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Crépel V, Estienne B, Regnault N. Variational Ansatz for an Abelian to Non-Abelian Topological Phase Transition in ν=1/2+1/2 Bilayers. PHYSICAL REVIEW LETTERS 2019; 123:126804. [PMID: 31633987 DOI: 10.1103/physrevlett.123.126804] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/2019] [Indexed: 06/10/2023]
Abstract
We propose a one-parameter variational ansatz to describe the tunneling-driven Abelian to non-Abelian transition in bosonic ν=1/2+1/2 fractional quantum Hall bilayers. This ansatz, based on exact matrix product states, captures the low-energy physics all along the transition and allows us to probe its characteristic features. The transition is continuous, characterized by the decoupling of antisymmetric degrees of freedom. We futhermore determine the tunneling strength above which non-Abelian statistics should be observed experimentally. Finally, we propose to engineer the interlayer tunneling to create an interface trapping a neutral chiral Majorana fermion. We microscopically characterize such an interface using a slightly modified model wave function.
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Affiliation(s)
- Valentin Crépel
- Laboratoire de Physique de l'École normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, 75005 Paris, France
| | - Benoit Estienne
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Énergies, LPTHE, F-75005 Paris, France
| | - Nicolas Regnault
- Laboratoire de Physique de l'École normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, 75005 Paris, France
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Ye BT, Han ZY, Mu LZ, Fan H. Investigating disordered many-body system with entanglement in momentum space. Sci Rep 2017; 7:16668. [PMID: 29192168 PMCID: PMC5709446 DOI: 10.1038/s41598-017-16889-y] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2017] [Accepted: 11/15/2017] [Indexed: 11/12/2022] Open
Abstract
We study the entanglement in momentum space of the ground state of a disordered one-dimensional fermion lattice model with attractive interaction. We observe two components in the entanglement spectrum, one of which is related to paired-fermion entanglement and contributes to the long-range correlation in position space. The vanishing point of it indicates the localization phenomenon in the ground state of this model. Additionally, by method of entanglement spectrum, we provide a new evidence to show the transition of two phases induced by interaction, and find that this phase transition is not influenced by the disorder. Our result show key characteristics in entanglement for different phases in the system, and provide a novel perspective to understand localization phenomena.
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Affiliation(s)
- Bing-Tian Ye
- School of Physics, Peking University, Beijing, 100871, China
| | - Zhao-Yu Han
- School of Physics, Peking University, Beijing, 100871, China
| | - Liang-Zhu Mu
- School of Physics, Peking University, Beijing, 100871, China.
| | - Heng Fan
- Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China.
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.
- Collaborative Innovation Center of Quantum Matter, Beijing, 100190, China.
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Lundgren R, Blair J, Greiter M, Läuchli A, Fiete GA, Thomale R. Momentum-space entanglement spectrum of bosons and fermions with interactions. PHYSICAL REVIEW LETTERS 2014; 113:256404. [PMID: 25554899 DOI: 10.1103/physrevlett.113.256404] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2014] [Indexed: 06/04/2023]
Abstract
We study the momentum space entanglement spectra of bosonic and fermionic formulations of the spin-1/2 XXZ chain with analytical methods and exact diagonalization. We investigate the behavior of the entanglement gaps, present in both formulations, across quantum phase transitions in the XXZ chain. In both cases, finite size scaling suggests that the entanglement gap closure does not occur at the physical transition points. For bosons, we find that the entanglement gap observed in Thomale et al. [Phys. Rev. Lett. 105, 116805 (2010)] depends on the scaling dimension of the conformal field theory as varied by the XXZ anisotropy. For fermions, the infinite entanglement gap present at the XX point persists well past the phase transition at the Heisenberg point. We elaborate on how these shifted transition points in the entanglement spectra may support the numerical study of phase transitions in the momentum space density matrix renormalization group.
