1
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Yang YB, Wang JH, Li K, Xu Y. Higher-order topological phases in crystalline and non-crystalline systems: a review. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2024; 36:283002. [PMID: 38574683 DOI: 10.1088/1361-648x/ad3abd] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/14/2023] [Accepted: 04/04/2024] [Indexed: 04/06/2024]
Abstract
In recent years, higher-order topological phases have attracted great interest in various fields of physics. These phases have protected boundary states at lower-dimensional boundaries than the conventional first-order topological phases due to the higher-order bulk-boundary correspondence. In this review, we summarize current research progress on higher-order topological phases in both crystalline and non-crystalline systems. We firstly introduce prototypical models of higher-order topological phases in crystals and their topological characterizations. We then discuss effects of quenched disorder on higher-order topology and demonstrate disorder-induced higher-order topological insulators. We also review the theoretical studies on higher-order topological insulators in amorphous systems without any crystalline symmetry and higher-order topological phases in non-periodic lattices including quasicrystals, hyperbolic lattices, and fractals, which have no crystalline counterparts. We conclude the review by a summary of experimental realizations of higher-order topological phases and discussions on potential directions for future study.
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Affiliation(s)
- Yan-Bin Yang
- Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong Special Administrative Region of China, People's Republic of China
- Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, People's Republic of China
| | - Jiong-Hao Wang
- Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, People's Republic of China
| | - Kai Li
- Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, People's Republic of China
| | - Yong Xu
- Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, People's Republic of China
- Hefei National Laboratory, Hefei 230088, People's Republic of China
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2
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Zhang JH, Mei F, Xiao L, Jia S. Dynamical Detection of Topological Spectral Density. PHYSICAL REVIEW LETTERS 2024; 132:036603. [PMID: 38307045 DOI: 10.1103/physrevlett.132.036603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2023] [Accepted: 12/18/2023] [Indexed: 02/04/2024]
Abstract
Local density of states (LDOS) is emerging as powerful means of exploring classical-wave topological phases. However, the current LDOS detection method remains rare and merely works for static situations. Here, we introduce a generic dynamical method to detect both the static and Floquet LDOS, based on an elegant connection between dynamics of chiral density and local spectral densities. Moreover, we find that the Floquet LDOS allows to measure out Floquet quasienergy spectra and identify topological π modes. As an example, we demonstrate that both the static and Floquet higher-order topological phase can be universally identified via LDOS detection, regardless of whether the topological corner modes are in energy gaps, bands, or continuous energy spectra without band gaps. Our study opens a new avenue utilizing dynamics to detect topological spectral densities and provides a universal approach of identifying static and Floquet topological phases.
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Affiliation(s)
- Jia-Hui Zhang
- State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China and Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
| | - Feng Mei
- State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China and Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
| | - Liantuan Xiao
- State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China and Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
| | - Suotang Jia
- State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China and Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
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3
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Ghosh AK, Nag T, Saha A. Generation of higher-order topological insulators using periodic driving. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2023; 36:093001. [PMID: 37983922 DOI: 10.1088/1361-648x/ad0e2d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2023] [Accepted: 11/20/2023] [Indexed: 11/22/2023]
Abstract
Topological insulators (TIs) are a new class of materials that resemble ordinary band insulators in terms of a bulk band gap but exhibit protected metallic states on their boundaries. In this modern direction, higher-order TIs (HOTIs) are a new class of TIs in dimensionsd > 1. These HOTIs possess(d-1)-dimensional boundaries that, unlike those of conventional TIs, do not conduct via gapless states but are themselves TIs. Precisely, annth orderd-dimensional higher-order TI is characterized by the presence of boundary modes that reside on itsdc=(d-n)-dimensional boundary. For instance, a three-dimensional second (third) order TI hosts gapless (localized) modes on the hinges (corners), characterized bydc=1(0). Similarly, a second-order TI (SOTI) in two dimensions only has localized corner states (dc=0). These higher-order phases are protected by various crystalline as well as discrete symmetries. The non-equilibrium tunability of the topological phase has been a major academic challenge where periodic Floquet drive provides us golden opportunity to overcome that barrier. Here, we discuss different periodic driving protocols to generate Floquet HOTIs while starting from a non-topological or first-order topological phase. Furthermore, we emphasize that one can generate the dynamical anomalousπ-modes along with the concomitant 0-modes. The former can be realized only in a dynamical setup. We exemplify the Floquet higher-order topological modes in two and three dimensions in a systematic way. Especially, in two dimensions, we demonstrate a Floquet SOTI (FSOTI) hosting 0- andπcorner modes. Whereas a three-dimensional FSOTI and Floquet third-order TI manifest one- and zero-dimensional hinge and corner modes, respectively.
