Some geometric features of Berry ́s phase

In this letter, we elaborate on the identification and construction of the differential geometric elements underlying Berry ́s phase. Berry bundles are built generally from the physical data of the quantum system under study. We apply this construction to typical and recently investigated systems presenting Berry ́s phase to explore their geometric features. Berry ́s phase [1] discovery for parameter-dependent quantum systems showed the existence of fundamental differential geometric features in quantum physics. In fact, the geometric nature of this phase leads to both, its theoretical importance and the ability to perform experiments in which this phase can be detected [2]. Because of this fact, it is important to have a suitable description of the underlying physical data of the system in terms of the geometric objects which lead to the phase shift under study. Usually [3] for non abelian phases, this geometric setting is modeled by means of the universal U(k) bundle over the grassmannian manifold GKm(H) [4] of K −dimensional subspaces of the total Hilbert space H, endowed with its canonical connection. When the parameter b varies within a parameter space B̃, the K−dimensional eigenspace F b ⊂ H of a given energy ǫ (b) also varies describing a curve in the grassmannian. Parallel transport along this curve captures the geometric Berry phase effect. The aim of this letter is to enlighten the fact that the geometry directly relevant for the study of the underlying physical problem is not that of the above mentioned universal bundle, but the one of the pull back [4] bundle along the induced map Parameter Space−→Grassmannian. Indeed, the space of physical parameters can be much smaller than or have a very different geometry from that of the grassmannian manifold. We remark this in the same sense that the geometry of a 2−sphere S2 →֒ R3, even though following form that of R3, is different and can be independently studied from that of the bigger ambient space R3. A very simple example of this is given by a large (s ≥ 1) spin s in an external magnetic field. In that case, the dimension of the grassmanninan can be huge (equal to 4s) while the space of physically accessible eigenspaces F b ⊂ H through the manipulation of the external magnetic field, is a submanifold of at most dimension 3 for every s (see examples below). On the other hand, considering the pull back bundle has the strategical advantage of allowing for a direct study on how the geometry of the physical parameter space B affects the resulting Berry phase. Moreover, this pull back bundle U(E) −→ B, that we shall refer to as Berry bundle, can be directly constructed from the natural physical inputs defining the quantum system: ∗ [email protected]