1
|
Yang Y, Yang B, Ma G, Li J, Zhang S, Chan CT. Non-Abelian physics in light and sound. Science 2024; 383:eadf9621. [PMID: 38386745 DOI: 10.1126/science.adf9621] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2022] [Accepted: 01/17/2024] [Indexed: 02/24/2024]
Abstract
Non-Abelian phenomena arise when the sequence of operations on physical systems influences their behaviors. By possessing internal degrees of freedom such as polarization, light and sound can be subjected to various manipulations, including constituent materials, structured environments, and tailored source conditions. These manipulations enable the creation of a great variety of Hamiltonians, through which rich non-Abelian phenomena can be explored and observed. Recent developments have constituted a versatile testbed for exploring non-Abelian physics at the intersection of atomic, molecular, and optical physics; condensed matter physics; and mathematical physics. These fundamental endeavors could enable photonic and acoustic devices with multiplexing functionalities. Our review aims to provide a timely and comprehensive account of this emerging topic. Starting from the foundation of matrix-valued geometric phases, we address non-Abelian topological charges, non-Abelian gauge fields, non-Abelian braiding, non-Hermitian non-Abelian phenomena, and their realizations with photonics and acoustics and conclude with future prospects.
Collapse
Affiliation(s)
- Yi Yang
- Department of Physics, The University of Hong Kong, Pokfulam, Hong Kong, China
- HK Institute of Quantum Science and Technology, The University of Hong Kong, Pokfulam, Hong Kong, China
| | - Biao Yang
- College of Advanced Interdisciplinary Studies, National University of Defense Technology, Changsha, China
- Hunan Provincial Key Laboratory of Novel Nano-Optoelectronic Information Materials and Devices, National University of Defense Technology, Changsha, China
- Nanhu Laser Laboratory, National University of Defense Technology, Changsha, China
| | - Guancong Ma
- Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China
| | - Jensen Li
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
| | - Shuang Zhang
- Department of Physics, The University of Hong Kong, Pokfulam, Hong Kong, China
- HK Institute of Quantum Science and Technology, The University of Hong Kong, Pokfulam, Hong Kong, China
- Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China
- New Cornerstone Science Laboratory, The University of Hong Kong, Pokfulam, Hong Kong, China
| | - C T Chan
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
| |
Collapse
|
2
|
Pan XH, Chen L, Liu DE, Zhang FC, Liu X. Majorana Zero Modes Induced by the Meissner Effect at Small Magnetic Field. PHYSICAL REVIEW LETTERS 2024; 132:036602. [PMID: 38307040 DOI: 10.1103/physrevlett.132.036602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2023] [Accepted: 11/28/2023] [Indexed: 02/04/2024]
Abstract
One key difficulty in realizing Majorana zero modes (MZMs) is the required high magnetic field, which causes serious issues, e.g., shrinks the superconducting gap, reduces topological region, and weakens their robustness against disorders. In this Letter, we propose that the Meissner effect can bring the topological superconducting phase to a superconductor/topological-insulator/superconductor (SC/TI/SC) hybrid system. Remarkably, the required magnetic field strength (<10 mT) to support MZMs has been reduced by several orders of magnitude compared to that (>0.5 T) in the previous schemes. Tuning the phase difference between the top and bottom superconductors can control the number and position of the MZMs. In addition, we account for the electrostatic potential in the superconductor/topological-insulator (SC/TI) interface through the self-consistent Schrödinger-Poisson calculation, which shows the experimental accessibility of our proposal. Our proposal only needs a small magnetic field of less than 10 mT and is robust against the chemical potential fluctuation, which makes the SC/TI/SC hybrid an ideal Majorana platform.
Collapse
Affiliation(s)
- Xiao-Hong Pan
- School of Physics and Institute for Quantum Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
- Hubei Key Laboratory of Gravitation and Quantum Physics and Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
- Wuhan Institute of Quantum Technology, Wuhan, Hubei 430074, China
| | - Li Chen
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China
| | - Dong E Liu
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China
| | - Fu-Chun Zhang
- Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
| | - Xin Liu
- School of Physics and Institute for Quantum Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
- Hubei Key Laboratory of Gravitation and Quantum Physics and Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
- Wuhan Institute of Quantum Technology, Wuhan, Hubei 430074, China
| |
Collapse
|
3
|
Phan HT, Koizumi K, Liu F, Wakabayashi K. Topological edge and corner states in biphenylene photonic crystal. OPTICS EXPRESS 2024; 32:2223-2234. [PMID: 38297757 DOI: 10.1364/oe.510595] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2023] [Accepted: 12/25/2023] [Indexed: 02/02/2024]
Abstract
The biphenylene network (BPN) has a unique two-dimensional atomic structure, where hexagonal unit cells are arranged on a square lattice. Inspired by such a BPN structure, we design a counterpart in the fashion of photonic crystals (PhCs), which we refer to as the BPN PhC. We study the photonic band structure using the finite element method and characterize the topological properties of the BPN PhC through the use of the Wilson loop. Our findings reveal the emergence of topological edge states in the BPN PhC, specifically in the zigzag edge and the chiral edge, as a consequence of the nontrivial Zak phase in the corresponding directions. In addition, we find the localization of electromagnetic waves at the corners formed by the chiral edges, which can be considered as second-order topological states, i.e., topological corner states.