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Affiliation(s)
- Arnob Kumar Ghosh
- Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
- Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden
| | - Tanay Nag
- Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden
- Department of Physics, BITS Pilani-Hyderabad Campus, Telangana 500078, India
| | - Arijit Saha
- Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
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4
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Xu X, Wang J, Dai J, Mao R, Cai H, Zhu SY, Wang DW. Floquet Superradiance Lattices in Thermal Atoms. PHYSICAL REVIEW LETTERS 2022; 129:273603. [PMID: 36638288 DOI: 10.1103/physrevlett.129.273603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/29/2022] [Revised: 11/07/2022] [Accepted: 12/08/2022] [Indexed: 06/17/2023]
Abstract
Floquet modulation has been widely used in optical lattices for coherent control of quantum gases, in particular for synthesizing artificial gauge fields and simulating topological matters. However, such modulation induces heating which can overwhelm the signal of quantum dynamics in ultracold atoms. Here we report that the thermal motion, instead of being a noise source, provides a new control knob in Floquet-modulated superradiance lattices, which are momentum-space tight-binding lattices of collectively excited states of atoms. The Doppler shifts combined with Floquet modulation provide effective forces along arbitrary directions in a lattice in frequency and momentum dimensions. Dynamic localization, dynamic delocalization, and chiral edge currents can be simultaneously observed from a single transport spectrum of superradiance lattices in thermal atoms. Our Letter paves a way for simulating Floquet topological matters in room-temperature atoms and facilitates their applications in photonic devices.
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Affiliation(s)
- Xingqi Xu
- Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device, School of Physics, Zhejiang University, Hangzhou 310027, Zhejiang Province, China
| | - Jiefei Wang
- Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device, School of Physics, Zhejiang University, Hangzhou 310027, Zhejiang Province, China
- College of Optical Science and Engineering, Zhejiang University, Hangzhou, 310027, China
| | - Jianhao Dai
- Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device, School of Physics, Zhejiang University, Hangzhou 310027, Zhejiang Province, China
| | - Ruosong Mao
- Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device, School of Physics, Zhejiang University, Hangzhou 310027, Zhejiang Province, China
| | - Han Cai
- Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device, School of Physics, Zhejiang University, Hangzhou 310027, Zhejiang Province, China
- College of Optical Science and Engineering, Zhejiang University, Hangzhou, 310027, China
| | - Shi-Yao Zhu
- Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device, School of Physics, Zhejiang University, Hangzhou 310027, Zhejiang Province, China
- Hefei National Laboratory, Hefei 230088, China
| | - Da-Wei Wang
- Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device, School of Physics, Zhejiang University, Hangzhou 310027, Zhejiang Province, China
- Hefei National Laboratory, Hefei 230088, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
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Vogl M, Chaudhary S, Fiete GA. Light driven magnetic transitions in transition metal dichalcogenide heterobilayers. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 35:095801. [PMID: 36560921 DOI: 10.1088/1361-648x/acab49] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/18/2022] [Accepted: 12/13/2022] [Indexed: 06/17/2023]
Abstract
Motivated by the recent excitement around the physics of twisted transition metal dichalcogenide (TMD) multilayer systems, we study strongly correlated phases of TMD heterobilayers under the influence of light. We consider both waveguide light and circularly polarized light. The former allows for longitudinally polarized light, which in the high frequency limit can be used to selectively modify interlayer hoppings in a tight-binding model. We argue based on quasi-degenerate perturbation theory that changes to the interlayer hoppings can be captured as a modulation to the strength of the moiré potential in a continuum model. As a consequence, waveguide light can be used to drive transitions between a myriad of different magnetic phases, including a transition from a 120∘Neel phase to a stripe ordered magnetic phase, or from a spin density wave phase to a paramagnetic phase, among others. When the system is subjected to circularly polarized light we find that the effective mass of the active TMD layer is modified by an applied electromagnetic field. By simultaneously applying waveguide light and circularly polarized light to a system, one has a high level of control in moving through the phase diagram in-situ. Lastly, we comment on the experimental feasibility of Floquet state preparation and argue that it is within reach of available techniques when the system is coupled to a judiciously chosen bath.
