Dynamical response and time correlation functions in random quantum systems

Time-dependent response and correlation functions are studied in random quantum systems composed of infinitely many parts without mutual interaction and defined with statistically independent random matrices. The latter are taken within the three Wigner–Dyson universality classes. In these systems, the response functions are shown to be exactly given by statistical averages over the random-matrix ensemble. Analytical results are obtained for the time dependence of the mean response and correlation functions at zero and positive temperatures. At long times, the mean correlation functions are shown to have a power-law decay for GOE at positive temperatures, but for GUE and GSE at zero temperature. Otherwise, the decay is much faster in time. In relation to these power-law decays, the associated spectral densities have a dip around zero frequency. The diagrammatic method is developed to obtain higher-order response functions and the third-order response function is explicitly calculated. The response to impulsive perturbations is also considered. In addition, the quantum fluctuations of the correlation function in individual members of the ensemble are characterised in terms of their probability distribution, which is shown to change with the temperature.