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>Wc, the typical renormalized hopping VLᵗʸᵖ≡e^ln VL decays exponentially in L as (ln VLᵗʸᵖ)-Lξloc and the localization length diverges as ξloc(W)~(W-Wc)^-νloc with a critical exponent of order νloc0.45. In the delocalized phase W<Wc, the renormalized hopping remains a finite random variable as L→∞ and the typical asymptotic value V_∞ᵗʸᵖ≡e^ln V_∞ presents an essential singularity (ln V_∞ᵗʸᵖ)~-(Wc-W)^-κ with an exponent of order κ~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.