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arXiv:2409.15943 · Physical review. B./Physical review. B

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

Maitri Ganguli, Sreemayee Aditya, Diptiman Sen

Published Jan 9, 2025• 9 citations
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Abstract

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.

DOI
10.1103/physrevb.111.045411

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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 VVV induces a spectral transition within the largest fragment from a weakly ETH-violating phase containing scars to a statistical bubble localized phase as VVV increases. In particular, the V\ensuremath→\ensuremath∞V\ensuremath{\rightarrow}\ensuremath{\infty}V\ensuremath→\ensuremath∞ limit produces an integrable model. We find that the addition of finite-VVV 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∞V=\ensuremath{\infty}V=\ensuremath∞ due to the emergence of a distinct fragmentation structure; we analytically explore this in a comprehensive manner. The infinite-VVV 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-VVV model as a particular limit of a special kind of t\ensuremath-Vt\text{\ensuremath{-}}Vt\ensuremath-V model with an on-site potential.
We introduce a one-dimensional correlated-hopping model of spinless fermions in which a particle can hop between two neighboring sites only if the sites to the left and right of those two sites have different particle numbers. Using a bond-to-site mapping, this model involving four-site terms can be mapped to an assisted pair-flipping model involving only three-site terms. This model shows strong Hilbert space fragmentation. We define irreducible strings (ISs) to label the different fragments, determine the number of fragments, and the sizes of fragments corresponding to some special ISs. In some classes of fragments, the Hamiltonian can be diagonalized completely, and in others it can be seen to have a structure characteristic of models which are not fully integrable. In the largest fragment in our model, the number of states grows exponentially with the system size, but the ratio of this number to the total Hilbert space size tends to zero exponentially in the thermodynamic limit. Within this fragment, we provide numerical evidence that only a weak version of the eigenstate thermalization hypothesis (ETH) remains valid; we call this subspace-restricted ETH. To understand the out-of-equilibrium dynamics of the model, we study the infinite-temperature time-dependent autocorrelation functions starting from a random initial state; we find that these exhibit a different behavior near the boundary compared to the bulk. Finally, we propose an experimental setup to realize our correlated-hopping model.
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-qqq 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>1q>1q>1, further revealing that increasing the range of constraints drives the transition to transpire at lower filling fractions as nc=1/(q+1){n}_{c}=1/(q+1)nc​=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.
Experimental observation of coherent oscillations in a Rydberg atom chain [H. Bernien et al., Nature (London) 551, 579 (2017)] has led to the discovery of quantum many-body scars (QMBSs) which is a new paradigm for ergodicity breaking. The experimental findings in the Rydberg chain can be well captured by a kinetically constrained model called the ``PXP'' model, which has been shown to host the eigenstate thermalization hypothesis (ETH)-violating scar states in the middle of the spectrum. Much effort has been put into identifying similar kinetically restricted systems that show a violation of ETH. In this paper, we study the QMBSs that can arise in one such model, namely the spin-1 Kitaev chain, where owing to some conserved quantities, the Hilbert space gets fragmented into unequal disconnected subspaces. Recently, You et al. [Phys. Rev. Res. 4, 013103 (2022)] showed that the ground state sector of this chain can be mapped exactly onto the prototypical PXP model and thus hosts QMBSs. Here, we demonstrate that the phenomenon of scarring is also present in other sectors, and in particular, we identify a sector that exhibits substantially more scarring than the ground state one. We propose an initial state and numerically demonstrate that its fidelity revivals are pronounced and longer lived than those in the PXP model.
The eigenstate thermalization hypothesis describes how isolated many-body quantum systems reach thermal equilibrium. However, quantum many-body scars and Hilbert space fragmentation violate this hypothesis and cause nonthermal behavior. We demonstrate that Hilbert space fragmentation may arise from lattice geometry in a spin-12\frac{1}{2}21​ model that conserves the number of domain walls. We generalize a known, one-dimensional, scarred model to larger dimensions and show that this model displays Hilbert space fragmentation on the Vicsek fractal lattice and the two-dimensional lattice. Using Monte Carlo methods, the model is characterized as strongly fragmented on the Vicsek fractal lattice when the number of domain walls is either small or close to the maximal value. On the two-dimensional lattice, the model is strongly fragmented when the density of domain walls is low and weakly fragmented when the density of domain walls is high. Furthermore, we show that the fragmentation persists at a finite density of domain walls in the thermodynamic limit for the Vicsek fractal lattice and the two-dimensional lattice. We also show that the model displays signatures similar to Hilbert space fragmentation on a section of the second-generation hexaflake fractal lattice and a modified two-dimensional lattice. We study the autocorrelation function of local observables and demonstrate that the model displays nonthermal dynamics.
We study Hilbert-space fragmentation and thermalization in a one-dimensional dipole-conserving Bose-Hubbard chain. By analyzing the structure of the Hamiltonian matrix in the Fock basis, we show that the system exhibits weak Hilbert-space fragmentation. We further construct an exponentially large family of frozen product states and derive analytical upper and lower bounds on their number. Using exact diagonalization, we examine the consequences of weak fragmentation for eigenstate half-chain entanglement, density relaxation dynamics, and level statistics. All these quantities reveal a transition from a weak eigenstate thermalization regime to a nonergodic regime with increasing on-site interaction strength. These results show that weak Hilbert-space fragmentation \textit{does not} preclude quantum chaos or thermalization, and provides a minimal platform for studying the interplay of dipole conservation, weak fragmentation, and ergodicity breaking.
Kinetic constraints in quantum many-body systems strongly restrict the accessible Hilbert space, giving rise to highly nontrivial dynamical behavior. In recent years, such systems have attracted growing interest as they provide insight into mechanisms of thermalization and into regimes where thermalization fails. In this work, we study a family of rule-based kinetically constrained models, including the celebrated Quantum Game of Life, from the perspective of quantum complexity, with a focus on entanglement, nonstabilizerness, and quantum chaos. Using spectral diagnostics such as level statistics and spectral form factors, we show that these models exhibit robust chaotic behavior while simultaneously supporting both strong and weak Hilbert-space fragmentation and quantum many-body scar states. Our results reveal that the nature of Hilbert-space fragmentation can be tuned qualitatively from strong to weak fragmentation and ultimately to its absence, within a single family of models through simple variations of the underlying kinetic rules. To further elucidate the structure of these fragmented subspaces, we characterize them through their ability to generate quantum resources. In particular, we show that resource-generation capacity does not necessarily correlate with the dimensionality of a fragmented sector, and that entanglement structure and the ability to generate nonstabilizerness provide effective diagnostics for distinguishing dynamically disconnected sectors, including those supporting nonthermal scarred dynamics. Our work thus places kinetically constrained models within a broader and more general framework, not limited to Rydberg blockade based constraints only, and demonstrates that simple variations in underlying rules can lead to qualitatively distinct static and dynamical regimes including chaotic, fragmented, and scarred phases.