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Long Y, Zhang B. Unsupervised Data-Driven Classification of Topological Gapped Systems with Symmetries. PHYSICAL REVIEW LETTERS 2023; 130:036601. [PMID: 36763386 DOI: 10.1103/physrevlett.130.036601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2022] [Revised: 11/18/2022] [Accepted: 12/20/2022] [Indexed: 06/18/2023]
Abstract
A remarkable breakthrough in topological phase classification is the establishment of the topological periodic table, which is mainly based on the classifying space analysis or K theory, but not based on concrete Hamiltonians that possess finite bands or arise in a lattice. As a result, it is still difficult to identify the topological phase of an arbitrary Hamiltonian; the common practice is, instead, to check the incomplete and still growing list of topological invariants one by one, very often by trial and error. Here, we develop unsupervised classifications of topological gapped systems with symmetries, and demonstrate the data-driven construction of the topological periodic table without a priori knowledge of topological invariants. This unsupervised data-driven strategy can take into account spatial symmetries, and further classify phases that were previously classified as trivial in the past. Our Letter introduces machine learning into topological phase classification and paves the way for intelligent explorations of new phases of topological matter.
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Affiliation(s)
- Yang Long
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
| | - Baile Zhang
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
- Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore 637371, Singapore
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2
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Mak KF, Shan J. Semiconductor moiré materials. NATURE NANOTECHNOLOGY 2022; 17:686-695. [PMID: 35836003 DOI: 10.1038/s41565-022-01165-6] [Citation(s) in RCA: 74] [Impact Index Per Article: 37.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2022] [Accepted: 05/26/2022] [Indexed: 06/15/2023]
Abstract
Moiré materials have emerged as a platform for exploring the physics of strong electronic correlations and non-trivial band topology. Here we review the recent progress in semiconductor moiré materials, with a particular focus on transition metal dichalcogenides. Following a brief overview of the general features in this class of materials, we discuss recent theoretical and experimental studies on Hubbard physics, Kane-Mele-Hubbard physics and equilibrium moiré excitons. We also comment on the future opportunities and challenges in the studies of transition metal dichalcogenide and other semiconductor moiré materials.
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Affiliation(s)
- Kin Fai Mak
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA.
- School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA.
- Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY, USA.
| | - Jie Shan
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA.
- School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA.
- Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY, USA.
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3
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Li T, Jiang S, Shen B, Zhang Y, Li L, Tao Z, Devakul T, Watanabe K, Taniguchi T, Fu L, Shan J, Mak KF. Quantum anomalous Hall effect from intertwined moiré bands. Nature 2021; 600:641-646. [PMID: 34937897 DOI: 10.1038/s41586-021-04171-1] [Citation(s) in RCA: 91] [Impact Index Per Article: 30.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/05/2021] [Accepted: 10/20/2021] [Indexed: 11/09/2022]
Abstract
Electron correlation and topology are two central threads of modern condensed matter physics. Semiconductor moiré materials provide a highly tuneable platform for studies of electron correlation1-12. Correlation-driven phenomena, including the Mott insulator2-5, generalized Wigner crystals2,6,9, stripe phases10 and continuous Mott transition11,12, have been demonstrated. However, non-trivial band topology has remained unclear. Here we report the observation of a quantum anomalous Hall effect in AB-stacked MoTe2 /WSe2 moiré heterobilayers. Unlike in the AA-stacked heterobilayers11, an out-of-plane electric field not only controls the bandwidth but also the band topology by intertwining moiré bands centred at different layers. At half band filling, corresponding to one particle per moiré unit cell, we observe quantized Hall resistance, h/e2 (with h and e denoting the Planck's constant and electron charge, respectively), and vanishing longitudinal resistance at zero magnetic field. The electric-field-induced topological phase transition from a Mott insulator to a quantum anomalous Hall insulator precedes an insulator-to-metal transition. Contrary to most known topological phase transitions13, it is not accompanied by a bulk charge gap closure. Our study paves the way for discovery of emergent phenomena arising from the combined influence of strong correlation and topology in semiconductor moiré materials.
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Affiliation(s)
- Tingxin Li
- School of Applied and Engineering Physics and Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA.,Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China
| | - Shengwei Jiang
- School of Applied and Engineering Physics and Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA.,Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China
| | - Bowen Shen
- School of Applied and Engineering Physics and Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA
| | - Yang Zhang
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Lizhong Li
- School of Applied and Engineering Physics and Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA
| | - Zui Tao
- School of Applied and Engineering Physics and Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA
| | - Trithep Devakul
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Kenji Watanabe
- National Institute for Materials Science, Tsukuba, Japan
| | | | - Liang Fu
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Jie Shan
- School of Applied and Engineering Physics and Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA. .,Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY, USA.
| | - Kin Fai Mak
- School of Applied and Engineering Physics and Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA. .,Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY, USA.
