Hilbert-Space Fragmentation

• Hilbert-space fragmentation is the division of a many-body Hilbert space into exponentially many disconnected Krylov subsectors, leading to strong memory retention and nonthermal dynamics.
• It arises from constrained kinetics, geometric frustration, and refined conservation laws that restrict state connectivity even within fixed global charge sectors.
• Experimental and theoretical diagnostics—such as quantum gas microscopy, graph connectivity analysis, and Mazur bounds—demonstrate its impact on sub-dimensional transport and equilibration.

Hilbert-space fragmentation is the decomposition of a many-body Hilbert space into exponentially many dynamically disconnected Krylov subsectors under the Hamiltonian, even after fixing standard conserved quantities such as particle number, energy, magnetization, or dipole moment. In a fragmented system, time evolution from one sector never reaches states in another, so closed dynamics can retain strong memory of initial conditions, suppress ordinary thermalization, and generate frozen states, sub-dimensional motion, and sector-restricted equilibration. Recent work has formulated this phenomenon through commutant algebras, statistically localized and local integrals of motion, generalized symmetries, graph connectivity, and direct experiments in constrained Bose-Hubbard systems and Rydberg platforms (Moudgalya et al., 2021, Adler et al., 2024, Rutkowski et al., 18 May 2026).

1. Definition and formal structure

A standard dynamical definition starts from the Krylov sector of an initial state,

$$\mathcal{K}(\lvert \psi_0\rangle)=\mathrm{span}\left\{\lvert \psi_0\rangle,H\lvert \psi_0\rangle,H^2\lvert \psi_0\rangle,\ldots\right\},$$

and identifies Hilbert-space fragmentation with exponential growth in the number of distinct Krylov sectors generated by a natural tensor-product basis as system size increases (Budde et al., 14 Apr 2026). In this sense, fragmentation is a statement about the connectivity graph induced by the Hamiltonian in a physically natural basis, not merely about the existence of a few symmetry sectors.

Several papers distinguish ordinary fragmentation from strong fragmentation. In the strongest form, even the largest Krylov sector grows asymptotically more slowly than the full Hilbert-space dimension, or becomes a negligible fraction of the relevant symmetry sector in the thermodynamic limit (Budde et al., 17 Apr 2026, Nicolau et al., 2023). This criterion excludes cases where the Hilbert space is split into many blocks but one block still dominates the dynamics.

An algebraic definition is provided by the commutant-algebra framework. For Hamiltonian families $H=\sum_j J_j \hat h_j$, the relevant object is the commutant $\mathcal C$ of operators commuting with every local term. Fragmentation is then defined as exponential growth of $\dim(\mathcal C)$ with system size, in contrast to conventional $U(1)$ or $SU(2)$ symmetries, whose commutants grow only polynomially (Moudgalya et al., 2021). The same framework distinguishes classical fragmentation, visible in a product-state basis and associated with an Abelian commutant, from quantum fragmentation, where the disconnected sectors are naturally resolved only in an entangled basis and the commutant is non-Abelian (Moudgalya et al., 2021).

This algebraic viewpoint has recently been related to the integer characteristic polynomial factorization method. For Hamiltonians with rational matrix representations, if the center of the commutant algebra has only rational eigenvalues, the fragmentation obtained by integer characteristic polynomial factorization is equal to or finer than the commutant-algebra fragmentation (Chen et al., 1 Jan 2026). This result does not eliminate the possibility of inequivalent decompositions, but it clarifies when two prominent identification schemes provably agree.

HSF is distinct from both disorder-induced many-body localization and ordinary integrability. In the two-dimensional tilted Bose-Hubbard setting, fragmentation arises in a clean system from kinetic constraints built into the dynamics itself, rather than from quenched randomness; in the $t$-$J_z$ setting, the fragmented blocks are not organized by an extensive set of ordinary local conserved charges (Adler et al., 2024, Lisiecki et al., 14 May 2025).

2. Microscopic mechanisms

The best-studied route to fragmentation is constrained kinetics. In the confining spin-$\tfrac12$ model of strict Ising-domain-wall confinement, two commuting $U(1)$ quantities—the total domain-wall number $H=\sum_j J_j \hat h_j$0 and the total magnetization $H=\sum_j J_j \hat h_j$1—restrict motion so strongly that many configurations become frozen, while other connected components are generated from specific root configurations (Yang et al., 2019). This mechanism yields exponentially many disconnected subsectors inside fixed $H=\sum_j J_j \hat h_j$2 sectors.

A different mechanism appears in flat-band lattices with commutative local symmetries. There, the equitable partition theorem separates compact localized states from extended states by parity, and on-site bosonic interactions generate a conserved local parity of compact-localized-state occupation in each unit cell. The fragmentation is hidden in the original site basis and becomes explicit only after rotating to the compact-localized-state/dispersive basis (Nicolau et al., 2023). The resulting sector structure is local and strong, and it survives a broad class of long-range density-density interactions.

