Bruno, P.

The Berry phase in magnetism and the anomalous Hall effect
Handbook of Magnetism and Advanced Magnetic Materials, Volume 1: Fundamentals and Theorypp 540-558 (Eds.) Kronmüller, H. and Parkin, S. S. P.,John Wiley & Sons Ltd, Chichester, UK 2007
In 1983, Berry made the surprising discovery that a quantum system adiabatically transported round a closed circuit C in the space of external parameters acquires, besides the familiar dynamical phase, a nonintegrable phase depending only on the geometry of the circuit C (Berry, 1984). This Berry phase, which had been overlooked for more than half a century, provides us a very deep insight into the geometric structure of quantum mechanics and gives rise to various observable effects. The concepts of the Berry phase has now become a central unifying concept in quantum mechanics, with applications in fields ranging from chemistry to condensed-matter physics (Shapere and Wilczek, 1989; Bohm et a., 2003).
The aim of this article is to give an elementary introduction to the Berry phase, and to discuss its various implications in the field of magnetism, where it plays an increasingly important role. The reader is referred to specialized textbooks (Shapere and Wilczek, 1989; Bohm et al., 2003) for a more comprehensive presentation of the field of geometrical phases. Particular emphasis will be given to the discussion of the anomalous Hall effect, the theory of which has been considerably renewed recently, on the basis of the concept of Berry phase.

TH-2007-20