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Xu H, Zhou J, Li J. Light-Induced Quantum Anomalous Hall Effect on the 2D Surfaces of 3D Topological Insulators. ADVANCED SCIENCE (WEINHEIM, BADEN-WURTTEMBERG, GERMANY) 2021; 8:e2101508. [PMID: 34216114 PMCID: PMC8425926 DOI: 10.1002/advs.202101508] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/12/2021] [Revised: 05/08/2021] [Indexed: 06/13/2023]
Abstract
Quantum anomalous Hall (QAH) effect generates quantized electric charge Hall conductance without external magnetic field. It requires both nontrivial band topology and time-reversal symmetry (TRS) breaking. In most cases, one can break the TRS of time-reversal invariant topological materials to yield QAH effect, which is essentially a topological phase transition. However, conventional topological phase transition induced by external field/stimulus usually needs a route along which the bandgap closes and reopens. Hence, the transition occurs only when the magnitude of field/stimulus is larger than a critical value. In this work the authors propose that using gapless systems, the transition can happen at an arbitrarily weak (but finite) external field strength. For such an unconventional topological phase transition, the bandgap closing is guaranteed by bulk-edge correspondence and symmetries, while the bandgap reopening is induced by external fields. This concept is demonstrated on the 2D surface states of 3D topological insulators like Bi2 Se3 , which become 2D QAH insulators once a circularly polarized light is turned on, according to the Floquet time crystal theory. The sign of quantized Chern number can be controlled via the chirality of the light. This provides a convenient and dynamic approach to trigger topological phase transitions and create QAH insulators.
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Affiliation(s)
- Haowei Xu
- Department of Nuclear Science and EngineeringMassachusetts Institute of TechnologyCambridgeMA02139USA
| | - Jian Zhou
- Department of Nuclear Science and EngineeringMassachusetts Institute of TechnologyCambridgeMA02139USA
| | - Ju Li
- Department of Nuclear Science and EngineeringMassachusetts Institute of TechnologyCambridgeMA02139USA
- Department of Materials Science and EngineeringMassachusetts Institute of TechnologyCambridgeMA02139USA
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5
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Bi S, He Y, Li P. Exact solution to sodium-iridate-BCS- Hubbard model along the symmetric line: non-trivial topology in the ferromagnetic order. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:145601. [PMID: 33494082 DOI: 10.1088/1361-648x/abdf94] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/11/2020] [Accepted: 01/25/2021] [Indexed: 06/12/2023]
Abstract
We study the sodium-iridates model on the honeycomb lattice with both BCS pairing potential and Hubbard interaction term. It is shown that this model can be exactly solved with appropriate choices of amplitude of pairing gaps, where the interacting terms are transformed to external field terms. The band structures of these exact solutions on both torus and cylinder geometry are discussed in great details. It is found that the ground state assumes an anti-ferromagnetic configuration, which breaks the time reversal symmetry spontaneously and renders the superconductor topologically trivial. On the other hand, the nontrivial topology is preserved with ferromagnetic configuration and can be characterized by the isospin Chern number.
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Affiliation(s)
- Shihao Bi
- College of Physics, Sichuan University, 610064, Chengdu, People's Republic of China
- Key Laboratory of High Energy Density Physics and Technology of Ministry of Education, Sichuan University, 610064, Chengdu, People's Republic of China
| | - Yan He
- College of Physics, Sichuan University, 610064, Chengdu, People's Republic of China
- Key Laboratory of High Energy Density Physics and Technology of Ministry of Education, Sichuan University, 610064, Chengdu, People's Republic of China
| | - Peng Li
- College of Physics, Sichuan University, 610064, Chengdu, People's Republic of China
- Key Laboratory of High Energy Density Physics and Technology of Ministry of Education, Sichuan University, 610064, Chengdu, People's Republic of China
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6
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Fedoseev AD. Corner excitations in the 2D triangle-shaped topological insulator with chiral superconductivity on the triangular lattice. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 32:405302. [PMID: 32554877 DOI: 10.1088/1361-648x/ab9e2f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2020] [Accepted: 06/18/2020] [Indexed: 06/11/2023]
Abstract
The 2D triangle-shapedC3-symmetric topological insulator with the chiral superconducting coupling on the triangular lattice is investigated. While such a system cannot provide the topologically protected corner excitations, we report the presence of the nontopological corner excitations with energy value to lie in the first-order edge spectrum gap. Though these excitations are not topologically protected, they appear for a rather wide range of the parameters values and are robust against the boundary defects and weak disorder. We reveal the presence of the Majorana corner states, which appear along the line in the parameter space.
