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Abstract
The topological properties of an object, associated with an integer called the topological invariant, are global features that cannot change continuously but only through abrupt variations, hence granting them intrinsic robustness. Engineered metamaterials (MMs) can be tailored to support highly nontrivial topological properties of their band structure, relative to their electronic, electromagnetic, acoustic and mechanical response, representing one of the major breakthroughs in physics over the past decade. Here, we review the foundations and the latest advances of topological photonic and phononic MMs, whose nontrivial wave interactions have become of great interest to a broad range of science disciplines, such as classical and quantum chemistry. We first introduce the basic concepts, including the notion of topological charge and geometric phase. We then discuss the topology of natural electronic materials, before reviewing their photonic/phononic topological MM analogues, including 2D topological MMs with and without time-reversal symmetry, Floquet topological insulators, 3D, higher-order, non-Hermitian and nonlinear topological MMs. We also discuss the topological aspects of scattering anomalies, chemical reactions and polaritons. This work aims at connecting the recent advances of topological concepts throughout a broad range of scientific areas and it highlights opportunities offered by topological MMs for the chemistry community and beyond.
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Affiliation(s)
- Xiang Ni
- Photonics Initiative, Advanced Science Research Center, City University of New York, New York, New York 10031, United States
- School of Physics and Electronics, Central South University, Changsha, Hunan 410083, China
| | - Simon Yves
- Photonics Initiative, Advanced Science Research Center, City University of New York, New York, New York 10031, United States
| | - Alex Krasnok
- Department of Electrical and Computer Engineering, Florida International University, Miami, Florida 33174, USA
| | - Andrea Alù
- Photonics Initiative, Advanced Science Research Center, City University of New York, New York, New York 10031, United States
- Department of Electrical Engineering, City College, The City University of New York, 160 Convent Avenue, New York, New York 10031, United States
- Physics Program, The Graduate Center, The City University of New York, 365 Fifth Avenue, New York, New York 10016, United States
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2
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Chen A, Brand H, Helbig T, Hofmann T, Imhof S, Fritzsche A, Kießling T, Stegmaier A, Upreti LK, Neupert T, Bzdušek T, Greiter M, Thomale R, Boettcher I. Hyperbolic matter in electrical circuits with tunable complex phases. Nat Commun 2023; 14:622. [PMID: 36739281 PMCID: PMC9899218 DOI: 10.1038/s41467-023-36359-6] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2022] [Accepted: 01/24/2023] [Indexed: 02/06/2023] Open
Abstract
Curved spaces play a fundamental role in many areas of modern physics, from cosmological length scales to subatomic structures related to quantum information and quantum gravity. In tabletop experiments, negatively curved spaces can be simulated with hyperbolic lattices. Here we introduce and experimentally realize hyperbolic matter as a paradigm for topological states through topolectrical circuit networks relying on a complex-phase circuit element. The experiment is based on hyperbolic band theory that we confirm here in an unprecedented numerical survey of finite hyperbolic lattices. We implement hyperbolic graphene as an example of topologically nontrivial hyperbolic matter. Our work sets the stage to realize more complex forms of hyperbolic matter to challenge our established theories of physics in curved space, while the tunable complex-phase element developed here can be a key ingredient for future experimental simulation of various Hamiltonians with topological ground states.
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Affiliation(s)
- Anffany Chen
- grid.17089.370000 0001 2190 316XDepartment of Physics, University of Alberta, Edmonton, AB T6G 2E1 Canada ,grid.17089.370000 0001 2190 316XTheoretical Physics Institute, University of Alberta, Edmonton, AB T6G 2E1 Canada
| | - Hauke Brand
- grid.8379.50000 0001 1958 8658Physikalisches Institut, Universität Würzburg, 97074 Würzburg, Germany
| | - Tobias Helbig
- grid.8379.50000 0001 1958 8658Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany
| | - Tobias Hofmann
- grid.8379.50000 0001 1958 8658Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany
| | - Stefan Imhof
- grid.8379.50000 0001 1958 8658Physikalisches Institut, Universität Würzburg, 97074 Würzburg, Germany
| | - Alexander Fritzsche
- grid.8379.50000 0001 1958 8658Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany ,grid.10493.3f0000000121858338Institut für Physik, Universität Rostock, 18059 Rostock, Germany
| | - Tobias Kießling
- grid.8379.50000 0001 1958 8658Physikalisches Institut, Universität Würzburg, 97074 Würzburg, Germany
| | - Alexander Stegmaier
- grid.8379.50000 0001 1958 8658Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany
| | - Lavi K. Upreti
- grid.8379.50000 0001 1958 8658Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany
| | - Titus Neupert
- grid.7400.30000 0004 1937 0650Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
| | - Tomáš Bzdušek
- grid.7400.30000 0004 1937 0650Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland ,grid.5991.40000 0001 1090 7501Condensed Matter Theory Group, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland
| | - Martin Greiter
- grid.8379.50000 0001 1958 8658Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany
| | - Ronny Thomale
- grid.8379.50000 0001 1958 8658Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany
| | - Igor Boettcher
