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arXiv:2605.17951

Graph-Theoretic Detection of Hilbert Space Fragmentation

A. Rutkowski, M. Mierzejewski, J. Herbrych

Published May 18, 2026• 0 citations
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Abstract

Hilbert-space fragmentation provides a mechanism for ergodicity breaking in quantum many-body systems even in the absence of disorder, leading to dynamically disconnected sectors and strong memory of initial conditions. However, identifying such structures is often challenging and typically relies on prior knowledge of conservation laws or model-specific analytical insight. Here we introduce an unbiased approach based on spectral graph theory and, within this framework, formulate the concept of nearly fragmented systems, in which perturbative processes couple otherwise fragmented sectors while preserving their dynamical imprint. By representing basis states as vertices and Hamiltonian matrix elements as edges, we map the connectivity structure of the many-body Hilbert space onto a graph and analyze it using tools such as the Laplacian spectrum, Fiedler vectors, and modularity. Exact fragmentation corresponds to disconnected graph components, while nearly fragmented systems manifest as weakly connected communities whose structure can still be resolved spectrally. Applying this framework to the one-dimensional $t$-$J$ model and its perturbations, we demonstrate that graph-theoretic diagnostics reliably identify both fragmented and nearly fragmented Hilbert-space structures and capture the hierarchy of dynamical time scales that governs the system's evolution. We further show that the method extends beyond kinetically constrained models by applying it directly to the Hubbard chain, where it reveals the emergence of nearly decoupled subspaces associated with doublon dynamics and spin configurations. Our results establish the spectral graph analysis as a general and scalable tool for diagnosing fragmentation and approximate dynamical constraints in complex quantum many-body systems.

DOI
10.48550/arxiv.2605.17951

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Hilbert-space fragmentation provides a mechanism for ergodicity breaking in quantum many-body systems even in the absence of disorder, leading to dynamically disconnected sectors and strong memory of initial conditions. However, identifying such structures is often challenging and typically relies on prior knowledge of conservation laws or model-specific analytical insight. Here we introduce an unbiased approach based on spectral graph theory and, within this framework, formulate the concept of nearly fragmented systems, in which perturbative processes couple otherwise fragmented sectors while preserving their dynamical imprint. By representing basis states as vertices and Hamiltonian matrix elements as edges, we map the connectivity structure of the many-body Hilbert space onto a graph and analyze it using tools such as the Laplacian spectrum, Fiedler vectors, and modularity. Exact fragmentation corresponds to disconnected graph components, while nearly fragmented systems manifest as weakly connected communities whose structure can still be resolved spectrally. Applying this framework to the one-dimensional ttt-JJJ model and its perturbations, we demonstrate that graph-theoretic diagnostics reliably identify both fragmented and nearly fragmented Hilbert-space structures and capture the hierarchy of dynamical time scales that governs the system's evolution. We further show that the method extends beyond kinetically constrained models by applying it directly to the Hubbard chain, where it reveals the emergence of nearly decoupled subspaces associated with doublon dynamics and spin configurations. Our results establish the spectral graph analysis as a general and scalable tool for diagnosing fragmentation and approximate dynamical constraints in complex quantum many-body systems.
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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The Lindblad equation for open quantum systems is central to our understanding of coherence and entanglement in the presence of Markovian dissipation. In closed quantum systems Hilbert-space fragmentation is an effective mechanism for slowing decoherence in the presence of constrained interactions. We develop a general mechanism for operator-space fragmentation of mixed states, undergoing Lindbladian evolution generated by frustration-free Hamiltonians and Pauli-string jump operators. The interplay of generator algebras of dissipative and unitary dynamics leads to a hierarchical partitioning of operator and real space into dynamically disconnected subspaces, which we elucidate using the bond and commutant algebras of superoperators. This fragmentation fundamentally constrains information spreading in open systems and provides new mechanisms to control highly entangled quantum states and dynamics. Our approach yields two key advances. Firstly, we introduce frustration graphs in operator space as a compact representation to construct effective non-Hermitian Hamiltonians in individual fragments and diagnose their free-fermion solvability. Secondly, using these methods we uncover a range of universal dynamical regimes in Pauli-Lindblad models, exhibiting symmetry enriched quantum chaos and integrability in operator-space fragments. Furthermore, we show dissipation-driven phase transitions corresponding to exceptional points in the Lindbladian spectrum whose signatures are captured by spectral statistics and operator dynamics. These results establish operator-space fragmentation as a fundamental principle for open quantum systems, with immediate implications for quantum error correction, where protected subspaces could emerge naturally from fragmentation. Our framework provides a systematic approach to discover and characterize novel non-equilibrium phases in open quantum many-body systems.
The Hamiltonian formulation of lattice gauge theories plays a central role in quantum simulations of gauge theories, and understanding their spectrum and other properties is expected to become crucial in the upcoming years. The relevant Hamiltonians in this framework possess local symmetry at each lattice site and may exhibit higher-form symmetries. There are then an exponentially large number of dynamically disconnected symmetry sectors, most of which are not translation-invariant. An exponential number of dynamically disconnected sectors, i.e., Hilbert space fragmentation, can also occur in systems in which no such symmetries have been identified. In this contribution, we describe an emergent gauge symmetry that is valid only in a subset of sectors of the fragmented S=1S=1S=1 dipole-conserving spin chain. These non-invertible symmetries can label exponentially many of the model's sectors. Simulating this Hamiltonian, which is not gauge-invariant, yields an exact quantum simulation of a gauge theory.
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.
We develop a systematic theory of spectral decimation for quantum many-body Hamiltonians and show that it provides a quantitative probe of emergent symmetries in statistically mixed spectra. Building on an analytical description of statistical mixtures, we derive an explicit expression for the size of a characteristic symmetry sector (CSS), defined as the largest subsequence of levels exhibiting non-Poissonian correlations. The CSS dimension is shown to be the size-biased average of the underlying symmetry sectors, establishing a direct link between spectral statistics and Hilbert-space structure. We apply this framework to two paradigmatic settings: Hilbert-space fragmentation and disorder-induced many-body localization (MBL). In fragmented systems, the CSS reproduces the mixture prediction and isolates correlated subsectors even when the full spectrum appears nearly Poissonian. In the disordered Heisenberg chain, spectral decimation reveals the gradual emergence of integrability through a shrinking CSS, whose statistics exhibit signatures consistent with local integrals of motion. We introduce a characteristic symmetry entropy (CSE) as a finite-size scaling observable and extract, within accessible system sizes, the crossover exponents. Our results establish spectral decimation as a controlled, unbiased and computationally inexpensive diagnostic of hidden structure in many-body spectra, capable of distinguishing between chaotic dynamics, statistical mixtures, and emergent integrability.