Direct product of random unitary matrices: Two-point correlations and fluctuations.

We study the ensembles of direct product of m random unitary matrices of size N drawn from a given circular ensemble. We calculate the statistical measures, viz. number variance and spacing distribution to investigate the level correlations and fluctuation properties of the eigenangle spectrum. Similar to the random unitary matrices, the level statistics is stationary for the ensemble constructed by their direct product. We find that the eigenangles are uncorrelated in the small spectral intervals. While, in large spectral intervals, the spectrum is rigid due to strong long-range correlations between the eigenangles. The analytical and numerical results are in good agreement. We also test our findings on the multipartite system of quantum kicked rotors.