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Selecta Mathematica

Remarks on eigenvalues and eigenvectors of Hermitian matrices, berry phase, adiabatic connections and quantum Hall effect

Vladimir Igorevich Arnol'd

Published Mar 1, 1995• 64 citations
Publisher
DOI
10.1007/bf01614072

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•Aug 24, 2009•3 citations
It is shown that the "geometrical phase factor" recently found by Berry in his study of the quantum adiabatic theorem is precisely the holonomy in a Hermitian line bundle since the adiabatic theorem naturally defines a connection in such a bundle. This not only takes the mystery out of Berry's phase factor and provides calculational simple formulas, but makes a connection between Berry's work and that of Thouless et al. This connection allows the author to use Berry's ideas to interpret the integers of Thouless et al. in terms of eigenvalue degeneracies.
In the standard (Hermitian) formalism of quantum mechanics, usually when a potential parameter is varied the crossing of energy levels with the same symmetry is avoided. However, within the framework of the non-Hermitian formalism it is possible that two (or even more) complex eigenvalues with the same symmetry will cross. At the crossing point the eigenvalues are degenerate and this is accompanied by the coalescence of the eigenfunctions (or eigenvectors). Therefore, we may term this special situation as a non-Hermitian degeneracy. This special situation is associated with a branch point in the complex energy plane which is commonly termed an “exceptional point” in the spectrum of the non-Hermitian Hamiltonian. With respect to the c-product defined for non-Hermitian operators (matrices) in Chapter 6, the degenerate eigenstate is self-orthogonal. Since a branch point in the spectrum is removed by any infinitesimally small external perturbation, it seems to be inaccessible experimentally and may be considered just as a mathematical object rather than a physical one. However, as we will show here, by varying the potential parameters the existence of a branch point is reflected in the measurement of the geometrical phases also known as Berry phases. It should be stressed here that while in our case the geometrical phase results from a coalescence of eigenfunctions of a non-Hermitian Hamiltonian, the so-called Berry phase phenomenon occurs also within the Hermitian formalism of quantum mechanics when the eigenvalues of the molecular Hamiltonian in the Born–Oppenheimer approximation are degenerate for specific geometry of the poly-atomic molecule.
Why Is Group Theory Useful in Quantum Mechanics? Irreducible Representations of SU(2) and SO(3), Rotation Matrices The Symmetric Groups The Lie Algebra of SO(4) and the Hydrogen Atom Geometric Phases: The Aharonov-Bohm Effect and the Magnetic Monopole The Berry Phase in Molecular Dynamics The Dynamic Phase: Riemann Surfaces in the Semiclassical Theory of Non-Adiabatic Collisions The Connection Is the Gauge Field and the Curvature Is the Force: Some Topology and Differential Geometry Topological Quantum (Chern) Numbers: The Integer Quantum Hall Effect Chern-Simons Theory: The Fractional Quantum Hall Effect and other chapters.