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Zhu Z, Yang L, Wu J, Meng Y, Xi X, Yan B, Chen J, Lu J, Huang X, Deng W, Shang C, Shum PP, Yang Y, Chen H, Xiang K, Liu GG, Liu Z, Gao Z. Brillouin Klein space and half-turn space in three-dimensional acoustic crystals. Sci Bull (Beijing) 2024:S2095-9273(24)00337-2. [PMID: 38782659 DOI: 10.1016/j.scib.2024.05.003] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2024] [Revised: 03/28/2024] [Accepted: 05/06/2024] [Indexed: 05/25/2024]
Abstract
The Bloch band theory and Brillouin zone (BZ) that characterize wave-like behaviors in periodic mediums are two cornerstones of contemporary physics, ranging from condensed matter to topological physics. Recent theoretical breakthrough revealed that, under the projective symmetry algebra enforced by artificial gauge fields, the usual two-dimensional (2D) BZ (orientable Brillouin two-torus) can be fundamentally modified to a non-orientable Brillouin Klein bottle with radically distinct manifold topology. However, the physical consequence of artificial gauge fields on the more general three-dimensional (3D) BZ (orientable Brillouin three-torus) was so far missing. Here, we theoretically discovered and experimentally observed that the fundamental domain and topology of the usual 3D BZ can be reduced to a non-orientable Brillouin Klein space or an orientable Brillouin half-turn space in a 3D acoustic crystal with artificial gauge fields. We experimentally identify peculiar 3D momentum-space non-symmorphic screw rotation and glide reflection symmetries in the measured band structures. Moreover, we experimentally demonstrate a novel stacked weak Klein bottle insulator featuring a nonzero Z2 topological invariant and self-collimated topological surface states at two opposite surfaces related by a nonlocal twist, radically distinct from all previous 3D topological insulators. Our discovery not only fundamentally modifies the fundamental domain and topology of 3D BZ, but also opens the door towards a wealth of previously overlooked momentum-space multidimensional manifold topologies and novel gauge-symmetry-enriched topological physics and robust acoustic wave manipulations beyond the existing paradigms.
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Affiliation(s)
- Zhenxiao Zhu
- State Key Laboratory of Optical Fiber and Cable Manufacture Technology, Department of Electronic and Electrical Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Linyun Yang
- State Key Laboratory of Optical Fiber and Cable Manufacture Technology, Department of Electronic and Electrical Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Jien Wu
- School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510640, China
| | - Yan Meng
- State Key Laboratory of Optical Fiber and Cable Manufacture Technology, Department of Electronic and Electrical Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Xiang Xi
- State Key Laboratory of Optical Fiber and Cable Manufacture Technology, Department of Electronic and Electrical Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Bei Yan
- State Key Laboratory of Optical Fiber and Cable Manufacture Technology, Department of Electronic and Electrical Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Jingming Chen
- State Key Laboratory of Optical Fiber and Cable Manufacture Technology, Department of Electronic and Electrical Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Jiuyang Lu
- School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510640, China; Key Laboratory of Artificial Micro- and Nanostructures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China
| | - Xueqin Huang
- School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510640, China
| | - Weiyin Deng
- School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510640, China; Key Laboratory of Artificial Micro- and Nanostructures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China
| | - Ce Shang
- King Abdullah University of Science and Technology (KAUST), Physical Science and Engineering Division (PSE), Thuwal 23955-6900, Saudi Arabia
| | - Perry Ping Shum
- State Key Laboratory of Optical Fiber and Cable Manufacture Technology, Department of Electronic and Electrical Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Yihao Yang
- Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, ZJU-Hangzhou Global Science and Technology Innovation Center, College of Information Science and Electronic Engineering, ZJU-UIUC Institute, Zhejiang University, Hangzhou 310027, China
| | - Hongsheng Chen
- Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, ZJU-Hangzhou Global Science and Technology Innovation Center, College of Information Science and Electronic Engineering, ZJU-UIUC Institute, Zhejiang University, Hangzhou 310027, China
| | - Kexin Xiang
- State Key Laboratory of Optical Fiber and Cable Manufacture Technology, Department of Electronic and Electrical Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Gui-Geng Liu
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore.
