Percolation in Fock space as a proxy for many-body localization

We study classical percolation models in Fock space as proxies for the quantum many-body localization (MBL) transition. Percolation rules are defined for two models of disordered quantum spin chains using their microscopic quantum Hamiltonians and the topologies of the associated Fock-space graphs. The percolation transition is revealed by the statistics of Fock-space cluster sizes, obtained by exact enumeration for finite-sized systems. As a function of disorder strength, the typical cluster size shows a transition from a volume law in Fock space to subvolume law, directly analogous to the behavior of eigenstate participation entropies across the MBL transition. Finite-size scaling analyses for several diagnostics of cluster size statistics yield mutually consistent critical properties. We show further that local observables averaged over Fock-space clusters also carry signatures of the transition, with their behavior across it in direct analogy to that of corresponding eigenstate expectation values across the MBL transition. The Fock-space clusters can be explored under a mapping to kinetically constrained models. Dynamics within this framework likewise show the ergodicity-breaking transition via Monte Carlo averaged local observables and yield critical properties consistent with those obtained from both exact cluster enumeration and analytic results derived in our recent work [arXiv:1812.05115]. This mapping allows access to system sizes two orders of magnitude larger than those accessible in exact enumerations. Simple physical pictures based on freezing of local real-space segments of spins are also presented and shown to give values for the critical disorder strength and correlation length exponent $\nu$ consistent with numerical studies.

Figure 1

Overview of the percolating delocalized phase (a) and the nonpercolating localized phase (b) on the Fock-space graph. The graph shown is the Fock space of a disordered Ising model [Eq. (3)] with ten spins, where each node represents a product state in the ${\sigma }_{\ell }^z=\pm 1$ basis. The different colors show disjoint clusters subject to the percolation criterion described in Sec. 2a. In the percolating phase, all nodes belong to the same cluster as indicated by the same color throughout the graph. By contrast, in the nonpercolating phase the Fock space splits up into many small clusters as indicated by different colors. In the particular disorder realization shown for the nonpercolating phase in (b), there are 10 clusters, labeled as ${\mathcal{C}}_1$ through ${\mathcal{C}}_{10}$, with the corresponding colors indicated in the legend.

Figure 2

Behavior of the three spin fractions, $f_a$ (always flippable), $f_n$ (never flippable), and $f_s$ (flippable under some configurations of neighbors), shown as a function of disorder strength $W$, with $W_c$ the critical disorder strength for the phase transition. The two vertical dotted lines at $W_c^{\pm }=\frac{J}{2}\pm 2J_z$ denote the bounds on the critical disorder. For $W<W_c^{-}$, the system is trivially percolating, whereas for $W>W_c^{+}$ the system is trivially localized.

Figure 3

Illustration showing that all nodes may belong to a single cluster, even though not all edges satisfy the percolation criterion. The spin at site $\ell$ (marked in blue) cannot flip under antiparallel orientations of its neighbors. This is indicated by the red arrows marked with a cross. The nodes $|I_i\rangle$ and $|I_f\rangle$ are therefore not directly connected. These nodes are however connected (and hence are part of the same cluster) via a different path, as depicted by the green arrows. This is due to the interactions enabling the spin flip at $\ell$ when both neighbors are down.

Figure 4

Illustration of freezing a local segment of spins in the TFI model. The fields at sites $\ell$ and $\ell +r$ satisfy $h_{\ell },h_{\ell +r}>J/2$, whereas the fields at intermediate sites $m$ satisfy the weaker condition $h_m>(J/2)-2J_z$. This results in the entire segment $[\ell ,\ell +r]$ being frozen irrespective of the spin configuration of the rest of the system.

Figure 5

Illustration of behavior in the four regimes of disorder strength, using the Fock-space hypercube for four spins. All bonds along a particular direction correspond to flipping a particular spin. Bonds connecting the inner cube to the outer correspond to the first spin, while bonds along $x$, $y$, and $z$ directions correspond to the second, third, and fourth spin, respectively. (a) For $W<W_c^{-}$, all edges satisfy the percolation criterion. (b) For $W_c^{-}<W<W_c$, not all edges satisfy the percolation criterion (some blue edges are missing), but all nodes belong to a single cluster. (c) For $W_c<W<W_c^{+}$ the system is in a localized phase with many clusters, as indicated by the different colors of the nodes. (d) For $W>W_c^{+}$ some spins completely freeze (the first and third in this example), and no bonds along the corresponding directions are active.

