to the final Hamiltonian. Interestingly, it has been shown that for all the phase boundaries, contrary to the equilibrium case, the saturation value of the EE increases logarithmically with the strength of impurity in a certain regime of λ_{d} and finally, for higher values of λ_{d}, it increases very slowly dictated by an exponential damping factor. The impurity-induced behavior of EE might bear some deep underlying connection to thermalization.
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Affiliation(s)
- Tanay Nag
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, Dresden 01187, Germany
| | - Atanu Rajak
- Department of Physics, Jack and Pearl Resnick Institute, Bar-Ilan University, Ramat-Gan 52900, Israel
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21
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Stanescu TD, Sitek A, Manolescu A. Robust topological phase in proximitized core-shell nanowires coupled to multiple superconductors. BEILSTEIN JOURNAL OF NANOTECHNOLOGY 2018; 9:1512-1526. [PMID: 29977684 PMCID: PMC6009546 DOI: 10.3762/bjnano.9.142] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/03/2017] [Accepted: 04/13/2018] [Indexed: 05/05/2023]
Abstract
We consider core-shell nanowires with prismatic geometry contacted with two or more superconductors in the presence of a magnetic field applied parallel to the wire. In this geometry, the lowest energy states are localized on the outer edges of the shell, which strongly inhibits the orbital effects of the longitudinal magnetic field that are detrimental to Majorana physics. Using a tight-binding model of coupled parallel chains, we calculate the topological phase diagram of the hybrid system in the presence of non-vanishing transverse potentials and finite relative phases between the parent superconductors. We show that having finite relative phases strongly enhances the stability of the induced topological superconductivity over a significant range of chemical potentials and reduces the value of the critical field associated with the topological quantum phase transition.
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Affiliation(s)
- Tudor D Stanescu
- Department of Physics and Astronomy, West Virginia University, Morgantown, WV 26506, USA
| | - Anna Sitek
- Department of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wroclaw University of Science and Technology, Wroclaw, 50-370, Poland
- School of Science and Engineering, Reykjavik University, Menntavegur 1, IS-101 Reykjavik, Iceland
| | - Andrei Manolescu
- School of Science and Engineering, Reykjavik University, Menntavegur 1, IS-101 Reykjavik, Iceland
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22
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Pérez M, Martínez G. Polynomial description of inhomogeneous topological superconducting wires. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:475503. [PMID: 29035275 DOI: 10.1088/1361-648x/aa93cd] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We present the universal features of the topological invariant for p-wave superconducting wires after the inclusion of spatial inhomogeneities. Three classes of distributed potentials are studied, a single-defect, a commensurate and an incommensurate model, using periodic site modulations. An analytic polynomial description is achieved by splitting the topological invariant into two parts; one part depends on the chemical potential and the other does not. For the homogeneous case, an elliptical region is found where the topological invariant oscillates. The zeros of these oscillations occur at points where the fermion parity switches for finite wires. The increase of these oscillations with the inhomogeneity strength leads to new isolated non-topological phases. We characterize these new phases according to each class of spatial distributions. Such phases could also be observed in the XY model, to which our model is dual.
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Affiliation(s)
- Marcos Pérez
- Instituto de Física, UFRGS, 91501-970 Porto Alegre-RS, Brazil
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23
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Rajak A, Nag T. Survival probability in a quenched Majorana chain with an impurity. Phys Rev E 2017; 96:022136. [PMID: 28950615 DOI: 10.1103/physreve.96.022136] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/26/2016] [Indexed: 11/07/2022]
Abstract
We investigate the dynamics of a one-dimensional p-wave superconductor with next-nearest-neighbor hopping and superconducting interaction derived from a three-spin interacting Ising model in transverse field by mapping to Majorana fermions. The next-nearest-neighbor hopping term leads to a new topological phase containing two zero-energy Majorana modes at each end of an open chain, compared to a nearest-neighbor p-wave superconducting chain. We study the Majorana survival probability (MSP) of a particular Majorana edge state when the initial Hamiltonian (H_{i}) is changed to the quantum critical as well as off-critical final Hamiltonian (H_{f}), which additionally contains an impurity term (H_{imp}) that breaks the time-reversal invariance. For the off-critical quenching inside the new topological phase with H_{f}=H_{i}+H_{imp}, and small impurity strength (λ_{d}), we observe a perfect oscillation of the MSP as a function of time with a single frequency (determined by the impurity strength λ_{d}) that can be analyzed from an equivalent two-level problem. On the other hand, the MSP shows a beating like structure with time for quenching to the phase boundary separating the topological phase (with two edge Majoranas at each edge) and the nontopological phase where the additional frequency is given by inverse of the system size. We attribute this behavior of the MSP to the modification of the energy levels of the final Hamiltonian due to the application of the impurity term.
