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arXiv:2510.16570 · Physical review. B./Physical review. B

Quantum complexity in rule-based constrained many-body models: Scars, fragmentation, and chaos

Arkaprava Sil, Sudipto Singha Roy

Published May 26, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/ps6c-c3s8

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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.
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Thermalization in closed quantum systems and its breakdown are central to understanding the emergence of statistical behavior in isolated many-body systems. Kinetic constraints can hinder thermalization by restricting the allowed dynamics and giving rise to non-ergodic behavior. While various aspects of ergodicity breaking in kinetically constrained models have been extensively studied in Hermitian systems, their behavior in non-Hermitian kinetically constrained models remains less explored. In particular, the inequivalence of left and right eigenstate ensembles in non-Hermitian systems raises the question of whether they exhibit qualitatively different dynamical and eigenstate properties. In this work, we study non-Hermitian generalizations of the paradigmatic PXP and PPXPP models, focusing on their Krylov complexity, eigenstate structure, and dynamics. By exploiting the Arnoldi's algorithm, we find distinct behavior of Krylov complexity in the left and right eigenstate ensembles as non-Hermiticity is increased: in the right ensemble, it exhibits persistent oscillations with an increasing period, while in the left ensemble, it gradually saturates. The characteristic arch-shaped profile of the Arnoldi coefficients is also modified differently in the two ensembles, with a Hermitian-like arch re-emerging in the left ensemble at large non Hermiticity. Additionally, in the non-Hermitian PXP model, the right ensemble retains the characteristic scar structure, while the left ensemble develops additional non-scar eigenstates with anomalously low entanglement and large overlap with the vacuum state. These findings reveal that non-Hermiticity can affect Krylov dynamics and eigenstate structure differently in the left and right ensembles, demonstrating the need to treat them separately when characterizing non-Hermitian kinetically constrained systems.
Quantum kinetically constrained models have recently attracted significant attention due to their anomalous dynamics and thermalization. In this work, we introduce a hitherto unexplored family of kinetically constrained models featuring conserved particle number and strong inversion-symmetry breaking due to facilitated hopping. We demonstrate that these models provide a generic example of so-called quantum Hilbert space fragmentation, that is manifested in disconnected sectors in the Hilbert space that are not apparent in the computational basis. Quantum Hilbert space fragmentation leads to an exponential in system size number of eigenstates with exactly zero entanglement entropy across several bipartite cuts. These eigenstates can be probed dynamically using quenches from simple initial product states. In addition, we study the particle spreading under unitary dynamics launched from the domain wall state, and find faster than diffusive dynamics at high particle densities, that crosses over into logarithmically slow relaxation at smaller densities. Using a classically simulable cellular automaton, we reproduce the logarithmic dynamics observed in the quantum case. Our work suggests that particle conserving constrained models with inversion symmetry breaking realize so far unexplored dynamical behavior and invite their further theoretical and experimental studies.
In a strongly interacting Rydberg atom array, the dynamics are often constrained to the decoupled Hilbert subspaces, representing an intriguing paradigm for nonergodicity. By considering a variable detuning of the global Rydberg coupling, we show that, not only is the existence of these Hilbert subspaces dependent on the interplay of detuning and interaction, but they are also strongly fragmented, with the fragment dimensions exhibiting various scaling behaviors with increasing system size. The resulting constrained dynamics of the system are thus governed by the dimension and connectivity of these fragments. We then adopt an auxiliary fermion description to reveal the underlying emergent kinetic constraints for the subspace fragmentation and fragment-confined dynamics. Our results provide a systematic understanding of Hilbert-space fragmentation in Rydberg arrays, and shed light on engineering nonergodic many-body dynamics beyond the PXP model.