Figure 6

Schematic example of how a domain of spins can freeze in the $XXZ$ model. In this example of a down-spin domain, the fields at the left end satisfy $|h_{\ell }-h_{\ell +1}|>J/2$ and $|h_{\ell }-h_{\ell +1}+2J_z|>J/2$. Similarly, those at the right end satisfy $|h_{m+1}-h_m|>J/2$ and $|h_{m+1}-h_m+2J_z|>J/2$. The spins within the domain are frozen irrespective of the values of $h_k$ for $\ell +1<k<m$ and irrespective of the spin configuration on sites outside the domain ($k<\ell$ or $k>m+1$).

Figure 7

Distributions for the disordered $XXZ$ model ($W_c=1.025$ for the parameters chosen, as is confirmed via finite-size scaling analyses). Top row: probability distributions of cluster sizes, $\tilde{P}(N_{\mathcal{C}}/N_{\mathcal{H}})=N_{\mathcal{H}}P\left(N_{\mathcal{C}}\right)$, for different system sizes $N$ and disorder $W$ (as indicated in panels). Bottom row: corresponding cumulative distributions $C\left(N_{\mathcal{C}}\right)$. For $W<W_c$ almost the entire weight of the distributions is on $N_{\mathcal{H}}/N_{\mathcal{C}}=1$, whereas for strong disorder the weight shifts towards $N_{\mathcal{H}}/N_{\mathcal{C}}\to 0$ as $N$ increases. For $W>W_c$, but not too deep inside the localized phase, the distribution for finite $N$ has a bimodal nature with its weight depleting from $N_{\mathcal{C}}=N_{\mathcal{H}}$ to clusters of sizes $N_{\mathcal{C}}\sim N_{\mathcal{H}}^{\alpha }$ with $\alpha <1$. However, the peak at $N_{\mathcal{C}}=N_{\mathcal{H}}$ loses weight with increasing $N$, suggesting that it vanishes in the thermodynamic limit. See text for discussion. For the numerics, $J_z=1$ and $J=4.1$, with statistics obtained from ${10}^5$ realizations.

Figure 8

Results for the disordered $XXZ$ chain. (a) ${\mathcal{S}}_{\mathrm{avg}}/N_{\mathcal{H}}$ and (b) ${\mathcal{S}}_{\mathrm{typ}}/N_{\mathcal{H}}$ versus disorder $W$, for system sizes $N=12–22$. Insets show raw data, whereas main panels show data collapsed onto a common function of $(W-W_c)N^{1/\nu }$, with $W_c$ constrained to $J/4$ and the extracted value of $\nu$ given in the panels. (c) ${\mathcal{S}}_{\mathrm{avg}}/N_{\mathcal{H}}$ as a function of $N_{\mathcal{H}}$ for various values of $W$ as indicated by the color scale. Specifically, the set of values are $\{0.25,0.75,1.75,2.25,2.75,3.25,3.75,4.25,4.75\}$. ${\mathcal{S}}_{\mathrm{avg}}/N_{\mathcal{H}}=1$ arises for $W<W_c$ (two such $W$ values are shown: 0.25 and 0.75; they are indistinguishable in this figure and both give $\alpha =1$). Circles are data points and dashed lines are fits to the form Eq. (30), with the extracted values of $\alpha$ shown in panel (d) for both ${\mathcal{S}}_{\mathrm{avg}}$ and ${\mathcal{S}}_{\mathrm{typ}}$. The inset in (d) shows the approach of $\alpha$ to 1 as the critical disorder is approached from the localized side. $J_z=1$ and $J=4.1$, with averaging over ${10}^5$ realizations. Statistical errors are calculated from 500 bootstrap resamplings and are not shown as they are smaller than the data markers.

Figure 9

Left panel: raw data for ${\mathcal{S}}_{\mathrm{fluc}}/N_{\mathcal{H}}$ [Eq. (31)] as a function of disorder $W$, for various system sizes $N$. Inset: zoom of the critical region, corresponding to the gray shaded region in the main panel, showing crossing of the data for various $N$. Right panel: the same data scale collapsed as a function of $(W-W_c)N^{1/\nu }$, with $\nu$ the finite-size scaling exponent.