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Affiliation(s)
- Michael Vogl
- Department of Physics, King Fahd University of Petroleum and Minerals, 31261 Dhahran, Saudi Arabia
| | - Swati Chaudhary
- Department of Physics, The University of Texas at Austin, Austin, TX 78712, United States of America
- Department of Physics, Northeastern University, Boston, MA 02115, United States of America
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States of America
| | - Gregory A Fiete
- Department of Physics, Northeastern University, Boston, MA 02115, United States of America
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States of America
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6
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Generating Many Majorana Corner Modes and Multiple Phase Transitions in Floquet Second-Order Topological Superconductors. Symmetry (Basel) 2022. [DOI: 10.3390/sym14122546] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/04/2022] Open
Abstract
A d-dimensional, nth-order topological insulator or superconductor has localized eigenmodes at its (d−n)-dimensional boundaries (n≤d). In this work, we apply periodic driving fields to two-dimensional superconductors, and obtain a wide variety of Floquet second-order topological superconducting (SOTSC) phases with many Majorana corner modes at both zero and π quasienergies. Two distinct Floquet SOTSC phases are found to be separated by three possible kinds of transformations, i.e., a topological phase transition due to the closing/reopening of a bulk spectral gap, a topological phase transition due to the closing/reopening of an edge spectral gap, or an entirely different phase in which the bulk spectrum is gapless. Thanks to the strong interplay between driving and intrinsic energy scales of the system, all the found phases and transitions are highly controllable via tuning a single hopping parameter of the system. Our discovery not only enriches the possible forms of Floquet SOTSC phases, but also offers an efficient scheme to generate many coexisting Majorana zero and π corner modes, which may find applications in Floquet quantum computation.
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7
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Minguzzi J, Zhu Z, Sandholzer K, Walter AS, Viebahn K, Esslinger T. Topological Pumping in a Floquet-Bloch Band. PHYSICAL REVIEW LETTERS 2022; 129:053201. [PMID: 35960575 DOI: 10.1103/physrevlett.129.053201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2022] [Accepted: 06/21/2022] [Indexed: 06/15/2023]
Abstract
Constructing new topological materials is of vital interest for the development of robust quantum applications. However, engineering such materials often causes technological overhead, such as large magnetic fields, spin-orbit coupling, or dynamical superlattice potentials. Simplifying the experimental requirements has been addressed on a conceptual level-by proposing to combine simple lattice structures with Floquet engineering-but there has been no experimental implementation. Here, we demonstrate topological pumping in a Floquet-Bloch band using a plain sinusoidal lattice potential and two-tone driving with frequencies ω and 2ω. We adiabatically prepare a near-insulating Floquet band of ultracold fermions via a frequency chirp, which avoids gap closings en route from trivial to topological bands. Subsequently, we induce topological pumping by slowly cycling the amplitude and the phase of the 2ω drive. Our system is well described by an effective Shockley model, establishing a novel paradigm to engineer topological matter from simple underlying lattice geometries. This approach could enable the application of quantized pumping in metrology, following recent experimental advances on two-frequency driving in real materials.
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Affiliation(s)
- Joaquín Minguzzi
- Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland
| | - Zijie Zhu
- Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland
| | - Kilian Sandholzer
- Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland
| | | | - Konrad Viebahn
- Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland
| | - Tilman Esslinger
- Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland
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8
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Peng Y. Topological Space-Time Crystal. PHYSICAL REVIEW LETTERS 2022; 128:186802. [PMID: 35594083 DOI: 10.1103/physrevlett.128.186802] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2022] [Revised: 04/09/2022] [Accepted: 04/14/2022] [Indexed: 06/15/2023]
Abstract
We introduce a new class of out-of-equilibrium noninteracting topological phases: the topological space-time crystals. These are time-dependent quantum systems that do not have discrete spatial translation symmetries but instead are invariant under discrete space-time translations. Similar to the Floquet-Bloch systems, the space-time crystals can be described by a frequency-domain-enlarged Hamiltonian, which is used to classify topologically distinct space-time crystals. We show that these space-time crystals can be engineered from conventional crystals with an additional time-dependent drive that behaves like a traveling wave moving across the crystal. Interestingly, we are able to construct 1D and 2D examples of topological space-time crystals based on tight-binding models that involve only one orbital, in contrast to the two-orbital minimal models for any previously discovered static or Floquet topological phases with crystalline structures.
