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

TL;DR

This work provides an analytic, combinatorics-driven characterization of Hilbert space fragmentation and the freezing transition in a family of one-dimensional East models with range-$q$ constraints. By identifying canonical root states and employing transfer matrices, it shows that the fragmentation follows generalized Catalan/Frey-Sellers sequences, with a universal critical filling $n_c=1/(q+1)$ marking the strong-to-weak fragmentation transition. The study reveals multiple, highly degenerate ground states for $q>1$ that are critical with logarithmic entanglement, and demonstrates rich, filling-dependent autocorrelation and transport behavior at intermediate times, including size-stretched exponential and anomalous dynamics. These results illuminate how constraint range and fragmentation shape thermalization, ground-state structure, and transport in constrained quantum many-body systems, with potential experimental relevance in Rydberg and cold-atom platforms.

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.

Figures (20)

• Figure 1: Schematic of the main results obtained from our analysis.
• Figure 2: Plot showing $D_{frag}$ vs $N_{f}$ in the range-2 and range-3 cases for $L=150$, obtained using the Frey-Sellers sequence. We observe that the largest fragment appears at $N_{f}=L/2+1$ (but $D_{frag}$ for $N_{f}=L/2$ is almost comparable) in $L\to\infty$ limit for the range-2 case, which does not significantly differ from the finite-size result. In the range-3 case, $D_{frag}$ turns out to be the largest at $N_{f}=L/2$ in $L\to\infty$ limit; this is identical to that observed for finite $L$'s.
• Figure 3: (a-b) Plot showing $D_{frag}$ vs $N_{f}$ for $L=80$ and $L=150$, respectively, for constraints with ranges, $q=7,14$ and $20$, obtained using root identification method and thereafter utilizing the Frey-Sellers formula with $m=(q+1)$. We see that the minimum fillings for the largest fragment appear at $N_{f}=L/2-2,L/2-6$, and $L/2-9$ for $q=7,10$ and $20$, respectively. This thus indicates no anomalous shift in the filling fraction with system sizes, unlike the range-1 constraint.
• Figure 4: (a-c): The variation of $D_{max}/D_{sum}$ vs $L$ for several filling fractions for models with range-1, range-2, and range-3 constraints, respectively, obtained followed by the identification of appropriate root states and afterward using the Catalan triangle ($q=1$) and Frey-Seller sequence with $m=3$ and $4$ for $q=2$ and $q=3$ cases, respectively. It is quite evident from Figs. (a-c) that the behavior of $D_{max}/D_{sum}$ switches from the exponential decay to a constant at filling fractions $n_{c}=1/2$, $1/3$, and $1/4$, respectively.
• Figure 5: (a-c): $D_{max}/D_{sum}$ vs $n_{f}$ in the models with range-1, range-2, and range-3 constraints in PBCs, respectively, for $L=12,14,16$ and $18$. In Figs. (a-c), it can be readily noticed that the behavior changes sharply at filling fractions $n_{c}\simeq1/2$, $1/3$, and $1/4$, respectively, which agrees well with the analytical behaviors obtained in OBCs.