| | - Zhengyou Liu
- Key Laboratory of Artificial Micro- and Nanostructures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China; Institute for Advanced Studies, Wuhan University, Wuhan 430072, China.
| | - Zhen Gao
- State Key Laboratory of Optical Fiber and Cable Manufacture Technology, Department of Electronic and Electrical Engineering, Southern University of Science and Technology, Shenzhen 518055, China.
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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.
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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
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3
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Zhang X, He T, Liu Y, Dai X, Liu G, Chen C, Wu W, Zhu J, Yang SA. Magnetic Real Chern Insulator in 2D Metal-Organic Frameworks. NANO LETTERS 2023; 23:7358-7363. [PMID: 37535707 DOI: 10.1021/acs.nanolett.3c01723] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 08/05/2023]
Abstract
Real Chern insulators have attracted great interest, but so far, their material realization is limited to nonmagnetic crystals and systems without spin-orbit coupling. Here, we reveal the magnetic real Chern insulator (MRCI) state in a recently synthesized metal-organic framework material Co3(HITP)2. Its ground state with in-plane ferromagnetic ordering hosts a nontrivial real Chern number, enabled by the C2zT symmetry and robustness against spin-orbit coupling. Distinct from previous nonmagnetic examples, the topological corner zero modes of MRCIs are spin-polarized. Furthermore, under small tensile strains, the material undergoes a topological phase transition from the MRCI to a magnetic double-Weyl semimetal phase, via a pseudospin-1 critical state. Similar physics can also be found in closely related materials Mn3(HITP)2 and Fe3(HITP)2, which also exist. Possible experimental detections and implications of an emerging magnetic flat band in the system are discussed.
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Affiliation(s)
- Xiaoming Zhang
- State Key Laboratory of Reliability and Intelligence of Electrical Equipment, Hebei University of Technology, Tianjin 300130, China
- School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, China
| | - Tingli He
- State Key Laboratory of Reliability and Intelligence of Electrical Equipment, Hebei University of Technology, Tianjin 300130, China
- School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, China
| | - Ying Liu
- State Key Laboratory of Reliability and Intelligence of Electrical Equipment, Hebei University of Technology, Tianjin 300130, China
- School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, China
| | - Xuefang Dai
- State Key Laboratory of Reliability and Intelligence of Electrical Equipment, Hebei University of Technology, Tianjin 300130, China
- School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, China
| | - Guodong Liu
- State Key Laboratory of Reliability and Intelligence of Electrical Equipment, Hebei University of Technology, Tianjin 300130, China
- School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, China
| | - Cong Chen
- Department of Physics, The University of Hong Kong, Hong Kong, China
- HKU-UCAS Joint Institute of Theoretical and Computational Physics at Hong Kong, Hong Kong, China
| | - Weikang Wu
- Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials (Ministry of Education), Shandong University, Jinan 250061, China
| | - Jiaojiao Zhu
- Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore
| | - Shengyuan A Yang
- Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore
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Zhang C, Chen ZY, Zhang Z, Zhao YX. General Theory of Momentum-Space Nonsymmorphic Symmetry. PHYSICAL REVIEW LETTERS 2023; 130:256601. [PMID: 37418718 DOI: 10.1103/physrevlett.130.256601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/17/2022] [Revised: 02/20/2023] [Accepted: 05/23/2023] [Indexed: 07/09/2023]
Abstract
As a fundamental concept of all crystals, space groups are partitioned into symmorphic groups and nonsymmorphic groups. Each nonsymmorphic group contains glide reflections or screw rotations with fractional lattice translations, which are absent in symmorphic groups. Although nonsymmorphic groups ubiquitously exist on real-space lattices, on the reciprocal lattices in momentum space, the ordinary theory only allows symmorphic groups. In this work, we develop a novel theory for momentum-space nonsymmorphic space groups (k-NSGs), utilizing the projective representations of space groups. The theory is quite general: Given any k-NSGs in any dimensions, it can identify the real-space symmorphic space groups (r-SSGs) and construct the corresponding projective representation of the r-SSG that leads to the k-NSG. To demonstrate the broad applicability of our theory, we show these projective representations and therefore all k-NSGs can be realized by gauge fluxes over real-space lattices. Our work fundamentally extends the framework of crystal symmetry, and therefore can accordingly extend any theory based on crystal symmetry, for instance, the classification crystalline topological phases.
