Geometric Phase Curvature Statistics
The probability distribution of the magnitude C of the curvature 2-form, that underlies the quantum geometric phase and the reaction force of geometric magnetism, is calculated for an ensemble of three-parameter Hamiltonians represented by the gaussian unitary ensemble of N × N matrices. The distr...
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| Published in | Journal of statistical physics Vol. 180; no. 1-6; pp. 297 - 303 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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01.09.2020
Springer Springer Nature B.V |
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| Abstract | The probability distribution of the magnitude
C
of the curvature 2-form, that underlies the quantum geometric phase and the reaction force of geometric magnetism, is calculated for an ensemble of three-parameter Hamiltonians represented by the gaussian unitary ensemble of
N
×
N
matrices. The distributions are determined analytically: exactly for
N
= 2 and approximately for
N
≥ 3, and compared with simulations. The distributions decay asymptotically as 1/
C
5/2
; this is a consequence of the codimension of energy-level degeneracies in the ensemble. |
|---|---|
| AbstractList | The probability distribution of the magnitude
C
of the curvature 2-form, that underlies the quantum geometric phase and the reaction force of geometric magnetism, is calculated for an ensemble of three-parameter Hamiltonians represented by the gaussian unitary ensemble of
N
×
N
matrices. The distributions are determined analytically: exactly for
N
= 2 and approximately for
N
≥ 3, and compared with simulations. The distributions decay asymptotically as 1/
C
5/2
; this is a consequence of the codimension of energy-level degeneracies in the ensemble. The probability distribution of the magnitude C of the curvature 2-form, that underlies the quantum geometric phase and the reaction force of geometric magnetism, is calculated for an ensemble of three-parameter Hamiltonians represented by the gaussian unitary ensemble of N x N matrices. The distributions are determined analytically: exactly for N = 2 and approximately for N [greater than or equal to] 3, and compared with simulations. The distributions decay asymptotically as 1/C.sup.5/2; this is a consequence of the codimension of energy-level degeneracies in the ensemble. The probability distribution of the magnitude C of the curvature 2-form, that underlies the quantum geometric phase and the reaction force of geometric magnetism, is calculated for an ensemble of three-parameter Hamiltonians represented by the gaussian unitary ensemble of N × N matrices. The distributions are determined analytically: exactly for N = 2 and approximately for N ≥ 3, and compared with simulations. The distributions decay asymptotically as 1/C5/2; this is a consequence of the codimension of energy-level degeneracies in the ensemble. |
| Audience | Academic |
| Author | Shukla, Pragya Berry, M. V. |
| Author_xml | – sequence: 1 givenname: M. V. orcidid: 0000-0001-7921-2468 surname: Berry fullname: Berry, M. V. email: asymptotico@bristol.ac.uk organization: H H Wills Physics Laboratory – sequence: 2 givenname: Pragya surname: Shukla fullname: Shukla, Pragya organization: Department of Physics, Indian Institute of Science, Department of Physics, Indian Institute of Science |
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| Cites_doi | 10.1016/0375-9601(87)90189-7 10.1515/9781400883875 10.1063/1.437734 10.1103/PhysRevLett.51.2167 10.1103/RevModPhys.64.51 10.1103/PhysRevLett.96.117208 10.1209/epl/i1998-00446-x 10.1088/1751-8121/aae5dd 10.1098/rspa.1984.0023 10.1103/PhysRevLett.122.146601 10.1088/0305-4470/10/12/015 10.1103/PhysRevLett.74.4055 10.1007/978-1-84628-723-7 10.1103/PhysRevLett.122.106601 10.1515/9781400881826 10.1016/0029-5582(63)90178-0 10.1088/2040-8986/ab14c4 10.1016/0003-4916(81)90189-5 10.1017/9781316662205 10.1103/PhysRevB.31.3372 10.1103/PhysRevB.90.125153 10.1086/676677 |
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| Keywords | Degeneracies 2-form Random-matrix Quantum statistics |
