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

Abstract

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 $\beta =2q_{n,n+1}^{EA}(N)/q_{n,n}^{EA}(N)$ in terms of the Edwards-Anderson matrix $q_{nm}^{EA}(N)\equiv \frac{1}{N}{\sum }_{i=1}^N|<{\varphi }_n|{\sigma }_i^z|{\varphi }_m>{|}^2$ 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 $\beta =2Y_{n,n+1}(N)/(Y_{n,n}(N)-Y_{n,n+1}(N))$ in terms of the Density Correlation matrix $Y_{nm}(N)\equiv {\sum }_{i=1}^N|<{\varphi }_n|i>{|}^2|<i|{\varphi }_m>{|}^2$ for consecutive eigenstates $m=n+1$, while the diagonal element $m=n$ corresponds to the Inverse Participation Ratio $Y_{nn}(N)\equiv {\sum }_{i=1}^N|<{\varphi }_n|i>{|}^4$ of the eigenstate $|{\varphi }_n>$.

22 pages

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• Local resonances and parametric level dynamics in the many-body localised phase
• Resonant energy scales and local observables in the many-body localised phase
• Localization transition in one dimension using Wegner flow equations
• Many Body Localization Transition in the strong disorder limit : entanglement entropy from the statistics of rare extensive resonances
• Efficient circular Dyson Brownian motion algorithm