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Affiliation(s)
- Yang Peng
- Department of Physics and Astronomy, California State University, Northridge, Northridge, California 91330, USA
- Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
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9
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Borja C, Gutiérrez E, López A. Emergence of Floquet edge states in the coupled Su-Schrieffer-Heeger model. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 34:205701. [PMID: 35203064 DOI: 10.1088/1361-648x/ac5865] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2022] [Accepted: 02/24/2022] [Indexed: 06/14/2023]
Abstract
The emergence of non equilibrium topological phases in low dimensional systems offers an interesting route for material properties engineering. We analyze the dynamical modulation of two coupled one-dimensional chains, described by the Su-Schrieffer-Heeger model. We find that the interplay of driving interactions and interchain coupling leads to the emergence of non-equilibrium edge states with nontrivial topological properties. Using an effective Hamiltonian approach, we quantify the emergent topological phases via the winding number and show that oscillations in the mean pseudospin polarization arise as a consequence of the periodic modulation. The patterns of these pseudospin oscillations are different for the static trivial and topological phases offering a dynamical means to distinguish both physical configurations. The system also exhibits non integer values of the winding number, which have been recently reported experimentally in connection to spin textures.
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Affiliation(s)
- Carla Borja
- School of Physical Sciences and Nanotechnology, Yachay Tech University, Urcuquí 100119, Ecuador
| | - Esther Gutiérrez
- Escuela Superior Politécnica del Litoral, ESPOL, Departamento de Física, Facultad de Ciencias Naturales y Matemáticas, Campus Gustavo Galindo Km. 30.5 Vía Perimetral, PO Box 09-01-5863, Guayaquil, Ecuador
| | - Alexander López
- Escuela Superior Politécnica del Litoral, ESPOL, Departamento de Física, Facultad de Ciencias Naturales y Matemáticas, Campus Gustavo Galindo Km. 30.5 Vía Perimetral, PO Box 09-01-5863, Guayaquil, Ecuador
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10
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Time-periodic corner states from Floquet higher-order topology. Nat Commun 2022; 13:11. [PMID: 35013180 PMCID: PMC8748824 DOI: 10.1038/s41467-021-27552-6] [Citation(s) in RCA: 19] [Impact Index Per Article: 9.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/02/2021] [Accepted: 11/24/2021] [Indexed: 11/24/2022] Open
Abstract
The recent discoveries of higher-order topological insulators (HOTIs) have shifted the paradigm of topological materials, previously limited to topological states at boundaries of materials, to include topological states at boundaries of boundaries, such as corners. So far, all HOTI realisations have been based on static systems described by time-invariant Hamiltonians, without considering the time-variant situation. There is growing interest in Floquet systems, in which time-periodic driving can induce unconventional phenomena such as Floquet topological phases and time crystals. Recent theories have attempted to combine Floquet engineering and HOTIs, but there has been no experimental realisation so far. Here we report on the experimental demonstration of a two-dimensional (2D) Floquet HOTI in a three-dimensional (3D) acoustic lattice, with modulation along a spatial axis serving as an effective time-dependent drive. Acoustic measurements reveal Floquet corner states with double the period of the underlying drive; these oscillations are robust, like time crystal modes, except that the robustness arises from topological protection. This shows that space-time dynamics can induce anomalous higher-order topological phases unique to Floquet systems. A combination of Floquet engineering and higher-order topological insulators has been lately only theoretically proposed. Here the authors experimentally realize an acoustic waveguides array to demonstrate a Floquet higherorder topological insulator extending its band topology from first order to higher order.
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11
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Yu J, Zhang RX, Song ZD. Dynamical symmetry indicators for Floquet crystals. Nat Commun 2021; 12:5985. [PMID: 34645782 PMCID: PMC8514516 DOI: 10.1038/s41467-021-26092-3] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/28/2021] [Accepted: 09/15/2021] [Indexed: 11/09/2022] Open
Abstract
Various exotic topological phases of Floquet systems have been shown to arise from crystalline symmetries. Yet, a general theory for Floquet topology that is applicable to all crystalline symmetry groups is still in need. In this work, we propose such a theory for (effectively) non-interacting Floquet crystals. We first introduce quotient winding data to classify the dynamics of the Floquet crystals with equivalent symmetry data, and then construct dynamical symmetry indicators (DSIs) to sufficiently indicate the inherently dynamical Floquet crystals. The DSI and quotient winding data, as well as the symmetry data, are all computationally efficient since they only involve a small number of Bloch momenta. We demonstrate the high efficiency by computing all elementary DSI sets for all spinless and spinful plane groups using the mathematical theory of monoid, and find a large number of different nontrivial classifications, which contain both first-order and higher-order 2+1D anomalous Floquet topological phases. Using the framework, we further find a new 3+1D anomalous Floquet second-order topological insulator (AFSOTI) phase with anomalous chiral hinge modes.