- grid.17089.370000 0001 2190 316XDepartment of Physics, University of Alberta, Edmonton, AB T6G 2E1 Canada ,grid.17089.370000 0001 2190 316XTheoretical Physics Institute, University of Alberta, Edmonton, AB T6G 2E1 Canada
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3
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Cui X, Zhang RY, Zhang ZQ, Chan CT. Photonic Z_{2} Topological Anderson Insulators. PHYSICAL REVIEW LETTERS 2022; 129:043902. [PMID: 35939009 DOI: 10.1103/physrevlett.129.043902] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2022] [Accepted: 06/28/2022] [Indexed: 06/15/2023]
Abstract
That disorder can induce nontrivial topology is a surprising discovery in topological physics. As a typical example, Chern topological Anderson insulators (TAIs) have been realized in photonic systems, where the topological phases exist without symmetry protection. In this Letter, by taking transverse magnetic and transverse electric polarizations as pseudospin degrees of freedom, we theoretically propose a scheme to realize disorder-induced symmetry-protected topological phase transitions in two-dimensional photonic crystals with a combined time-reversal, mirror, and duality symmetry T_{f}=TM_{z}D. In particular, we demonstrate that the disorder-induced symmetry-protected topological phase persists even without pseudospin conservation, thereby realizing a photonic Z_{2} TAI, in contrast to a Z-classified quantum spin Hall (QSH) TAI with decoupled spins. By formulating a new scattering approach, we show that the topology of both the QSH and Z_{2} TAIs can be manifested by the accumulated spin rotations of the reflected waves from the photonic crystals. Using a transmission structure, we also illustrate the trivialization of a disordered QSH phase with an even integer topological index caused by spin coupling.
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Affiliation(s)
- Xiaohan Cui
- Department of Physics, The Hong Kong University of Science and Technology, Hong Kong, China
| | - Ruo-Yang Zhang
- Department of Physics, The Hong Kong University of Science and Technology, Hong Kong, China
| | - Zhao-Qing Zhang
- Department of Physics, The Hong Kong University of Science and Technology, Hong Kong, China
| | - C T Chan
- Department of Physics, The Hong Kong University of Science and Technology, Hong Kong, China
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4
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Chen J, Qin Q, Peng C, Li ZY. Robust topological one-way edge states in radius-fluctuated photonic Chern topological insulators. OPTICS EXPRESS 2022; 30:21621-21633. [PMID: 36224877 DOI: 10.1364/oe.457593] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2022] [Accepted: 05/19/2022] [Indexed: 06/16/2023]
Abstract
Recent developments in topological photonics have shown that the introduction of disorders can yield the innovative and striking transport phenomena. Here, we theoretically investigate topological one-way edge states in radius-fluctuated photonic Chern topological insulators (PCTIs), which are composed of two-dimensional gyromagnetic photonic crystals with cylinder site fixed but with cylinder radius fluctuated. We use a fluctuation index to characterize the degree of radius fluctuation, employ two empirical parameters to inspect the evolution of topological one-way edge states, and verify the stability of topological one-way edge states by calculating massive samples with various random numbers. We find that as the radius-fluctuation strength increases, there arises a competition between topological one-way edge state, Anderson localization state and trivial bulk state. We reveal that the Anderson localization state appears far more easily in the radius-fluctuation PCTI with even a weak strength compared with the position-perturbed PCTI with a strong randomness. We also demonstrate that the topological one-way edge states are protected against a strong fluctuation much larger than the fabrication errors in practical experiments. Our results show that the PCTIs consisting of gyromagnetic photonic crystals have a high-tolerance for the material and sample fabrication errors, and this would provide a deeper understanding of fundamental topology physics.
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Xu M, He Q, Pu M, Zhang F, Li L, Sang D, Guo Y, Zhang R, Li X, Ma X, Luo X. Emerging Long-Range Order from a Freeform Disordered Metasurface. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2022; 34:e2108709. [PMID: 34997941 DOI: 10.1002/adma.202108709] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2021] [Revised: 12/20/2021] [Indexed: 06/14/2023]
Abstract
Recently, disordered metasurfaces have attracted considerable interest due to their potential applications in imaging, holography, and wavefront shaping. However, how to emerge long-range ordered phase distribution in disordered metasurfaces remains an outstanding problem. Here, a general framework is proposed to generate a spatially homogeneous in-plane phase distribution from a disordered metasurface, by engineering disorder parameters together with topology optimization. As a proof-of-concept demonstration, an all-dielectric disordered supercell metasurface with relatively homogeneous in-plane phase fluctuation is designed by disorder parameter engineering, manifesting as polarization conversion-dependent random scattering or unidirectional transmission. Then, a topology optimization approach is utilized to overcome the lattice coupling effect and to further improve the homogeneity of complex electric field fluctuation. In comparison with the initial supercell metasurface, both the phase fluctuation range and the relative efficiency of the topology-optimized freeform metasurface are significantly improved, leading to a long-range ordered electric field distribution. Moreover, three experimental realizations are performed, all of which agree well with the theoretical results. This methodology may inspire more exotic optical phenomena and find more promising applications in disordered metasurfaces and disordered optics.
