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Zhuo D, Zhou L, Zhao YF, Zhang R, Yan ZJ, Wang AG, Chan MHW, Liu CX, Chen CZ, Chang CZ. Engineering Plateau Phase Transition in Quantum Anomalous Hall Multilayers. NANO LETTERS 2024; 24:6974-6980. [PMID: 38829211 DOI: 10.1021/acs.nanolett.4c01313] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2024]
Abstract
The plateau phase transition in quantum anomalous Hall (QAH) insulators corresponds to a quantum state wherein a single magnetic domain gives way to multiple domains and then reconverges back to a single magnetic domain. The layer structure of the sample provides an external knob for adjusting the Chern number C of the QAH insulators. Here, we employ molecular beam epitaxy to grow magnetic topological insulator multilayers and realize the magnetic field-driven plateau phase transition between two QAH states with odd Chern number change ΔC. We find that critical exponents extracted for the plateau phase transitions with ΔC = 1 and ΔC = 3 in QAH insulators are nearly identical. We construct a four-layer Chalker-Coddington network model to understand the consistent critical exponents for the plateau phase transitions with ΔC = 1 and ΔC = 3. This work will motivate further investigations into the critical behaviors of plateau phase transitions with different ΔC in QAH insulators.
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Affiliation(s)
- Deyi Zhuo
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Lingjie Zhou
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Yi-Fan Zhao
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Ruoxi Zhang
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Zi-Jie Yan
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Annie G Wang
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Moses H W Chan
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Chao-Xing Liu
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
| | - Chui-Zhen Chen
- Institute for Advanced Study and School of Physical Science and Technology, Soochow University, Suzhou 215006, China
| | - Cui-Zu Chang
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States
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de Castro SG, Lopes JMVP, Ferreira A, Bahamon DA. Fast Fourier-Chebyshev Approach to Real-Space Simulations of the Kubo Formula. PHYSICAL REVIEW LETTERS 2024; 132:076302. [PMID: 38427886 DOI: 10.1103/physrevlett.132.076302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/16/2023] [Accepted: 01/19/2024] [Indexed: 03/03/2024]
Abstract
The Kubo formula is a cornerstone in our understanding of near-equilibrium transport phenomena. While conceptually elegant, the application of Kubo's linear-response theory to interesting problems is hindered by the need for algorithms that are accurate and scalable to large lattice sizes beyond one spatial dimension. Here, we propose a general framework to numerically study large systems, which combines the spectral accuracy of Chebyshev expansions with the efficiency of divide-and-conquer methods. We use the hybrid algorithm to calculate the two-terminal conductance and the bulk conductivity tensor of 2D lattice models with over 10^{7} sites. By efficiently sampling the microscopic information contained in billions of Chebyshev moments, the algorithm is able to accurately resolve the linear-response properties of complex systems in the presence of quenched disorder. Our results lay the groundwork for future studies of transport phenomena in previously inaccessible regimes.
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Affiliation(s)
- Santiago Giménez de Castro
- School of Engineering, Mackenzie Presbyterian University, São Paulo - 01302-907, Brazil
- MackGraphe - Graphene and Nanomaterials Research Institute, Mackenzie Presbyterian University, São Paulo -01302-907, Brazil
| | - João M Viana Parente Lopes
- Centro de Física das Universidades do Minho e Porto, LaPMET, Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal
| | - Aires Ferreira
- School of Physics, Engineering and Technology and York Centre for Quantum Technologies, University of York, York YO10 5DD, United Kingdom
| | - D A Bahamon
- School of Engineering, Mackenzie Presbyterian University, São Paulo - 01302-907, Brazil
- MackGraphe - Graphene and Nanomaterials Research Institute, Mackenzie Presbyterian University, São Paulo -01302-907, Brazil
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3
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Dresselhaus EJ, Sbierski B, Gruzberg IA. Scaling Collapse of Longitudinal Conductance near the Integer Quantum Hall Transition. PHYSICAL REVIEW LETTERS 2022; 129:026801. [PMID: 35867470 DOI: 10.1103/physrevlett.129.026801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2022] [Accepted: 05/09/2022] [Indexed: 06/15/2023]
Abstract
Within the mature field of Anderson transitions, the critical properties of the integer quantum Hall transition still pose a significant challenge. Numerical studies of the transition suffer from strong corrections to scaling for most observables. In this Letter, we suggest to overcome this problem by using the longitudinal conductance g of the network model as the scaling observable, which we compute for system sizes nearly 2 orders of magnitude larger than in previous studies. We show numerically that the sizable corrections to scaling of g can be accounted for in a remarkably simple form, which leads to an excellent scaling collapse. Surprisingly, the scaling function turns out to be indistinguishable from a Gaussian. We propose a cost-function-based approach and estimate ν=2.609(7) for the localization length exponent, consistent with previous results, but considerably more precise than in most works on this problem. Extending previous approaches for Hamiltonian models, we also confirm our finding using integrated conductance as a scaling variable.
