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Lotrič T, Simon SH. Chern-Simons Modified RPA-Eliashberg Theory of the ν=1/2+1/2 Quantum Hall Bilayer. PHYSICAL REVIEW LETTERS 2024; 132:176502. [PMID: 38728712 DOI: 10.1103/physrevlett.132.176502] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/28/2023] [Revised: 01/31/2024] [Accepted: 04/04/2024] [Indexed: 05/12/2024]
Abstract
The ν=1/2+1/2 quantum Hall bilayer has been previsously modeled using Chern-Simons-RPA-Eliashberg (CSRPAE) theory to describe pairing between the two layers. However, these approaches are troubled by a number of divergences and ambiguities. By using a "modified" RPA approximation to account for mass renormalization, we can work in a limit where the cyclotron frequency is taken to infinity, effectively projecting to a single Landau level. This, surprisingly, controls the important divergences and removes ambiguities found in prior attempts at CSRPAE. Examining BCS pairing of composite fermions we find that the angular momentum channel l=+1 dominates for all distances d between layers and at all frequency scales. Examining BCS pairing of composite fermion electrons in one layer with composite fermion holes in the opposite layer, we find the l=0 pairing channel dominates for all d and all frequencies. The strength of the pairing in these two different descriptions of the same phase of matter is found to be almost identical. This agrees well with our understanding that these are two different but dual descriptions of the same phase of matter.
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Affiliation(s)
- Tevž Lotrič
- Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford, OX1 3PU, United Kingdom
| | - Steven H Simon
- Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford, OX1 3PU, United Kingdom
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2
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Wagner G, Nguyen DX, Simon SH, Halperin BI. s-Wave Paired Electron and Hole Composite Fermion Trial State for Quantum Hall Bilayers with ν=1. PHYSICAL REVIEW LETTERS 2021; 127:246803. [PMID: 34951796 DOI: 10.1103/physrevlett.127.246803] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2021] [Accepted: 11/12/2021] [Indexed: 06/14/2023]
Abstract
We introduce a new variational wave function for a quantum Hall bilayer at total filling ν_{T}=1, which is based on s-wave BCS pairing between electron composite fermions in one layer and hole composite fermions in the other. In addition, we reexamine a trial wave function based on p-wave BCS pairing between electron composite fermions in both layers. We compute the overlap of the optimized trial functions with the ground state from exact diagonalization calculations of up to 14 electrons in a spherical geometry, and we find excellent agreement over the entire range of values of the ratio between the layer separation and the magnetic length. The s-wave trial wave function naturally allows for charge imbalance between the layers and provides important insights into how the physics at large interlayer separations crosses over to that at small separations in a fashion analogous to the BEC-BCS crossover.
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Affiliation(s)
- Glenn Wagner
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom
- Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
| | - Dung X Nguyen
- Brown Theoretical Physics Center, Brown University, Providence, Rhode Island 02912, USA
| | - Steven H Simon
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom
| | - Bertrand I Halperin
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
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3
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Eisenstein JP, Pfeiffer LN, West KW. Precursors to Exciton Condensation in Quantum Hall Bilayers. PHYSICAL REVIEW LETTERS 2019; 123:066802. [PMID: 31491172 DOI: 10.1103/physrevlett.123.066802] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2019] [Indexed: 06/10/2023]
Abstract
Tunneling spectroscopy reveals evidence for interlayer electron-hole correlations in quantum Hall bilayer two-dimensional electron systems at layer separations near, but above, the transition to the incompressible exciton condensate at total Landau level filling ν_{T}=1. These correlations are manifested by a nonlinear suppression of the Coulomb pseudogap which inhibits low energy interlayer tunneling in weakly coupled bilayers. The pseudogap suppression is strongest at ν_{T}=1 and grows rapidly as the critical layer separation for exciton condensation is approached from above.
