Many-body localization in Fock space: A local perspective

A canonical model for many-body localization (MBL) is studied, of interacting spinless fermions on a lattice with uncorrelated quenched site disorder. The model maps onto a tight-binding model on a ‘Fock-space (FS) lattice’ of many-body states, with an extensive local connectivity. We seek to understand some aspects of MBL from this perspective, via local propagators for the FS lattice and their self-energies (SE's), focusing on the SE probability distributions, over disorder and FS sites. A probabilistic mean-field theory (MFT) is first developed, centered on self-consistent determination of the geometric mean of the distribution. Despite its simplicity this captures some key features of the problem, including recovery of an MBL transition, and predictions for the forms of the SE distributions. The problem is then studied numerically in $1d$ by exact diagonalization, free from MFT assumptions. The geometric mean indeed appears to act as a suitable order parameter for the transition. Throughout the MBL phase the appropriate SE distribution is confirmed to have a universal form, with long-tailed Lévy behavior as predicted by MFT. In the delocalized phase for weak disorder, SE distributions are clearly log-normal, while on approaching the transition they acquire an intermediate Lévy-tail regime, indicative of the incipient MBL phase.

Figure 1

The averaged $\langle {\tilde{\mathrm{\Delta }}}_I\rangle$ vs disorder $W/t$ for band center states ($\tilde{\omega }=0$) in $d=1$; shown as a function of system size for $N=10$ (red), 12 (blue), 14 (green), 16 (purple), as obtained by exact diagonalization. Results are for a box $P\left(\epsilon \right)=\theta (\frac{W}{2}-|\epsilon \left|\right)$, at half filling, and for $V/t=2$. $\langle {\tilde{\mathrm{\Delta }}}_I\rangle$ converges rapidly with increasing $N$, and is well converged by $N\simeq 12$. The Fermi golden rule result [Eq. (19)] is shown for comparison (dashed line). Inset: Corresponding system size evolution of $\langle {\mathrm{\Delta }}_I\left(\omega \right)\rangle$ ($={\mu }_E{\tilde{\mathrm{\Delta }}}_I\left(\tilde{\omega }\right)$). Discussion in text.

Figure 2

Evolution with disorder ($W/t$) of the self-consistent mean-field ${\tilde{\mathrm{\Delta }}}_{\mathrm{t}}$ (delocalized/ergodic phase) and ${\tilde{\mathrm{\Delta }}}_{\mathrm{t}}/\tilde{\eta }$ (MBL phase), for the band center $\tilde{\omega }=0$. Shown for a box site-energy distribution, for $d=1$ at half filling with $V/t=2$. The dashed line indicates the transition point. Inset: Mean-field phase diagram in the $(W/t,V/t)$ plane [with constants $a,b$ given by Eq. (47)]. Discussion in text.

Figure 3

Mean-field $F\left({\tilde{\mathrm{\Delta }}}_I^{}\right)$ vs ${\tilde{\mathrm{\Delta }}}_I^{}$ [Eq. (43)] in the delocalized phase, for the same parameters as Figs. 1 and 2 and shown for $W/t=3,5,7$ ($W_{\mathrm{crit}}/t\simeq 8.6$). With increasing disorder, while remaining in the delocalized phase, a Lévy form for $F\left({\tilde{\mathrm{\Delta }}}_I^{}\right)$ emerges (shown for the $W/t=7$ case, dashed line). Discussion in text.

Figure 4

Geometric mean ${\tilde{\mathrm{\Delta }}}_{\mathrm{t}}$ (circles) of $F\left({\tilde{\mathrm{\Delta }}}_I\right)$ vs disorder $W/t$, for system sizes $N=12$ (blue), 14 (green), 16 (purple). Arithmetic means $\langle {\tilde{\mathrm{\Delta }}}_I\rangle$ (crosses) for the same sizes are also shown (as in Fig. 1) and are well converged in $N$ by $N\simeq 12$. Inset: Mean ratio $r$ (squares) of consecutive eigenvalue spacings vs $W/t$ for the same system sizes. Discussion in text.

Figure 5

Numerical results in the MBL phase for $W/t=20$, showing the distribution $\tilde{F}\left(y\right)$ vs $y$ ($={\tilde{\mathrm{\Delta }}}_I/\tilde{\eta }$), for $N=10$ (red), 12 (blue), 14 (green), and 16 (purple). The dashed line is a fit of the $N=16$ results to a (full) Lévy distribution.

