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

Abstract

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 $V_L^{typ}\equiv e^{\overline{\ln V_L}}$ decays exponentially in $L$ as $(\ln V_L^{typ})\simeq -\frac{L}{{\xi }_{loc}}$ and the localization length diverges as ${\xi }_{loc}(W)\sim (W-W_c{)}^{-{\nu }_{loc}}$ with a critical exponent of order ${\nu }_{loc}\simeq 0.5$. In the delocalized phase $W<W_c$, the renormalized hopping remains a finite random variable as $L\to \infty$, and the typical asymptotic value $V_{\infty }^{typ}\equiv e^{\overline{\ln V_{\infty }}}$ presents an essential singularity $(\ln V_{\infty }^{typ})\sim -(W_c-W{)}^{-\kappa }$ with an exponent of order $\kappa \sim 1.4$. 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.

v2 revised version with new material on real-space correlation (11 pages, 11 figures)

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