Laboratoire de Physique Théorique
Toulouse - UMR 5152

At the Many-Body-Localization (MBL) Transition, the statistics of energy levels and the entanglement entropy of eigenstates change : the Many-Body-Localized phase is characterized by the Poisson level statistics and an area-law for the entanglement, while the Delocalized phase is characterized by the Wigner-Dyson level statistics and a volume-law for the entanglement. In the first part of the talk, the Dyson Brownian Motion approach for quantum spin Hamiltonians with random fields will be described : the statistics of energy levels can be studied via Langevin and Fokker-Planck equations, and one obtains the level repulsion exponent $\beta$ in terms of the Edwards-Anderson matrix elements. In the second part of the talk, we will consider the strong disorder limit of the MBL transition, where the critical level statistics is close to the Poisson statistics, in order to determine the statistical properties of the rare extensive resonances that are needed to escape from the area-law entanglement of the Localized phase. At criticality, the entanglement entropy can grow with an exponent $0<\alpha < 1$ anywhere between the area law $\alpha=0$ and the volume law $\alpha=1$, as a function of the resonances properties. In addition, the correlation length exponent takes the simple value $\nu=1$. Independently of this strong disorder limit, we will explain why for the Many-Body-Localization transition concerning individual eigenstates, the correlation length exponent $\nu$ is not constrained by the usual Harris inequality $\nu \geq 2/d$.