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Affiliation(s)
- Chen Zhang
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
- The University of Hong Kong Shenzhen Institute of Research and Innovation, Shenzhen 518057, China
| | - Z Y Chen
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
- The University of Hong Kong Shenzhen Institute of Research and Innovation, Shenzhen 518057, China
| | - Zheng Zhang
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
- The University of Hong Kong Shenzhen Institute of Research and Innovation, Shenzhen 518057, China
| | - Y X Zhao
- Department of Physics and HKU-UCAS Joint Institute for Theoretical and Computational Physics at Hong Kong, The University of Hong Kong, Pokfulam Road, Hong Kong, China
- HK Institute of Quantum Science and Technology, The University of Hong Kong, Pokfulam Road, Hong Kong, China
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5
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Jiang C, Song Y, Li X, Lu P, Ke S. Photonic Möbius topological insulator from projective symmetry in multiorbital waveguides. OPTICS LETTERS 2023; 48:2337-2340. [PMID: 37126268 DOI: 10.1364/ol.488210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
The gauge fields dramatically alter the algebraic structure of spatial symmetries and make them projectively represented, giving rise to novel topological phases. Here, we propose a photonic Möbius topological insulator enabled by projective translation symmetry in multiorbital waveguide arrays, where the artificial π gauge flux is aroused by the inter-orbital coupling between the first (s) and third (d) order modes. In the presence of π flux, the two translation symmetries of rectangular lattices anti-commute with each other. By tuning the spatial spacing between two waveguides to break the translation symmetry, a topological insulator is created with two Möbius twisted edge bands appearing in the bandgap and featuring 4π periodicity. Importantly, the Möbius twists are accompanied by discrete diffraction in beam propagation, which exhibit directional transport by tuning the initial phase of the beam envelope according to the eigenvalues of translation operators. This work manifests the significance of gauge fields in topology and provides an efficient approach to steering the direction of beam transmission.
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Classification of time-reversal-invariant crystals with gauge structures. Nat Commun 2023; 14:743. [PMID: 36765052 PMCID: PMC9918504 DOI: 10.1038/s41467-023-36447-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2022] [Accepted: 01/26/2023] [Indexed: 02/12/2023] Open
Abstract
A peculiar feature of quantum states is that they may embody so-called projective representations of symmetries rather than ordinary representations. Projective representations of space groups-the defining symmetry of crystals-remain largely unexplored. Despite recent advances in artificial crystals, whose intrinsic gauge structures necessarily require a projective description, a unified theory is yet to be established. Here, we establish such a unified theory by exhaustively classifying and representing all 458 projective symmetry algebras of time-reversal-invariant crystals from 17 wallpaper groups in two dimensions-189 of which are algebraically non-equivalent. We discover three physical signatures resulting from projective symmetry algebras, including the shift of high-symmetry momenta, an enforced nontrivial Zak phase, and a spinless eight-fold nodal point. Our work offers a theoretical foundation for the field of artificial crystals and opens the door to a wealth of topological states and phenomena beyond the existing paradigms.
