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 RG procedure for Anderson localization one-particle models [H. Aoki, J. Phys. C13, 3369 (1980)]. We focus on the statistical properties of the renormalized hopping $V_L$ 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 towards the existence of a many-body localization transition at a finite disorder strength $W_c$. In the localized phase $W>W_c$, the typical renormalized hopping $V_L^{typ} \equiv e^{\bar{\ln V_L}}$ decays exponentially in $L$ as $(\ln V_L^{typ}) \simeq - \frac{L}{ξ_{loc}}$ and the localization length diverges as $ξ_{loc}(W) \sim (W-W_c)^{-ν_{loc}}$ with a critical exponent of order $ν_{loc} \simeq 0.5$. In the delocalized phase $W<W_c$, the renormalized hopping remains a finite random variable as $L \to \infty$, and the typical asymptotic value $V_{\infty}^{typ} \equiv e^{\bar{\ln V_{\infty}}}$ presents an essential singularity $(\ln V_{\infty}^{typ}) \sim - (W_c-W)^{-κ}$ with an exponent of order $κ\sim 1.4$. 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.

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