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 $V_L$ 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 toward 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 $V_L^{typ}\equiv e^{\overline{\text{ln}{V}_L}}$ decays exponentially in $L$ as $\left(\text{ln}{V}_L^{typ}\right)\simeq -\frac{L}{{\xi }_{loc}}$ and the localization length diverges as ${\xi }_{loc}\left(W\right)\sim {\left(W-W_c\right)}^{-{\nu }_{loc}}$ with a critical exponent of order ${\nu }_{loc}\simeq 0.45$. In the delocalized phase $W<W_c$, the renormalized hopping remains a finite random variable as $L\to \infty$ and the typical asymptotic value $V_{\infty }^{typ}\equiv e^{\overline{\text{ln}{V}_{\infty }}}$ presents an essential singularity $\left(\text{ln}{V}_{\infty }^{typ}\right)\sim -{\left(W_c-W\right)}^{-\kappa }$ 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.

Figure 1

(Color online) Statistics of the renormalized hopping $V_L$ in the localized phase (data for the disorder strength $W=20$): (a) probability distribution $P_L\left(\text{ln}{V}_L\right)$ of the logarithm of renormalized hopping $V_L$ for the sizes $L=4,6,8,10,12$. (b) Same data for the rescaled variable $x$ of Eq. (11) for the sizes $L=6,8,10,12$: the convergence toward a fixed rescaled distribution $\tilde{P}\left(x\right)$ is rapid (we have only excluded the smallest size $L=4$ that was too different).

Figure 2

(Color online) Exponential decay of the typical renormalized hopping $V_L^{typ}\equiv \overline{e^{\text{ln}{V}_L}}$ in the localized phase: (a) linear decay of $\overline{\text{ln}{V}_L}$ as a function of $L$ [see Eq. (12)]. (b) Behavior of the slope $1/{\xi }_{loc}\left(W\right)$ [inverse of the localization length ${\xi }_{loc}\left(W\right)$ of Eq. (12)] as a function of the disorder strength $W$.

Figure 3

(Color online) Statistics of the renormalized hopping $V_L$ in the delocalized phase (a) for the disorder strength $W=2$, the probability distribution $P_L\left(\text{ln}{V}_L\right)$ of the logarithm of renormalized hopping $V_L$ remains the same for the sizes $L=8,10,12$ (we have excluded the smallest sizes $L=4,6$ that were a bit different). This should be compared with Fig. 1a corresponding to the localized phase. (b) Behavior of the typical asymptotic renormalized hopping $V_{\infty }^{typ}\equiv e^{\overline{\text{ln}{V}_{\infty }}}$: $\text{ln}{V}_{\infty }^{typ}=\overline{\text{ln}{V}_{\infty }}$ as a function of the disorder strength $W$.

Figure 4

Critical behaviors obtained for a critical point at the value $W_c=5.6$ (a) localized phase $W>W_c$: the plot of $\text{ln}\left[1/{\xi }_{loc}\left(W\right)\right]$ as a function of $\text{ln}\left(W-W_c\right)$ corresponds to the slope ${\nu }_{loc}\simeq 0.45$ [see Eq. (13)]. (b) Delocalized phase $W<W_c$: the plot of $\text{ln}\left(-\overline{\text{ln}{V}_{\infty }}\right)$ as a function of $\text{ln}\left(W_c-W\right)$ corresponds to the slope $\kappa \simeq 1.4$ [see Eq. (17)].

Figure 5

(Color online) Finite-size scaling analysis of our numerical data for the sizes $L=8$ (triangles), $L=10$ (squares), and $L=12$ (circles) according to the form of Eq. (22) with the values $\rho =0.76$ [see Eq. (28)] and ${\nu }_{FS}=1.85$ [see Eq. (25)]: this plot is rather convincing at criticality and in the localized phase $W>W_c$, whereas stronger corrections to scaling seem to be present in the delocalized phase $W<W_c$.

Figure 6

(Color online) Exponential decay of the typical real-space two-point correlation function $C^{typ}\left(L\right)\equiv e^{\overline{\text{ln}C\left(L\right)}}$ [see Eq. (29)] in the localized phase: (a) linear decay of $\overline{\text{ln}{C}_L}$ as a function of $L$ [see Eq. (12)]. (b) Behavior of the slope $1/{\xi }_C\left(W\right)$ [inverse of the localization length ${\xi }_C\left(W\right)$ of Eq. (30)] as a function of the disorder strength $W$ (circles), as compared to $1/{\xi }_{loc}\left(W\right)$ (squares) found previously for the renormalized hopping in configuration space [see Eq. (12) and Fig. 2].