Russian Journal of Mathematical Physics

, Volume 24, Issue 3, pp 326–335 | Cite as

Magneto-dimensional resonance. Pseudospin phase and hidden quantum number

  • M. V. KarasevEmail author


The Schrödinger operator for a spinless charge inside a layer with parabolic confinement profile and homogeneous magnetic field is considered. The Lorentz (cyclotron) and the confinement frequencies are assumed to be equal to each other. After inclination of the layer normal from the magnetic field direction there appears a pseudospin su(2)-field removing the resonance degeneracy of Landau levels. Under deviations of the layer surface from the plane shape, a longitudinal geometric current is created. In circulations around surface warping, there is a nontrivial quantum phase transition generated by an element of the π1-homotopy group and a hidden degree of freedom (spectral degeneracy) associated with a “charge” of geometric poles on the layer. The quantization rule contains an additional parity index related to the algebraic number of geometric poles and the Landau level number. The resonance pseudospin phase-shift represents an example of general Aharonov–Bohm type topologic phenomena in quantum (semiclassical or adiabatic) systems with delta-function singularities in symplectic structure.


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  1. 1.
    P. Maraner, “Landau Ground State on Riemannian Surfaces,” Mod. Phys. Lett. 7, 2555–2558 (1992).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    R. Iengo and D. Li, “Quantum Mechanics and Quantum Hall Effect on Riemann Surfaces,” Nucl. Phys. B 413, 735–753 (1994).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    P. Maraner, “A Complete Perturbative Expansion for Quantum Mechanics with Constraints,” J. Phys. A: Math. Gen. 28 (10), 2939 (1995).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    P. Schuster and R. Jaffe, “Quantum Mechanics on Manifolds Embedded in Euclidean Space,” Ann. Phys. 307 (1), 132–143 (2003).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    P. Gosselin, H. Boumrar, and H. Mohrbach, “Semiclassical Quantization of Electrons in Magnetic Fields: the Generalized Peierls Substitution,” Europhysics Letters (EPL) 84, 50002 (2008).ADSCrossRefGoogle Scholar
  6. 6.
    G. Ferrari and G. Cuoght, “Schrödinger Equation for a Particle on a Curved Surface in an Electric and Magnetic Field,” Phys. Rev. Lett. 100, 230403 (2008).ADSCrossRefGoogle Scholar
  7. 7.
    M. V. Karasev, “Magneto-Metric Hamiltonians on Quantum Surfaces in the Configuration Space,” Russ. J. Math. Phys. 14 (1), 57–65 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 7a.
    M. V. Karasev, “Internal Geometric Current and the Maxwell Equation as a Hamiltonian System on Configuration Surfaces,” Russ. J. Math. Phys. 14 (2), 134–141 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 7b.
    M. V. Karasev, “Quantum Geometry of Nano-Space,” Russ. J. Math. Phys. 15 (3), 417–420 (2008).Google Scholar
  10. 7c.
    M. V. Karasev, “Graphene as a Quantum Surface with Curvature-Strain Preserving Dynamics,” Russ. J. Math. Phys. 18 (1), 25–32 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 7d.
    M. V. Karasev, “Hall quantum Hamiltonians and electric curvature,” Russ. J. Math. Phys. 19 (3), 299–306 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 8.
    A. V. Caplik and R. H. Blick, “On Geometric Potentials in Quantum-Electromechanical Circuits,” New J. Phys. 6 (1), 33 (2004).ADSCrossRefGoogle Scholar
  13. 9.
