Configuration-controlled many-body localization and the mobility emulsion

We uncover a new nonergodic phase, distinct from the many-body localized (MBL) phase, in a disordered two-leg ladder of interacting hardcore bosons. The dynamics of this emergent phase, which has no single-particle analog and exists only for strong disorder and finite interaction, is determined by the many-body configuration of the initial state. Remarkably, this phase features the coexistence of localized and extended many-body states at fixed energy density and thus does not exhibit a many-body mobility edge, nor does it reduce to a model with a single-particle mobility edge in the noninteracting limit. We show that eigenstates in this phase can be described in terms of interacting emergent Ising spin degrees of freedom (“singlons”) suspended in a mixture with inert charge degrees of freedom (“doublons” and “holons”) and thus dub it a mobility emulsion (ME). We argue that grouping eigenstates by their doublon/holon density reveals a transition between localized and extended states that is invisible as a function of energy density. We further demonstrate that the dynamics of the system following a quench may exhibit either delocalizing or localized behavior depending on the doublon/holon density of the initial product state. Intriguingly, the ergodicity of the ME is thus tuned by the initial state of the many-body system. These results establish a new paradigm for using many-body configurations as a tool to study and control the MBL transition. The ME phase may be observable in suitably prepared cold atom optical lattices.

Figure 1

Schematic depiction of the mobility emulsion. The solid black bonds represent entanglement between emergent Ising spins (singlons), cf. Fig. 2. (a) Above the critical doublon/holon density, a typical eigenstate consists of a sparse network of singlons suspended in a background of localized doublons and holons. The system supports clusters of interacting singlons, but these clusters are typically much farther apart than their typical size, so that the network of singlons is many-body localized. Processes admixing these clusters with doublons and holons are far off resonance at strong disorder, and the “emulsion” is stable. (b) Below the critical doublon/holon density, a typical eigenstate consists of a sparse set of doublons and holons suspended in a delocalizing bath of singlons. Doublons exchange energy with this bath and undergo variable-range hopping (dashed bonds), thereby mediating the transport of charge.

Figure 2

Two-leg ladder with identical (mirror-symmetric) disorder potentials on the two legs. For strong disorder, mirror-related sites that share one boson form emergent Ising spin degrees of freedom (red arrows) suspended in a mixture of doublons (doubly occupied rungs) and holons (empty rungs) denoted by full and open circles, respectively.

Figure 3

The dynamical phase diagram of Eq. (2.1) at $\mathrm{\Delta }=0.5$. Here and in the remainder of the paper, we work in units such that the intraleg hopping amplitude $J=1$. In addition to the ETH and MBL phases, we find a new phase—the mobility emulsion (ME)—that predominates at strong disorder. We determine transitions using the mirror-glass order parameter $q_n$ [Eq. (2.4)] for the MBL/ME phase boundary line and the doublon correlator $p_n$ [Eq. (2.6)] for the ETH/ME phase boundary line. See Sec. 2b for representative examples of the exact-diagonalization data used to obtain the points on the above phase diagram.

Figure 4

Representative plots of the mirror-glass order parameter (2.4) and the doublon correlator (2.6) used to determine the horizontal and vertical phase boundary lines, respectively, in Fig. 3. (Top) The energy- and disorder-averaged mirror-glass order parameter (2.4) in the zero doublon/holon sector, rescaled by the system size $L$, at $W=4$. For all data shown, we set $\mathrm{\Delta }=0.5$ and $J=1$. The inset shows the scaling collapse of the data near the transition point $J_{\perp }^{*}$, for $a\approx 1$, $\nu \approx 2/3$, and $J_{\perp }^{*}\approx 0.049$. The approximate location of the transition is shown as a grey vertical line in the main plot. (Bottom) The infinite-temperature disorder-averaged doublon correlator (2.6), rescaled by the system size $L$, at $J_{\perp }=0.1$. The inset shows the scaling collapse of the data near the transition point $W^{*}$, for $a\approx 1$, $\nu \approx 2/3$, and $W^{*}\approx 1.4$.

Figure 5

Bipartite entanglement entropy in the effective model (3.5) describing states in the all-singlon sector $n_{\mathrm{DH}}=0$. We fix $W=10$ and, as in Figs. 3 and 4, we work at $\mathrm{\Delta }=0.5$ and $J=1$. At small $J_{\perp }$ (blue curves), the entanglement entropy exhibits a very weak dependence on system size and appears to saturate to a value of order $ln2$. At larger $J_{\perp }$ (red curves), it increases monotonically with system size. To obtain the dashed curves, a weak uniform “longitudinal field” $h=J_{\perp }/10$, which breaks the mirror symmetry $M$, was added. This field does not change the qualitative behavior, indicating that the ME phase is not protected by mirror symmetry.

Figure 6

Schematic depiction of the localized and delocalizing limits discussed in Secs. 3b1 and 3b2. (a) Eigenstates containing a sparse distribution of singlons remain localized because the self-generated random field $\delta J_{\perp }$ is parametrically stronger at strong disorder than the intersinglon interaction $J_r$, which arises at a higher order in perturbation theory when the intersinglon distance $r$ is sufficiently large. (b) Eigenstates containing a sparse distribution of doublons and holons delocalize because the doublons and holons undergo variable-range hopping mediated by the delocalizing bath of singlons. A doublon absorbing energy $E$ from the bath can hop a distance that scales as $W/E$, but the amplitude for this process is exponentially suppressed by the decay of its wave function. This leads to an optimal hopping distance and rate that is calculated in the Appendix.