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Affiliation(s)
- Rex Lundgren
- Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
| | - Jonathan Blair
- Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
| | - Martin Greiter
- Insitute for Theoritical Physics, Univesity of Würzburg, D-97074 Würzburg, Germany
| | - Andreas Läuchli
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria
| | - Gregory A Fiete
- Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
| | - Ronny Thomale
- Insitute for Theoritical Physics, Univesity of Würzburg, D-97074 Würzburg, Germany
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Hsieh TH, Fu L. Bulk entanglement spectrum reveals quantum criticality within a topological state. PHYSICAL REVIEW LETTERS 2014; 113:106801. [PMID: 25238375 DOI: 10.1103/physrevlett.113.106801] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2013] [Indexed: 06/03/2023]
Abstract
A quantum phase transition is usually achieved by tuning physical parameters in a Hamiltonian at zero temperature. Here, we show that the ground state of a topological phase itself encodes critical properties of its transition to a trivial phase. To extract this information, we introduce an extensive partition of the system into two subsystems both of which extend throughout the bulk in all directions. The resulting bulk entanglement spectrum has a low-lying part that resembles the excitation spectrum of a bulk Hamiltonian, which allows us to probe a topological phase transition from a single wave function by tuning either the geometry of the partition or the entanglement temperature. As an example, this remarkable correspondence between the topological phase transition and the entanglement criticality is rigorously established for integer quantum Hall states.
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Affiliation(s)
- Timothy H Hsieh
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Liang Fu
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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Chandran A, Khemani V, Sondhi SL. How universal is the entanglement spectrum? PHYSICAL REVIEW LETTERS 2014; 113:060501. [PMID: 25148308 DOI: 10.1103/physrevlett.113.060501] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2014] [Indexed: 05/10/2023]
Abstract
It is now commonly believed that the ground state entanglement spectrum (ES) exhibits universal features characteristic of a given phase. In this Letter, we show that this belief is false in general. Most significantly, we show that the entanglement Hamiltonian can undergo quantum phase transitions in which its ground state and low-energy spectrum exhibit singular changes, even when the physical system remains in the same phase. For broken symmetry problems, this implies that the low-energy ES and the Rényi entropies can mislead entirely, while for quantum Hall systems, the ES has much less universal content than assumed to date.
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Affiliation(s)
- Anushya Chandran
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA and Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo, Ontario N2L 2Y5, Canada
| | - Vedika Khemani
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - S L Sondhi
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
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Kim EH. Characterizing topological order in superconductors via entanglement. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2014; 26:205602. [PMID: 24786467 DOI: 10.1088/0953-8984/26/20/205602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We investigate/characterize topological order in superconducting systems using entanglement. Demonstrating/utilizing the directional dependence of the entanglement, results are shown for two-dimensional [Formula: see text] and [Formula: see text] +idxy spin-singlet superconductors, comparing the results to those for a spin-polarized px+ipy superconductor; we also discuss the topological properties of a one-dimensional spin-polarized p-wave superconductor threaded by a magnetic flux. In carrying out these investigations, we establish an efficient approach to investigate entanglement in superconductors or, more generally, in quadratic fermionic Hamiltonians with pairing interactions.
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Affiliation(s)
- E H Kim
- Department of Physics, University of Windsor, Windsor, Ontario N9B 3P4, Canada
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Alba V, Haque M, Läuchli AM. Entanglement spectrum of the two-dimensional Bose-Hubbard model. PHYSICAL REVIEW LETTERS 2013; 110:260403. [PMID: 23848849 DOI: 10.1103/physrevlett.110.260403] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/08/2013] [Indexed: 06/02/2023]
Abstract
We study the entanglement spectrum (ES) of the Bose-Hubbard model on the two-dimensional square lattice at unit filling, both in the Mott insulating and in the superfluid phase. In the Mott phase, we demonstrate that the ES is dominated by the physics at the boundary between the two subsystems. On top of the boundary-local (perturbative) structure, the ES exhibits substructures arising from one-dimensional dispersions along the boundary. In the superfluid phase, the structure of the ES is qualitatively different, and reflects the spontaneously broken U(1) symmetry of the phase. We attribute the basic low-lying structure to the "tower of states" Hamiltonian of the model. We then discuss how these characteristic structures evolve across the superfluid to Mott insulator transition and their influence on the behavior of the entanglement entropies. We briefly outline the implications of the ES structure on the efficiency of matrix-product-state based algorithms in two dimensions.
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Affiliation(s)
- Vincenzo Alba
- Department of Physics and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, D-80333 München, Germany
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