Level repulsion exponent β for many-body localization transitions and for Anderson localization transitions via Dyson Brownian motion. (18th March 2016) - Monthus, Cécile
Abstract: The generalization of the Dyson Brownian motion approach of random matrices to Anderson localization (AL) models (Chalker et al 1996 Phys. Rev. Lett .77 554) and to many-body localization (MBL) Hamiltonians (Serbyn and Moore 2015 arXiv:1508.07293) is revisited to extract the level repulsion exponent β, where in the delocalized phase governed by the Wigner–Dyson statistics, , in the localized phase governed by the Poisson statistics, and 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 then be studied via Langevin and Fokker–Planck equations. For the MBL quantum spin Hamiltonian with random fields h i, we obtain in terms of the Edwards–Anderson matrix 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 in terms of the density correlation matrix for consecutive eigenstates m = n + 1, while the diagonal element m = n corresponds to the inverse participation ratio of the eigenstate .