A Toy Model for Thermalization in Systems with Many-body Localization Phase Transition

The transition from ergodic to many-body localized (MBL) phase remains a challenge from both analytical and numerical perspective. The system thermalization involves an exponential number of degrees of freedom encoded in many-body quantum correlations. This leads to the enormous computational complexity of ergodic phase description and limits the ability of numerical simulations. To address this challenge, we propose a phenomenological model for thermalization of generic many-body systems with local many-body interactions. The method allows reducing the number of degrees of freedom from exponential to polynomial in system size. We construct an effective Hamiltonian and show numerically that the model exhibits a transition from Wigner-Dyson to Poisson level statistics with an increase of the disorder strength. Also, we propose a method to derive the growth the bipartite quantum correlations in the model after a quench. Finally, we apply the results to MBL phase transition problem of interacting spinless fermions in the one-dimensional lattice.