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Affiliation(s)
- J P Eisenstein
- Institute for Quantum Information and Matter, Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
| | - L N Pfeiffer
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - K W West
- Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
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Zhu Z, Fu L, Sheng DN. Numerical Study of Quantum Hall Bilayers at Total Filling ν_{T}=1: A New Phase at Intermediate Layer Distances. PHYSICAL REVIEW LETTERS 2017; 119:177601. [PMID: 29219462 DOI: 10.1103/physrevlett.119.177601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/02/2017] [Indexed: 06/07/2023]
Abstract
We study the phase diagram of quantum Hall bilayer systems with total filing ν_{T}=1/2+1/2 of the lowest Landau level as a function of layer distances d. Based on numerical exact diagonalization calculations, we obtain three distinct phases, including an exciton superfluid phase with spontaneous interlayer coherence at small d, a composite Fermi liquid at large d, and an intermediate phase for 1.1<d/l_{B}<1.8 (l_{B} is the magnetic length). The transition from the exciton superfluid to the intermediate phase is identified by (i) a dramatic change in the Berry curvature of the ground state under twisted boundary conditions on the two layers and (ii) an energy level crossing of the first excited state. The transition from the intermediate phase to the composite Fermi liquid is identified by the vanishing of the exciton superfluid stiffness. Furthermore, from our finite-size study, the energy cost of transferring one electron between the layers shows an even-odd effect and possibly extrapolates to a finite value in the thermodynamic limit, indicating the enhanced intralayer correlation. Our identification of an intermediate phase and its distinctive features shed new light on the theoretical understanding of the quantum Hall bilayer system at total filling ν_{T}=1.
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Affiliation(s)
- Zheng Zhu
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Liang Fu
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - D N Sheng
- Department of Physics and Astronomy, California State University, Northridge, California 91330, USA
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Confinement-deconfinement transition due to spontaneous symmetry breaking in quantum Hall bilayers. Nat Commun 2016; 7:10462. [PMID: 26804790 PMCID: PMC4737752 DOI: 10.1038/ncomms10462] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/14/2015] [Accepted: 12/12/2015] [Indexed: 11/18/2022] Open
Abstract
Band-inverted electron-hole bilayers support quantum spin Hall insulator and exciton condensate phases. Interest in quantum spin Hall effect in these systems has recently put them in the spotlight. We investigate such a bilayer in an external magnetic field. We show that the interlayer correlations lead to formation of a helical quantum Hall exciton condensate state. Existence of the counterpropagating edge modes in this system results in formation of a ground state spin-texture not supporting gapless single-particle excitations. The charged edge excitations in a sufficiently narrow Hall bar are confined: a charge on one of the edges always gives rise to an opposite charge on the other edge. Magnetic field and gate voltages allow the control of a confinement-deconfinement transition of charged edge excitations, which can be probed with nonlocal conductance. Confinement-deconfinement transitions are of great interest, not least because of their possible significance in shedding light on the confinement problem of quarks. Electronic systems with inverted band structures can support exotic topological insulator and exciton condensate states. Here, the authors demonstrate the formation of a helical exciton condensate in quantum Hall bilayers, and a quark-like quasiparticle confinement-deconfinement transition.
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Barlas Y, Côté R, Lambert J, Macdonald AH. Anomalous exciton condensation in graphene bilayers. PHYSICAL REVIEW LETTERS 2010; 104:096802. [PMID: 20367001 DOI: 10.1103/physrevlett.104.096802] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2009] [Indexed: 05/29/2023]
Abstract
In ordinary semiconductor bilayers, exciton condensates appear at total Landau-level filling factor nu{T}=1. We predict that similar states will occur in Bernal stacked graphene bilayers at many nonzero integer filling factors. For nu{T}=-3, 1 we find that the superfluid density of the exciton condensate vanishes and that a finite-temperature fluctuation-induced first order isotropic-smectic phase transition occurs when the layer densities are not balanced. These anomalous properties of bilayer graphene exciton condensates are due to the degeneracy of Landau levels with n=0 and n=1 orbital character.
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Affiliation(s)
- Yafis Barlas
- National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306, USA
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Roostaei B, Mullen KJ, Fertig HA, Simon SH. Theory of activated transport in bilayer quantum Hall systems. PHYSICAL REVIEW LETTERS 2008; 101:046804. [PMID: 18764355 DOI: 10.1103/physrevlett.101.046804] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2008] [Indexed: 05/26/2023]
Abstract
We analyze the transport properties of bilayer quantum Hall systems at total filling factor nu=1 in drag geometries as a function of interlayer bias, in the limit where the disorder is sufficiently strong to unbind meron-antimeron pairs, the charged topological defects of the system. We compute the typical energy barrier for these objects to cross incompressible regions within the disordered system using a Hartree-Fock approach, and show how this leads to multiple activation energies when the system is biased. We then demonstrate using a bosonic Chern-Simons theory that in drag geometries current in a single layer directly leads to forces on only two of the four types of merons, inducing dissipation only in the drive layer. Dissipation in the drag layer results from interactions among the merons, resulting in very different temperature dependences for the drag and drive layers, in qualitative agreement with experiment.