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Affiliation(s)
- Jiabin Yu
- Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, MD, USA.
| | - Rui-Xing Zhang
- Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, MD, USA
- Joint Quantum Institute, University of Maryland, College Park, MD, USA
| | - Zhi-Da Song
- Department of Physics, Princeton University, Princeton, NJ, USA
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12
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Wang G, Li C, Cappellaro P. Observation of Symmetry-Protected Selection Rules in Periodically Driven Quantum Systems. PHYSICAL REVIEW LETTERS 2021; 127:140604. [PMID: 34652183 DOI: 10.1103/physrevlett.127.140604] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2021] [Revised: 08/09/2021] [Accepted: 09/09/2021] [Indexed: 06/13/2023]
Abstract
Periodically driven (Floquet) quantum systems have recently been a focus of nonequilibrium physics by virtue of their rich dynamics. Time-periodic systems not only exhibit symmetries that resemble those in spatially periodic systems, but also display novel behavior that arises from symmetry breaking. Characterization of such dynamical symmetries is crucial, but often challenging due to limited driving strength and lack of an experimentally accessible characterization technique. Here, we show how to reveal dynamical symmetries, namely, parity, rotation, and particle-hole symmetries, by observing symmetry-induced Floquet selection rules. Notably, we exploit modulated driving to reach the strong light-matter coupling regime, and we introduce a protocol to experimentally extract the transition matrix elements between Floquet states from the system coherent evolution. By using nitrogen-vacancy centers in diamond as an experimental test bed, we execute our protocol to observe symmetry-protected dark states and dark bands, and coherent destruction of tunneling. Our work shows how one can exploit the quantum control toolkit to study dynamical symmetries that arise in the topological phases of strongly driven Floquet systems.
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Affiliation(s)
- Guoqing Wang
- Research Laboratory of Electronics and Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Changhao Li
- Research Laboratory of Electronics and Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Paola Cappellaro
- Research Laboratory of Electronics and Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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13
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Zhang RX, Das Sarma S. Anomalous Floquet Chiral Topological Superconductivity in a Topological Insulator Sandwich Structure. PHYSICAL REVIEW LETTERS 2021; 127:067001. [PMID: 34420352 DOI: 10.1103/physrevlett.127.067001] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2020] [Accepted: 06/24/2021] [Indexed: 06/13/2023]
Abstract
We show that Floquet chiral topological superconductivity arises naturally in Josephson junctions made of magnetic topological insulator-superconductor sandwich structures. The Josephson phase modulation associated with an applied bias voltage across the junction drives the system into the anomalous Floquet chiral topological superconductor hosting chiral Majorana edge modes in the quasienergy spectrum, with the bulk Floquet bands carrying zero Chern numbers. The bias voltage acts as a tuning parameter enabling novel Floquet topological quantum phase transitions driving the system into a myriad of exotic Majorana-carrying Floquet topological superconducting phases. Our theory establishes a new paradigm for realizing Floquet chiral topological superconductivity in solid-state systems, which should be experimentally directly accessible.
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Affiliation(s)
- Rui-Xing Zhang
- Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
| | - S Das Sarma
- Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
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14
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Li CA, Zhang SB, Li J, Trauzettel B. Higher-Order Fabry-Pérot Interferometer from Topological Hinge States. PHYSICAL REVIEW LETTERS 2021; 127:026803. [PMID: 34296912 DOI: 10.1103/physrevlett.127.026803] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2021] [Revised: 04/23/2021] [Accepted: 06/09/2021] [Indexed: 06/13/2023]
Abstract
We propose an intrinsic three-dimensional Fabry-Pérot type interferometer, coined "higher-order interferometer," that is based on the chiral hinge states of second-order topological insulators and cannot be mapped to an equivalent two-dimensional setting because of higher-order topological obstructions. Quantum interference patterns in the two-terminal conductance of this interferometer are controllable not only by tuning the strength but also, particularly, by rotating the direction of the magnetic field applied perpendicularly to the transport direction. Remarkably, the conductance exhibits a characteristic beating pattern with multiple frequencies depending on the field strength and direction in a unique fashion. Our novel interferometer thus provides feasible and robust magnetotransport signatures for hinge states of higher-order topological insulators.
