BROKEN TIME-REVERSAL SYMMETRY AND BERRY PHASE

Berry's phase is typically realized in a system consisting of fast and slow variables, when the trajectory of the slow variable makes a clos ed loop. The quantum mechanical phase picked up by the fast variable w hile the slow variable traverses the loop has turned out to produce re al physical effects through quantum interference. In this article, we investigate origins of Berry's geometric phase and show that they are in general attributable to the broken time-reversal symmetry of the sy stem. Our analysis leads to the classification of Berry's phase for Ha miltonian systems in terms of symmetry properties under time-reversal operations. Spontaneous time-reversal symmetry-breaking of state vecto rs is shown to give rise to Berry's phase as exemplified by a quantum- mechanical rotated hoop. A system with an explicitly time-reversal sym metry-breaking Hamiltonian is also demonstrated to exhibit nontrivial Berry's phase. The quantization of the geometric phase associated with the real two-dimensional Hamiltonian having topological singularity i s explained within the same framework. The unique role of the time-rev ersal operator among general antiunitary operators is also discussed.