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Affiliation(s)
- B Roostaei
- Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106, USA
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Abedinpour SH, Polini M, MacDonald AH, Tanatar B, Tosi MP, Vignale G. Theory of the pseudospin resonance in semiconductor bilayers. PHYSICAL REVIEW LETTERS 2007; 99:206802. [PMID: 18233174 DOI: 10.1103/physrevlett.99.206802] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2007] [Indexed: 05/25/2023]
Abstract
The pseudospin degree of freedom in a semiconductor bilayer gives rise to a collective mode analogous to the ferromagnetic-resonance mode of a ferromagnet. We present a many-body theory of the dependence of the energy and the damping of this mode on layer separation d. Based on these results, we discuss the possibilities of realizing transport-current driven pseudospin-transfer oscillators in semiconductors, and of using the pseudospin-transfer effect as an experimental probe of intersubband plasmons.
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9
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Ye J, Jiang L. Quantum phase transitions in bilayer quantum hall systems at a total filling factor nuT=1. PHYSICAL REVIEW LETTERS 2007; 98:236802. [PMID: 17677927 DOI: 10.1103/physrevlett.98.236802] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/24/2006] [Indexed: 05/16/2023]
Abstract
We construct a quantum Ginsburg-Landau theory to study the quantum phase transition from the excitonic superfluid to a possible pseudospin density wave (PSDW) at some intermediate distances driven by the magnetoroton minimum collapsing at a finite wave vector. We explicitly show that the PSDW takes a square lattice structure. We suggest the existence of zero-point quantum fluctuation generated vacancies in the PSDW and that correlated hopping of vacancies in the active and passive layers in the PSDW state leads to very large and temperature dependent drag consistent with the experimental data. Comparisons with previous numerical calculations are made. Further experimental implications are given.
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Affiliation(s)
- Jinwu Ye
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania, 16802, USA
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10
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Ye J. Fractional charges and quantum phase transitions in imbalanced bilayer quantum Hall systems. PHYSICAL REVIEW LETTERS 2006; 97:236803. [PMID: 17280224 DOI: 10.1103/physrevlett.97.236803] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/03/2004] [Indexed: 05/13/2023]
Abstract
We extend the composite boson theory to study slightly imbalanced bilayer quantum Hall systems. In the global U(1) symmetry breaking excitonic superfluid side, as the imbalance increases, the system supports continuously changing fractional charges. In the translational symmetry breaking pseudospin density wave (PSDW) side, there are two quantum phase transitions from the commensurate PSDW to an incommensurate PSDW and then to the excitonic superfluid state. We compare our theory with experimental data and also the previous microscopic calculations.
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Affiliation(s)
- Jinwu Ye
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
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11
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Stern A, Halperin BI. Strong enhancement of drag and dissipation at the weak- to strong-coupling phase transition in a bilayer system at a total Landau level filling nu = 1. PHYSICAL REVIEW LETTERS 2002; 88:106801. [PMID: 11909380 DOI: 10.1103/physrevlett.88.106801] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/29/2001] [Indexed: 05/23/2023]
Abstract
We consider a bilayer electronic system at a total Landau level filling factor nu = 1, and focus on the transition from the regime of weak interlayer coupling to that of the strongly coupled (1,1,1) phase (or "quantum Hall ferromagnet"). Making the assumption that in the transition region the system is made of puddles of the (1,1,1) phase embedded in a bulk of the weakly coupled state, we show that the transition is accompanied by a strong increase in longitudinal Coulomb drag that reaches a maximum of approximately h/2e(2). In that regime the longitudinal drag increases with decreasing temperature.
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Affiliation(s)
- Ady Stern
- Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
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12
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Schliemann J, Girvin SM, MacDonald AH. Strong correlation to weak correlation phase transition in bilayer quantum Hall systems. PHYSICAL REVIEW LETTERS 2001; 86:1849-1852. [PMID: 11290264 DOI: 10.1103/physrevlett.86.1849] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2000] [Indexed: 05/23/2023]
Abstract
At small layer separations, the ground state of a nu = 1 bilayer quantum Hall system exhibits spontaneous interlayer phase coherence. The evolution of this state with increasing layer separation d has been a matter of controversy. We report on small system exact diagonalization calculations which suggest that a single-phase transition, likely of first order, separates incompressible states with strong interlayer correlations from compressible states with weak interlayer correlations. We find a dependence of the phase boundary on d and interlayer tunneling amplitude that is in very good agreement with recent experiments.