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Affiliation(s)
- Chang-An Li
- Institute for Theoretical Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany
| | - Song-Bo Zhang
- Institute for Theoretical Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany
| | - Jian Li
- School of Science, Westlake University, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province, China
- Institute of Natural Sciences, Westlake Institute for Advanced Study, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province, China
| | - Björn Trauzettel
- Institute for Theoretical Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany
- Würzburg-Dresden Cluster of Excellence ct.qmat, Germany
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15
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Zhou L. Floquet Second-Order Topological Phases in Momentum Space. NANOMATERIALS (BASEL, SWITZERLAND) 2021; 11:1170. [PMID: 33947026 PMCID: PMC8146154 DOI: 10.3390/nano11051170] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/09/2021] [Revised: 04/04/2021] [Accepted: 04/26/2021] [Indexed: 11/30/2022]
Abstract
Higher-order topological phases (HOTPs) are characterized by symmetry-protected bound states at the corners or hinges of the system. In this work, we reveal a momentum-space counterpart of HOTPs in time-periodic driven systems, which are demonstrated in a two-dimensional extension of the quantum double-kicked rotor. The found Floquet HOTPs are protected by chiral symmetry and characterized by a pair of topological invariants, which could take arbitrarily large integer values with the increase of kicking strengths. These topological numbers are shown to be measurable from the chiral dynamics of wave packets. Under open boundary conditions, multiple quartets Floquet corner modes with zero and π quasienergies emerge in the system and coexist with delocalized bulk states at the same quasienergies, forming second-order Floquet topological bound states in the continuum. The number of these corner modes is further counted by the bulk topological invariants according to the relation of bulk-corner correspondence. Our findings thus extend the study of HOTPs to momentum-space lattices and further uncover the richness of HOTPs and corner-localized bound states in continuum in Floquet systems.
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Affiliation(s)
- Longwen Zhou
- Department of Physics, College of Information Science and Engineering, Ocean University of China, Qingdao 266100, China
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16
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Bhat RV, Bera S. Out of equilibrium chiral higher order topological insulator on a π-flux square lattice. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:164005. [PMID: 33752196 DOI: 10.1088/1361-648x/abf0c3] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/26/2020] [Accepted: 03/22/2021] [Indexed: 06/12/2023]
Abstract
One of the hallmarks of bulk topology is the existence of robust boundary localized states. For instance, a conventionalddimensional topological system hostsd- 1 dimensional surface modes, which are protected by non-spatial symmetries. Recently, this idea has been extended to higher order topological phases with boundary modes that are localized in lower dimensions such as in the corners or in one dimensional hinges of the system. In this work, we demonstrate that a higher order topological phase can be engineered in a nonequilibrium state when the time-independent model does not possess any symmetry protected topological states. The higher order topology is protected by an emerging chiral symmetry, which is generated through the Floquet driving. Using both the exact numerical method and an effective high-frequency Hamiltonian obtained from the Brillouin-Wigner (BW) perturbation theory, we verify the emerging topological phase on aπ-flux square lattice. We show that the localized corner modes in our model are robust against a chiral symmetry preserving perturbation and can be classified as 'extrinsic' higher order topological phase. Finally, we identify a two dimensional topological invariant from the winding number of the corresponding sublattice symmetric one dimensional model. The latter model belongs to class AIII of ten-fold symmetry classification of topological matter.
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Affiliation(s)
- Ruchira V Bhat
- Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India
| | - Soumya Bera
- Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India
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Engelhardt G, Cao J. Dynamical Symmetries and Symmetry-Protected Selection Rules in Periodically Driven Quantum Systems. PHYSICAL REVIEW LETTERS 2021; 126:090601. [PMID: 33750178 DOI: 10.1103/physrevlett.126.090601] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2020] [Revised: 12/07/2020] [Accepted: 02/09/2021] [Indexed: 06/12/2023]
Abstract
In recent experiments, the light-matter interaction has reached the ultrastrong coupling limit, which can give rise to dynamical generalizations of spatial symmetries in periodically driven systems. Here, we present a unified framework of dynamical-symmetry-protected selection rules based on Floquet response theory. Within this framework, we study rotational, parity, particle-hole, chiral, and time-reversal symmetries and the resulting selection rules in spectroscopy, including symmetry-protected dark states (spDS), symmetry-protected dark bands, and symmetry-induced transparency. Specifically, dynamical rotational and parity symmetries establish spDS and symmetry-protected dark band conditions. A particle-hole symmetry introduces spDSs for symmetry-related Floquet states and also a symmetry-induced transparency at quasienergy crossings. Chiral symmetry and time-reversal symmetry alone do not imply spDS conditions but can be combined to define a particle-hole symmetry. These symmetry conditions arise from destructive interference due to the synchronization of symmetric quantum systems with the periodic driving. Our predictions reveal new physical phenomena when a quantum system reaches the strong light-matter coupling regime, which is important for superconducting qubits, atoms and molecules in optical or plasmonic field cavities, and optomechanical systems.