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Affiliation(s)
- Mingfeng Xu
- State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, 610209, China
- Division of Frontier Science and Technology, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, 610209, China
| | - Qiong He
- State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, 610209, China
| | - Mingbo Pu
- State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, 610209, China
- School of Optoelectronics, University of Chinese Academy of Sciences, Beijing, 100049, China
| | - Fei Zhang
- State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, 610209, China
| | - Ling Li
- Tianfu Xinglong Lake Laboratory, Chengdu, 610299, China
| | - Di Sang
- State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, 610209, China
- Division of Frontier Science and Technology, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, 610209, China
- College of Electronic Science and Technology, National University of Defense Technology, Changsha, 410072, China
| | - Yinghui Guo
- State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, 610209, China
- School of Optoelectronics, University of Chinese Academy of Sciences, Beijing, 100049, China
| | - Renyan Zhang
- State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, 610209, China
- Division of Frontier Science and Technology, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, 610209, China
| | - Xiong Li
- State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, 610209, China
- School of Optoelectronics, University of Chinese Academy of Sciences, Beijing, 100049, China
| | - Xiaoliang Ma
- State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, 610209, China
- School of Optoelectronics, University of Chinese Academy of Sciences, Beijing, 100049, China
| | - Xiangang Luo
- State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, 610209, China
- School of Optoelectronics, University of Chinese Academy of Sciences, Beijing, 100049, China
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6
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Observation of topological phase with critical localization in a quasi-periodic lattice. Sci Bull (Beijing) 2021; 66:2175-2180. [PMID: 36654108 DOI: 10.1016/j.scib.2021.07.025] [Citation(s) in RCA: 15] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/22/2021] [Revised: 07/01/2021] [Accepted: 07/14/2021] [Indexed: 01/20/2023]
Abstract
Disorder and localization have dramatic influence on the topological properties of a quantum system. While strong disorder can close the band gap thus depriving topological materials of topological features, disorder may also induce topology from trivial band structures, wherein topological invariants are shared by completely localized states. Here we experimentally investigate a fundamentally distinct scenario where topology is identified in a critically localized regime, with eigenstates neither fully extended nor completely localized. Adopting the technique of momentum-lattice engineering for ultracold atoms, we implement a one-dimensional, generalized Aubry-André model with both diagonal and off-diagonal quasi-periodic disorder in momentum space, and characterize its localization and topological properties through dynamic observables. We then demonstrate the impact of interactions on the critically localized topological state, as a first experimental endeavor toward the clarification of many-body critical phase, the critical analogue of the many-body localized state.
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7
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Mukherjee A, Nandy A, Sil S, Chakrabarti A. Engineering topological phase transition and Aharonov-Bohm caging in a flux-staggered lattice. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 33:035502. [PMID: 32992300 DOI: 10.1088/1361-648x/abbc9a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2020] [Accepted: 09/29/2020] [Indexed: 06/11/2023]
Abstract
A tight binding network of diamond shaped unit cells trapping a staggered magnetic flux distribution is shown to exhibit a topological phase transition under a controlled variation of the flux trapped in a cell. A simple real space decimation technique maps a binary flux staggered network into an equivalent Su-Shrieffer-Heeger (SSH) model. In this way, dealing with a subspace of the full degrees of freedom, we show that a topological phase transition can be initiated by tuning the applied magnetic field that eventually simulates an engineering of the numerical values of the overlap integrals in the paradigmatic SSH model. Thus one can use an external agent, rather than monitoring the intrinsic property of a lattice to control the topological properties. This is advantageous from an experimental point of view. We also provide an in-depth description and analysis of the topologically protected edge states, and discuss how, by tuning the flux from outside one can enhance the spatial extent of the Aharonov-Bohm caging of single particle states for any arbitrary period of staggering. This feature can be useful for the study of transport of quantum information. Our results are exact.