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Affiliation(s)
- Atanu Rajak
- CMP Division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700 064, India and Department of Physics, Jack and Pearl Resnick Institute, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Tanay Nag
- Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208 016, India
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24
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Qin W, Xiao D, Chang K, Shen SQ, Zhang Z. Disorder-induced topological phase transitions in two-dimensional spin-orbit coupled superconductors. Sci Rep 2016; 6:39188. [PMID: 27991541 PMCID: PMC5171876 DOI: 10.1038/srep39188] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2016] [Accepted: 11/02/2016] [Indexed: 11/09/2022] Open
Abstract
Normal superconductors with Rashba spin-orbit coupling have been explored as candidate systems of topological superconductors. Here we present a comparative theoretical study of the effects of different types of disorder on the topological phases of two-dimensional Rashba spin-orbit coupled superconductors. First, we show that a topologically trivial superconductor can be driven into a chiral topological superconductor upon diluted doping of isolated magnetic disorder, which close and reopen the quasiparticle gap of the paired electrons in a nontrivial manner. Secondly, the superconducting nature of a topological superconductor is found to be robust against Anderson disorder, but the topological nature is not, converting the system into a topologically trivial state even in the weak scattering limit. These topological phase transitions are distinctly characterized by variations in the topological invariant. We discuss the central findings in connection with existing experiments, and provide new schemes towards eventual realization of topological superconductors.
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Affiliation(s)
- Wei Qin
- International Center for Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for Physical Sciences at Microscale (HFNL), and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui, 230026, China
| | - Di Xiao
- Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
| | - Kai Chang
- SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P. O. Box 912, Beijing 100083, China
| | - Shun-Qing Shen
- Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China
| | - Zhenyu Zhang
- International Center for Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for Physical Sciences at Microscale (HFNL), and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui, 230026, China
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25
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Gao Y, Zhou T, Huang H, Huang R. Majorana zero modes in the hopping-modulated one-dimensional p-wave superconducting model. Sci Rep 2015; 5:17049. [PMID: 26586205 PMCID: PMC4653629 DOI: 10.1038/srep17049] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2015] [Accepted: 10/23/2015] [Indexed: 11/23/2022] Open
Abstract
We investigate the one-dimensional p-wave superconducting model with
periodically modulated hopping and show that under time-reversal symmetry, the
number of the Majorana zero modes (MZMs) strongly depends on the modulation period.
If the modulation period is odd, there can be at most one MZM. However if the period
is even, the number of the MZMs can be zero, one and two. In addition, the MZMs will
disappear as the chemical potential varies. We derive the condition for the
existence of the MZMs and show that the topological properties in this model are
dramatically different from the one with periodically modulated potential.
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Affiliation(s)
- Yi Gao
- Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 210023, China
| | - Tao Zhou
- College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China
| | - Huaixiang Huang
- Department of Physics, Shanghai University, Shanghai, 200444, China
| | - Ran Huang
- Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 210023, China
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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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27
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Rajak A, Nag T, Dutta A. Possibility of adiabatic transport of a Majorana edge state through an extended gapless region. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042107. [PMID: 25375438 DOI: 10.1103/physreve.90.042107] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2014] [Indexed: 06/04/2023]
Abstract
In the context of slow quenching dynamics of a p-wave superconducting chain, it has been shown that a Majorana edge state cannot be adiabatically transported from one topological phase to the other separated by a quantum critical line. On the other hand, the inclusion of a phase factor in the hopping term, which breaks the effective time-reversal invariance, results in an extended gapless region between two topological phases. We show that for a finite chain with an open boundary condition, there exists a nonzero probability that a Majorana edge state can be adiabatically transported from one topological phase to the other across this gapless region following a slow quench of the superconducting term; this happens for an optimum transit time, which is proportional to the system size and diverges for a thermodynamically large chain. We attribute this phenomenon to the mixing of the Majorana only with low-lying inverted bulk states. Remarkably, the Majorana state always persists with the same probability even after the quenching is stopped. For a periodic chain, on the other hand, we find a Kibble-Zurek scaling of the defect density with a renormalized rate of quenching.