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Affiliation(s)
| | - Björn Sbierski
- Department of Physics and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 Munich, Germany
- Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany
| | - Ilya A Gruzberg
- Department of Physics, Ohio State University, 191 West Woodruff Avenue, Columbus, Ohio 43210, USA
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Liu Y, Qi S, Fang J, Sun J, Liu C, Liu Y, Qi J, Xing Y, Liu H, Lin X, Wang L, Xue QK, Xie XC, Wang J. Observation of In-Plane Quantum Griffiths Singularity in Two-Dimensional Crystalline Superconductors. PHYSICAL REVIEW LETTERS 2021; 127:137001. [PMID: 34623853 DOI: 10.1103/physrevlett.127.137001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/02/2020] [Revised: 01/27/2021] [Accepted: 08/09/2021] [Indexed: 06/13/2023]
Abstract
Quantum Griffiths singularity (QGS) reveals the profound influence of quenched disorder on the quantum phase transitions, characterized by the divergence of the dynamical critical exponent at the boundary of the vortex glasslike phase, named as quantum Griffiths phase. However, in the absence of vortices, whether the QGS can exist under a parallel magnetic field remains a puzzle. Here, we study the magnetic field induced superconductor-metal transition in ultrathin crystalline PdTe_{2} films grown by molecular beam epitaxy. Remarkably, the QGS emerges under both perpendicular and parallel magnetic field in four-monolayer PdTe_{2} films. The direct activated scaling analysis with a new irrelevant correction has been proposed, providing important evidence of QGS. With increasing film thickness to six monolayers, the QGS disappears under perpendicular field but persists under parallel field, and this discordance may originate from the differences in microscopic processes. Our work demonstrates the universality of parallel field induced QGS and can stimulate further investigations on novel quantum phase transitions under parallel magnetic field.
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Affiliation(s)
- Yi Liu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Department of Physics, Renmin University of China, Beijing 100872, China
| | - Shichao Qi
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
| | - Jingchao Fang
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
| | - Jian Sun
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
| | - Chong Liu
- State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China
| | - Yanzhao Liu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
| | - Junjie Qi
- Beijing Academy of Quantum Information Sciences, Beijing 100193, China
| | - Ying Xing
- Department of Materials Science and Engineering, College of New Energy and Materials, China University of Petroleum, Beijing 102249, China
| | - Haiwen Liu
- Center for Advanced Quantum Studies, Department of Physics, Beijing Normal University, Beijing 100875, China
| | - Xi Lin
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Beijing Academy of Quantum Information Sciences, Beijing 100193, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
| | - Lili Wang
- State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China
| | - Qi-Kun Xue
- State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China
- Beijing Academy of Quantum Information Sciences, Beijing 100193, China
| | - X C Xie
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Beijing Academy of Quantum Information Sciences, Beijing 100193, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
| | - Jian Wang
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Beijing Academy of Quantum Information Sciences, Beijing 100193, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
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Li H, Jiang H, Chen CZ, Xie XC. Critical Behavior and Universal Signature of an Axion Insulator State. PHYSICAL REVIEW LETTERS 2021; 126:156601. [PMID: 33929241 DOI: 10.1103/physrevlett.126.156601] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2020] [Accepted: 03/12/2021] [Indexed: 06/12/2023]
Abstract
Recently, the search for an axion insulator state in the ferromagnetic-3D topological insulator (TI) heterostructure and MnBi_{2}Te_{4} has attracted intense interest. However, its detection remains difficult in experiments. We systematically investigate the disorder-induced phase transition of the axion insulator state in a 3D TI with antiparallel magnetization alignment surfaces. It is found that there exists a 2D disorder-induced phase transition on the surfaces of the 3D TI which shares the same universality class with the quantum Hall plateau to plateau transition. Then, we provide a phenomenological theory which maps the random mass Dirac Hamiltonian of the axion insulator state into the Chalker-Coddington network model. Therefore, we propose probing the axion insulator state by investigating the universal signature of such a phase transition in the ferromagnetic-3D TI heterostructure and MnBi_{2}Te_{4}. Our findings not only show a global phase diagram of the axion insulator state, but also stimulate further experiments to probe it.
