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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 V₋ 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 W₂. In the localized phase W>W₂, the typical renormalized hopping V₋^type^ln {V₋} decays exponentially in L as (ln V₋^typ) -L{₋₎₂} and the localization length diverges as ₋₎₂ (W) (W-{W₂) }^-{₋₎₂} with a critical exponent of order ₋₎₂0. 45. In the delocalized phase W<W₂, the renormalized hopping remains a finite random variable as L and the typical asymptotic value V_^type^ln {V_{}} presents an essential singularity (ln V_^typ) - ({W₂-W) }^- with an exponent of order 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.
Monthus et al. (Wed,) studied this question.