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Affiliation(s)
- Amrita Mukherjee
- Department of Physics, University of Kalyani, Kalyani, West Bengal-741 235, India
| | - Atanu Nandy
- Department of Physics, Kulti College, Kulti, Paschim Bardhaman, West Bengal-713 343, India
| | - Shreekantha Sil
- Department of Physics, Visva-Bharati, Santiniketan, West Bengal-731 235, India
| | - Arunava Chakrabarti
- Department of Physics, Presidency University, 86/1 College Street, Kolkata, West Bengal-700 073, India
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Yang B, Zhang H, Shi Q, Wu T, Ma Y, Lv Z, Xiao X, Dong R, Yan X, Zhang X. Details of the topological state transition induced by gradually increased disorder in photonic Chern insulators. OPTICS EXPRESS 2020; 28:31487-31498. [PMID: 33115121 DOI: 10.1364/oe.405820] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/19/2020] [Accepted: 09/26/2020] [Indexed: 06/11/2023]
Abstract
Using two well-defined empirical parameters, we numerically investigate the details of the disorder-induced topological state transition (TST) in photonic Chern insulators composed of two-dimensional magnetic photonic crystals (MPCs). The TST undergoes a gradual process, accompanied with some interesting phenomena as the disorder of rod positions in MPCs increases gradually. This kind of TST is determined by the competition among the topologically protected edge state, disorder-induced wave localizations and bulk states in the system. More interestingly, the disorder-induced wave localizations almost have no influence on the one-way propagation of the original photonic topological states (PTSs), and the unidirectional nature of the PTSs at the edge area can survive even when the bulk states arise at stronger disorders. Our results provide detailed demonstrations for the deep understanding of fundamental physics underlying topology and disorder and are also of practical significance in device fabrication with PTSs.
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Yang Y, Gao Z, Feng X, Huang YX, Zhou P, Yang SA, Chong Y, Zhang B. Ideal Unconventional Weyl Point in a Chiral Photonic Metamaterial. PHYSICAL REVIEW LETTERS 2020; 125:143001. [PMID: 33064518 DOI: 10.1103/physrevlett.125.143001] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2020] [Accepted: 09/03/2020] [Indexed: 06/11/2023]
Abstract
Unconventional Weyl points (WPs), carrying topological charge 2 or higher, possess interesting properties different from ordinary charge-1 WPs, including multiple Fermi arcs that stretch over a large portion of the Brillouin zone. Thus far, such WPs have been observed in chiral materials and acoustic metamaterials, but there has been no clean demonstration in photonics in which the unconventional photonic WPs are separated from trivial bands. We experimentally realize an ideal symmetry-protected photonic charge-2 WP in a three-dimensional topological chiral microwave metamaterial. We use field mapping to directly observe the projected bulk dispersion, as well as the two long surface arcs that form a noncontractible loop wrapping around the surface Brillouin zone. The surface states span a record-wide frequency window of around 22.7% relative bandwidth. We demonstrate that the surface states exhibit a novel topological self-collimation property and are robust against disorder. This work provides an ideal photonic platform for exploring fundamental physics and applications of unconventional WPs.
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Affiliation(s)
- Yihao Yang
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
- Centre for Disruptive Photonic Technologies, The Photonics Institute, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
| | - Zhen Gao
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
| | - Xiaolong Feng
- Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore
| | - Yue-Xin Huang
- Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore
| | - Peiheng Zhou
- National Engineering Research Center of Electromagnetic Radiation Control Materials, State Key Laboratory of Electronic Thin Film and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China
| | - Shengyuan A Yang
- Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore
| | - Yidong Chong
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
- Centre for Disruptive Photonic Technologies, The Photonics Institute, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
| | - Baile Zhang
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
- Centre for Disruptive Photonic Technologies, The Photonics Institute, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
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10
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Liu GG, Yang Y, Ren X, Xue H, Lin X, Hu YH, Sun HX, Peng B, Zhou P, Chong Y, Zhang B. Topological Anderson Insulator in Disordered Photonic Crystals. PHYSICAL REVIEW LETTERS 2020; 125:133603. [PMID: 33034499 DOI: 10.1103/physrevlett.125.133603] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2020] [Revised: 07/22/2020] [Accepted: 08/17/2020] [Indexed: 06/11/2023]
Abstract
Recent studies have revealed the counterintuitive possibility that increasing disorder can turn a topologically trivial insulator into a nontrivial insulator, called a topological Anderson insulator (TAI). Here, we propose and experimentally demonstrate a photonic TAI in a two-dimensional disordered gyromagnetic photonic crystal in the microwave regime. We directly observe the disorder-induced topological phase transition from a trivial insulator to a TAI with robust chiral edge states. We also demonstrate topological heterostructures that host edge states at interfaces between domains with different disorder parameters.