Figure 10

Distribution of local magnetization of the cluster, $P_m\left(m\right)$, for four different disorder values, $W$, and different system sizes $N$ (specified in panels). For weak disorder $P_m\left(m\right)$ is strongly peaked at $m=0$ and becomes sharper with increasing $N$. On increasing $W$, additional peaks at $m\pm 1$ appear and eventually dominate the distribution at strong disorder.

Figure 11

Top row: cumulative distribution $C_m\left(m\right)$ corresponding to $P_m\left(m\right)$ shown for two values of $W$, one in the percolating phase and the other in the localized phase. The former has a finite jump only at $m=0$, indicating the presence of a component $\delta \left(m\right)$ in $P_m\left(m\right)$. In the latter there are additional jumps at $m=\pm 1$, indicating the presence of $\delta \left(\right|m|-1)$ components in $P_m\left(m\right)$. Bottom row: the deviation of the weights of the $\delta$ functions $a_0$ and $a_1$ [see Eq. (33)] from 1 and zero, as a function of disorder $W$, also take on finite values near $W_c\simeq 1$.

Figure 12

Standard deviation of $P_m\left(m\right)$, ${\mu }_m$, shown as a function of disorder $W$ for various system sizes $N$. Left panel shows the raw data, whereas the right panel shows data rescaled by $N^{-\zeta /\nu }$ for different system sizes and collapsed onto a common function of $(W-W_c)N^{1/\nu }$. The scaling analysis with $W_c$ and $\nu$ constrained to the values obtained earlier yields the exponent $\zeta =0.51$.

Figure 13

Top: for the TFI model, the probability distribution $P_{\mathrm{MC}}(m,t_{\mathrm{MC}})$ of the Monte Carlo averaged local spin expectation $m$, Eq. (40), as a function of Monte Carlo time, $t_{\mathrm{MC}}$, shown as a color map. In the percolating phase (left), the distribution becomes unimodal with a peak at $m=0$, whereas in the nonpercolating phase (right), the peaks in $P_{\mathrm{MC}}$ at $m=\pm 1$ persist even at very large $t_{\mathrm{MC}}$, as indicated via red arrows. Bottom: dynamics of ${\mu }_{\mathrm{MC}}\left(t_{\mathrm{MC}}\right)$ defined in Eq. (41). Different colors and symbols represent different values of disorder strength $W$ and system size $N$, respectively. In the ergodic phase, $W<W_c$, ${\mu }_{\mathrm{MC}}$ shows a decay as $t_{\mathrm{MC}}^{-1/2}$. In the nonergodic phase, $W>W_c$, it saturates to finite values, which grow with increasing $W$. Note that the data for $W\le W_c=2.05$ lie on top of each other. Results are obtained over ${10}^5$ disorder realizations, with statistical errors determined via a standard bootstrap with 500 resamplings.

Figure 14

(a) Collapse of the data for ${\mu }_{\mathrm{MC}}\left(t_{\mathrm{MC}}\right)$ for various $N$ and $W$ of the TFI model suggests a scaling form concomitant with that conjectured in Eq. (42). Different colors and symbols indicate different disorder strengths $W$ and system sizes $N$, respectively. (b) ${\tau }_{\mathrm{MC}}^{-1/2}$ as function of $W$ for various $N$, showing that it is a suitable diagnostic of the phase transition. (c) Finite-size scaling analysis of ${\tau }_{\mathrm{MC}}^{-1/2}$ shows that the curves for various $N$ can be collapsed onto a common function of $(W-W_c)N^{1/\nu }$ when rescaled with $N^{-\zeta /\nu }$, yielding the displayed values for the critical disorder strength $W_c$ and exponents $\nu$ and $\zeta$.

Figure 15

Dynamics of ${\mu }_{\mathrm{MC}}\left(t_{\mathrm{MC}}\right)$ for the $XXZ$ model [compare with Fig. 13 (bottom) for the TFI model]. We use $J_z=1$ and $J=4.1$, for which $W_c=1.025$ for the $XXZ$ model as predicted earlier. Results are obtained from $5\times {10}^4$ disorder realizations, with statistical errors obtained via a standard bootstrap with 500 resamplings.