Fluctuations and ergodicity of the form factor of quantum propagators and random unitary matrices

Abstract

We consider the spectral form factor of random unitary matrices as well as of Floquet matrices of kicked tops, as given by the (squared moduli of) the traces tn = tr Fn with the integer `time' n = 0,±1,±2, ... . For a typical matrix F the time dependence of the form factor |tn|2 looks erratic; only after a local time average over a suitably large time window n does a systematic time dependence become manifest. For matrices drawn from the circular unitary ensemble we prove ergodicity: in the limits of large matrix dimension and time window n the local time average has vanishingly small ensemble fluctuations and may be identified with the ensemble average. By numerically diagonalizing Floquet matrices of kicked tops with a globally chaotic classical limit we find the same ergodicity. As a byproduct we find that the traces tn of random matrices from the circular ensembles behave very much like independent Gaussian random numbers. Again, Floquet matrices of chaotic tops share that universal behaviour. It becomes clear that the form factor of chaotic dynamical systems can be fully faithful to random-matrix theory, not only in its locally time-averaged systematic time dependence but also in its fluctuations.