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Meng Y, Lin S, Shi BJ, Wei B, Yang L, Yan B, Zhu Z, Xi X, Wang Y, Ge Y, Yuan SQ, Chen J, Liu GG, Sun HX, Chen H, Yang Y, Gao Z. Spinful Topological Phases in Acoustic Crystals with Projective PT Symmetry. PHYSICAL REVIEW LETTERS 2023; 130:026101. [PMID: 36706409 DOI: 10.1103/physrevlett.130.026101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2022] [Accepted: 12/09/2022] [Indexed: 06/18/2023]
Abstract
For the classification of topological phases of matter, an important consideration is whether a system is spinless or spinful, as these two classes have distinct symmetry algebra that gives rise to fundamentally different topological phases. However, only recently has it been realized theoretically that in the presence of gauge symmetry, the algebraic structure of symmetries can be projectively represented, which possibly enables the switch between spinless and spinful topological phases. Here, we report the experimental demonstration of this idea by realizing spinful topological phases in "spinless" acoustic crystals with projective space-time inversion symmetry. In particular, we realize a one-dimensional topologically gapped phase characterized by a 2Z winding number, which features double-degenerate bands in the entire Brillouin zone and two pairs of degenerate topological boundary modes. Our Letter thus overcomes a fundamental constraint on topological phases by spin classes.
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Affiliation(s)
- Yan Meng
- Department of Electrical and Electronic Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Shuxin Lin
- Department of Electrical and Electronic Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Bin-Jie Shi
- Research Center of Fluid Machinery Engineering and Technology, School of Physics and Electronics Engineering, Jiangsu University, Zhenjiang 212013, China
| | - Bin Wei
- SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China
- Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
| | - Linyun Yang
- Department of Electrical and Electronic Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Bei Yan
- Department of Electrical and Electronic Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Zhenxiao Zhu
- Department of Electrical and Electronic Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Xiang Xi
- Department of Electrical and Electronic Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Yin Wang
- Research Center of Fluid Machinery Engineering and Technology, School of Physics and Electronics Engineering, Jiangsu University, Zhenjiang 212013, China
| | - Yong Ge
- Research Center of Fluid Machinery Engineering and Technology, School of Physics and Electronics Engineering, Jiangsu University, Zhenjiang 212013, China
| | - Shou-Qi Yuan
- Research Center of Fluid Machinery Engineering and Technology, School of Physics and Electronics Engineering, Jiangsu University, Zhenjiang 212013, China
| | - Jingming Chen
- Department of Electrical and Electronic Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Gui-Geng Liu
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
| | - Hong-Xiang Sun
- Research Center of Fluid Machinery Engineering and Technology, School of Physics and Electronics Engineering, Jiangsu University, Zhenjiang 212013, China
- State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China
| | - Hongsheng Chen
- Interdisciplinary Center for Quantum Information, State Key Laboratory of Extreme Photonics and Instrumentation, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Zhejiang University, Hangzhou 310027, China; International Joint Innovation Center, The Electromagnetics Academy at Zhejiang University, Zhejiang University, Haining 314400, China; Key Lab. of Advanced Micro/Nano Electronic Devices & Smart Systems of Zhejiang, Jinhua Institute of Zhejiang University, Zhejiang University, Jinhua 321099, China; Shaoxing Institute of Zhejiang University, Zhejiang University, Shaoxing 312000, China
| | - Yihao Yang
