Local Description of Band Rearrangements
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Local Description of Band Rearrangements Comparison of Semi-quantum and Full Quantum Approach Toshihiro Iwai · Boris Zhilinskii
Received: 16 July 2014 / Accepted: 7 November 2014 © Springer Science+Business Media Dordrecht 2014
Abstract Rearrangement of rotation-vibration energy bands in isolated molecules within semi-quantum approach is characterized by delta-Chern invariants, each of which is associated to a locally approximated semi-quantum Hamiltonian valid in a small neighborhood of a degeneracy point for the initial semi-quantum Hamiltonian and also valid in a small neighborhood of a critical point corresponding to the crossing of the boundary between iso-Chern domains in the control parameter space. For a full quantum model, a locally approximated Hamiltonian is assumed to take the form of a Dirac operator together with a specific boundary condition. It is demonstrated that the crossing of the boundary along a path with a delta-Chern invariant equal to ±1 corresponds to the transfer of one quantum level from a subspaces of quantum states to the other subspace associated with respective positive and negative energy eigenvalues of the local Dirac Hamiltonian. Keywords Energy band · Chern number · Dirac operator Mathematics Subject Classification 53C80 · 81Q70 · 81V55
1 Introduction Redistribution of energy levels between energy bands in rotation-vibration structure of isolated molecules under the variation of a control parameter was a subject of a number of publications [5, 9, 11, 12, 14]. In semi-quantum models [16, 17, 20], rotational and vibrational variables are treated as classical and quantum ones, respectively, and the semi-quantum Hamiltonian takes the form of a Hermitian matrix depending on classical variables. With
T. Iwai Kyoto University, 606-8501 Kyoto, Japan e-mail: [email protected]
B
B. Zhilinskii ( ) Université du Littoral Côte d’Opale, 189A av. M. Schumann, Dunkerque 59140, France e-mail: [email protected]
T. Iwai, B. Zhilinskii
each non-degenerate eigenvalue of that Hamiltonian, there are associated an eigen-line bundle, whose base space is a classical phase space for the rotational variables and fibers represent vibrational quantum states [5, 9]. The Chern numbers of respective eigen-line bundles are topological invariants characterizing the band structure. The qualitative rearrangement of energy bands is associated with the formation of degeneracy points of eigenvalues of the matrix Hamiltonian. The critical points in the control parameter space, for which the Hamiltonian has degeneracy points on the classical phase space, form boundaries between iso-Chern domains. For a one-parameter family of Hamiltonians attached with a path in the control parameter space, the crossing of the boundary between iso-Chern domains can be characterized by a local delta-Chern, which is a homotopical invariant associated with a locally approximated Hamiltonian defined in a small neighborhood of a degeneracy point for the Hamiltonian and also defined in a small neighborhood of t
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