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

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 $0<\beta_c<1$ at the critical point. The idea is that the Gaussian disorder variables $h_i$ are promoted to Gaussian stationary processes $h_i(t)$ in order to sample the disorder stationary distribution with some time correlation $\tau$. The statistics of energy levels can be then studied via Langevin and Fokker-Planck equations. For the MBL quantum spin Hamiltonian with random fields $h_i$, we obtain $\beta =2q^{EA}_{n,n+1}(N)/q^{EA}_{n,n}(N)$ in terms of the Edwards-Anderson matrix $q^{EA}_{nm}(N) \equiv \frac{1}{N} \sum_{i=1}^N | < \phi_n | \sigma_i^z | \phi_m> |^2$ for the same eigenstate $m=n$ and for consecutive eigenstates $m=n+1$. For the Anderson Localization tight-binding Hamiltonian with random on-site energies $h_i$, we find $\beta =2 Y_{n,n+1}(N)/(Y_{n,n}(N)-Y_{n,n+1}(N))$ in terms of the Density Correlation matrix $Y_{nm}(N) \equiv \sum_{i=1}^N | < \phi_n | i> |^2 | <i | \phi_m> |^2$ 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 $| \phi_n>$.