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Affiliation(s)
- J Schliemann
- Department of Physics, Indiana University, Bloomington, Indiana 47405-7105, USA
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13
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Brey L, Fertig HA, Côté R, MacDonald AH. Charged pseudospin textures in double-layer quantum Hall systems: Bimerons and meron crystals. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:16888-16902. [PMID: 9985817 DOI: 10.1103/physrevb.54.16888] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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14
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Yang K, Moon K, Belkhir L, Mori H, Girvin SM, MacDonald AH, Zheng L, Yoshioka D. Spontaneous interlayer coherence in double-layer quantum Hall systems: Symmetry-breaking interactions, in-plane fields, and phase solitons. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:11644-11658. [PMID: 9984954 DOI: 10.1103/physrevb.54.11644] [Citation(s) in RCA: 35] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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15
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Côté R, MacDonald AH. Spin-ordering and magnon collective modes for two-dimensional electron lattices in strong magnetic fields. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:10019-10029. [PMID: 9982567 DOI: 10.1103/physrevb.53.10019] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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16
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Zheng L. Coulomb scattering lifetime of a two-dimensional electron gas. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:9964-9967. [PMID: 9982561 DOI: 10.1103/physrevb.53.9964] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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17
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Zheng L, Fertig HA. Wigner-crystal states for the two-dimensional electron gas in a double-quantum-well system. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:12282-12290. [PMID: 9980371 DOI: 10.1103/physrevb.52.12282] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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18
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Narasimhan S, Ho TL. Wigner-crystal phases in bilayer quantum Hall systems. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:12291-12306. [PMID: 9980372 DOI: 10.1103/physrevb.52.12291] [Citation(s) in RCA: 37] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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19
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Brey L, Fertig HA, Côté R, MacDonald AH. Skyrme crystal in a two-dimensional electron gas. PHYSICAL REVIEW LETTERS 1995; 75:2562-2565. [PMID: 10059343 DOI: 10.1103/physrevlett.75.2562] [Citation(s) in RCA: 49] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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20
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Zheng L, Fertig HA. The Hofstadter spectrum of the Wigner crystal. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 52:R2321-R2324. [PMID: 9981401 DOI: 10.1103/physrevb.52.r2321] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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21
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Yang K, MacDonald AH. Charged pseudospin textures in double-layer quantum Hall systems with spontaneous interlayer coherence. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:17247-17250. [PMID: 9978748 DOI: 10.1103/physrevb.51.17247] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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22
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Côté R, Brey L, Fertig H, MacDonald AH. Collective modes of soliton-lattice states in double-quantum-well systems. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:13475-13490. [PMID: 9978151 DOI: 10.1103/physrevb.51.13475] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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23
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Moon K, Mori H, Yang K, Girvin SM, MacDonald AH, Zheng L, Yoshioka D, Zhang SC. Spontaneous interlayer coherence in double-layer quantum Hall systems: Charged vortices and Kosterlitz-Thouless phase transitions. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:5138-5170. [PMID: 9979391 DOI: 10.1103/physrevb.51.5138] [Citation(s) in RCA: 234] [Impact Index Per Article: 8.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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24
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Arovas DP, Renn SR. Bogoliubov theory of Coulomb-coupled. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:15408-15411. [PMID: 9975897 DOI: 10.1103/physrevb.50.15408] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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25
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MacDonald AH, Zhang SC. Collective excitations in double-layer quantum Hall systems. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 49:17208-17215. [PMID: 10010900 DOI: 10.1103/physrevb.49.17208] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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26
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Falko VI. Optical branch of magnetophonons in a double-layer Wigner crystal. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 49:7774-7777. [PMID: 10009529 DOI: 10.1103/physrevb.49.7774] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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27
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Yang K, Moon K, Zheng L, MacDonald AH, Girvin SM, Yoshioka D, Zhang SC. Quantum ferromagnetism and phase transitions in double-layer quantum Hall systems. PHYSICAL REVIEW LETTERS 1994; 72:732-735. [PMID: 10056509 DOI: 10.1103/physrevlett.72.732] [Citation(s) in RCA: 36] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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28
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Vasko FT, Vasilopoulos P. Magneto-oscillatory phenomena in double quantum wells with asymmetric scattering. II. Longitudinal transport coefficients. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 48:18095-18100. [PMID: 10008448 DOI: 10.1103/physrevb.48.18095] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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29
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Vasko FT, Vasilopoulos P. Magneto-oscillatory phenomena in double quantum wells with asymmetric scattering. I. Density of states and thermodynamic quantities. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 48:18088-18094. [PMID: 10008447 DOI: 10.1103/physrevb.48.18088] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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30
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Brey L. Many-body effects on the symmetric-antisymmetric gap in double quantum wells in strong magnetic fields. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 47:4585-4591. [PMID: 10006606 DOI: 10.1103/physrevb.47.4585] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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