Energy bands: Chern numbers and symmetry

Energy bands formed by rotation–vibrational states of molecules in the presence of symmetry and their qualitative modifications under variation of some control parameters are studied within the semi-quantum model. Rotational variables are treated as classical whereas a finite set of vibrational states is considered as quantum. In the two-state approximation the system is described in terms of a fiber bundle with the base space being a two-dimensional sphere, the classical phase space for rotational variables. Generically this rank 2 complex vector bundle can be decomposed into two complex line bundles characterized by a topological invariant, the first Chern class. A general method of explicit calculation of Chern classes and of their possible modifications under variation of control parameters in the presence of symmetry is suggested. The construction of iso-Chern diagrams which split the space of control parameters into connected domains with fixed Chern numbers is suggested. A detailed analysis of the rovibrational model Hamiltonian for a D3D3 invariant molecule possessing two vibrational states transforming according to the two-dimensional irreducible representation is done to illustrate non-trivial restrictions imposed by symmetry on possible values of Chern classes.

► Complex line bundles associated with eigenvalues of 2×2 Hermitian matrix Hamiltonians.
► Hamiltonians are defined on the 2-sphere and invariant under symmetry groups.
► Symmetry permits only some special integers as Chern numbers.
► For SO(2)SO(2) symmetry, the possible values are 0, ±K±K, where KK is an index of the representation.
► For D3D3 symmetry, the possible values are ±2, ±4 within our model Hamiltonians.