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Affiliation(s)
- A D Fedoseev
- Federal Research Center KSC SB RAS, Kirensky Institute of Physics, Akademgorodok 50/38, 660036 Krasnoyarsk, Russia
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7
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Ezawa M. Second-order topological insulators and loop-nodal semimetals in Transition Metal Dichalcogenides XTe 2 (X = Mo, W). Sci Rep 2019; 9:5286. [PMID: 30918317 PMCID: PMC6437301 DOI: 10.1038/s41598-019-41746-5] [Citation(s) in RCA: 34] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/09/2018] [Accepted: 03/04/2019] [Indexed: 11/22/2022] Open
Abstract
Transition metal dichalcogenides XTe2 (X = Mo, W) have been shown to be second-order topological insulators based on first-principles calculations, while topological hinge states have been shown to emerge based on the associated tight-binding model. The model is equivalent to the one constructed from a loop-nodal semimetal by adding mass terms and spin-orbit interactions. We propose to study a chiral-symmetric model obtained from the original Hamiltonian by simplifying it but keeping almost identical band structures and topological hinge states. A merit is that we are able to derive various analytic formulas because of chiral symmetry, which enables us to reveal basic topological properties of transition metal dichalcogenides. We find a linked loop structure where a higher linking number (even 8) is realized. We construct second-order topological semimetals and two-dimensional second-order topological insulators based on this model. It is interesting that topological phase transitions occur without gap closing between a topological insulator, a topological crystalline insulator and a second-order topological insulator. We propose to characterize them by symmetry detectors discriminating whether the symmetry is preserved or not. They differentiate topological phases although the symmetry indicators yield identical values to them. We also show that topological hinge states are controllable by the direction of magnetization. When the magnetization points the z direction, the hinges states shift, while they are gapped when it points the in-plane direction. Accordingly, the quantized conductance is switched by controlling the magnetization direction. Our results will be a basis of future topological devices based on transition metal dichalcogenides.
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Affiliation(s)
- Motohiko Ezawa
- Department of Applied Physics, University of Tokyo, Hongo, 7-3-1, 113-8656, Japan.
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8
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Zhu Q, Tu MWY, Tong Q, Yao W. Gate tuning from exciton superfluid to quantum anomalous Hall in van der Waals heterobilayer. SCIENCE ADVANCES 2019; 5:eaau6120. [PMID: 30746454 PMCID: PMC6357754 DOI: 10.1126/sciadv.aau6120] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2018] [Accepted: 12/04/2018] [Indexed: 05/31/2023]
Abstract
Van der Waals heterostructures of two-dimensional (2D) materials provide a powerful approach toward engineering various quantum phases of matter. Examples include topological matter such as quantum spin Hall (QSH) insulator and correlated matter such as exciton superfluid. It can be of great interest to realize these vastly different quantum phases matter on a common platform; however, their distinct origins tend to restrict them to material systems of incompatible characters. Here, we show that heterobilayers of 2D valley semiconductors can be tuned through interlayer bias between an exciton superfluid, a quantum anomalous Hall insulator, and a QSH insulator. The tunability between these distinct phases results from the competition of Coulomb interaction with the interlayer quantum tunneling that has a chiral form in valley semiconductors. Our findings point to exciting opportunities for harnessing both protected topological edge channels and bulk superfluidity in an electrically configurable platform.
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9
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Rachel S. Interacting topological insulators: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2018; 81:116501. [PMID: 30057370 DOI: 10.1088/1361-6633/aad6a6] [Citation(s) in RCA: 53] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/02/2023]
Abstract
The discovery of the quantum spin Hall effect and topological insulators more than a decade ago has revolutionized modern condensed matter physics. Today, the field of topological states of matter is one of the most active and fruitful research areas for both experimentalists and theorists. The physics of topological insulators is typically well described by band theory and systems of non-interacting fermions. In contrast, several of the most fascinating effects in condensed matter physics merely exist due to electron-electron interactions, examples include unconventional superconductivity, the Kondo effect, and the Mott-Hubbard transition. The aim of this review article is to give an overview of the manifold directions which emerge when topological bandstructures and correlation physics interfere and compete. These include the study of the stability of topological bandstructures and correlated topological insulators. Interaction-induced topological phases such as the topological Kondo insulator provide another exciting topic. More exotic states of matter such as topological Mott insulator and fractional Chern insulators only exist due to the interplay of topology and strong interactions and do not have any bandstructure analogue. Eventually the relation between topological bandstructures and frustrated quantum magnetism in certain transition metal oxides is emphasized.
