Topological properties of Berry’s phase

By using a second quantized formulation of level crossing, which does not assume adiabatic approximation, a convenient formula for geometric terms including offdiagonal terms is derived. The analysis of geometric phases is reduced to a simple diagonalization of the Hamiltonian in the present formulation. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial for any finite time interval T . The topological interpretation of Berry’s phase such as the topological proof of phase-change rule thus fails in the practical Born-Oppenheimer approximation, where a large but finite ratio of two time scales is involved. The geometric phases are mostly defined in the framework of adiabatic approximation [1]-[6], though a non-adiabatic treatment has been considered in, for example, [7] and the (non-adiabatic) correction to the geometric phases has been analyzed [8]. One may then wonder if some of the characteristic properties generally attributed to the geometric phases are the artifacts of the approximation. We here show that the topological properties of the geometric phases associated with level crossing are the artifacts of the adiabatic approximation which assumes the infinite time interval T → ∞ [2]. To substantiate this statement, we start with the exact definition of geometric terms associated with level crossing. The level crossing problem is neatly formulated by using the second quantization technique without assuming adiabatic approximation. The analysis of phase factors in this formulation is reduced to a diagonalization of the Hamiltonian, and thus it is formulated both in the path integral and in the operator formulation. We thus derive a convenient formula for geometric terms [1] and their off-diagonal counter parts. (As for off-diagonal geometric phases, see [9] where the off-diagonal geometric phases in the framework of an adiabatic picture in the first quantization have been proposed, and their properties have been analyzed in [10, 11, 12].) Our formula allows us to analyze the topological properties of the geometric terms precisely in the infinitesimal neighborhood of level crossing. At the level crossing point, the conventional energy eigenvalues become degenerate but the degeneracy is lifted if one diagonalizes the geometric terms. It is then shown that the geometric phases become trivial (and thus no monopole singularity) in the infinitesimal neighborhood of level crossing for any finite time interval T . This is proved independently of the adiabatic approximation. The topological interpretation [3, 1] of geometric phases such as the topological proof of Longuet-Higgins’ phase-change rule [4] thus