- Interdisciplinary Center for Quantum Information, State Key Laboratory of Extreme Photonics and Instrumentation, ZJU-Hangzhou Global Scientific and Technological Innovation Center, Zhejiang University, Hangzhou 310027, China; International Joint Innovation Center, The Electromagnetics Academy at Zhejiang University, Zhejiang University, Haining 314400, China; Key Lab. of Advanced Micro/Nano Electronic Devices & Smart Systems of Zhejiang, Jinhua Institute of Zhejiang University, Zhejiang University, Jinhua 321099, China; Shaoxing Institute of Zhejiang University, Zhejiang University, Shaoxing 312000, China
| | - Zhen Gao
- Department of Electrical and Electronic Engineering, Southern University of Science and Technology, Shenzhen 518055, China
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Brillouin Klein bottle from artificial gauge fields. Nat Commun 2022; 13:2215. [PMID: 35468905 PMCID: PMC9038716 DOI: 10.1038/s41467-022-29953-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/08/2021] [Accepted: 04/10/2022] [Indexed: 11/23/2022] Open
Abstract
A Brillouin zone is the unit for the momentum space of a crystal. It is topologically a torus, and distinguishing whether a set of wave functions over the Brillouin torus can be smoothly deformed to another leads to the classification of various topological states of matter. Here, we show that under \documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb{Z}}}_{2}$$\end{document}Z2 gauge fields, i.e., hopping amplitudes with phases ±1, the fundamental domain of momentum space can assume the topology of a Klein bottle. This drastic change of the Brillouin zone theory is due to the projective symmetry algebra enforced by the gauge field. Remarkably, the non-orientability of the Brillouin Klein bottle corresponds to the topological classification by a \documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb{Z}}}_{2}$$\end{document}Z2 invariant, in contrast to the Chern number valued in \documentclass[12pt]{minimal}
\usepackage{amsmath}
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\begin{document}$${\mathbb{Z}}$$\end{document}Z for the usual Brillouin torus. The result is a novel Klein bottle insulator featuring topological modes at two edges related by a nonlocal twist, radically distinct from all previous topological insulators. Our prediction can be readily achieved in various artificial crystals, and the discovery opens a new direction to explore topological physics by gauge-field-modified fundamental structures of physics. Topological states are exploited based on crystalline symmetry, but under artificial gauge fields, symmetries may satisfy projective algebras, which remains less studied. Here, the authors reveal that projective symmetry algebra leads to momentum-space nonsymmorphic symmetry, resulting in new topological states over a momentum-space Klein bottle.
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Li T, Du J, Zhang Q, Li Y, Fan X, Zhang F, Qiu C. Acoustic Möbius Insulators from Projective Symmetry. PHYSICAL REVIEW LETTERS 2022; 128:116803. [PMID: 35362999 DOI: 10.1103/physrevlett.128.116803] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2021] [Accepted: 01/25/2022] [Indexed: 06/14/2023]
Abstract
In the presence of gauge symmetry, common but not limited to artificial crystals, the algebraic structure of crystalline symmetries needs to be projectively represented, giving rise to unprecedented topological physics. Here, we demonstrate this novel idea by exploiting a projective translation symmetry and constructing a variety of Möbius-twisted topological phases. Experimentally, we realize two Möbius insulators in acoustic crystals for the first time: a two-dimensional one of first-order band topology and a three-dimensional one of higher-order band topology. We observe unambiguously the peculiar Möbius edge and hinge states via real-space visualization of their localiztions, momentum-space spectroscopy of their 4π periodicity, and phase-space winding of their projective translation eigenvalues. Not only does our work open a new avenue for artificial systems under the interplay between gauge and crystalline symmetries, but it also initializes a new framework for topological physics from projective symmetry.