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| References_xml | – volume: 125 start-page: 476 year: 1987 ident: 2400_CR5 publication-title: Phys. Lett. A doi: 10.1016/0375-9601(87)90189-7 contributor: fullname: B Zygelman – volume-title: The Topology of Fibre Bundles year: 1951 ident: 2400_CR3 doi: 10.1515/9781400883875 contributor: fullname: N Steenrod – volume: 70 start-page: 2284 year: 1979 ident: 2400_CR6 publication-title: J. Chem. Phys. doi: 10.1063/1.437734 contributor: fullname: CA Mead – volume-title: Statistical Theories of Spectra: Fluctuations year: 1965 ident: 2400_CR27 contributor: fullname: CE Porter – volume: 51 start-page: 2167 year: 1983 ident: 2400_CR2 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.51.2167 contributor: fullname: B Simon – ident: 2400_CR11 – volume: 8 start-page: 1 issue: 6719 year: 2018 ident: 2400_CR13 publication-title: Sci. Rep. contributor: fullname: I Makhfudz – volume: 64 start-page: 51 year: 1992 ident: 2400_CR7 publication-title: Rev. Mod. Phys. doi: 10.1103/RevModPhys.64.51 contributor: fullname: CA Mead – volume: 96 start-page: 117208 year: 2006 ident: 2400_CR20 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.96.117208 contributor: fullname: P Bruno – volume: 44 start-page: 131 year: 1998 ident: 2400_CR12 publication-title: Europhys. Lett. doi: 10.1209/epl/i1998-00446-x contributor: fullname: P Pradhan – volume: 51 start-page: 475101 year: 2018 ident: 2400_CR21 publication-title: J. Phys. A doi: 10.1088/1751-8121/aae5dd contributor: fullname: MV Berry – volume: 24 start-page: 213202 year: 2012 ident: 2400_CR9 publication-title: J. Phys.: Condens. Matter contributor: fullname: M Gradhand – ident: 2400_CR25 – volume: 392 start-page: 45 year: 1984 ident: 2400_CR4 publication-title: Proc. R. Soc. Lond. A doi: 10.1098/rspa.1984.0023 contributor: fullname: MV Berry – volume: 122 start-page: 146601 year: 2019 ident: 2400_CR17 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.122.146601 contributor: fullname: K Kudo – volume-title: Geometric Phases in Physics year: 1989 ident: 2400_CR1 contributor: fullname: A Shapere – volume: 10 start-page: 2061 year: 1977 ident: 2400_CR26 publication-title: J. Phys. A doi: 10.1088/0305-4470/10/12/015 contributor: fullname: MV Berry – volume: 74 start-page: 4055 year: 1995 ident: 2400_CR18 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.74.4055 contributor: fullname: PN Walker – volume-title: Fundamentals of computerized tomography: image reconstruction from projection year: 2009 ident: 2400_CR24 doi: 10.1007/978-1-84628-723-7 contributor: fullname: GT Herman – volume: 122 start-page: 106601 year: 2019 ident: 2400_CR14 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.122.106601 contributor: fullname: MA Werner – volume-title: Characteristic Classes year: 1974 ident: 2400_CR19 doi: 10.1515/9781400881826 contributor: fullname: JW Milnor – volume: 48 start-page: 385 year: 1963 ident: 2400_CR29 publication-title: Nucl. Phys. doi: 10.1016/0029-5582(63)90178-0 contributor: fullname: PB Kahn – volume: 30 start-page: 467 year: 1929 ident: 2400_CR23 publication-title: Phys. Z. contributor: fullname: J Von Neumann – volume: 21 start-page: 064002 year: 2019 ident: 2400_CR22 publication-title: J. Opt. doi: 10.1088/2040-8986/ab14c4 contributor: fullname: MV Berry – volume: 131 start-page: 163 year: 1981 ident: 2400_CR28 publication-title: Ann. Phys. doi: 10.1016/0003-4916(81)90189-5 contributor: fullname: MV Berry – volume-title: Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators year: 2018 ident: 2400_CR8 doi: 10.1017/9781316662205 contributor: fullname: D Vanderbilt – volume: 31 start-page: 3372 year: 1985 ident: 2400_CR15 publication-title: Phys. Rev. B doi: 10.1103/PhysRevB.31.3372 contributor: fullname: Q Niu – volume: 90 start-page: 125153 year: 2014 ident: 2400_CR16 publication-title: Phys. Rev. B doi: 10.1103/PhysRevB.90.125153 contributor: fullname: R Bianco – volume: 81 start-page: 349 year: 2014 ident: 2400_CR10 publication-title: Philos. Sci. doi: 10.1086/676677 contributor: fullname: RW Batterman |
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| Snippet | The probability distribution of the magnitude
C
of the curvature 2-form, that underlies the quantum geometric phase and the reaction force of geometric... The probability distribution of the magnitude C of the curvature 2-form, that underlies the quantum geometric phase and the reaction force of geometric... |
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| SubjectTerms | Curvature Magnetism Mathematical and Computational Physics Physical Chemistry Physics Physics and Astronomy Quantum Physics Statistical Physics and Dynamical Systems Statistics Theoretical |
| Title | Geometric Phase Curvature Statistics |
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