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Affiliation(s)
- Gui-Geng Liu
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
- Centre for Disruptive Photonic Technologies, The Photonics Institute, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
| | - Yihao Yang
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
- Centre for Disruptive Photonic Technologies, The Photonics Institute, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
| | - Xin Ren
- National Engineering Research Center of Electromagnetic Radiation Control Materials, State Key Laboratory of Electronic Thin Film and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China
| | - Haoran Xue
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
| | - Xiao Lin
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
| | - Yuan-Hang Hu
- National Engineering Research Center of Electromagnetic Radiation Control Materials, State Key Laboratory of Electronic Thin Film and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China
| | - Hong-Xiang Sun
- Research Center of Fluid Machinery Engineering and Technology, School of Physics and Electronic Engineering, Jiangsu University, Zhenjiang, China
| | - Bo Peng
- National Engineering Research Center of Electromagnetic Radiation Control Materials, State Key Laboratory of Electronic Thin Film and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China
| | - Peiheng Zhou
- National Engineering Research Center of Electromagnetic Radiation Control Materials, State Key Laboratory of Electronic Thin Film and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China
| | - Yidong Chong
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
- Centre for Disruptive Photonic Technologies, The Photonics Institute, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
| | - Baile Zhang
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
- Centre for Disruptive Photonic Technologies, The Photonics Institute, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
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11
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Yu S, Piao X, Park N. Topological Hyperbolic Lattices. PHYSICAL REVIEW LETTERS 2020; 125:053901. [PMID: 32794858 DOI: 10.1103/physrevlett.125.053901] [Citation(s) in RCA: 25] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/19/2020] [Accepted: 07/07/2020] [Indexed: 06/11/2023]
Abstract
Non-Euclidean geometry, discovered by negating Euclid's parallel postulate, has been of considerable interest in mathematics and related fields for the description of geographical coordinates, Internet infrastructures, and the general theory of relativity. Notably, an infinite number of regular tessellations in hyperbolic geometry-hyperbolic lattices-are expected to extend Euclidean Bravais lattices and the consequent wave phenomena to non-Euclidean geometry. However, topological states of matter in hyperbolic lattices have yet to be reported. Here we investigate topological phenomena in hyperbolic geometry, exploring how the quantized curvature and edge dominance of the geometry affect topological phases. We report a recipe for the construction of a Euclidean photonic platform that inherits the topological band properties of a hyperbolic lattice under a uniform, pseudospin-dependent magnetic field, realizing a non-Euclidean analog of the quantum spin Hall effect. For hyperbolic lattices with different quantized curvatures, we examine the topological protection of helical edge states and generalize Hofstadter's butterfly, by employing two empirical parameters that measure the edge confinement and defect immunity. We demonstrate that the proposed platforms exhibit the unique spectral-magnetic sensitivity of topological immunity in highly curved hyperbolic planes. Our approach is applicable to general non-Euclidean geometry and enables the exploitation of infinite lattice degrees of freedom for band theory.
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Affiliation(s)
- Sunkyu Yu
- Photonic Systems Laboratory, Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Korea
| | - Xianji Piao
- Photonic Systems Laboratory, Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Korea
| | - Namkyoo Park
- Photonic Systems Laboratory, Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Korea
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Zhou P, Liu GG, Ren X, Yang Y, Xue H, Bi L, Deng L, Chong Y, Zhang B. Photonic amorphous topological insulator. LIGHT, SCIENCE & APPLICATIONS 2020; 9:133. [PMID: 32728433 PMCID: PMC7381680 DOI: 10.1038/s41377-020-00368-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/30/2020] [Revised: 07/06/2020] [Accepted: 07/08/2020] [Indexed: 06/11/2023]
Abstract
The current understanding of topological insulators and their classical wave analogs, such as photonic topological insulators, is mainly based on topological band theory. However, standard band theory does not apply to amorphous phases of matter, which are formed by non-crystalline lattices with no long-range positional order but only short-range order, exhibiting unique phenomena such as the glass-to-liquid transition. Here, we experimentally investigate amorphous variants of a Chern number-based photonic topological insulator. By tuning the disorder strength in the lattice, we demonstrate that photonic topological edge states can persist into the amorphous regime prior to the glass-to-liquid transition. After the transition to a liquid-like lattice configuration, the signatures of topological edge states disappear. This interplay between topology and short-range order in amorphous lattices paves the way for new classes of non-crystalline topological photonic bandgap materials.
