Level repulsion exponent $\beta$ for Many-Body Localization Transitions and for Anderson Localization Transitions via Dyson Brownian Motion

Overview & Synopsis

The generalization of the Dyson Brownian Motion approach of random matrices to Anderson Localization (AL) models [Chalker, Lerner and Smith PRL 77, 554 (1996)] and to Many-Body Localization (MBL) Hamiltonians [Serbyn and Moore arxiv:1508.07293] is revisited to extract the level repulsion exponent $\beta$, where $\beta=1$ in the delocalized phase governed by the Wigner-Dyson statistics, $\beta=0$ in the localized phase governed by the Poisson statistics, and $0 |^2 | |^2 $ for consecutive eigenstates $m=n+1$, while the diagonal element $m=n$ corresponds to the Inverse Participation Ratio $Y_{nn}(N) \equiv \sum_{i=1}^N | < \phi_n | i> |^4 $ of the eigenstate $| \phi_n>$.