Diagnostics of Hilbert space fragmentation, freezing transition, and its effects in the family of quantum East models involving varying range of constraints

This paper explores the effect of the strong-to-weak fragmentation transition, namely, the freezing transition, and its rich characteristics in a family of one-dimensional spinless fermionic models with facilitated hoppings involving short-to-long-range East constraints. Focusing on this family of models with range-$q$ constraints, our investigation furnishes an exhaustive understanding of the fragmented Hilbert space utilizing the enumerative combinatorics and transfer-matrix methods. This further allows us to get insight into the freezing transition in this family of models with the help of the generalization of Catalan numbers introduced by Frey and Sellers for $q>1$, further revealing that increasing the range of constraints drives the transition to transpire at lower filling fractions as ${n}_{c}=1/(q+1)$. This distinct fragmentation structure also yields the emergence of ground states at multiple fillings, which can be explained from the root structure of the largest fragment within the full Hilbert space. Furthermore, the ground state exhibits signatures of criticality with logarithmic scaling of entanglement entropy. Thereafter, our investigation exemplifies that the above transition has a profound impact on the thermalization of bulk and boundary autocorrelators at long times, which includes intricate filling-dependent inhomogeneous long-time autocorrelation profiles across the chain in open boundary conditions (OBCs). Finally, we probe the effect of the strong-to-weak fragmentation transition on the transport at intermediate times in periodic boundary conditions (PBCs), restricting ourselves to models up to range-3 constraints. This investigation discloses a vast range of anomalous transport possibilities, ranging from size-stretched exponential relaxation through superdiffusive to subdiffusive behaviors, akin to the fragmentation structure supported by the filling fraction and range of constraints. In brevity, our paper reveals intriguing possibilities conspired by constraints with varying ranges, and those-induced Hilbert-space fragmentation and freezing transition, thus offering exciting avenues for future explorations.