Aspects of Hilbert space fragmentation in the quantum East model: Fragmentation, subspace-restricted quantum scars, and effects of density-density interactions

We investigate a one-dimensional correlated-hopping model of spinless fermions with an East constraint. We first analytically unravel the complete fragmentation structure of this model by labeling each fragment by a unique root configuration and utilizing the transfer matrix method. We show that the growth of the size of each fragment of the model follows the widely studied Dyck sequence and is therefore analytically tractable with the help of Catalan triangles. While the eigenstate thermalization hypothesis (ETH) does not hold within the full Hilbert space which exhibits Poisson statistics of the energy level spacing, an examination of various quantities restricted to the largest fragments shows that a weaker version of the subspace-restricted thermalization holds. This weaker violation of the ETH within the largest fragments is supported by the presence of subspace-restricted quantum many-body scars due to quantum fragmentation. Our analysis further reveals that the filling fraction at which the model has the largest fragment shifts with increasing system sizes from half-filling due to the distinct fragmentation structure. Next, we show that the inclusion of a nearest-neighbor density-density interaction with strength $V$ induces a spectral transition within the largest fragment from a weakly ETH-violating phase containing scars to a statistical bubble localized phase as $V$ increases. In particular, the $V\ensuremath{\rightarrow}\ensuremath{\infty}$ limit produces an integrable model. We find that the addition of finite-$V$ stabilizes the ground state near half-filling while keeping intact the fragmentation structure of the East model. However, this behavior abruptly changes exactly at $V=\ensuremath{\infty}$ due to the emergence of a distinct fragmentation structure; we analytically explore this in a comprehensive manner. The infinite-$V$ model has many interesting properties, among which the appearance of the ground state and the largest fragment at two different filling fractions is specially noteworthy. Finally, we propose an experimental setup to realize the infinite-$V$ model as a particular limit of a special kind of $t\text{\ensuremath{-}}V$ model with an on-site potential.