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Affiliation(s)
- Georg Engelhardt
- Beijing Computational Science Research Center, Beijing 100193, People's Republic of China
| | - Jianshu Cao
- Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
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Zhang L, Zhang L, Liu XJ. Unified Theory to Characterize Floquet Topological Phases by Quench Dynamics. PHYSICAL REVIEW LETTERS 2020; 125:183001. [PMID: 33196215 DOI: 10.1103/physrevlett.125.183001] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2020] [Revised: 08/26/2020] [Accepted: 09/24/2020] [Indexed: 05/22/2023]
Abstract
The conventional characterization of periodically driven systems usually necessitates the time-domain information beyond Floquet bands, hence lacking universal and direct schemes of measuring Floquet topological invariants. Here we propose a unified theory, based on quantum quenches, to characterize generic d-dimensional Floquet topological phases in which the topological invariants are constructed with only minimal information of the static Floquet bands. For a d-dimensional phase that is initially static and trivial, we introduce the quench dynamics by suddenly turning on the periodic driving. We show that the quench dynamics exhibits emergent topological patterns in (d-1)-dimensional momentum subspaces where Floquet bands cross, from which the Floquet topological invariants are directly obtained. This result provides a simple and unified characterization in which one can extract the number of conventional and anomalous Floquet boundary modes and identify the topologically protected singularities in the phase bands. These applications are illustrated with one- and two-dimensional models that are readily accessible in cold-atom experiments. Our study opens a new framework for the characterization of Floquet topological phases.
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Affiliation(s)
- Long Zhang
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
| | - Lin Zhang
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
| | - Xiong-Jun Liu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
- Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
- Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China
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Crowley PJD, Martin I, Chandran A. Half-Integer Quantized Topological Response in Quasiperiodically Driven Quantum Systems. PHYSICAL REVIEW LETTERS 2020; 125:100601. [PMID: 32955337 DOI: 10.1103/physrevlett.125.100601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2019] [Revised: 12/19/2019] [Accepted: 08/04/2020] [Indexed: 06/11/2023]
Abstract
A spin strongly driven by two harmonic incommensurate drives can pump energy from one drive to the other at a quantized average rate, in close analogy with the quantum Hall effect. The pumping rate is a nonzero integer in the topological regime, while the trivial regime does not pump. The dynamical transition between the regimes is sharp in the zero-frequency limit and is characterized by a Dirac point in a synthetic band structure. We show that the pumping rate is half-integer quantized at the transition and present universal Kibble-Zurek scaling functions for energy transfer processes. Our results adapt ideas from quantum phase transitions, quantum information, and topological band theory to nonequilibrium dynamics, and identify qubit experiments to observe the universal linear and nonlinear response of a Dirac point in synthetic dimensions.
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Affiliation(s)
- P J D Crowley
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - I Martin
- Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
| | - A Chandran
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
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Zhao G, Mu H, Liu F, Wang Z. Folding Graphene into a Chern Insulator with Light Irradiation. NANO LETTERS 2020; 20:5860-5865. [PMID: 32658490 DOI: 10.1021/acs.nanolett.0c01758] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Recently, the precise folding of flexible graphene is reported experimentally [ Science, 2019, 365, 1036-1040], demonstrating an efficient approach to manipulate its electronic and optoelectronic properties. Here, we propose a light-induced high-Chern-number Chern insulator (CI) in the folded graphene. Along both armchair and zigzag folding directions, we demonstrate that there are two-handedness-dependent chiral interface states localized at the curved region. Physically, they can be attributed to the light-induced mass-term inversion across the folded graphene. Most remarkably, by rationally designing the folding processes, 2D and 3D CIs are also realizable in a single-wall carbon nanotube and periodic folded graphene, respectively, illustrating a high tunability of the folding degree of freedom. We envision that this intriguing form of "foldtronics" will provide a new platform for investigating the topological state in 2D materials to draw immediate experimental attention.