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Affiliation(s)
- Tianzi Li
- Key Laboratory of Artificial Micro- and Nano-Structures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China
| | - Juan Du
- Key Laboratory of Artificial Micro- and Nano-Structures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China
| | - Qicheng Zhang
- Key Laboratory of Artificial Micro- and Nano-Structures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China
| | - Yitong Li
- Key Laboratory of Artificial Micro- and Nano-Structures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China
| | - Xiying Fan
- Key Laboratory of Artificial Micro- and Nano-Structures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China
| | - Fan Zhang
- Department of Physics, University of Texas at Dallas, Richardson, Texas 75080, USA
| | - Chunyin Qiu
- Key Laboratory of Artificial Micro- and Nano-Structures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China
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Xue H, Wang Z, Huang YX, Cheng Z, Yu L, Foo YX, Zhao YX, Yang SA, Zhang B. Projectively Enriched Symmetry and Topology in Acoustic Crystals. PHYSICAL REVIEW LETTERS 2022; 128:116802. [PMID: 35363008 DOI: 10.1103/physrevlett.128.116802] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2021] [Accepted: 01/27/2022] [Indexed: 06/14/2023]
Abstract
Symmetry plays a key role in modern physics, as manifested in the revolutionary topological classification of matter in the past decade. So far, we seem to have a complete theory of topological phases from internal symmetries as well as crystallographic symmetry groups. However, an intrinsic element, i.e., the gauge symmetry in physical systems, has been overlooked in the current framework. Here, we show that the algebraic structure of crystal symmetries can be projectively enriched due to the gauge symmetry, which subsequently gives rise to new topological physics never witnessed under ordinary symmetries. We demonstrate the idea by theoretical analysis, numerical simulation, and experimental realization of a topological acoustic lattice with projective translation symmetries under a Z_{2} gauge field, which exhibits unique features of rich topologies, including a single Dirac point, Möbius topological insulator, and graphenelike semimetal phases on a rectangular lattice. Our work reveals the impact when gauge and crystal symmetries meet together with topology and opens the door to a vast unexplored land of topological states by projective symmetries.
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Affiliation(s)
- Haoran Xue
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
| | - Zihao Wang
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
| | - Yue-Xin Huang
- Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore
| | - Zheyu Cheng
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
| | - Letian Yu
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
| | - Y X Foo
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
| | - Y X Zhao
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
- Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - Shengyuan A Yang
- Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore
| | - Baile Zhang
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
- Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore 637371, Singapore
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11
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Deng Y, Jing Y. Acoustic Crystals with a Möbius Twist. PHYSICS 2022. [DOI: 10.1103/physics.15.36] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/04/2022]
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12
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Encyclopedia of emergent particles in three-dimensional crystals. Sci Bull (Beijing) 2021; 67:375-380. [DOI: 10.1016/j.scib.2021.10.023] [Citation(s) in RCA: 32] [Impact Index Per Article: 10.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2021] [Revised: 10/25/2021] [Accepted: 10/26/2021] [Indexed: 10/19/2022]
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Shao LB, Liu Q, Xiao R, Yang SA, Zhao YX. Gauge-Field Extended k·p Method and Novel Topological Phases. PHYSICAL REVIEW LETTERS 2021; 127:076401. [PMID: 34459642 DOI: 10.1103/physrevlett.127.076401] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/16/2021] [Accepted: 07/01/2021] [Indexed: 06/13/2023]
Abstract
Although topological artificial systems, like acoustic and photonic crystals and cold atoms in optical lattices were initially motivated by simulating topological phases of electronic systems, they have their own unique features such as the spinless time-reversal symmetry and tunable Z_{2} gauge fields. Hence, it is fundamentally important to explore new topological phases based on these features. Here, we point out that the Z_{2} gauge field leads to two fundamental modifications of the conventional k·p method: (i) The little co-group must include the translations with nontrivial algebraic relations. (ii) The algebraic relations of the little co-group are projectively represented. These give rise to higher-dimensional irreducible representations and therefore highly degenerate Fermi points. Breaking the primitive translations can transform the Fermi points to interesting topological phases. We demonstrate our theory by two models: a rectangular π-flux model exhibiting graphenelike semimetal phases, and a graphite model with interlayer π flux that realizes the real second-order nodal-line semimetal phase with hinge helical modes. Their physical realizations with a general bright-dark mechanism are discussed. Our finding opens a new direction to explore novel topological phases unique to crystalline systems with gauge fields and establishes the approach to analyze these phases.
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Affiliation(s)
- L B Shao
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
- Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - Q Liu
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
| | - R Xiao
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
| | - Shengyuan A Yang
- Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore
| | - Y X Zhao
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
- Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
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