Diabolic Points, Geometric Phases, and Quantum Chaos

  • Dirk Dubbers
  • Hans-Jürgen Stöckmann
Part of the Graduate Texts in Physics book series (GTP)


Quantum systems have a number of peculiar features that can all be traced back to one common origin, namely the universal “diabolic” shapes of energy surfaces in the vicinity of a degeneracy. The first such feature is the avoided-crossing effect, when energy levels do not cross but seem to repel each other when plotted in dependence of an external parameter. The next feature is the occurrence of geometric phases, also known as Berry phases, which accumulate during excursions in parameter space. We discuss experimental examples of such phases, both in the space of magnetic fields and in what we call the space of shapes. The Aharonov–Bohm phase is closely related to such geometric phases. Finally, in a short section we show that the hallmarks of quantum chaos can be understood in terms of locally defined 2×2 matrices and their avoided level crossings.


Chaotic System Current Sheet Pure Spin Quadrupole Interaction Geometric Phase 
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  1. Aharonov, Y., Anandan, J.: Phase change during a cyclic quantum evolution. Phys. Rev. Lett. 58, 1593–1596 (1987) MathSciNetADSCrossRefGoogle Scholar
  2. Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in quantum theory. Phys. Rev. 115, 485–491 (1959) MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. Bayh, W.: Messung der kontinuierlichen Phasenschiebung von Elektronenwellen im kraftfeldfreien Raum durch das magnetische Vektorpotential einer Wolfram-Wendel. Z. Phys. 169, 492–510 (1962) ADSCrossRefGoogle Scholar
  4. Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984) ADSzbMATHCrossRefGoogle Scholar
  5. Bitter, T., Dubbers, D.: Manifestation of Berry’s topological phase in neutron spin rotation. Phys. Rev. Lett. 59, 251–254 (1987) ADSCrossRefGoogle Scholar
  6. Bohigas, O., Giannoni, M.J., Schmit, C.: Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52, 1–4 (1984) MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. Gutzwiller, M.C.: Phase-integral approximation in momentum space and the bound states of an atom. J. Math. Phys. 8, 1979–2000 (1967) ADSCrossRefGoogle Scholar
  8. Haake, F.: Quantum Signatures of Chaos, 3rd edn. Springer, Berlin (2010) zbMATHCrossRefGoogle Scholar
  9. Lauber, H.-M., Weidenhammer, P., Dubbers, D.: Geometric phases and hidden symmetries in simple resonators. Phys. Rev. Lett. 72, 1004–1007 (1994) ADSCrossRefGoogle Scholar
  10. Manini, N., Pistolesi, F.: Off-diagonal Berry phases. Phys. Rev. Lett. 85, 3067–3071 (2000) ADSCrossRefGoogle Scholar
  11. Nakamura, K., Thomas, H.: Quantum billiard in a magnetic field: chaos and diamagnetism. Phys. Rev. Lett. 61, 247–250 (1988) MathSciNetADSCrossRefGoogle Scholar
  12. Simon, B.: Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys. Rev. Lett. 51, 2167–2170 (1983) MathSciNetADSCrossRefGoogle Scholar
  13. Stöckmann, H.-J.: Quantum Chaos—An Introduction. University Press, Cambridge (1999) zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Dirk Dubbers
    • 1
  • Hans-Jürgen Stöckmann
    • 2
  1. 1.Fak. Physik und Astronomie, Physikalisches InstitutUniversität HeidelbergHeidelbergGermany
  2. 2.Fachbereich Physik Philipps-Universität MarburgMarburgGermany

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