Mobility edges through inverted quantum many-body scarring

We show that the rainbow state, which has volume-law entanglement entropy for most choices of bipartitions, can be embedded in a many-body localized spectrum. For a broad range of disorder strengths in the resulting model, we numerically find a narrow window of highly entangled states in the spectrum, embedded in a sea of area law entangled states. The construction hence embeds mobility edges in many-body localized systems. This can be thought of as the complement to many-body scars, an “inverted quantum many-body scar,” providing a further type of setting where the eigenstate thermalization hypothesis is violated.

Figure 1

(a) Disorder averaged half-chain entanglement entropy divided by the Page [4] value $S_{\mathrm{Page}}=[2Nln\left(2\right)-1]/2$ for the eigenstate closest to the energy density $\epsilon =0.4$ plotted against the disorder strength $\delta$ for different system sizes. The transition from thermal behavior at weak disorder to MBL behavior at strong disorder is seen. The number of disorder realizations is ${10}^4$ for $N\in \{4,5,6\}$, 5000 for $N=7$, and 1500 for $N=8$. (b) The standard deviation of the half-chain entanglement entropy ${\mathrm{\Delta }}_s$ computed for the same set of data shows a peak at the transition point. (c) The finite-size scaling collapse for ${\mathrm{\Delta }}_s$ suggests that the transition happens at $\delta \sim 8$ for large systems.

Figure 2

(a) Adjacent gap ratio $r_{\mathrm{ave}}$ at strong disorder $\delta =10$ computed for the 13, 50, 100, or 800 energy levels closest to the considered energy density for $N=4$, 5, 6, or 7, respectively, and averaged over 3000 disorder realizations. As the system size $2N$ increases, $r_{\mathrm{ave}}$ gets close to the Poisson (POI) value, which signals that the majority of the states in the spectrum are many-body localized. (b) The same data, but plotted as a function of system size for different energy densities.

Figure 3

(a) The disorder averaged half-chain entanglement entropy $\langle S_N\rangle$ as a function of disorder strength $\delta$ and energy density relative to the rainbow state $\epsilon -{\epsilon }_{\text{RB}}^i$ for $N=6$. We average states with the same value of $n-n_{\text{RB}}^i$ over 2000 disorder realizations, where $n_{\text{RB}}^i$ denotes the index of the rainbow state which lies at energy density ${\epsilon }_{\text{RB}}^i=(E_{\text{RB}}-E_{min}^i)/(E_{max}^i-E_{min}^i)$ for the $i\mathrm{th}$ disorder realization. The dark horizontal line at zero is produced by the rainbow state, and other highly entangled states are seen in its vicinity. Within the strongly disordered regime where the rest of the spectrum is many-body localized, these highly entangled states produce a mobility edge. (b) A zoom of panel (a) showing the band of high-entropy states. Most of the high-entropy states have energies below the rainbow state, but a few of them are at energies higher than the rainbow state. (c) $\langle S_N\rangle$ for a fixed $\delta =7$ as a function of additional disorder of strength ${\delta }_p$ on the second half of the chain only. The high-entropy states disappear for ${\delta }_p\approx \delta /300$. [The symmetric case ${\delta }_p=0$ also provides a magnified version of the mobility edges at the cut denoted by the short, vertical, green line in panel (b).]

Figure 4

Scaling of the number of atypical eigenstates with high entanglement entropy as a function of system size for $\delta =10$. The number of atypical eigenstates is obtained by counting the number of eigenstates with entropy higher than a certain cutoff value $S_c=f_{c}Nln\left(2\right)$ in each disorder realization, and this number is then averaged over 2000 disorder realizations for $N=3,4,5,6$, or 1090 for $N=7$. For all considered $f_c$, the number of atypical eigenstates scales exponentially with $N$, and the dashed lines indicate the best fit with the function $2^{aN+b}$. The inset shows $a$ as a function of $f_c$. Note that $a<2$ for all considered $f_c$, which means that the fraction of atypical states approaches zero for large system sizes.