Collapse
|
4
|
Zhao W, Yang M, Xu R, Du X, Li Y, Zhai K, Peng C, Pei D, Gao H, Li Y, Xu L, Han J, Huang Y, Liu Z, Yao Y, Zhuang J, Du Y, Zhou J, Chen Y, Yang L. Topological electronic structure and spin texture of quasi-one-dimensional higher-order topological insulator Bi 4Br 4. Nat Commun 2023; 14:8089. [PMID: 38062024 PMCID: PMC10703900 DOI: 10.1038/s41467-023-43882-z] [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: 04/21/2023] [Accepted: 11/22/2023] [Indexed: 03/25/2024] Open
Abstract
The notion of topological insulators (TIs), characterized by an insulating bulk and conducting topological surface states, can be extended to higher-order topological insulators (HOTIs) hosting gapless modes localized at the boundaries of two or more dimensions lower than the insulating bulk. In this work, by performing high-resolution angle-resolved photoemission spectroscopy (ARPES) measurements with submicron spatial and spin resolution, we systematically investigate the electronic structure and spin texture of quasi-one-dimensional (1D) HOTI candidate Bi4Br4. In contrast to the bulk-state-dominant spectra on the (001) surface, we observe gapped surface states on the (100) surface, whose dispersion and spin-polarization agree well with our ab-initio calculations. Moreover, we reveal in-gap states connecting the surface valence and conduction bands, which is a signature of the hinge states inside the (100) surface gap. Our findings provide compelling evidence for the HOTI phase of Bi4Br4. The identification of the higher-order topological phase promises applications based on 1D spin-momentum locked current in electronic and spintronic devices.
Collapse
Affiliation(s)
- Wenxuan Zhao
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China
| | - Ming Yang
- School of Physics, Beihang University, Beijing, 100191, China
- Centre of Quantum and Matter Sciences, International Research Institute for Multidisciplinary Science, Beihang University, Beijing, 100191, China
| | - Runzhe Xu
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China
| | - Xian Du
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China
| | - Yidian Li
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China
| | - Kaiyi Zhai
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China
| | - Cheng Peng
- Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK
| | - Ding Pei
- Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK
| | - Han Gao
- School of Physical Science and Technology, ShanghaiTech University and CAS-Shanghai Science Research Center, Shanghai, 201210, China
| | - Yiwei Li
- School of Physical Science and Technology, ShanghaiTech University and CAS-Shanghai Science Research Center, Shanghai, 201210, China
| | - Lixuan Xu
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China
| | - Junfeng Han
- Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics, Beijing Institute of Technology, Beijing, 100081, China
- Yangtze Delta Region Academy of Beijing Institute of Technology, Jiaxing, 314001, Zhejiang province, China
| | - Yuan Huang
- Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics, Beijing Institute of Technology, Beijing, 100081, China
- Yangtze Delta Region Academy of Beijing Institute of Technology, Jiaxing, 314001, Zhejiang province, China
| | - Zhongkai Liu
- School of Physical Science and Technology, ShanghaiTech University and CAS-Shanghai Science Research Center, Shanghai, 201210, China
- ShanghaiTech Laboratory for Topological Physics, Shanghai, 200031, China
| | - Yugui Yao
- Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics, Beijing Institute of Technology, Beijing, 100081, China
- Yangtze Delta Region Academy of Beijing Institute of Technology, Jiaxing, 314001, Zhejiang province, China
| | - Jincheng Zhuang
- School of Physics, Beihang University, Beijing, 100191, China
- Centre of Quantum and Matter Sciences, International Research Institute for Multidisciplinary Science, Beihang University, Beijing, 100191, China
| | - Yi Du
- School of Physics, Beihang University, Beijing, 100191, China.
- Centre of Quantum and Matter Sciences, International Research Institute for Multidisciplinary Science, Beihang University, Beijing, 100191, China.
| | - Jinjian Zhou
- Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics, Beijing Institute of Technology, Beijing, 100081, China.
| | - Yulin Chen
- Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK.
- School of Physical Science and Technology, ShanghaiTech University and CAS-Shanghai Science Research Center, Shanghai, 201210, China.
- ShanghaiTech Laboratory for Topological Physics, Shanghai, 200031, China.
| | - Lexian Yang
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China.
- Frontier Science Center for Quantum Information, Beijing, 100084, China.