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Affiliation(s)
- Hailong Li
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
| | - Hua Jiang
- School of Physical Science and Technology, Soochow University, Suzhou 215006, China
- Institute for Advanced Study, Soochow University, Suzhou 215006, China
| | - Chui-Zhen Chen
- School of Physical Science and Technology, Soochow University, Suzhou 215006, China
- Institute for Advanced Study, Soochow University, Suzhou 215006, China
| | - X C Xie
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Beijing Academy of Quantum Information Sciences, Beijing 100193, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
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Sbierski B, Dresselhaus EJ, Moore JE, Gruzberg IA. Criticality of Two-Dimensional Disordered Dirac Fermions in the Unitary Class and Universality of the Integer Quantum Hall Transition. PHYSICAL REVIEW LETTERS 2021; 126:076801. [PMID: 33666465 DOI: 10.1103/physrevlett.126.076801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/31/2020] [Accepted: 01/22/2021] [Indexed: 06/12/2023]
Abstract
Two-dimensional (2D) Dirac fermions are a central paradigm of modern condensed matter physics, describing low-energy excitations in graphene, in certain classes of superconductors, and on surfaces of 3D topological insulators. At zero energy E=0, Dirac fermions with mass m are band insulators, with the Chern number jumping by unity at m=0. This observation lead Ludwig et al. [Phys. Rev. B 50, 7526 (1994)PRBMDO0163-182910.1103/PhysRevB.50.7526] to conjecture that the transition in 2D disordered Dirac fermions (DDF) and the integer quantum Hall transition (IQHT) are controlled by the same fixed point and possess the same universal critical properties. Given the far-reaching implications for the emerging field of the quantum anomalous Hall effect, modern condensed matter physics, and our general understanding of disordered critical points, it is surprising that this conjecture has never been tested numerically. Here, we report the results of extensive numerics on the phase diagram and criticality of 2D DDF in the unitary class. We find a critical line at m=0, with an energy-dependent localization length exponent. At large energies, our results for the DDF are consistent with state-of-the-art numerical results ν_{IQH}=2.56-2.62 from models of the IQHT. At E=0, however, we obtain ν_{0}=2.30-2.36 incompatible with ν_{IQH}. This result challenges conjectured relations between different models of the IQHT, and several interpretations are discussed.
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Affiliation(s)
- Björn Sbierski
- Department of Physics, University of California, Berkeley, California 94720, USA
| | | | - Joel E Moore
- Department of Physics, University of California, Berkeley, California 94720, USA
- Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
| | - Ilya A Gruzberg
- Department of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, Ohio 43210, USA
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Huang KS, Raghu S, Kumar P. Numerical Study of a Dual Representation of the Integer Quantum Hall Transition. PHYSICAL REVIEW LETTERS 2021; 126:056802. [PMID: 33605754 DOI: 10.1103/physrevlett.126.056802] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/25/2020] [Accepted: 01/13/2021] [Indexed: 06/12/2023]
Abstract
We study the critical properties of the noninteracting integer quantum Hall to insulator transition (IQHIT) in a "dual" composite-fermion (CF) representation. A key advantage of the CF representation over electron coordinates is that at criticality CF states are delocalized at all energies. The CF approach thus enables us to study the transition from a new vantage point. Using a lattice representation of CF mean-field theory, we compute the critical and multifractal exponents of the IQHIT. We obtain ν=2.56±0.02 and η=0.51±0.01, both of which are consistent with the predictions of the Chalker-Coddington network model formulated in the electron representation.
