Multifractal Scalings Across the Many-Body Localization Transition

In contrast with Anderson localization where a genuine localization is observed in real space, the many-body localization (MBL) problem is much less understood in Hilbert space, the support of the eigenstates. In this Letter, using exact diagonalization techniques we address the ergodicity properties in the underlying $\mathcal{N}$-dimensional complex networks spanned by various computational bases for up to $L=24$ spin-$1/2$ particles (i.e., Hilbert space of size $\mathcal{N}\simeq 2.7\times {10}^6$). We report fully ergodic eigenstates in the delocalized phase (irrespective of the computational basis), while the MBL regime features a generically (basis-dependent) multifractal behavior, delocalized but nonergodic. The MBL transition is signaled by a nonuniversal jump of the multifractal dimensions.

Figure 1

Schematic picture of the multifractal properties of eigenstates in Fock and spin configuration bases across the MBL transition for Eq. (2). Two typical eigenstates of Eq. (2) on a small $L=14$ system for ETH ($h=0.5$, blue) and MBL ($h=10$, green) regimes are graphically represented, with circle sizes proportional to $|{\psi }_{\alpha }{|}^2$ in the spin configuration basis.

Figure 2

Finite size scaling of the disorder average PEs $\overline{S_q}$ shown for both Fock [left: (a), (c)] and configuration spaces [right: (b), (d)] and a few representative values of disorder strength $h$ in both ETH (a), (b) and MBL (c), (d) regimes for $q=1$, 2. Lines are of the form $D_q\ln \mathcal{N}+b_q$ with $D_q=1$ and $b_q<0$ in the ETH regime (guides to the eye), while $D_q<1$ and $b_q>0$ in the MBL regime (fits).

Figure 3

Scaled PE $\overline{S_q}/\ln \mathcal{N}$ plotted against disorder $h$ in both Fock (a) and configuration spaces (d), (e). Crossing points signal a sign change of the subleading correction $b_q$ in the PE scalings $D_q\ln \mathcal{N}+b_q$, which occurs in the vicinity of the ETH-MBL transition point $h_c\sim 3.8$ (gray shaded region). Insets (b), (c) show histograms $P(S_1^{\text{Fock}}/\ln \mathcal{N})$ of PE in the Fock basis.

Figure 4

Scaling curves for the PE following Eq. (4) across the full ETH-MBL regimes with varying disorder strengths (colored symbols). Data for $q=2$ are displayed for both Fock (left) and configuration (right) spaces, after subtracting the critical PE data $\overline{S_{2,c}}$ taken at $h=3.8$. For $h<h_c$ (top panels) a volumic scaling ${\mathcal{G}}_{\mathrm{vol}}(\mathcal{N}/\mathrm{\Lambda })$ gives the best data collapse. The nonergodicity volumes $\mathrm{\Lambda }$ plotted in insets (a) show similar (universal) divergences $\ln \mathrm{\Lambda }\sim (h_c-h{)}^{-a}$ with $a\sim 0.45$ for Fock ($q=2$) and configuration ($q=1$ and $q=2$) spaces, as indicated on the plot. Bottom panels show data collapses in the MBL regime for $h>h_c$ following a linear scaling ${\mathcal{G}}_{\mathrm{lin}}(\ln \mathcal{N}/\xi )$. The length scale $\xi$ plotted in insets (b) diverges as $(h-h_c{)}^{-a^{\prime }}$ with $a^{\prime }\sim 0.76$. Note that small deviations to scaling Eq. (5) are expected due to nonuniversal subleading constant corrections $b_q>0$, but are difficult to observe in this strong disorder regime.