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Affiliation(s)
- Stephan Rachel
- School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
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10
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Zhang W, Chen M, Dai J, Wang X, Zhong Z, Cheong SW, Wu W. Topological Phase Transition with Nanoscale Inhomogeneity in (Bi 1- xIn x) 2Se 3. NANO LETTERS 2018; 18:2677-2682. [PMID: 29582663 DOI: 10.1021/acs.nanolett.8b00597] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Topological insulators are a class of band insulators with nontrivial topology, a result of band inversion due to the strong spin-orbit coupling. The transition between topological and normal insulator can be realized by tuning the spin-orbit coupling strength and has been observed experimentally. However, the impact of chemical disorders on the topological phase transition was not addressed in previous studies. Herein, we report a systematic scanning tunneling microscopy/spectroscopy and first-principles study of the topological phase transition in single crystals of In-doped Bi2Se3. Surprisingly, no band gap closure was observed across the transition. Furthermore, our spectroscopic-imaging results reveal that In defects are extremely effective "suppressors" of the band inversion, which leads to microscopic phase separation of topological-insulator-like and normal-insulator-like nano regions across the "transition". The observed topological electronic inhomogeneity demonstrates the significant impact of chemical disorders in topological materials, shedding new light on the fundamental understanding of topological phase transition.
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Affiliation(s)
- Wenhan Zhang
- Department of Physics and Astronomy , Rutgers University , Piscataway , New Jersey 08854 , United States
| | - Mingxing Chen
- College of Physics and Information Science , Hunan Normal University , Changsha , Hunan 410081 , China
| | - Jixia Dai
- Department of Physics and Astronomy , Rutgers University , Piscataway , New Jersey 08854 , United States
| | - Xueyun Wang
- Department of Physics and Astronomy , Rutgers University , Piscataway , New Jersey 08854 , United States
- School of Aerospace Engineering , Beijing Institute of Technology , Beijing 100081 , China
| | - Zhicheng Zhong
- Key Laboratory of Magnetic Materials and Devices , Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences , Ningbo 315201 , China
| | - Sang-Wook Cheong
- Department of Physics and Astronomy , Rutgers University , Piscataway , New Jersey 08854 , United States
- Rutgers Center for Emergent Materials , Rutgers University , Piscataway , New Jersey 08854 , United States
| | - Weida Wu
- Department of Physics and Astronomy , Rutgers University , Piscataway , New Jersey 08854 , United States
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11
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Anisotropic Friedel oscillations in graphene-like materials: The Dirac point approximation in wave-number dependent quantities revisited. Sci Rep 2018; 8:2667. [PMID: 29422619 PMCID: PMC5805790 DOI: 10.1038/s41598-018-19730-2] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2017] [Accepted: 12/15/2017] [Indexed: 11/08/2022] Open
Abstract
Friedel oscillations of the graphene-like materials are investigated theoretically for low and intermediate Fermi energies. Numerical calculations have been performed within the random phase approximation. It was demonstrated that for intra-valley transitions the contribution of the different Dirac points in the wave-number dependent quantities is determined by the orientation of the wave-number in k-space. Therefore, identical contribution of the different Dirac points is not automatically guaranteed by the degeneracy of the Hamiltonian at these points. Meanwhile, it was shown that the contribution of the inter-valley transitions is always anisotropic even when the Dirac points coincide with the Fermi level (E F = 0). This means that the Dirac point approximation based studies could give the correct physics only at long wave length limit. The anisotropy of the static dielectric function reveals different contribution of the each Dirac point. Additionally, the anisotropic k-space dielectric function results in anisotropic Friedel oscillations in graphene-like materials. Increasing the Rashba interaction strength slightly modifies the Friedel oscillations in this family of materials. Anisotropy of the dielectric function in k-space is the clear manifestation of band anisotropy in the graphene-like systems.
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12
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Mandal PS, Springholz G, Volobuev VV, Caha O, Varykhalov A, Golias E, Bauer G, Rader O, Sánchez-Barriga J. Topological quantum phase transition from mirror to time reversal symmetry protected topological insulator. Nat Commun 2017; 8:968. [PMID: 29042565 PMCID: PMC5645419 DOI: 10.1038/s41467-017-01204-0] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2016] [Accepted: 08/24/2017] [Indexed: 11/10/2022] Open
Abstract
Topological insulators constitute a new phase of matter protected by symmetries. Time-reversal symmetry protects strong topological insulators of the Z2 class, which possess an odd number of metallic surface states with dispersion of a Dirac cone. Topological crystalline insulators are merely protected by individual crystal symmetries and exist for an even number of Dirac cones. Here, we demonstrate that Bi-doping of Pb1-x Sn x Se (111) epilayers induces a quantum phase transition from a topological crystalline insulator to a Z2 topological insulator. This occurs because Bi-doping lifts the fourfold valley degeneracy and induces a gap at [Formula: see text], while the three Dirac cones at the [Formula: see text] points of the surface Brillouin zone remain intact. We interpret this new phase transition as caused by a lattice distortion. Our findings extend the topological phase diagram enormously and make strong topological insulators switchable by distortions or electric fields.Transitions between topological phases of matter protected by different symmetries remain rare. Here, Mandal et al. report a quantum phase transition from a topological crystalline insulator to a Z2 topological insulator by doping Bi into Pb1-x Sn x Se (111) thin films.