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Affiliation(s)
- Gan Zhao
- Hefei National Laboratory for Physical Sciences at the Microscale, CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Haimen Mu
- Hefei National Laboratory for Physical Sciences at the Microscale, CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Feng Liu
- Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112, United States
| | - Zhengfei Wang
- Hefei National Laboratory for Physical Sciences at the Microscale, CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
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Huang B, Liu WV. Floquet Higher-Order Topological Insulators with Anomalous Dynamical Polarization. PHYSICAL REVIEW LETTERS 2020; 124:216601. [PMID: 32530681 DOI: 10.1103/physrevlett.124.216601] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/17/2018] [Accepted: 05/11/2020] [Indexed: 06/11/2023]
Abstract
Higher-order topological insulators (HOTIs) have emerged as a new class of phases, whose robust in-gap "corner" modes arise from the bulk higher-order multipoles beyond the dipoles in conventional topological insulators. Here, we incorporate Floquet driving into HOTIs, and report for the first time a dynamical polarization theory with anomalous nonequilibrium multipoles. Further, a proposal to detect not only corner states but also their dynamical origin in cold atoms is demonstrated, with the latter one never achieved before. Experimental determination of anomalous Floquet corner modes is also proposed.
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Affiliation(s)
- Biao Huang
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh PA 15260, USA
| | - W Vincent Liu
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh PA 15260, USA
- Wilczek Quantum Center, School of Physics and Astronomy and T. D. Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
- Shanghai Research Center for Quantum Sciences, Shanghai 201315, China
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Hu H, Huang B, Zhao E, Liu WV. Dynamical Singularities of Floquet Higher-Order Topological Insulators. PHYSICAL REVIEW LETTERS 2020; 124:057001. [PMID: 32083918 DOI: 10.1103/physrevlett.124.057001] [Citation(s) in RCA: 24] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/16/2019] [Accepted: 01/15/2020] [Indexed: 05/22/2023]
Abstract
We propose a versatile framework to dynamically generate Floquet higher-order topological insulators by multistep driving of topologically trivial Hamiltonians. Two analytically solvable examples are used to illustrate this procedure to yield Floquet quadrupole and octupole insulators with zero- and/or π-corner modes protected by mirror symmetries. Furthermore, we introduce dynamical topological invariants from the full unitary return map and show its phase bands contain Weyl singularities whose topological charges form dynamical multipole moments in the Brillouin zone. Combining them with the topological index of a Floquet Hamiltonian gives a pair of Z_{2} invariant ν_{0} and ν_{π} which fully characterize the higher-order topology and predict the appearance of zero- and π-corner modes. Our work establishes a systematic route to construct and characterize Floquet higher-order topological phases.
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Affiliation(s)
- Haiping Hu
- Department of Physics and Astronomy, George Mason University, Fairfax, Virginia 22030, USA
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
| | - Biao Huang
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
| | - Erhai Zhao
- Department of Physics and Astronomy, George Mason University, Fairfax, Virginia 22030, USA
- Quantum Materials Center, George Mason University, Fairfax, Virginia 22030, USA
| | - W Vincent Liu
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
- Wilczek Quantum Center, School of Physics and Astronomy and T.D. Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
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Yan Z. Higher-Order Topological Odd-Parity Superconductors. PHYSICAL REVIEW LETTERS 2019; 123:177001. [PMID: 31702245 DOI: 10.1103/physrevlett.123.177001] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2019] [Indexed: 06/10/2023]
Abstract
The topological property of a gapped odd-parity superconductor is jointly determined by its pairing nodes and Fermi surfaces in the normal state. We reveal that the contractibility of Fermi surfaces without crossing any time-reversal invariant momentum and the presence of nontrivial Berry phase on Fermi surfaces are two key conditions for the realization of higher-order topological odd-parity superconductors. When the normal state is a normal metal, we reveal the necessity of removable Dirac pairing nodes and provide a general and simple principle to realize higher-order topological odd-parity superconductors. Our findings can be applied to design new platforms of higher-order topological superconductors, as well as higher-order topological insulators owing to their direct analogy in Hamiltonian description.
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Affiliation(s)
- Zhongbo Yan
- School of Physics, Sun Yat-Sen University, Guangzhou 510275, China
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