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Affiliation(s)
- Peiheng Zhou
- National Engineering Research Center of Electromagnetic Radiation Control Materials, University of Electronic Science and Technology of China, 610054 Chengdu, 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
| | - Xin Ren
- National Engineering Research Center of Electromagnetic Radiation Control Materials, University of Electronic Science and Technology of China, 610054 Chengdu, China
| | - Yihao Yang
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371 Singapore
- Centre for Disruptive Photonic Technologies, The Photonics Institute, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798 Singapore
| | - Haoran Xue
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371 Singapore
| | - Lei Bi
- National Engineering Research Center of Electromagnetic Radiation Control Materials, University of Electronic Science and Technology of China, 610054 Chengdu, China
| | - Longjiang Deng
- National Engineering Research Center of Electromagnetic Radiation Control Materials, University of Electronic Science and Technology of China, 610054 Chengdu, China
| | - Yidong Chong
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371 Singapore
- Centre for Disruptive Photonic Technologies, The Photonics Institute, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798 Singapore
| | - Baile Zhang
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371 Singapore
- Centre for Disruptive Photonic Technologies, The Photonics Institute, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798 Singapore
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13
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Kim M, Jacob Z, Rho J. Recent advances in 2D, 3D and higher-order topological photonics. LIGHT, SCIENCE & APPLICATIONS 2020; 9:130. [PMID: 32704363 PMCID: PMC7371865 DOI: 10.1038/s41377-020-0331-y] [Citation(s) in RCA: 70] [Impact Index Per Article: 17.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2020] [Revised: 04/21/2020] [Accepted: 05/07/2020] [Indexed: 05/25/2023]
Abstract
Over the past decade, topology has emerged as a major branch in broad areas of physics, from atomic lattices to condensed matter. In particular, topology has received significant attention in photonics because light waves can serve as a platform to investigate nontrivial bulk and edge physics with the aid of carefully engineered photonic crystals and metamaterials. Simultaneously, photonics provides enriched physics that arises from spin-1 vectorial electromagnetic fields. Here, we review recent progress in the growing field of topological photonics in three parts. The first part is dedicated to the basics of topological band theory and introduces various two-dimensional topological phases. The second part reviews three-dimensional topological phases and numerous approaches to achieve them in photonics. Last, we present recently emerging fields in topological photonics that have not yet been reviewed. This part includes topological degeneracies in nonzero dimensions, unidirectional Maxwellian spin waves, higher-order photonic topological phases, and stacking of photonic crystals to attain layer pseudospin. In addition to the various approaches for realizing photonic topological phases, we also discuss the interaction between light and topological matter and the efforts towards practical applications of topological photonics.
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Affiliation(s)
- Minkyung Kim
- Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Pohang, 37673 Republic of Korea
| | - Zubin Jacob
- School of Electrical and Computer Engineering, Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47906 USA
| | - Junsuk Rho
- Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), Pohang, 37673 Republic of Korea
- Department of Chemical Engineering, Pohang University of Science and Technology (POSTECH), Pohang, 37673 Republic of Korea
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14
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Longhi S. Topological Anderson phase in quasi-periodic waveguide lattices. OPTICS LETTERS 2020; 45:4036-4039. [PMID: 32667348 DOI: 10.1364/ol.399742] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2020] [Accepted: 06/25/2020] [Indexed: 06/11/2023]
Abstract
The topological trivial band of a lattice can be driven into a topological phase by disorder in the system. This so-called topological Anderson phase has been predicted and observed for uncorrelated static disorder, while in the presence of correlated disorder, conflicting results are found. Here we consider a Su-Schrieffer-Heeger waveguide lattice in the trivial topological phase and show that quasi-periodic disorder in the coupling constants can drive the lattice into a topological nontrivial phase. A method to detect the emergence of the topological Anderson phase, based on light dynamics at the edge of a quasi-periodic waveguide lattice, is suggested.
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15
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Park J, Oh I, Jin MJ, Jo J, Choe D, Yun HD, Lee SW, Lee Z, Kwon SY, Jin H, Chung SB, Yoo JW. Observation of spin-polarized Anderson state around charge neutral point in graphene with Fe-clusters. Sci Rep 2020; 10:4784. [PMID: 32179819 PMCID: PMC7076017 DOI: 10.1038/s41598-020-61481-6] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2019] [Accepted: 02/17/2020] [Indexed: 11/09/2022] Open
Abstract
The pristine graphene described with massless Dirac fermion could bear topological insulator state and ferromagnetism via the band structure engineering with various adatoms and proximity effects from heterostructures. In particular, topological Anderson insulator state was theoretically predicted in tight-binding honeycomb lattice with Anderson disorder term. Here, we introduced physi-absorbed Fe-clusters/adatoms on graphene to impose exchange interaction and random lattice disorder, and we observed Anderson insulator state accompanying with Kondo effect and field-induced conducting state upon applying the magnetic field at around a charge neutral point. Furthermore, the emergence of the double peak of resistivity at ν = 0 state indicates spin-splitted edge state with high effective exchange field (>70 T). These phenomena suggest the appearance of topological Anderson insulator state triggered by the induced exchange field and disorder.