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Affiliation(s)
- Partha S Mandal
- Helmholtz-Zentrum Berlin für Materialien und Energie, Albert-Einstein Strasse 15, 12489, Berlin, Germany.,Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht Street 24/25, 14476, Potsdam, Germany
| | - Gunther Springholz
- Institute for Semiconductor and Solid State Physics, Johannes Kepler Universität, Altenberger Strasse 69, 4040, Linz, Austria
| | - Valentine V Volobuev
- Institute for Semiconductor and Solid State Physics, Johannes Kepler Universität, Altenberger Strasse 69, 4040, Linz, Austria.,National Technical University "Kharkiv Polytechnic Institute", Frunze Street 21, 61002, Kharkiv, Ukraine
| | - Ondrej Caha
- Department of Condensed Matter Physics, Masaryk University, Kotlářská 267/2, 61137, Brno, Czech Republic
| | - Andrei Varykhalov
- Helmholtz-Zentrum Berlin für Materialien und Energie, Albert-Einstein Strasse 15, 12489, Berlin, Germany
| | - Evangelos Golias
- Helmholtz-Zentrum Berlin für Materialien und Energie, Albert-Einstein Strasse 15, 12489, Berlin, Germany
| | - Günther Bauer
- Institute for Semiconductor and Solid State Physics, Johannes Kepler Universität, Altenberger Strasse 69, 4040, Linz, Austria
| | - Oliver Rader
- Helmholtz-Zentrum Berlin für Materialien und Energie, Albert-Einstein Strasse 15, 12489, Berlin, Germany
| | - Jaime Sánchez-Barriga
- Helmholtz-Zentrum Berlin für Materialien und Energie, Albert-Einstein Strasse 15, 12489, Berlin, Germany.
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13
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Mohanta N, Kampf AP, Kopp T. Emergent Momentum-Space Skyrmion Texture on the Surface of Topological Insulators. Sci Rep 2017; 7:45664. [PMID: 28378779 PMCID: PMC5381213 DOI: 10.1038/srep45664] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/18/2016] [Accepted: 03/02/2017] [Indexed: 11/09/2022] Open
Abstract
The quantum anomalous Hall effect has been theoretically predicted and experimentally verified in magnetic topological insulators. In addition, the surface states of these materials exhibit a hedgehoglike “spin” texture in momentum space. Here, we apply the previously formulated low-energy model for Bi2Se3, a parent compound for magnetic topological insulators, to a slab geometry in which an exchange field acts only within one of the surface layers. In this sample set up, the hedgehog transforms into a skyrmion texture beyond a critical exchange field. This critical field marks a transition between two topologically distinct phases. The topological phase transition takes place without energy gap closing at the Fermi level and leaves the transverse Hall conductance unchanged and quantized to e2/2h. The momentum-space skyrmion texture persists in a finite field range. It may find its realization in hybrid heterostructures with an interface between a three-dimensional topological insulator and a ferromagnetic insulator.
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Affiliation(s)
- Narayan Mohanta
- Center for Electronic Correlations and Magnetism, Theoretical Physics III, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany
| | - Arno P Kampf
- Center for Electronic Correlations and Magnetism, Theoretical Physics III, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany
| | - Thilo Kopp
- Center for Electronic Correlations and Magnetism, Experimental Physics VI, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany
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14
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Liu X, Agarwal GS. The New Phases due to Symmetry Protected Piecewise Berry Phases; Enhanced Pumping and Non-reciprocity in Trimer Lattices. Sci Rep 2017; 7:45015. [PMID: 28337994 PMCID: PMC5364478 DOI: 10.1038/srep45015] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/19/2016] [Accepted: 02/20/2017] [Indexed: 11/09/2022] Open
Abstract
Finding new phase of matter is a fundamental task in physics. Generally, various phases or states of matter (for instance solid/liquid/gas phases) have different symmetries, the phase transitions among them can be explained by Landau’s symmetry breaking theory. The topological phases discovered in recent years show that different phases may have the same symmetry. The different topological phases are characterized by different integer values of the Berry phases. By studying one dimensional (1D) trimer lattices we report new phases beyond topological phases. The new phases that we find are characterized by piecewise continuous Berry phases with the discontinuity occurring at the transition point. With time-dependent changes in trimer lattices, we can generate two dimensional (2D) phases, which are characterized by the Berry phase of half period. This half-period Berry phase changes smoothly within one state of the system while changes discontinuously at the transition point. We further demonstrate the existence of adiabatic pumping for each phase and gain assisted enhanced pumping. The non reciprocity of the pumping process makes the system a good optical diode.