- Collaborative Innovation Center of Quantum Matter, Beijing, China.
| |
Collapse
|
5
|
Tang W, Ding K, Ma G. Experimental realization of non-Abelian permutations in a three-state non-Hermitian system. Natl Sci Rev 2022; 9:nwac010. [PMID: 36523566 PMCID: PMC9746695 DOI: 10.1093/nsr/nwac010] [Citation(s) in RCA: 11] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2021] [Revised: 12/28/2021] [Accepted: 12/29/2021] [Indexed: 07/31/2023] Open
Abstract
Eigenstates of a non-Hermitian system exist on complex Riemannian manifolds, with multiple sheets connecting at branch cuts and exceptional points (EPs). These eigenstates can evolve across different sheets-a process that naturally corresponds to state permutation. Here, we report the first experimental realization of non-Abelian permutations in a three-state non-Hermitian system. Our approach relies on the stroboscopic encircling of two different exceptional arcs (EAs), which are smooth trajectories of order-2 EPs appearing from the coalescence of two adjacent states. The non-Abelian characteristics are confirmed by encircling the EAs in opposite sequences. A total of five non-trivial permutations are experimentally realized, which together comprise a non-Abelian group. Our approach provides a reliable way of investigating non-Abelian state permutations and the related exotic winding effects in non-Hermitian systems.
Collapse
Affiliation(s)
- Weiyuan Tang
- Department of Physics, Hong Kong Baptist University, Hong Kong, China
| | - Kun Ding
- Department of Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Fudan University, Shanghai200438, China
| | - Guancong Ma
- Department of Physics, Hong Kong Baptist University, Hong Kong, China
| |
Collapse
|
6
|
Wu Y, Jiang H, Chen H, Liu H, Liu J, Xie XC. Non-Abelian Braiding in Spin Superconductors Utilizing the Aharonov-Casher Effect. PHYSICAL REVIEW LETTERS 2022; 128:106804. [PMID: 35333073 DOI: 10.1103/physrevlett.128.106804] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/28/2021] [Revised: 11/16/2021] [Accepted: 02/04/2022] [Indexed: 06/14/2023]
Abstract
Spin superconductor (SSC) is an exciton condensate state where the spin-triplet exciton superfluidity is charge neutral while spin 2(ℏ/2). In analogy to the Majorana zero mode (MZM) in topological superconductors, the interplay between SSC and band topology will also give rise to a specific kind of topological bound state obeying non-Abelian braiding statistics. Remarkably, the non-Abelian geometric phase here originates from the Aharonov-Casher effect of the "half-charge" other than the Aharonov-Bohm effect. Such topological bound state of SSC is bound with the vortex of electric flux gradient and can be experimentally more distinct than the MZM for being electrically charged. This theoretical proposal provides a new avenue investigating the non-Abelian braiding physics without the assistance of MZM and charge superconductor.
Collapse
Affiliation(s)
- Yijia Wu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
| | - Hua Jiang
- School of Physical Science and Technology, Soochow University, Suzhou 215006, China
- Institute for Advanced Study, Soochow University, Suzhou 215006, China
| | - Hua Chen
- Department of Physics, Zhejiang Normal University, Jinhua 321004, China
| | - Haiwen Liu
- Center for Advanced Quantum Studies, Department of Physics, Beijing Normal University, Beijing 100875, China
| | - Jie Liu
- Department of Applied Physics, School of Science, Xian Jiaotong University, Xian 710049, China
| | - X C Xie
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
| |
Collapse
|
7
|
Wu Y, Jiang H, Liu J, Liu H, Xie XC. Non-Abelian Braiding of Dirac Fermionic Modes Using Topological Corner States in Higher-Order Topological Insulator. PHYSICAL REVIEW LETTERS 2020; 125:036801. [PMID: 32745393 DOI: 10.1103/physrevlett.125.036801] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/09/2020] [Revised: 04/13/2020] [Accepted: 06/17/2020] [Indexed: 06/11/2023]
Abstract
We numerically demonstrate that the topological corner states residing in the corners of higher-order topological insulator possess non-Abelian braiding properties. Such topological corner states are Dirac fermionic modes other than Majorana zero modes. We claim that Dirac fermionic modes protected by nontrivial topology also support non-Abelian braiding. An analytical description on such non-Abelian braiding is conducted based on the vortex-induced Dirac-type fermionic modes. Finally, the braiding operators for Dirac fermionic modes, especially their explicit matrix forms, are analytically derived and compared with the case of Majorana zero modes.
Collapse
Affiliation(s)
- Yijia Wu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
| | - Hua Jiang
- School of Physical Science and Technology, Soochow University, Suzhou 215006, China
- Institute for Advanced Study, Soochow University, Suzhou 215006, China
| | - Jie Liu
- Department of Applied Physics, School of Science, Xian Jiaotong University, Xian 710049, China
| | - Haiwen Liu
- Center for Advanced Quantum Studies, Department of Physics, Beijing Normal University, Beijing 100875, China
| | - X C Xie
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Beijing Academy of Quantum Information Sciences, Beijing 100193, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
| |
Collapse
|