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A study of energy band rearrangement in isolated molecules by means of the Dirac oscillator approximation. (English) Zbl 1458.37063

Summary: Energy band rearrangement along a control parameter in isolated molecules is studied through axially symmetric Hamiltonians describing the coupling of two angular momenta \(\mathbf{S}\) and \(\mathbf{L}\) of fixed amplitude. We focus our attention on the case \(S=1\) which, albeit nongeneric, describes the global rearrangement of a system of energy bands between two well-defined limits corresponding to uncoupled and coupled momenta. The redistribution of energy levels between bands is closely related to the degeneracy of the eigenvalues of the corresponding semiquantum Hamiltonian at isolated points of the three-dimensional Cartesian product of the two-dimensional phase space and the one-dimensional control parameter space. The present paper shows that the band rearrangement for the full quantum system can be quantitatively, rather than qualitatively, reproduced with Dirac oscillator approximations. We also interpret the energy band rearrangement by comparing the evolution of the joint spectra of commuting observables (i.e., energy and axial angular momentum) with that of the image of the energy-momentum map of the completely classical limit of the Dirac oscillator approximations.

MSC:

37J39 Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.)
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
53D20 Momentum maps; symplectic reduction
58K65 Topological invariants on manifolds
81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
81V55 Molecular physics
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
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[1] Alonso, J.; Dullin, H. R.; Hohloch, S., Symplectic Classification of Coupled Angular Momenta, Nonlinearity, 33, 1, 417-468 (2020) · Zbl 1435.37081 · doi:10.1088/1361-6544/ab4e05
[2] Alonso, J. and Hohloch, S., Survey on Recent Developments in Semitoric Systems, arXiv:1901.10433 (2019).
[3] Arnold, V. I., Remarks on Eigenvalues and Eigenvectors of Hermitian Matrices, Berry Phase, Adiabatic Connections and Quantum Hall Effect, Selecta Math., 1, 1, 1-19 (1995) · Zbl 0841.58008 · doi:10.1007/BF01614072
[4] Arnold, V. I.; Arnold, V.; Atiyah, M.; Lax, P.; Mazur, B., Polymathematics: Is Mathematics a Single Science or a Set of Arts?, Mathematics: Frontiers and Perspectives, 403-416 (1999), Providence, R.I.: AMS, Providence, R.I. · Zbl 0980.00004
[5] Avron, J. E.; Sadun, L.; Segert, J.; Simon, B., Chern Numbers, Quaternions, and Berry’s Phases in Fermi Systems, Commun. Math. Phys., 124, 4, 595-627 (1989) · Zbl 0830.57020 · doi:10.1007/BF01218452
[6] Bernevig, B. A.; Hughes, T. L., Topological Insulators and Topological Superconductors (2013), Princeton: Princeton Univ. Press, Princeton · Zbl 1269.82001
[7] Berry, M. V., Quantal Phase Factors Accompanying Adiabatic Changes, Proc. Roy. Soc. London Ser. A, 392, 1802, 45-57 (1984) · Zbl 1113.81306
[8] Colin de Verdière, Y., The Level Crossing Problem in Semi-Classical Analysis: I. The Symmetric Case, Ann. Inst. Fourier (Grenoble), 53, 4, 1023-1054 (2003) · Zbl 1113.35151 · doi:10.5802/aif.1973
[9] Colin de Verdière, Y., The Level Crossing Problem in Semi-Classical Analysis: II. The Hermitian Case, Ann. Inst. Fourier (Grenoble), 54, 5, 1423-1441 (2004) · Zbl 1067.35162 · doi:10.5802/aif.2054