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Affiliation(s)
- Xuele Liu
- 120 W Miller Ave, Stillwater, Oklahoma 74078, USA
| | - G S Agarwal
- Institute for Quantum Science and Engineering, Department of Biological and Agricultural Engineering, Texas A&M University, College Station, TX 77845, USA.,The Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
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15
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Cheng Q, Zhang Y, Zhang K, Jin B, Zhang C. Symmetry analysis of transport properties in helical superconductor junctions. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:085601. [PMID: 28081002 DOI: 10.1088/1361-648x/aa5383] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We study the discrete symmetries satisfied by helical p-wave superconductors with the d-vectors [Formula: see text] or [Formula: see text] and the transformations brought by symmetry operations to ferromagnet and spin-singlet superconductors, which show intimate associations with the transport properties in heterojunctions, including helical superconductors. In particular, the partial symmetries of the Hamiltonian under spin-rotation and gauge-rotation operations are responsible for the novel invariances of the conductance in tunnel junctions and the new selection rules for the lowest current and peculiar phase diagrams in Josephson junctions, which were reported recently. The symmetries of constructed free energies for Josephson junctions are also analyzed, and are consistent with the results from the Hamiltonian.
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Affiliation(s)
- Qiang Cheng
- School of Science, Qingdao Technological University, Qingdao, Shandong 266520, People's Republic of China
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16
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Rachel S. Quantum phase transitions of topological insulators without gap closing. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:405502. [PMID: 27530509 DOI: 10.1088/0953-8984/28/40/405502] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We consider two-dimensional Chern insulators and time-reversal invariant topological insulators and discuss the effect of perturbations breaking either particle-number conservation or time-reversal symmetry. The appearance of trivial mass terms is expected to cause quantum phase transitions into trivial phases when such a perturbation overweighs the topological term. These phase transitions are usually associated with a bulk-gap closing. In contrast, the chiral Chern insulator is unaffected by particle-number breaking perturbations. Moreover, the [Formula: see text] topological insulator undergoes phase transitions into topologically trivial phases without bulk-gap closing in the presence of any of such perturbations. In certain cases, these phase transitions can be circumvented and the protection restored by another U(1) symmetry, e.g. due to spin conservation. These findings are discussed in the context of interacting topological insulators.
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Affiliation(s)
- Stephan Rachel
- Institute for Theoretical Physics, TU Dresden, 01062 Dresden, Germany
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17
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Sun SJ, Chung CH, Chang YY, Tsai WF, Zhang FC. Helical Majorana fermions in d(x2-y2) + id(xy)-wave topological superconductivity of doped correlated quantum spin Hall insulators. Sci Rep 2016; 6:24102. [PMID: 27064108 PMCID: PMC4827071 DOI: 10.1038/srep24102] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2015] [Accepted: 03/16/2016] [Indexed: 11/09/2022] Open
Abstract
There has been growing interest in searching for exotic self-conjugate, charge-neutral low-energy fermionic quasi-particles, known as Majorana fermions (MFs) in solid state systems. Their signatures have been proposed and potentially observed at edges of topological superconcuctors with non-trivial topological invariant in the bulk electronic band structure. Much effort have been focused on realizing MFs in odd-parity superconductors made of strong spin-orbit coupled materials in proximity to conventional superconductors. In this paper, we propose a novel mechanism for realizing MFs in 2D spin-singlet topological superconducting state induced by doping a correlated quantum spin Hall (Kane-Mele) insulator. Via a renormalized mean-field approach, the system is found to exhibits time-reversal symmetry (TRS) breaking d(x2-y2) + id(xy)-wave (chiral d-wave) superconductivity near half-filling in the limit of large on-site repulsion. Surprisingly, however, at large spin-orbit coupling, the system undergoes a topological phase transition and enter into a new topological phase protected by a pseudo-spin Chern number, which can be viewed as a persistent extension of the quantum spin Hall phase upon doping. From bulk-edge correspondence, this phase is featured by the presence of two pairs of counter-propagating helical Majorana modes per edge, instead of two chiral propagating edge modes in the d + id' superconductors.
