Many-body localization transition in a lattice model of interacting fermions: Statistics of renormalized hoppings in configuration space

We consider the one-dimensional lattice model of interacting fermions with disorder studied previously by Oganesyan and Huse [Phys. Rev. B 75, 155111 (2007)]. To characterize a possible many-body localization transition as a function of the disorder strength W, we use an exact renormalization procedure in configuration space that generalizes the Aoki real-space renormalization procedure for Anderson localization one-particle models [H. Aoki, J. Phys. C 13, 3369 (1980)]. We focus on the statistical properties of the renormalized hopping VL between two configurations separated by a distance L in configuration space (distance being defined as the minimal number of elementary moves to go from one configuration to the other). Our numerical results point toward the existence of a many-body localization transition at a finite disorder strength Wc. In the localized phase W > W-c, the typical renormalized hopping V-L(typ) e((lnVL) over bar) decays exponentially in L as (ln V-L(typ)) similar or equal to -L/xi(loc) and the localization length diverges as xi(loc) (W) similar to (W - W-c)-(nu loc) with a critical exponent of order nu(loc) similar or equal to 0.45. In the delocalized phase W < W-c, the renormalized hopping remains a finite random variable as L -> infinity and the typical asymptotic value V-infinity(typ) e((lnV infinity) over bar) presents an essential singularity (ln V-infinity(typ)) similar to -(W-c - W)-(kappa) with an exponent of order kappa similar to 1.4. Finally, we show that this analysis in configuration space is compatible with the localization properties of the simplest two-point correlation function in real space.