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Level repulsion exponent $β$ 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 $β$, where $β=1$ in the delocalized phase governed by the Wigner-Dyson statistics, $β=0$ in the localized phase governed by the Poisson statistics, and $0<β_c<1$ 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 $τ$. 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 $β=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 | < ϕ_n | σ_i^z | ϕ_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 $β=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 | < ϕ_n | i> |^2 | <i | ϕ_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 | < ϕ_n | i> |^4 $ of the eigenstate $| ϕ_n>$.