Topological Chern Indices in Molecular Spectra

Faure, Frédéric; Zhilinskii, B.I.

doi:10.1103/physrevlett.85.960

Cited by 66 publications

(94 citation statements)

References 21 publications

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“…In the case of rotational structure of vibrational states, classical rotational variables form a phase space which is a twodimensional sphere and the number of quantum states taken into account corresponds to a number of analyzed vibrational states counted with their degeneracies. Further studies have shown that generically the redistribution of energy levels in the rotational band structure consists in the transfer of one energy level and is associated with the modification of Chern number by one [5,6].…”

Section: Introductionmentioning

confidence: 99%

Rearrangement of Energy Bands: Chern Numbers in the Presence of Cubic Symmetry

Iwai

Zhilinskii

2012

Acta Appl Math

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Formation of energy bands in the system of rotation-vibration quantum states of molecules is described within semi-quantum models under the presence of a symmetry group characterizing the equilibrium molecular configuration. Effective rotation-vibration Hamiltonians are written in two-quantum state models with rotational variables treated as classical ones. Eigen-line bundles associated with eigenvalues of 2 × 2 Hermitian matrix defined on rotational classical phase space which is a two-dimensional sphere are characterized by the first Chern class. Explicit procedure for the calculation of Chern numbers are suggested and realized for a family of Hamiltonians depending on extra control parameters in the presence of symmetry. Effective Hamiltonians for two vibrational states transforming according to some representations of the cubic symmetry group are studied. Chern numbers are evaluated for respective model Hamiltonians. The iso-Chern diagrams are introduced which split the parameter space into regions with fixed Chern numbers.

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Section: Introductionmentioning

confidence: 99%

Rearrangement of Energy Bands: Chern Numbers in the Presence of Cubic Symmetry

Iwai

Zhilinskii

2012

Acta Appl Math

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“…The relation between quantum monodromy and the phenomenon of redistribution of quantum states between branches in the energy spectrum was observed on several occasions [27,12,13] in the context of study of the same problem in pure quantum, semi-quantum, and purely classical approaches. A mathematically rigorous interpretation of such relation is still lacking.…”

Section: Discussionmentioning

confidence: 96%

Interpretation of Quantum Hamiltonian Monodromy in Terms of Lattice Defects

Zhilinskii

2005

Acta Appl Math

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“…This study is beyond the scope of the present basic RE analysis. We mention only that each band can be assigned a topological index (Chern index) [82], which gives the difference between the number of states 2J + 1 of an isolated rotational multiplet and the number of states in the band. The sum of these indexes over all components of the polyad equals zero.…”

Section: Combined Systems With Several Dynamical Integralsmentioning

confidence: 99%

Analysis of Rotation--Vibration Relative Equilibria on the Example of a Tetrahedral Four Atom Molecule

Efstathiou¹,

Sadovskií²,

Zhilinskii³

2004

SIAM J. Appl. Dyn. Syst.

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Abstract. We study relative equilibria (RE) of a nonrigid molecule, which vibrates about a well-defined equilibrium configuration and rotates as a whole. Our analysis unifies the theory of rotational and vibrational RE. We rely on the detailed study of the symmetry group action on the initial and reduced phase space of our system and consider the consequences of this action for the dynamics of the system. We develop our approach on the concrete example of a four-atomic molecule A4 with tetrahedral equilibrium configuration, a dynamical system with six vibrational degrees of freedom. 1. Introduction. This paper unifies modern methods of classical theory of symmetric Hamiltonian dynamical systems and quantum theory of molecules (and other isolated finiteparticle systems). Considerable progress was achieved in both directions in the last decades and deep relations between these seemingly distant theories became evident. Significant effort by mathematicians and molecular physicists to converge the two fields resulted in the qualitative theory of highly excited quantum molecular systems based on recent mathematical developments. We join the two approaches and demonstrate what kind of concrete results can be immediately obtained in molecular systems [1,2,3,4] by applying powerful methods of symmetric Hamiltonian systems [5,6,7,8,9,10,11]. We choose a concrete problem of rotation-vibration of a four-atomic molecule with tetrahedral equilibrium configuration [12,13] in order to explain the details of our approach.

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Topological Chern Indices in Molecular Spectra

Cited by 66 publications

References 21 publications

Rearrangement of Energy Bands: Chern Numbers in the Presence of Cubic Symmetry

Rearrangement of Energy Bands: Chern Numbers in the Presence of Cubic Symmetry

Interpretation of Quantum Hamiltonian Monodromy in Terms of Lattice Defects

Analysis of Rotation--Vibration Relative Equilibria on the Example of a Tetrahedral Four Atom Molecule

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