Disorder enhanced quantum many-body scars in Hilbert hypercubes

We consider a model arising in facilitated Rydberg chains with positional disorder which features a Hilbert space with the topology of a $d$-dimensional hypercube. This allows for a straightforward interpretation of the many-body dynamics in terms of a single-particle one on the Hilbert space and provides an explicit link between the many-body and single-particle scars. Exploiting this perspective, we show that an integrability-breaking disorder enhances the scars followed by inhibition of the dynamics due to strong localization of the eigenstates in the large disorder limit. Next, mapping the model to the spin-1/2 XX Heisenberg chain offers a simple geometrical perspective on the recently proposed Onsager scars [Phys. Rev. Lett. 124, 180604 (2020)], which can be identified with the scars on the edge of the Hilbert space. This makes apparent the origin of their insensitivity to certain types of disorder perturbations.

Figure 1

(a) Hilbert space structure for $M=5$ in the $N_{\mathrm{cl}}=1$ sector. The (blue, red) boxes highlight the respective phases ($-,+$) of basis states constituting a specific sparse eigenvector (a scar). (b),(c) The occupation Eq. (9) with $(\mathrm{\Omega }{\tau }_0,\mathrm{\Omega }{\tau }_1)=(175,700)$ for a quench from the initial state Eq. (8) with $w=2$ and the initial momenta $\mathbf{p}$ and positions ${\overline{\mathbf{x}}}_0$ indicated in the insets.

Figure 2

(a) An example of the autocorrelation $A\left(t\right)=|\langle {\psi }_G|\psi \left(t\right)\rangle {|}^2$ for $M=11$. (b) The threshold time $t_c$ vs disorder strength for various system sizes $M$. (c)–(e) Examples of the occupation Eq. (9) for various disorder strengths indicated by circle, cross, and triangle, respectively, in panel (f). (f) $F_r$ (blue) and the $r$ statistics (orange) vs disorder strength $d$ [here $\alpha =6$, $s_0=0.03,\epsilon =9$, $(\mathrm{\Omega }{\tau }_0,\mathrm{\Omega }{\tau }_1)=(50,250)$ and $V_{\mathrm{NN}}/\mathrm{\Omega }=4]$.

Figure 3

(a) Evolution of half-chain entanglement entropy $S$ for $d=0.12$ and $M=25$ following a quench from $|{\psi }_{\mathrm{G}}\rangle$ (blue), $|{\psi }_{\mathrm{rand}}\rangle$ (green), and $|{\psi }_{\mathrm{mid}}\rangle$ (orange). The vertical dashed lines indicate the (scaled) times ${\tau }_0,{\tau }_1$ used in Fig. 2 and the inset shows the detail of the late-time evolution. (b) The standard deviation of the saturated $S$ vs $d$. Data obtained with 10 realizations of the initial conditions (a) and 300 realizations (b), where static disorder was considered for numerical reasons, yielding a value of the average saturated entropy compatible with (a) within $\mathrm{std}\left[S\right(t\to \infty \left)\right]$.