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Affiliation(s)
- Shih-Jye Sun
- Department of Applied Physics, National University of Kaohsiung, Kaohsiung, Taiwan, R.O.C.
| | - Chung-Hou Chung
- Electrophysics Department, National Chiao-Tung University, HsinChu, Taiwan, 300, R.O.C.
- Physics Division, National Center for Theoretical Sciences, HsinChu, Taiwan, 300 R.O.C.
| | - Yung-Yeh Chang
- Electrophysics Department, National Chiao-Tung University, HsinChu, Taiwan, 300, R.O.C.
| | - Wei-Feng Tsai
- Department of Physics, National Sun Yat-Sen University, Kaohsiung, Taiwan, R.O.C.
| | - Fu-Chun Zhang
- Department of Physics, Zhejiang University, Hangzhou, China
- Collaborative Innovation Center of Advanced Microstructures, Nanjing, China
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18
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Timonin PN, Chitov GY. Exploring percolative landscapes: Infinite cascades of geometric phase transitions. Phys Rev E 2016; 93:012102. [PMID: 26871019 DOI: 10.1103/physreve.93.012102] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2015] [Indexed: 11/07/2022]
Abstract
The evolution of many kinetic processes in 1+1 (space-time) dimensions results in 2D directed percolative landscapes. The active phases of these models possess numerous hidden geometric orders characterized by various types of large-scale and/or coarse-grained percolative backbones that we define. For the patterns originated in the classical directed percolation (DP) and contact process we show from the Monte Carlo simulation data that these percolative backbones emerge at specific critical points as a result of continuous phase transitions. These geometric transitions belong to the DP universality class and their nonlocal order parameters are the capacities of corresponding backbones. The multitude of conceivable percolative backbones implies the existence of infinite cascades of such geometric transitions in the kinetic processes considered. We present simple arguments to support the conjecture that such cascades of transitions are a generic feature of percolation as well as of many other transitions with nonlocal order parameters.
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Affiliation(s)
- P N Timonin
- Physics Research Institute, Southern Federal University, 344090, Stachki 194, Rostov-on-Don, Russia
| | - Gennady Y Chitov
- Department of Physics, Laurentian University, Sudbury, Ontario, Canada P3E 2C6
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19
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Lau A, Ortix C, van den Brink J. Topological Edge States with Zero Hall Conductivity in a Dimerized Hofstadter Model. PHYSICAL REVIEW LETTERS 2015; 115:216805. [PMID: 26636866 DOI: 10.1103/physrevlett.115.216805] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2015] [Indexed: 06/05/2023]
Abstract
The Hofstadter model is a simple yet powerful Hamiltonian to study quantum Hall physics in a lattice system, manifesting its essential topological states. Lattice dimerization in the Hofstadter model opens an energy gap at half filling. Here we show that even if the ensuing insulator has a Chern number equal to zero, concomitantly a doublet of edge states appear that are pinned at specific momenta. We demonstrate that these states are topologically protected by inversion symmetry in specific one-dimensional cuts in momentum space, define and calculate the corresponding invariants, and identify a platform for the experimental detection of these novel topological states.
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Affiliation(s)
- Alexander Lau
- Institute for Theoretical Solid State Physics, IFW Dresden, 01171 Dresden, Germany
| | - Carmine Ortix
- Institute for Theoretical Solid State Physics, IFW Dresden, 01171 Dresden, Germany
- Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, Netherlands
| | - Jeroen van den Brink
- Institute for Theoretical Solid State Physics, IFW Dresden, 01171 Dresden, Germany
- Department of Physics, TU Dresden, 01062 Dresden, Germany
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20
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Zhang G, Song Z. Topological Characterization of Extended Quantum Ising Models. PHYSICAL REVIEW LETTERS 2015; 115:177204. [PMID: 26551140 DOI: 10.1103/physrevlett.115.177204] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/01/2015] [Indexed: 06/05/2023]
Abstract
We show that a class of exactly solvable quantum Ising models, including the transverse-field Ising model and anisotropic XY model, can be characterized as the loops in a two-dimensional auxiliary space. The transverse-field Ising model corresponds to a circle and the XY model corresponds to an ellipse, while other models yield cardioid, limacon, hypocycloid, and Lissajous curves etc. It is shown that the variation of the ground state energy density, which is a function of the loop, experiences a nonanalytical point when the winding number of the corresponding loop changes. The winding number can serve as a topological quantum number of the quantum phases in the extended quantum Ising model, which sheds some light upon the relation between quantum phase transition and the geometrical order parameter characterizing the phase diagram.