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Affiliation(s)
- Jungmin Park
- School of Materials Science and Engineering, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea
- Center for Scientific Instrumentation, Division of Scientific Instrumentation & Management, Korea Basic Science Institute, Daejeon, 34133, Korea
| | - Inseon Oh
- School of Materials Science and Engineering, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea
| | - Mi-Jin Jin
- School of Materials Science and Engineering, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea
| | - Junhyeon Jo
- School of Materials Science and Engineering, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea
| | - Daeseong Choe
- School of Materials Science and Engineering, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea
| | - Hyung Duk Yun
- School of Materials Science and Engineering, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea
| | - Suk Woo Lee
- School of Materials Science and Engineering, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea
- Center for Multidimensional Carbon Materials, Institute for Basic Science (IBS), Ulsan, 44919, Republic of Korea
| | - Zonghoon Lee
- School of Materials Science and Engineering, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea
- Center for Multidimensional Carbon Materials, Institute for Basic Science (IBS), Ulsan, 44919, Republic of Korea
| | - Soon-Yong Kwon
- School of Materials Science and Engineering, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea
| | - Hosub Jin
- Department of Physics, Ulsan National Institute of Science and Technology, Ulsan, 44919, Korea
| | - Suk Bum Chung
- Department of Physics, University of Seoul, Seoul, 02504, Korea
| | - Jung-Woo Yoo
- School of Materials Science and Engineering, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea.
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16
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You JW, Lan Z, Panoiu NC. Four-wave mixing of topological edge plasmons in graphene metasurfaces. SCIENCE ADVANCES 2020; 6:eaaz3910. [PMID: 32258407 PMCID: PMC7101229 DOI: 10.1126/sciadv.aaz3910] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/04/2019] [Accepted: 01/06/2020] [Indexed: 05/14/2023]
Abstract
We study topologically protected four-wave mixing (FWM) interactions in a plasmonic metasurface consisting of a periodic array of nanoholes in a graphene sheet, which exhibits a wide topological bandgap at terahertz frequencies upon the breaking of time reversal symmetry by a static magnetic field. We demonstrate that due to the significant nonlinearity enhancement and large life time of graphene plasmons in specific configurations, a net gain of FWM interaction of plasmonic edge states located in the topological bandgap can be achieved with a pump power of less than 10 nW. In particular, we find that the effective nonlinear edge-waveguide coefficient is about γ ≃ 1.1 × 1013 W-1 m-1, i.e., more than 10 orders of magnitude larger than that of commonly used, highly nonlinear silicon photonic nanowires. These findings could pave a new way for developing ultralow-power-consumption, highly integrated, and robust active photonic systems at deep-subwavelength scale for applications in quantum communications and information processing.
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17
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Hou J, Li Z, Luo XW, Gu Q, Zhang C. Topological Bands and Triply Degenerate Points in Non-Hermitian Hyperbolic Metamaterials. PHYSICAL REVIEW LETTERS 2020; 124:073603. [PMID: 32142317 DOI: 10.1103/physrevlett.124.073603] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/31/2018] [Accepted: 02/04/2020] [Indexed: 06/10/2023]
Abstract
Hyperbolic metamaterials (HMMs), an unusual class of electromagnetic metamaterials, have found important applications in various fields due to their distinctive properties. A surprising feature of HMMs is that even continuous HMMs can possess topological edge modes. However, previous studies based on equal-frequency surface (analogy of Fermi surface) may not correctly capture the topology of entire bands. Here we develop a topological band description for continuous HMMs that can be described by a non-Hermitian Hamiltonian formulated from Maxwell's equations. We find two types of three-dimensional non-Hermitian triply degenerate points with complex linear dispersions and topological charges ±2 and 0 induced by chiral and gyromagnetic effects. Because of the photonic nature, the vacuum band plays an important role for topological edge states and bulk-edge correspondence in HMMs. The topological band results are numerically confirmed by direct simulation of Maxwell's equations. Our work presents a general non-Hermitian topological band treatment of continuous HMMs, paving the way for exploring interesting topological phases in photonic continua and device implementations of topological HMMs.
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Affiliation(s)
- Junpeng Hou
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA
| | - Zhitong Li
- Department of Electrical and Computer Engineering, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA
| | - Xi-Wang Luo
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA
| | - Qing Gu
- Department of Electrical and Computer Engineering, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA
| | - Chuanwei Zhang
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA
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18
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Meier EJ, An FA, Dauphin A, Maffei M, Massignan P, Hughes TL, Gadway B. Observation of the topological Anderson insulator in disordered atomic wires. Science 2018; 362:929-933. [DOI: 10.1126/science.aat3406] [Citation(s) in RCA: 152] [Impact Index Per Article: 25.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/19/2018] [Accepted: 09/27/2018] [Indexed: 11/02/2022]
Abstract
Topology and disorder have a rich combined influence on quantum transport. To probe their interplay, we synthesized one-dimensional chiral symmetric wires with controllable disorder via spectroscopic Hamiltonian engineering, based on the laser-driven coupling of discrete momentum states of ultracold atoms. Measuring the bulk evolution of a topological indicator after a sudden quench, we observed the topological Anderson insulator phase, in which added disorder drives the band structure of a wire from topologically trivial to nontrivial. In addition, we observed the robustness of topologically nontrivial wires to weak disorder and measured the transition to a trivial phase in the presence of strong disorder. Atomic interactions in this quantum simulation platform may enable realizations of strongly interacting topological fluids.