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Affiliation(s)
- Atanu Rajak
- Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700 064, India
| | - Tanay Nag
- Indian Institute of Technology Kanpur, Kanpur 208 016, India
| | - Amit Dutta
- Indian Institute of Technology Kanpur, Kanpur 208 016, India
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28
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Cai X. Quantum phase transitions and phase diagram for a one-dimensional p-wave superconductor with an incommensurate potential. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2014; 26:155701. [PMID: 24675766 DOI: 10.1088/0953-8984/26/15/155701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
The effect of the incommensurate potential is studied for the one-dimensional p-wave superconductor. It is determined by analyzing various properties, such as the superconducting gap, the long-range order of the correlation function, the inverse participation ratio and the Z2 topological invariant, etc. In particular, two important aspects of the effect are investigated: (1) as disorder, the incommensurate potential destroys the superconductivity and drives the system into the Anderson localized phase; (2) as a quasi-periodic potential, the incommensurate potential causes band splitting and turns the system with certain chemical potential into the band insulator phase. A full phase diagram is also presented in the chemical potential-incommensurate potential strength plane.
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Affiliation(s)
- X Cai
- State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, People's Republic of China
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29
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Rajak A, Dutta A. Survival probability of an edge Majorana in a one-dimensional p-wave superconducting chain under sudden quenching of parameters. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:042125. [PMID: 24827211 DOI: 10.1103/physreve.89.042125] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/08/2014] [Indexed: 06/03/2023]
Abstract
We consider the temporal evolution of a zero-energy edge Majorana of a spinless p-wave superconducting chain following a sudden change of a parameter of the Hamiltonian. Starting from one of the topological phases that has an edge Majorana, the system is suddenly driven to the other topological phase or to the (topologically) trivial phases and to the quantum critical points (QCPs) separating these phases. The survival probability of the initial edge Majorana as a function of time is studied following the quench. Interestingly when the chain is quenched to the QCP, we find a nearly perfect oscillation of the survival probability, indicating that the Majorana travels back and forth between two ends, with a time period that scales with the system size. We also generalize to the situation when there is a next-nearest-neighbor hopping in a superconducting chain and there results in a pair of edge Majorana at each end of the chain in the topological phase. We show that the frequency of oscillation of the survival probability gets doubled in this case. We also perform an instantaneous quenching of the Hamiltonian (with two Majorana modes at each end of the chain) to an another Hamiltonian which has only one Majorana mode in equilibrium; the MSP shows oscillations as a function of time with a noticeable decay in the amplitude. On the other hand for a quenching which is reverse to the previous one, the MSP decays rapidly and stays close to zero with fluctuations in amplitude.
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Affiliation(s)
- Atanu Rajak
- CMP Division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700 064, India
| | - Amit Dutta
- Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India
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30
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Reynoso AA, Frustaglia D. Complex band structure eigenvalue method adapted to Floquet systems: topological superconducting wires as a case study. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2014; 26:035301. [PMID: 24353216 DOI: 10.1088/0953-8984/26/3/035301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
For systems that can be modeled as a single-particle lattice extended along a privileged direction, such as, for example, quantum wires, the so-called eigenvalue method provides full information about the propagating and evanescent modes as a function of energy. This complex band structure method can be applied either to lattices consisting of an infinite succession of interconnected layers described by the same local Hamiltonian or to superlattices: systems in which the spatial periodicity involves more than one layer. Here, for time-dependent systems subject to a periodic driving, we present an adapted version of the superlattice scheme capable of obtaining the Floquet states and the Floquet quasienergy spectrum. Within this scheme the time periodicity is treated as existing along a spatial dimension added to the original system. The solutions at a single energy for the enlarged artificial system provide the solutions of the original Floquet problem. The method is suited for arbitrary periodic excitations, including strong and anharmonic drivings. We illustrate the capabilities of the methods for both time-independent and time-dependent systems by discussing: (a) topological superconductors in multimode quantum wires with spin-orbit interaction and (b) microwave driven quantum dots in contact with a topological superconductor.
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Affiliation(s)
- Andres A Reynoso
- ARC Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, NSW 2006, Australia
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Cai X, Lang LJ, Chen S, Wang Y. Topological superconductor to Anderson localization transition in one-dimensional incommensurate lattices. PHYSICAL REVIEW LETTERS 2013; 110:176403. [PMID: 23679750 DOI: 10.1103/physrevlett.110.176403] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/25/2012] [Indexed: 06/02/2023]
Abstract
We study the competition of disorder and superconductivity for a one-dimensional p-wave superconductor in incommensurate potentials. With the increase in the strength of the incommensurate potential, the system undergoes a transition from a topological superconducting phase to a topologically trivial localized phase. The phase boundary is determined both numerically and analytically from various aspects and the topological superconducting phase is characterized by the presence of Majorana edge fermions in the system with open boundary conditions. We also calculate the topological Z2 invariant of the bulk system and find it can be used to distinguish the different topological phases even for a disordered system.
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Affiliation(s)
- Xiaoming Cai
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
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