[10] Cushman, R. H.; Bates, L. M., Global Aspects of Classical Integrable Systems (2015), Basel: Birkhäuser, Basel · Zbl 1321.70001
[11] Cushman, R.; Duistermaat, J. J., The Quantum Mechanical Spherical Pendulum, Bull. Amer. Math. Soc. (N. S.), 19, 2, 475-479 (1988) · Zbl 0658.58039 · doi:10.1090/S0273-0979-1988-15705-9
[12] Dhont, G.; Iwai, T.; Zhilinskii, B., Topological Phase Transition in a Molecular Hamiltonian with Symmetry and Pseudo-Symmetry, Studied through Quantum, Semi-Quantum and Classical Models, SIGMA Symmetry Integrability Geom. Methods Appl., 13 (2017) · Zbl 1433.81164
[13] Duistermaat, J. J., On Global Action-Angle Coordinates, Comm. Pure Appl. Math., 33, 6, 687-706 (1980) · Zbl 0439.58014 · doi:10.1002/cpa.3160330602
[14] Dullin, H.; Giacobbe, A.; Cushman, R., Monodromy in the Resonant Swing Spring, Phys. D, 190, 1-2, 15-37 (2004) · Zbl 1098.70520 · doi:10.1016/j.physd.2003.10.004
[15] Faure, F.; Zhilinskii, B. I., Topological Chern Indices in Molecular Spectra, Phys. Rev. Lett., 85, 5, 960-963 (2000) · doi:10.1103/PhysRevLett.85.960
[16] Faure, F.; Zhilinskii, B. I., Topological Properties of the Born - Oppenheimer Approximation and Implications for the Exact Spectrum, Lett. Math. Phys., 55, 3, 219-238 (2001) · Zbl 0981.81028 · doi:10.1023/A:1010912815438
[17] Fermanian Kammerer, C.; Lasser, C., Propagation through Generic Level Crossings: A Surface Hopping Semigroup, SIAM J. Math. Anal., 40, 1, 103-133 (2008) · Zbl 1168.35043 · doi:10.1137/070686810
[18] Hagedorn, G. A., Molecular Propagation through Electron Energy Level Crossings (1994), Providence, R.I.: AMS, Providence, R.I. · Zbl 0833.92025
[19] Haldane, F. D. M., Model for a Quantum Hall Effect without Landau Levels: Condensed - Matter Realization of the “Parity Anomaly”, Phys. Rev. Lett., 61, 18, 2015-2018 (1988) · doi:10.1103/PhysRevLett.61.2015
[20] Hasan, M. Z.; Kane, C. L., Colloquium: Topological Insulators, Rev. Mod. Phys., 82, 4, 3045-3067 (2010) · doi:10.1103/RevModPhys.82.3045
[21] Herzberg, G.; Longuet-Higgins, H. C., Intersection of Potential Energy Surfaces in Polyatomic Molecules, Discuss. Faraday Soc., 35, 77-82 (1963) · doi:10.1039/df9633500077
[22] Itô, D.; Mori, K.; Carriere, E., An Example of Dynamical Systems with Linear Trajectory, Il Nuovo Cimento A, 51, 4, 1119-1121 (1967) · doi:10.1007/BF02721775
[23] Iwai, T., Sadovskii, D. A., and Zhilinskii, B. I., Angular Momentum Coupling, Dirac Oscillators, and Quantum Band Rearrangements in the Presence of Momentum Reversal Symmetries, J. Geom. Mech., 2020, doi:10.3934/jgm.2020021. · Zbl 1508.81860
[24] Iwai, T.; Zhilinskii, B. I., Energy Bands: Chern Numbers and Symmetry, Ann. Phys. (NY), 326, 12, 3013-3066 (2011) · Zbl 1236.81217 · doi:10.1016/j.aop.2011.07.002
[25] Iwai, T.; Zhilinskii, B. I., Qualitative Feature of the Rearrangement of Molecular Energy Spectra from a “Wall-Crossing” Perspective, Phys. Lett. A, 377, 38, 2481-2486 (2013) · doi:10.1016/j.physleta.2013.07.043
[26] Iwai, T.; Zhilinskii, B., Topological Phase Transitions in the Vibration-Rotation Dynamics of an Isolated Molecule, Theor. Chem. Acc., 133, 7 (2014) · doi:10.1007/s00214-014-1501-x
[27] Iwai, T.; Zhilinskii, B. I., Chern Number Modification in Crossing the Boundary between Different Band Structures: Three-Band Model with Cubic Symmetry, Rev. Math. Phys., 29, 2 (2017) · Zbl 1364.53073 · doi:10.1142/S0129055X17500040
[28] Iwai, T.; Zhilinskii, B., The 2D Kramers - Dirac Oscillator, Phys. Lett. A, 383, 13, 1389-1395 (2019) · Zbl 1470.81030 · doi:10.1016/j.physleta.2019.01.062
