Statistical Properties of the Eigenvalues of Random Matrices from Generalized Hermite Ensembles

What is it about?

We analytically compute the density of states, nearest neighbor spacing and its ratio, therefore quantifying the short-range energy correlations for the real and complex Rosenzweig-Porter ensemble (RPE) with system size, N = 2 and 3. Consequently, we can quantify the degree of level repulsion in RPE and observe a crossover from integrability to chaos as we vary the relative strength of diagonal and off-diagonal disorder. We verify the analytical expressions using extensive numerical simulations. We also provide a Coulomb gas analogy for the joint density of energy levels of RPE for general N, hence generalizing Dyson's approach. Such an analogy helps us understand the phase diagram in the two-parameter phase plane, where the system parameters are disorder in the diagonal and off-diagonal elements, respectively.

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Photo by Compare Fibre on Unsplash

Photo by Compare Fibre on Unsplash

Why is it important?

To study intermediate dynamics, mostly phenomenological models like Brody distributions are used, which have several limitations. Dropping canonical invariance leads to physically relevant way of generalizing Wigner ensembles. It is known that multifractal nature of the eigenfunctions of such generalized ensembles (e.g. Rosenzweig-Porter ensemble) may indicate non-ergodic extended states in crossover from Many Body Localization to thermal phases. While studying this generalized ensemble, we have observed more interesting features, e.g. rank two matrices show very different statistics than any higher ranked matrices. In thermodynamic limit (i.e. very large matrices), we have obtained a more general condition for fully chaotic systems.

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