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Affiliation(s)
- Kevin S Huang
- Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305, USA
| | - S Raghu
- Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305, USA
- Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - Prashant Kumar
- Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305, USA
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
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8
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Wu X, Xiao D, Chen CZ, Sun J, Zhang L, Chan MHW, Samarth N, Xie XC, Lin X, Chang CZ. Scaling behavior of the quantum phase transition from a quantum-anomalous-Hall insulator to an axion insulator. Nat Commun 2020; 11:4532. [PMID: 32913228 PMCID: PMC7483742 DOI: 10.1038/s41467-020-18312-z] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2020] [Accepted: 08/18/2020] [Indexed: 11/08/2022] Open
Abstract
The phase transitions from one plateau to the next plateau or to an insulator in quantum Hall and quantum anomalous Hall (QAH) systems have revealed universal scaling behaviors. A magnetic-field-driven quantum phase transition from a QAH insulator to an axion insulator was recently demonstrated in magnetic topological insulator sandwich samples. Here, we show that the temperature dependence of the derivative of the longitudinal resistance on magnetic field at the transition point follows a characteristic power-law that indicates a universal scaling behavior for the QAH to axion insulator phase transition. Similar to the quantum Hall plateau to plateau transition, the QAH to axion insulator transition can also be understood by the Chalker-Coddington network model. We extract a critical exponent κ ~ 0.38 ± 0.02 in agreement with recent high-precision numerical results on the correlation length exponent of the Chalker-Coddington model at ν ~ 2.6, rather than the generally-accepted value of 2.33.
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Affiliation(s)
- Xinyu Wu
- International Center for Quantum Materials, Peking University, Beijing, 100871, China
| | - Di Xiao
- Department of Physics, The Pennsylvania State University, University Park, PA, 16802, USA
| | - Chui-Zhen Chen
- Institute for Advanced Study and School of Physical Science and Technology, Soochow University, Suzhou, 215006, China
| | - Jian Sun
- International Center for Quantum Materials, Peking University, Beijing, 100871, China
| | - Ling Zhang
- Department of Physics, The Pennsylvania State University, University Park, PA, 16802, USA
| | - Moses H W Chan
- Department of Physics, The Pennsylvania State University, University Park, PA, 16802, USA
| | - Nitin Samarth
- Department of Physics, The Pennsylvania State University, University Park, PA, 16802, USA
| | - X C Xie
- International Center for Quantum Materials, Peking University, Beijing, 100871, China
- Beijing Academy of Quantum Information Sciences, Beijing, 100193, China
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing, 100190, China
| | - Xi Lin
- International Center for Quantum Materials, Peking University, Beijing, 100871, China.
- Beijing Academy of Quantum Information Sciences, Beijing, 100193, China.
- CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing, 100190, China.
| | - Cui-Zu Chang
- Department of Physics, The Pennsylvania State University, University Park, PA, 16802, USA.
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Ippoliti M, Bhatt RN. Dimensional Crossover of the Integer Quantum Hall Plateau Transition and Disordered Topological Pumping. PHYSICAL REVIEW LETTERS 2020; 124:086602. [PMID: 32167341 DOI: 10.1103/physrevlett.124.086602] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2019] [Accepted: 02/11/2020] [Indexed: 06/10/2023]
Abstract
We study the quantum Hall plateau transition on rectangular tori. As the aspect ratio of the torus is increased, the two-dimensional critical behavior, characterized by a subthermodynamic number of topological states in a vanishing energy window around a critical energy, changes drastically. In the thin-torus limit, the entire spectrum is Anderson localized; however, an extensive number of states retain a Chern number C≠0. We resolve this apparent paradox by mapping the thin-torus quantum Hall system onto a disordered Thouless pump, where the Chern number corresponds to the winding number of an electron's path in real space during a pump cycle. We then characterize quantitatively the crossover between the one- and two-dimensional regimes for finite torus thickness, where the average Thouless conductance also shows anomalous scaling.
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Affiliation(s)
- Matteo Ippoliti
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - R N Bhatt
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
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