[29] Jaynes, E. T.; Cummings, F. W., Comparison of Quantum and Semiclassical Radiation Theories with Application to the Beam Maser, Proc. IEEE, 51, 1, 89-109 (1963) · doi:10.1109/PROC.1963.1664
[30] Kitaev, A., Periodic Table for Topological Insulators and Superconductors, AIP Conf. Proc., 1134, 1, 22-30 (2009) · Zbl 1180.82221 · doi:10.1063/1.3149495
[31] Kohmoto, M., Topological Invariant and the Quantization of the Hall Conductance, Ann. Phys. (NY), 160, 2, 343-354 (1985) · doi:10.1016/0003-4916(85)90148-4
[32] Langer, R. E., On the Connection Formulas and the Solutions of the Wave Equation, Phys. Rev., 51, 8, 669-676 (1937) · Zbl 0017.01705 · doi:10.1103/PhysRev.51.669
[33] Mead, C. A., Molecular Kramers Degeneracy and non-Abelian Adiabatic Phase Factors, Phys. Rev. Lett., 59, 2, 161-164 (1987) · doi:10.1103/PhysRevLett.59.161
[34] Michel, L.; Zhilinskií, B. I., Symmetry, Invariants, Topology: Basic Tools, Phys. Rep., 341, 1-6, 11-84 (2001) · Zbl 0971.22500 · doi:10.1016/S0370-1573(00)00088-0
[35] Moshinsky, M.; Szczepaniak, A., The Dirac Oscillator, J. Phys. A, 22, 17, 817-819 (1989) · doi:10.1088/0305-4470/22/17/002
[36] Nekhoroshev, N. N., Action-Angle Variables and Their Generalization, Trans. Moscow Math. Soc., 26, 180-198 (1972) · Zbl 0284.58009
[37] Ortega, J.-P.; Ratiu, T. S., Momentum Maps and Hamiltonian Reduction (2004), Boston, Mass.: Birkhäuser, Boston, Mass. · Zbl 1241.53069
[38] Pavlov-Verevkin, V. B.; Sadovskii, D. A.; Zhilinskii, B. I., On the Dynamical Meaning of the Diabolic Points, Europhys. Lett., 6, 7, 573-578 (1988) · doi:10.1209/0295-5075/6/7/001
[39] Pelayo, Á.; Vu Ngoc, S., Hamiltonian Dynamics and Spectral Theory for Spin-Oscillators, Commun. Math. Phys., 309, 1, 123-154 (2012) · Zbl 1263.70022 · doi:10.1007/s00220-011-1360-4
[40] Sadovskii, D. A.; Zhilinskii, B. I., Monodromy, Diabolic Points, and Angular Momentum Coupling, Phys. Lett. A, 256, 4, 235-244 (1999) · Zbl 0934.81005 · doi:10.1016/S0375-9601(99)00229-7
[41] Schnyder, A. P.; Ryu, S.; Furusaki, A.; Ludwig, A. W. W., Classification of Topological Insulators and Superconductors in Three Spatial Dimensions, Phys. Rev. B, 78, 19 (2008) · doi:10.1103/PhysRevB.78.195125
[42] Schwarz, G. W., Smooth Functions Invariant under the Action of a Compact Lie Group, Topology, 14, 1, 63-68 (1975) · Zbl 0297.57015 · doi:10.1016/0040-9383(75)90036-1
[43] Shapere, A.; Wilczek, F., Geometric Phases in Physics (1989), Teaneck, N.J.: World Sci., Teaneck, N.J. · Zbl 0914.00014
[44] Simon, B., The Classical Limit of Quantum Partition Functions, Commun. Math. Phys., 71, 3, 247-276 (1980) · Zbl 0436.22012 · doi:10.1007/BF01197294
[45] Thouless, D. J., Topological Quantum Numbers in Nonrelativistic Physics (1998), Singapore: World Sci., Singapore · Zbl 1346.81004
[46] Thouless, D. J.; Kohmoto, M.; Nightingale, M. P.; den Nijs, M., Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett., 49, 6, 405-408 (1982) · doi:10.1103/PhysRevLett.49.405
[47] von Neumann, J.; Wigner, E. P., Über das Verhalten von Eigenwerten bei adiabatischen Prozessen, Phys. Z., 30, 467-470 (1929) · JFM 55.0520.05
[48] Vu Ngoc, S., Moment Polytopes for Symplectic Manifolds with Monodromy, Adv. Math., 208, 2, 909-934 (2007) · Zbl 1118.53051 · doi:10.1016/j.aim.2006.04.004
[49] Zhilinskií, B. I., Symmetry, Invariants, and Topology in Molecular Models, Phys. Rep., 341, 1-6, 85-171 (2001) · Zbl 0971.81593 · doi:10.1016/S0370-1573(00)00089-2
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