Figure 6

The distribution $f\left(x\right)$ vs $x$ of $x=y/y_{\mathrm{t}}$ for $W/t=20$, and $N=10$ (red), 12 (blue), 14 (green), and 16 (purple). The dashed line is the estimated large-$N$ limiting form of $f\left(x\right)$.

Figure 7

$f\left(x\right)$ vs $x=y/y_{\mathrm{t}}$ shown for $W/t=24$ (left panel) and 28 (right panel), again for $N=10$ (red), 12 (blue), 14 (green), and 16 (purple). The dashed line is the estimated limiting form of $f\left(x\right)$ obtained from the $W/t=20$ results of Fig. 6. Results shown are barely distinguishable from those for $W/t=20$ in Fig. 6. See text for discussion.

Figure 8

For 1BL, geometric means $y_{\mathrm{t}}={\mathrm{\Delta }}_{\mathrm{t}}/\eta$ vs system size $N$ for $W/t=8$ (green), 4 (blue), and 1 (red).

Figure 9

Upper panel: For 1BL with $W/t=4$, numerical distributions $\tilde{F}\left(y\right)$ vs $y$ ($={\mathrm{\Delta }}_I/\eta$), for system sizes $N$ ranging from $N=20$ to 5000 as indicated. The dashed line is a fit of the $N=5000$ results to a full Lévy distribution Eq. (54). Lower panel: same data, showing fits (dashed lines) to the inverse Gaussian distribution Eq. (53) appropriate to nonzero $\eta$, which clearly capture the departures from the leading Lévy tails. The same fit parameters ($\xi$ and $c$) were used throughout, the only $N$ dependence arising in $\eta \propto 1/N$.

Figure 10

Numerical MBL distributions $\tilde{F}\left(y\right)$ vs $y$ ($={\tilde{\mathrm{\Delta }}}_I/\tilde{\eta }$) for $W/t=20$, and system sizes $N=10–16$ as indicated. Dashed lines show one-parameter fits to the inverse Gaussian distribution Eq. (53), discussed in text.

Figure 11

Distributions $F\left({\tilde{\mathrm{\Delta }}}_I\right)$ vs ${\tilde{\mathrm{\Delta }}}_I={\mathrm{\Delta }}_I/{\mu }_E$ for $W/t=1$ and system sizes $N=10$ (red), 12 (blue), 14 (green), 16 (purple). Inset: corresponding distributions for ${\mathrm{\Delta }}_I$. See text for discussion.

Figure 12

Distributions $F\left({\tilde{\mathrm{\Delta }}}_I\right)$ vs ${\tilde{\mathrm{\Delta }}}_I$ for $W/t=1$ (red), 2 (blue), and 4 (green), compared to fits (dashed lines) to the log-normal form Eq. (56). Data for $N=16$. Inset: For $W/t=2$, convergence with increasing $N$ of the distribution $\mathcal{F}\left(x\right)$ of $x=[ln({\tilde{\mathrm{\Delta }}}_I/{\tilde{\mathrm{\Delta }}}_{\mathrm{t}})]/\sigma$, to a standard normal distribution (dashed line), shown for $N=12$ (blue), 14 (green), 16 (purple). Results for different $N$ are practically indistinguishable.

Figure 13

Distributions $F\left({\tilde{\mathrm{\Delta }}}_I\right)$ vs ${\tilde{\mathrm{\Delta }}}_I$ for $W/t=8$ (red), 10 (blue), 12 (green), and 14 (purple), with $N=16$. Inset: same data on log-log scale, with the dotted line showing an emergent ${\tilde{\mathrm{\Delta }}}_I^{-3/2}$ component to the tail of $F\left({\tilde{\mathrm{\Delta }}}_I\right)$.

Figure 14

For $W/t=12$, numerical $F\left({\tilde{\mathrm{\Delta }}}_I\right)$ vs ${\tilde{\mathrm{\Delta }}}_I$ (red line) for $N=16$, on a log-log scale. Fits to both an inverse Gaussian (green line) and a Lévy distribution (dotted line) are shown. Inset: same data on a linear scale, showing also the fit to a LN distribution (blue line). Discussion in text.