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Xie BY, Wang HF, Zhu XY, Lu MH, Wang ZD, Chen YF. Photonics meets topology. OPTICS EXPRESS 2018; 26:24531-24550. [PMID: 30469568 DOI: 10.1364/oe.26.024531] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/06/2018] [Accepted: 08/20/2018] [Indexed: 06/09/2023]
Abstract
The topological phases in materials have been studied in recent decades for their unique boundary states and transport properties. Photonic systems with band structures embrace the topological phases closely, where they not only provide platforms to testify the topological band theory, but also shed light on designing novel optical devices. In this review, we present exciting developments, supported by brief descriptions of prominent milestones of topological phases in photonic systems in recent years. These studies may sustain further developments of optical devices and offer novel methods for light manipulations.
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20
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Hu C, Li Z, Tong R, Wu X, Xia Z, Wang L, Li S, Huang Y, Wang S, Hou B, Chan CT, Wen W. Type-II Dirac Photons at Metasurfaces. PHYSICAL REVIEW LETTERS 2018; 121:024301. [PMID: 30085689 DOI: 10.1103/physrevlett.121.024301] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/23/2018] [Indexed: 06/08/2023]
Abstract
Topological characteristics of energy bands, such as Dirac and Weyl nodes, have attracted substantial interest in condensed matter systems as well as in classical wave systems. Among these energy bands, the type-II Dirac point is a nodal degeneracy with tilted conical dispersion, leading to a peculiar crossing dispersion in the constant-energy plane. Such nodal points have recently been found in electronic materials. The analogous topological feature in photonic systems remains a theoretical curiosity, with experimental realization expected to be challenging. Here, we experimentally realize the type-II Dirac point using a planar metasurface architecture, where the band degeneracy point is protected by the underlying mirror symmetry of the metasurface. Gapless edge modes are found and measured at the boundary between the different domains of the symmetry-broken metasurface. Our Letter shows that metasurfaces are simple and practical platforms for realizing electromagnetic type-II Dirac points, and their planar structure is a distinct advantage that facilitates applications in two-dimensional topological photonics.
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Affiliation(s)
- Chuandeng Hu
- Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 518000, China
| | - Zhenyu Li
- College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
- Key Laboratory of Modern Optical Technologies of Ministry of Education & Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province, Suzhou 215006, China
| | - Rui Tong
- Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 518000, China
| | - Xiaoxiao Wu
- Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 518000, China
| | - Zengzilu Xia
- Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 518000, China
| | - Li Wang
- Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 518000, China
| | - Shanshan Li
- College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
- Key Laboratory of Modern Optical Technologies of Ministry of Education & Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province, Suzhou 215006, China
| | - Yingzhou Huang
- Chongqing Key Laboratory of Soft Condensed Matter Physics and Smart Materials, College of Physics, Chongqing University, Chongqing 401331, China
| | - Shuxia Wang
- Chongqing Key Laboratory of Soft Condensed Matter Physics and Smart Materials, College of Physics, Chongqing University, Chongqing 401331, China
| | - Bo Hou
- College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
- Key Laboratory of Modern Optical Technologies of Ministry of Education & Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province, Suzhou 215006, China
| | - C T Chan
- Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 518000, China
| | - Weijia Wen
- Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 518000, China
- Materials Genome Institute, Shanghai University, Shanghai 200444, China
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Jiang J, Guo Z, Ding Y, Sun Y, Li Y, Jiang H, Chen H. Experimental demonstration of the robust edge states in a split-ring-resonator chain. OPTICS EXPRESS 2018; 26:12891-12902. [PMID: 29801322 DOI: 10.1364/oe.26.012891] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2018] [Accepted: 05/02/2018] [Indexed: 06/08/2023]
Abstract
One of the fascinating topological phenomena is the edge state in one-dimensional system. In this work, the topological photonics in the dimer chains composed by the split ring resonators are revealed based on the Su-Schrieffer-Heeger model. The topologically protected photonic edge state is observed directly with the in situ measurements of the local density of states in the topological nontrivial chain. Moreover, we experimentally demonstrate that the edge state localized at both ends is robust against a varied of perturbations, such as losses and disorder. Our results not only provide a versatile platform to study the topological physics in photonics but also may have potential applications in the robust power transfer.
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