Lattice geometry can itself induce fragmentation. In the domain-wall-conserving spin-$H=\sum_j J_j \hat h_j$3 model generalized to higher dimensions and fractal graphs, sites with higher coordination act as blockades for domain-wall motion. On the Vicsek fractal lattice, the model is strongly fragmented when the number of domain walls is either small or close to the maximal value; on the two-dimensional open square lattice, it is strongly fragmented at low domain-wall density and weakly fragmented at high density (Harkema et al., 2024). The same work reports signatures similar to fragmentation on a section of the second-generation hexaflake fractal lattice and on a looped modified two-dimensional lattice, indicating that loops can reduce fragmentation.

Geometric frustration produces yet another route. In kagome XXZ limits, local ice-rule and three-coloring constraints define macroscopically degenerate low-energy manifolds, but the effective quantum moves fail to connect all allowed configurations. In the easy-axis Balents-Fisher-Girvin regime, the ice manifold splits into four topological sectors and includes 16 isolated triangular pinwheel states; in the easy-plane three-coloring regime, the effective Kempe dynamics yields exponentially many fragments with a hierarchical structure controlled by loop length (Lee et al., 2020). This produces broad distributions of relaxation times and glassy state dependence.

Rydberg dressing provides a perturbative realization of constrained transport. In the large-detuning regime, the Rydberg Ising chain maps to a generalized folded XXZ model with constrained flip-flop processes that preserve both the total excitation number and the number of nearest-neighbor dimers, equivalently the number of domain walls. Many configurations with the same $H=\sum_j J_j \hat h_j$4 remain dynamically disconnected, giving strong fragmentation in the largest symmetry sector (Yang et al., 2024).

3. Conserved structures, sector labels, and symmetry organization

A recurring theme is that fragmented sectors can be labeled by conserved quantities more refined than standard global charges. In the $H=\sum_j J_j \hat h_j$5-$H=\sum_j J_j \hat h_j$6 chain, the disconnected blocks are labeled by statistically localized integrals of motion (SLIOMs), each associated with the spin of the $H=\sum_j J_j \hat h_j$7-th particle. These operators are not strictly local, but they behave as local in typical basis states and control both block structure and dynamics (Lisiecki et al., 14 May 2025).

In the nearest-neighbor pair-hopping model, fragmentation is accompanied by genuine local integrals of motion. Beyond the exact LIOMs associated with dipole moment and even-sublattice density, an infinite family of frozen long-wavelength density modes emerges in the thermodynamic limit. When longer-range pair hopping is added, these exact LIOMs are lost except for the dipole moment, and the previously frozen density modes relax subdiffusively with $H=\sum_j J_j \hat h_j$8 (Łydżba et al., 2024). This establishes a sharp distinction between fragmentation labeled by nonlocal SLIOMs and fragmentation that also carries strictly local conserved operators.

The interplay of fragmentation with ordinary symmetries leads to symmetry fragmentation. In charge-conserving systems with charge conjugation and translation, a symmetry may exchange two disconnected Krylov sectors rather than preserve each one. Projectors onto such sector pairs, together with the symmetry action, generate an SU(2)-like Pauli algebra and encode degenerate logical qubits (Iadecola, 7 Oct 2025). In the kinetically constrained XNOR model, the number of such encoded qubits can itself grow exponentially with system size because the number of exchanged Krylov-sector pairs is exponential.

Generalized symmetries broaden the landscape further. Gauge, higher-form, and subsystem symmetries can generate exponentially many symmetry sectors, and non-invertible symmetries can generate additional fragmentation within a chosen symmetry sector (Budde et al., 14 Apr 2026). This leads to a conceptual qualification: the presence of exponentially many Krylov sectors does not by itself imply ergodicity breaking in a deeper sense, because some instances of apparent fragmentation can be reinterpreted as generalized-symmetry resolution rather than symmetry-free dynamical shattering (Budde et al., 14 Apr 2026).

A closely related development concerns emergent gauge symmetry. In the fragmented $H=\sum_j J_j \hat h_j$9 dipole-conserving chain, effective local conserved quantities exist only in an exponentially large family of Krylov sectors, where projected operators become non-invertible symmetry operators and label a subset of sectors with emergent $\mathcal C$0 gauge structure (Budde et al., 17 Apr 2026). A plausible implication is that fragmentation can function not only as a source of nonergodicity but also as a sector-selective organizational principle for exact gauge-theory simulation.

