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Topological properties of Berry’s phase

Kazuo Fujikawa

Published Jan 1, 2004• 18 citations
OpenAlex

Abstract

By using a second quantized formulation of level crossing, which does not assume adiabatic approximation, the exact formula for geometric terms including off-diagonal terms is derived. 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 for any finite T. 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]. 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 approximation which assumes the infinite time interval T → ∞. To substantiate this statement, we start with the exact definition of geometric terms associated with level crossing. The level crossing problem is formulated by using the second quantization technique

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By using a second quantized formulation of level crossing, which does not assume adiabatic approximation, the exact formula for geometric terms including off-diagonal terms is derived. 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 for any finite T. 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]. 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 approximation which assumes the infinite time interval T → ∞. To substantiate this statement, we start with the exact definition of geometric terms associated with level crossing. The level crossing problem is formulated by using the second quantization technique
By using a second quantized formulation of level crossing, which does not assume adiabatic approximation, a convenient formula for geometric terms including off-diagonal 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 level crossing problem is neatly formulated by the second quantized formulation, which exhibits a hidden local gauge symmetry. The analysis of geometric phases is reduced to a simple diagonalization of the Hamiltonian. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial (and thus no monopole singularity) for arbitrarily large but finite time interval TTT. The topological proof of the Longuet-Higgins' phase-change rule, for example, thus fails in the practical Born-Oppenheimer approximation where TTT is identified with the period of the slower system. The crucial difference between the Aharonov-Bohm phase and the geometric phase is explained. It is also noted that the gauge symmetries involved in the adiabatic and non-adiabatic geometric phases are quite different.
The level crossing problem and associated geometric terms are neatly formulated by the second-quantized formulation. This formulation exhibits a hidden local gauge symmetry related to the arbitrariness of the phase choice of the complete orthonormal basis set. By using this second-quantized formulation, which does not assume adiabatic approximation, a convenient exact formula for the geometric terms including off-diagonal geometric terms is derived. The analysis of geometric phases is then reduced to a simple diagonalization of the Hamiltonian, and it is analyzed both in the operator and path-integral formulations. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial (and thus no monopole singularity) for arbitrarily large but finite time interval TTT. The integrability of Schr\"odinger equation and the appearance of the seemingly nonintegrable phases are thus consistent. The topological proof of the Longuet-Higgins' phase-change rule, for example, fails in the practical Born-Oppenheimer approximation where a large but finite ratio of two time scales is involved and TTT is identified with the period of the slower system. The difference and similarity between the geometric phases associated with level crossing and the exact topological object such as the Aharonov-Bohm phase become clear in the present formulation. A crucial difference between the quantum anomaly and the geometric phases is also noted.
The level crossing problem and associated geometric terms are neatly formulated by using the second quantization technique both in the operator and path integral formulations. The analysis of geometric phases is then reduced to the familiar diagonalization of the Hamiltonian. If one diagonalizes the Hamiltonian in one specific limit, one recovers the conventional formula for geometric phases. On the other hand, if one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial (and thus no monopole singularity) for arbitrarily large but finite time interval T. The topological proof of the Longuet-Higgins' phase-change rule, for example, thus fails in the practical Born-Oppenheimer approximation where a large but finite ratio of two time scales is involved and T is identified with the period of the slower system.
By analyzing an exactly solvable model in the second quantized formulation which allows a unified treatment of adiabatic and non-adiabatic geometric phases, it is shown that the topology of the adiabatic Berry's phase, which is characterized by the singularity associated with possible level crossing, is trivial in a precise sense. This topology of the geometric phase is quite different from the topology of the Aharonov-Bohm effect, where the topology is specified by the external local gauge field and it is exact for the slow as well as for the fast motion of the electron.
The analysis of geometric phases associated with level crossing is reduced to the familiar diagonalization of the Hamiltonian in the second quantized formulation. A hidden local gauge symmetry, which is associated with the arbitrariness of the phase choice of a complete orthonormal basis set, becomes explicit in this formulation (in particular, in the adiabatic approximation) and specifies physical observables. The choice of a basis set which specifies the coordinate in the functional space is arbitrary in the second quantization, and a subclass of coordinate transformations, which keeps the form of the action invariant, is recognized as the gauge symmetry. We discuss the implications of this hidden local gauge symmetry in detail by analyzing geometric phases for cyclic and noncyclic evolutions. It is shown that the hidden local symmetry provides a basic concept alternative to the notion of holonomy to analyze geometric phases and that the analysis based on the hidden local gauge symmetry leads to results consistent with the general prescription of Pancharatnam. We however note an important difference between the geometric phases for cyclic and noncyclic evolutions. We also explain a basic difference between our hidden local gauge symmetry and a gauge symmetry (or equivalence class) used by Aharonov and Anandan in their definition of generalized geometric phases.