Multiscale entanglement clusters at the many-body localization phase transition

We numerically study the formation of entanglement clusters across the many-body localization transition. We observe a crossover from strong many-body entanglement in the ergodic phase to weak local correlations in the localized phase, with continuous clusters throughout the phase diagram. Critical states close to the transition have a structure compatible with fractal or multiscale-entangled states, characterized by entanglement at multiple levels: small strongly entangled clusters are weakly entangled together to form larger clusters. The critical point therefore features subthermal entanglement and a power-law distributed cluster size. Upon entering the localized phase, the power-law distribution seems to persist with a varying power that crosses over into a stretched exponent before eventually becoming exponential deep in the localized phase. These results are in agreement with some of the recently proposed phenomenological renormalization-group schemes characterizing the many-body localized critical point, and serve to constrain other such schemes.

Figure 1

Schematics for different types of entanglement structures obtained in different RG proposals. Black arrows represent spins, while the solid links mark entanglement between them. A set of spins connected by these links forms a single ergodic cluster, with different colors associated to different clusters. MBL clusters correspond to continuous sets of unconnected spins. The dotted lines mark a weaker entanglement. Though some residual entanglement survives in a MBL cluster, for simplicity, it is not represented here. (a) Ergodic and MBL connected clusters simply alternate. (b) One sparse ergodic cluster spans the entire chain. (c) Multiscale entanglement with small strongly entangled clusters (full lines) weakly entangled at a higher scale (dotted line) to form a larger cluster.

Figure 2

Schematic description of the recursive algorithm used to identify entanglement clusters. The indexed circles represent physical sites, while links illustrate entanglement between sites. Dotted lines represent weaker correlations and the dashed lines the splitting into different clusters. The arrows mark one step of the algorithm, wherein the continuous bipartition that minimizes the mutual information is identified. We then apply the same process recursively to each so-obtained cluster.

Figure 3

(a) We illustrate how the blue sparse cluster in the state depicted in the top row is extracted even when considering only bipartitions in continuous segments. In the first step, the system is partitioned into two sets, depicted in the second row, separating the three central spins $\{3,4,5\}$ from the rest of the system $\mathcal{A}=\{1,2,6,7,8,9,10\}$. When we further partition $\mathcal{A}$, the three central spins $\{3,4,5\}$ are no longer part of the current partition branch and are considered removed, and the set $\mathcal{A}$ is taken to be continuous and periodic. The bipartition of $\mathcal{A}$, depicted in the third row, now gives the blue cluster $\{2,6,7\}$, which is now considered continuous, and the rest $\{1,8,9,10\}$. (b) An example of an entanglement configuration that is not captured by our approximate algorithm. The two intertwined clusters $\{1,2,6,8\}$ and $\{3,4,5,7,9,10\}$ cannot be separated as both of them are sparse over the same support.

Figure 4

Average number of sites $n_{\text{av}}$ in a cluster as a function of mutual information cutoff for different system sizes for disorder strength $W=1.5$ (a) and $W=15$ (b). For $W=1.5$, the system is ergodic and below the thermal value of $i_{\mathrm{cut}}$, which is renormalized by finite size, the complete system forms a single cluster; above this value, clusters become single site. The abrupt transition marks true many-body entanglement and is a sign of ergodicity. For $W=15$, the system is many-body localized. Even at low values of the entanglement cutoff, the system is fragmented into smaller clusters. With increasing mutual information cutoff, these clusters become smaller, with some resonating pairs surviving even at high cutoff.

Figure 5

(a) Fraction of clusters that consist of a pair of spins and (b) fraction of pairs that are separated by $d$ sites for $L=16$, as a function of disorder strength. Resonant pairs are still present at high disorder strength (while the fraction of pairs decreases with system size, the number of clusters increases faster), but are short range with at most next-nearest neighbors. (c) Fraction of continuous clusters as a function of disorder strength. (d) Fraction of clusters with holes (here taken to be $l-n$); there are up to $4\%$ with a single hole, less than $1\%$ with two holes, and practically no clusters with a larger number of discontinuities. All data are obtained for $i_{\text{cut}}=0.35$.

Figure 6

(a) Mean length of minimal clusters as a function of disorder strength and (b) scaling collapse to the universal scaling form $n_{\mathrm{av}}=L^{-1}f\left((W-W_c)L^{1/\nu }\right)$. Similar results are obtained for the mean range. The collapse suggests that these quantities act as good order parameters for the MBL phase transition.

Figure 7

Left panel shows the average length rescaled with logarithm of system size $(n_{\mathrm{av}}-1)/lnL$ (top dotted line) and the logarithm of the typical length $ln\left(n_{\mathrm{typ}}\right)/10$ (bottom full line) at the critical point as a function of the cutoff $i_{\mathrm{cut}}$. The logarithm of the typical length has been rescaled for convenience, and the lines are guides to the eye. We observe a slowly divergent average length of a cluster, while the typical length is itself constant. Right panel shows the distribution of the cluster length for $i_{\mathrm{cut}}=0.2$, with a decay as $l^{-\alpha }$, until finite-size effects kick in. The dotted line is the best power-law fit. All graphs are taken at the estimated critical point $W_c\approx 3.8$. In inset, we show the evolution of $\alpha$ when varying the cutoff for $L=16$.

