The geometric phase for chaotic systems

Abstract The geometric phase acquired by the eigenstates of cycled quantum systems is given by the flux of a two-form through a surface in the system’s parameter space. We obtain the classical limit of this two-form in a form applicable to systems whose classical dynamics is chaotic. For integrable systems the expression is equivalent to the Hannay two-form. We discuss various properties of the classical two-form, derive semiclassical corrections to it (associated with classical periodic orbits), and consider implications for the semiclassical density of degeneracies.