2015/10/31 by Cecile Monthus
Physics and Astronomy · #cond-mat.dis-nn
paper · pdf · doi:10.1088/1742-5468/2016/03/033113
published as J. Stat. Mech. (2016) 033113 · 22 pages
arxiv created 2016/03/21 · arxiv updated 2016/03/22
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<βc<1 at the critical point. The idea is that the Gaussian disorder variables hi are promoted to Gaussian stationary processes hi(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 hi, we obtain β=2qEAn,n+1(N)/qEAn,n(N) in terms of the Edwards-Anderson matrix qEAnm(N) ≡ (1)/(N) ∑i=1N | < ϕn | σiz | ϕm> |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 hi, we find β=2 Yn,n+1(N)/(Yn,n(N)-Yn,n+1(N)) in terms of the Density Correlation matrix Ynm(N) ≡ ∑i=1N | < ϕn | i> |2 | <i | ϕm> |2 for consecutive eigenstates m=n+1, while the diagonal element m=n corresponds to the Inverse Participation Ratio Ynn(N) ≡ ∑i=1N | < ϕn | i> |4 of the eigenstate | ϕn>.