Symmetry breaking and localization in quantum chaotic systems.

Abstract

Breaking an antiunitary symmetry modifies the localization length. In analogy to results obtained recently in the context of Anderson localization in disordered solids, we establish that the localization length \ensuremath{\lambda} of dynamically localizing quantum chaotic systems depends on the invariance properties of the system under antiunitary symmetry operations. We consider a Hamiltonian system---a modified version of the kicked rotor---which, by tuning its parameters, can change its invariance properties in a manner similar to the (Gaussian orthogonal ensemble)\ensuremath{\rightarrow}(Gaussian unitary ensemble)\ensuremath{\rightarrow}(Gausian symplectic ensemble) transitions (\ensuremath{\beta}=1\ensuremath{\rightarrow}\ensuremath{\beta}=2\ensuremath{\rightarrow}\ensuremath{\beta}=4) in Dyson's theory random matrices. We find that \ensuremath{\lambda} depends on the universality class according to \ensuremath{\lambda}(\ensuremath{\beta})=\ensuremath{\beta}\ensuremath{\lambda}(\ensuremath{\beta}-1). This relation holds as long as the corresponding classical diffusion constant is kept at a fixed value. Based on semiclassical arguments substantiated by an extensive numerical study of the symplectic kicked rotor, we show that the trasnsition between different universality classes is a smooth function of the symmetry-breaking interaction. For the transition from the orthogonal to the unitary class (\ensuremath{\beta}=1\ensuremath{\rightarrow}\ensuremath{\beta}=2), the semiclassical theory provides an approximate expression for the transition function as well as the critical strength of the symmetry-breaking interaction necessary to achieve the full factor-of-2 increase of the localization length. Universal scaling functions describe the crossover between the diffusive regime and the Anderson localized regime for all three universality classes. The three functions are very similar and, within a few percent, can be reproduced by a single function.