Figure 8

Variation of the distribution of cluster lengths as a function of disorder strength, for $i_{\mathrm{cut}}=0.2$ (similar results are obtained for $0.1\le i_{\mathrm{cut}}\le 0.7$) and for $L=16$. (a) Around the estimated critical point, we observe a large region compatible with a power-law probability distribution $P\left(n\right)\propto n^{-\alpha }$, with exponents that vary slowly with disorder strength. Note that a regression with $P\left(n\right)\propto exp(-\beta \sqrt{n})$ also fits decently the intermediate cluster lengths. (b) At larger disorder strengths, the distribution becomes a stretched exponential $P\left(n\right)\propto exp(-\beta n^{\alpha })$, with $\alpha$ going to 1 as disorder increases.

Figure 9

The different characteristic lengths for random states of length $L=18$, generated according the structures described in Fig. 1. In (a), ergodic clusters survive high entanglement cutoff, while the localized ones are broken down quickly. In (b), the sparse ergodic cluster leads to a significant discrepancy between maximum range and maximum size. In (c) and (d), multiscale entanglement leads to a progressive separation of clusters into smaller ones, and the difference between average and maximum cluster length is much smaller. The difference between (c) and (d) lies in the probability distribution of wave function coefficients; see text for further details on how the random states are generated.

Figure 10

Comparison of the different length scales—range in (a) and number of sites in (b)—between simulations where we evaluate the entropy on all bipartitions of the system (solid lines) and simulations where we only consider continuous partitions in each subset (symbols). We take $i_{\text{cut}}=0.2$. We observe a good agreement for all disorder strengths and all entanglement cutoffs we consider.

Figure 11

In this figure, we compare the structure of the clusters obtained for small system sizes when checking all possible partitions (lines) to the one obtained using our main algorithm (markers). (a) Fraction of clusters that consist of a pair of spins and (b) fraction of pairs that are separated by $d$ sites for $L=16$. (c) Fraction of continuous clusters as a function of disorder strength. (d) Fraction of clusters with holes (here taken to be $l-n$). Only the fraction of clusters with one hole is underestimated by less than $0.5\%$, while all other measures coincide. All data are obtained for $i_{\text{cut}}=0.35$.

Figure 12

In this figure, we compare the probability distribution of the clusters obtained for small system sizes when checking all possible partitions (lines) to the one obtained using our main algorithm (markers). Both panel shows the distribution of the cluster length for $i_{\mathrm{cut}}=0.2$. On the left, the distribution is taken for $W=3.8$ and decays as a power law until finite-size effects kick in. On the right, the distribution is taken for $W=8$ and decays as a stretched exponential. For both disorder realizations, the two schemes agree.

Figure 13

Example of a decomposition into clusters by our algorithm (with $L=10$ and $W=3.7$). On the left, the binary tree we obtain, with each node corresponding to a set of spins labeled by the sites it includes (here from 0 to 9). Sets are divided into subsets following the edges of the tree. The label on the edges corresponds to the normalized mutual information between its two descendants. On the right, a schematic picture of the progressive appearance of clusters when varying the mutual information cutoff $i_{\text{cut}}$, applied to the binary tree on the left. The indexed circles correspond to sites and the cluster decomposition is represented by the dashed lines. Each arrow corresponds to a change in the structure at some precise value of the mutual information cutoff. As a concrete example, we obtain the second decomposition for $0.19\le i_{\text{cut}}<0.25$.

Figure 14

Probability distribution of the minimal mutual information for different disorder strengths and system sizes. Deep in the ergodic phase, the probability distribution peaks around the (renormalized) thermal entropy. For $W=2.5$, strong finite-size effects are visible, which correspond to the onset of the transition in small systems. At the phase transition, we observe a decent collapse at intermediate values, though the low-entanglement realizations vastly differ. The average minimal mutual information $\langle i_{min}\rangle$ decreases with system size as $L^{-0.5\pm 0.2}$.

Figure 15

Statistical average of the mean length of the minimal clusters for two values of the interaction strength $\mathrm{\Delta }$, away from the usual Heisenberg limit. (a) and (b) are taken for $\mathrm{\Delta }=0.5$, while (c) and (d) correspond to $\mathrm{\Delta }=2$. (b) and (d) are fits to the universal scaling form $n_{\mathrm{av}}=L^{-1}f\left((W-W_c)L^{1/\nu }\right)$. The exponents obtained are similar to the one for $\mathrm{\Delta }=1$.

Figure 16

Three typical examples of the hierarchical multiscale structure obtained for states close to the critical point, with $W=3.7$ and $L=16$. Each node corresponds to a cluster, labeled by the sites it includes (here from 1 to 16). The label on the edges corresponds to the mutual information cutoff $i_{\text{cut}}$ required to break it into its descendants. For ease of visualization, the data is no longer a binary tree; we have only kept the nodes that appear at the given value of the cutof and removed all that are less entangled than their parents (and therefore never appear as a proper cluster). As is readily seen, thermal subclusters (strongly entangled clusters that break into single-site clusters) are present. Clusters can themselves be entangled at larger scales, with a weaker internal entanglement ($\left[134561213141516\right]$ in the first example, or $\left[789101112\right]$ in the third one are good examples of such structures).