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Affiliation(s)
- G Zhang
- School of Physics, Nankai University, Tianjin 300071, China
| | - Z Song
- School of Physics, Nankai University, Tianjin 300071, China
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21
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Amaricci A, Budich JC, Capone M, Trauzettel B, Sangiovanni G. First-order character and observable signatures of topological quantum phase transitions. PHYSICAL REVIEW LETTERS 2015; 114:185701. [PMID: 26001010 DOI: 10.1103/physrevlett.114.185701] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/05/2014] [Indexed: 06/04/2023]
Abstract
Topological quantum phase transitions are characterized by changes in global topological invariants. These invariants classify many-body systems beyond the conventional paradigm of local order parameters describing spontaneous symmetry breaking. For noninteracting electrons, it is well understood that such transitions are continuous and always accompanied by a gap closing in the energy spectrum, given that the symmetries protecting the topological phase are maintained. Here, we demonstrate that a sufficiently strong electron-electron interaction can fundamentally change the situation: we discover a topological quantum phase transition of first-order character in the genuine thermodynamic sense that occurs without a gap closing. Our theoretical study reveals the existence of a quantum critical endpoint associated with an orbital instability on the transition line between a 2D topological insulator and a trivial band insulator. Remarkably, this phenomenon entails unambiguous signatures related to the orbital occupations that can be detected experimentally.
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Affiliation(s)
- A Amaricci
- Democritos National Simulation Center, Consiglio Nazionale delle Ricerche, Istituto Officina dei Materiali (IOM) and Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, 34136 Trieste, Italy
| | - J C Budich
- Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, 6020 Innsbruck, Austria
| | - M Capone
- Democritos National Simulation Center, Consiglio Nazionale delle Ricerche, Istituto Officina dei Materiali (IOM) and Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, 34136 Trieste, Italy
| | - B Trauzettel
- Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
| | - G Sangiovanni
- Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
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Mizushima T, Tsutsumi Y, Sato M, Machida K. Symmetry protected topological superfluid (3)He-B. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:113203. [PMID: 25730099 DOI: 10.1088/0953-8984/27/11/113203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Owing to the richness of symmetry and well-established knowledge of bulk superfluidity, the superfluid (3)He has offered a prototypical system to study intertwining of topology and symmetry. This article reviews recent progress in understanding the topological superfluidity of (3)He in a multifaceted manner, including symmetry considerations, the Jackiw-Rebbi's index theorem, and the quasiclassical theory. Special focus is placed on the symmetry protected topological superfuidity of the (3)He-B confined in a slab geometry. The (3)He-B under a magnetic field is separated to two different sub-phases: the symmetry protected topological phase and non-topological phase. The former phase is characterized by the existence of symmetry protected Majorana fermions. The topological phase transition between them is triggered by the spontaneous breaking of a hidden discrete symmetry. The critical field is quantitatively determined from the microscopic calculation that takes account of magnetic dipole interaction of the (3)He nucleus. It is also demonstrated that odd-frequency even-parity Cooper pair amplitudes are emergent in low-lying quasiparticles. The key ingredients, symmetry protected Majorana fermions and odd-frequency pairing, bring an important consequence that the coupling of the surface states to an applied field is prohibited by the hidden discrete symmetry, while the topological phase transition with the spontaneous symmetry breaking is accompanied by anomalous enhancement and anisotropic quantum criticality of surface spin susceptibility. We also illustrate common topological features between topological crystalline superconductors and symmetry protected topological superfluids, taking UPt3 and Rashba superconductors as examples.
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Ezawa M. Antiferromagnetic topological superconductor and electrically controllable Majorana fermions. PHYSICAL REVIEW LETTERS 2015; 114:056403. [PMID: 25699460 DOI: 10.1103/physrevlett.114.056403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/22/2014] [Indexed: 06/04/2023]
Abstract
We investigate the realization of a topological superconductor in a generic bucked honeycomb system equipped with four types of mass-generating terms, where the superconductor gap is introduced by attaching the honeycomb system to an s-wave superconductor. Constructing the topological phase diagram, we show that Majorana modes are formed in the phase boundary. In particular, we analyze the honeycomb system with antiferromagnetic order in the presence of perpendicular electric field E(z). It becomes topological for |E(z)|>E(z)(cr) and trivial for |E(z)|<E(z)(cr), with E(z)(cr) a certain critical field. It is possible to create a topological spot in a trivial superconductor by controlling applied electric field. One Majorana zero-energy bound state appears at the phase boundary. We can arbitrarily control the position of the Majorana fermion by moving the spot of applied electric field, which will be made possible by a scanning tunneling microscope probe.
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Affiliation(s)
- Motohiko Ezawa
- Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan
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