    Y. V. Pershin and C. Piermarocchi, “Persistent and Radiation-Induced Currents in Distorted Quantum Rings,” Phys. Rev. B 72, 125348 (2005).ADSCrossRefGoogle Scholar
  14. 10.
    M. Szelag and M. Szopa, “Persistent Currents in Distorted Quantum Rings,” J. Phys. Conf. Ser. 104 (1), 012006 (2008).CrossRefGoogle Scholar
  15. 11.
    H. Taira and H. Shima, “Torsion-Induced Persistent Current in a Twisted Quantum Ring,” J. Phys. Condensed Matter 22 (7), 075301 (2010).ADSCrossRefGoogle Scholar
  16. 12.
    M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets. Geometry and Quantization (Nauka, Moscow, 1991; Amer. Math. Soc., Providence, 1993).zbMATHGoogle Scholar
  17. 13.
    B. Simon, “Holonomy, the Quantum Adiabatic Theorem and Berry’s Phase,” Phys. Rev. Lett. 5 (1), 2167–2170, 1983.ADSMathSciNetCrossRefGoogle Scholar
  18. 14.
    M. V. Berry, “Quantal Phase Factors Accompanying Adiabatic Changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 15.
    Geometric Phases in Physics, Ed. by F. Wilczek and A. Shapere (World Scientific, Singapore, 1989).Google Scholar
  20. 16.
    P. Gosselin, A. Bérard, and H. Mohrbach, “Semiclassical Diagonalization of Quantum Hamiltonian and Equations of Motion with Berry Phase Corrections,” Eur. Phys. Jour. B 58, 137 (2007).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 17.
    P. Gosselin, J. Hanssen, and H. Mohrbach, “Recursive Diagonalization of Quantum Hamiltonians to All Orders in h,” Phys. Rev. D 77, 085008 (2008).ADSCrossRefGoogle Scholar
  22. 18.
    M. V. Karasev, “New Global Asymptotics and Anomalies in the Problem of Quantization of the Adiabatic Invariant,” Funktsional. Anal. i Prilozhen 24 (2), 24–36 (1990) [Functional Anal. Appl. 24, 104–114 (1990)].MathSciNetCrossRefzbMATHGoogle Scholar
  23. 19.
    M. Karasev, “Quantization and Coherent States over Lagrangian Submanifolds,” Russ. J. Math. Phys. 3, 393–400 (1985).MathSciNetzbMATHGoogle Scholar
  24. 20.
    Y. Aharonov and A. Casher, “Ground State of a Spin-1/2 Charged Particle in a Two-Dimensional Magnetic Field,” Phys. Rev. 19 (6), 2461–2462 (1979)ADSMathSciNetCrossRefGoogle Scholar
  25. 20a.
    Y. Aharonov and A. Casher, “Topological quantum effects for neutral particles,” Phys. Rev. Lett. 53 (4), 319–321(1984).ADSMathSciNetCrossRefGoogle Scholar
  26. 21.
    P. Maraner and C. Destri, “Geometry-Induced Yang–Mills Fields in Constrained Quantum Mechanics,” Mod. Phys. Lett. A08 (9), 861–868 (1993).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 22.
    P. Maraner, “Monopole Gauge Field and Quantum Potentials Induced by the Geometry in Simple Dynamical Systems,” Ann. Phys. 246 (2), 325–346 (1996).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 23.
    L. Magarill, D. Romanov, and A. Chaplik, “Electron Energy Spectrum and the Persistent Current in an Elliptical Quantum Ring,” JETP 83 (2), 361 (1996).ADSGoogle Scholar
  29. 24.
    A. Bruno-Alfonso and A. Latge, “The Aharonov–Bohm Oscillations in a Quantum Ring: Eccentricity and Electric-Field Effects,” Phys. Rev. B 71, 125312 (2005).ADSCrossRefGoogle Scholar
  30. 25.
    E. H. Semiromi, “The Aharonov–Bohm Oscillations and the Energy Spectrum in Two-Dimensional Elliptical Quantum Ring Nanotstructures,” Phys Ser. 85 (3), 035706 (2012).ADSCrossRefzbMATHGoogle Scholar
  31. 26.
    N. S. Manton and P. M. Sutcliffe, “Topological Solitons,” Cambridge University Press (2004).CrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.National Research University “Higher School of Economics”MoscowRussia
  2. 2.Laboratory for Mathematical Methods in Natural ScienceMoscowRussia

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