Figure 7

Histogram of the bipartite entanglement entropy $S_A$ and many-body energy density $\epsilon$ accumulated from every eigenstate of 100 disorder realizations. The color scale indicates the log of the number of counts in each bin in order to make outliers more visible. In both panels $J=1,\mathrm{\Delta }=0.5,W=1000$, and $L=8$ (16 sites). (a) In the MBL mirror-glass phase, the strong peaks of $S_A$ near the values 0 and $ln2$ result from fully-localized paramagnetic states and symmetry-broken cat states, respectively. (b) In the ME phase, the distribution of entanglement entropy spreads out to much larger values, indicating a stronger tendency towards delocalization. This is particularly evident for high-entanglement states near $\epsilon \approx 0.5$, which coexist with the band of localized states near $S_A\approx 0$. The strong increase of entanglement in the ME phase is not accompanied by a many-body mobility edge, as a nonzero fraction of localized states exist and violate ETH for all $\epsilon$ [compare top panels of (a) and (b)].

Figure 8

Histogram of the bipartite entanglement and doublon/holon density $n_{\mathrm{DH}}$ using the same data set and color scale as in Fig. 7. The distribution of entanglement organizes into well-separated doublon/holon density bands labeled by quantized values of $n_{\mathrm{DH}}=2k/L$ ($L=8$ shown) with $k=0,1,\cdots ,L/2$ [see top panels of (a) and (b)]. (a) In the MBL phase, the mean of the distribution of $S_A$ in a given density band (red dots) grows with decreasing $n_{\mathrm{DH}}$ but saturates to $\sim ln2$ for $n_{\mathrm{DH}}\lesssim 0.5$. (b) In the ME phase, $S_A$ grows monotonically without saturating as $n_{\mathrm{DH}}$ decreases. This indicates that the degree of localization of a given eigenstate is configuration-controlled.

Figure 9

Average bipartite entanglement entropy $S_A$ in each doublon/holon density sector at $L=6$ and 8 (for $L=8$, we use the same data sets as in Figs. 7 and 8). Error bars represent one standard deviation of the distribution of $S_A$ (horizontal error bars are smaller than the point size). (Top) In the MBL phase, $S_A$ saturates to $ln2$ for $n_{\mathrm{DH}}\lesssim 1/2$ and is insensitive to the system size. (Bottom) In the ME phase, $S_A$ is nearly independent of $n_{\mathrm{DH}}^{}$ for $n_{\mathrm{DH}}^{}\ge 2/3$ and increases with decreasing $n_{\mathrm{DH}}^{}$ for $n_{\mathrm{DH}}^{}<2/3$. For $n_{\mathrm{DH}}\lesssim 1/2$, $S_A$ clearly increases with $L$, suggesting delocalization below some critical doublon/holon density.

Figure 10

Dynamics of the local density autocorrelator (5.1) in the ETH (top), MBL (middle), and ME (bottom) phases at $J=1,\mathrm{\Delta }=0.5,$ and $L=8$. In each case, we compare the dynamics starting from two initial states: the all-doublon state $|•\circ •\circ •\circ •\circ \rangle$ and the all-singlon state $|\uparrow \uparrow \downarrow \downarrow \uparrow \uparrow \downarrow \downarrow \rangle$. In the ETH and MBL phases, the dynamics of the autocorrelator does not depend on the choice of initial state; in the former case, it quickly decays to zero, whereas in the latter case it remains frozen near its initial value. In the ME phase, however, the autocorrelator does not decay when the system is initialized in the all-doublon state (similar to its behavior in the MBL phase), while it does when initialized in the all-singlon state (similar to its behavior in the ETH phase). Note that the data for the MBL and ME phases were obtained using the same disorder realization—the only change in going from the middle to the bottom panel is the increase in $J_{\perp }$.

Figure 11

Dynamics of the moving average of the imbalance ${\mathcal{C}}_{\mathrm{S}}$ as a function of $J_{\perp }$ at $J=1,\mathrm{\Delta }=0.5,$ and $L=7$. The data shown were generated using a single disorder realization at $W=6$. The initial state is taken to be the all-singlon configuration $|\uparrow \uparrow \downarrow \downarrow \uparrow \uparrow \downarrow \rangle$. In the MBL phase $J_{\perp }\lesssim 0.03$ the imbalance has a nonzero late time average, while in the ME phase, $J_{\perp }\gtrsim 0.03$, the late time average vanishes.

Figure 12

Dynamics of the moving average of the imbalance ${\mathcal{C}}_{\mathrm{DH}}$ as a function of disorder strength at $J=1,\mathrm{\Delta }=0.5,J_{\perp }=0.14,$ and $L=7$. The initial state is taken to be the all-doublon/holon configuration $|•\circ •\circ •\circ •\rangle$. To investigate the dependence of the late-time average of ${\mathcal{C}}_{\mathrm{DH}}$ on the disorder strength, we fix the disorder potential to be the one used in Fig. 5, and globally rescale it to change the effective value of $W$. In the ME phase, $W\gtrsim 1.4$, the imbalance has a nonzero late-time average, while in the ETH phase, $W\lesssim 1.4$, the late time average vanishes. The gray horizontal indicates the value $1/L=0.14\cdots$, below which the late-time average should be viewed as indistinguishable from zero. Note that the parameters and disorder potential used for the purple curve at $W=6$ coincides with those of the purple curve in Fig. 11; the only difference between the two curves (besides the quantity being measured) is the initial state used.