Nonexponential dephasing in a local random matrix model

Deviations from exponential decay dynamics have been proposed for a wide variety of systems in high-energy, atomic, and molecular physics. This work examines the quantum dynamics of a simple hierarchical local random matrix model. The hierarchical structure is imposed by two physically motivated constraints: an exponential size scaling of matrix elements, and a quantum number ``triangle rule,'' which introduces correlations in the quantum-state space by mimicking the nodal structure of wave functions in a coordinate Hamiltonian. These correlations lead to a systematic slowing of dephasing dynamics compared to exponential decays. A generalized Lorentzian line shape is introduced as the Fourier transform of a polynomial survival amplitude to describe the average behavior of these decays. The model is brought into a representation that can be compared directly with the golden rule. In this representation, the deviations from exponentiality arise from energy-dependent correlations among the coupling matrix elements that persist even for large systems. Finally, the effects of relaxing the size scaling and ``triangle rule'' constraints are studied. Sparsity of the random matrix alone is not sufficient to produce slow asymptotic dynamics; both types of constraints are required.