Diagnostics of Hilbert space fragmentation, freezing transition and its effects in the family of quantum East models involving varying ranges of facilitated hopping

Abstract: This paper explores the effect of strong-to-weak fragmentation transition, namely freezing transition, and its rich characteristics in a family of one-dimensional spinless fermionic models involving short-to-long-range facilitated hoppings with an East constraint. Focusing on this family of models with range-$q$ terms, our investigation furnishes an exhaustive understanding of the fractured 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; further, 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 an intricate filling-dependent inhomogeneous long-time autocorrelation profiles across the chain in OBCs. Finally, we probe the effect of the same on the transport at intermediate times in 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 an intriguing interplay between constraints with varying ranges, fragmentation structure, and freezing transition.