4. Dynamical signatures and diagnostics

The most direct signature of fragmentation is that states with the same global quantum numbers relax differently. In the two-dimensional tilted Bose-Hubbard experiment, chequerboard, dimer, and squares product states can belong to the same global $\mathcal C$1 sector yet exhibit strikingly different dynamics. The imbalance

$\mathcal C$2

remains large for the chequerboard state, decays rapidly for the dimer state, and shows intermediate behavior for the squares state (Adler et al., 2024). This is the operational meaning of fragmentation: global conservation laws alone do not determine dynamics once the accessible state space splits into disconnected pieces.

The same experiment resolves fractonic consequences of fragmentation. Defects inserted on a frozen chequerboard background spread only along one-dimensional equipotential lines in a two-dimensional lattice, and interfaces between localized and thermalizing regions evolve differently depending on whether they are parallel or orthogonal to the equipotential direction (Adler et al., 2024). These observations connect HSF to sub-dimensional transport and orientation-dependent domain-wall dynamics.

Several diagnostics quantify the onset or gradual destruction of fragmentation. In the tunable $\mathcal C$3-$\mathcal C$4 ladder, adding rungs removes SLIOMs two at a time, progressively merging blocks rather than restoring ergodicity in a single step. The resulting extended critical regime is diagnosed by multiple peaks in the regularized average fidelity susceptibility, each associated with a change in the number of SLIOMs and a corresponding ultra-slow relaxation of local observables (Lisiecki et al., 14 May 2025).

Autocorrelation functions and Mazur bounds provide complementary information. In fragmented models, projectors onto Krylov sectors can be inserted into Mazur-type bounds, yielding strictly positive late-time memory for local observables. This logic is used both in the commutant-algebra treatment, which derives improved bounds for local autocorrelations in fragmented chains, and in the lattice-geometry analysis, where positive long-time autocorrelation demonstrates nonthermal dynamics on the Vicsek fractal, the square lattice, and related geometries (Moudgalya et al., 2021, Harkema et al., 2024).

A recent graph-theoretic reformulation makes the sector structure itself a diagnostic object. Basis states are treated as graph vertices and Hamiltonian matrix elements as edges; exact fragmentation corresponds to disconnected graph components, while nearly fragmented systems appear as weakly connected communities detected through Laplacian spectra, Fiedler vectors, and modularity (Rutkowski et al., 18 May 2026). In this framework, the weighted adjacency $\mathcal C$5 also controls the short-time decay of a subspace-projected Loschmidt echo, tying graph connectivity directly to dynamical timescale separation (Rutkowski et al., 18 May 2026).

5. Experimental realizations and engineered platforms

The first direct observation of HSF beyond one dimension was reported in a two-dimensional tilted Bose-Hubbard system using quantum gas microscopy. Operating near the resonant condition $\mathcal C$6 in the strong-interaction, strong-tilt regime, the experiment realized effective first-order constrained dynamics, imaged bulk states, interfaces, and defects, and directly observed fractonic excitations through highly anisotropic defect motion (Adler et al., 2024). The work established a two-dimensional platform for microscopic studies of constrained transport.

Rydberg platforms support both coherent and dissipative realizations. In the large-detuning dressing regime, the Rydberg Ising chain maps to a generalized folded XXZ model whose transport asymmetry between magnons and holes allows tuning from an integrable folded-XXZ regime, through a Krylov-restricted thermal phase, to a statistical bubble localization regime (Yang et al., 2024). The same setting admits nonlocal interaction terms and position disorder, leading respectively to enriched fragmentation patterns and a symmetry-selective many-body localization transition in which hole sectors localize more readily than magnon sectors (Yang et al., 2024).

A driven-dephasing Rydberg chain yields an explicitly open-system form of fragmentation. In the dephasing PXP model, the Liouvillian decomposes into disconnected invariant subspaces, each supporting a maximally mixed zero mode $\mathcal C$7. The sectors are labeled by the consecutive double-excitation addressing operator

$\mathcal C$8

and the number of zero modes grows with chain length according to a modified Fibonacci recursion (Yan et al., 28 Nov 2025). The observed metastable plateaus in magnetization and purity are therefore sector-resolved consequences of open-system fragmentation.

HSF has also been proposed as a resource. In a two-dimensional transverse-field Ising model with strong spatially inhomogeneous Ising couplings, emergent fragmentation creates frozen ancillary regions that dynamically decouple a GHZ-encoded probe subset from the rest of the lattice. With suitable compensation of induced longitudinal fields, the probe spins evolve effectively under the target transverse field alone, enabling Heisenberg-limited sensitivity with respect to the number of probe spins while avoiding thermalization on the relevant timescale (Yoshinaga et al., 2022).

6. Extensions and conceptual frontiers

Open-system generalizations sharpen the distinction between classical and quantum fragmentation. In the Temperley-Lieb model, a dephasing bath with jump operators $\mathcal C$9 collapses quantum fragmentation to classical fragmentation and yields a separable steady state, while a structure-preserving bath with $\dim(\mathcal C)$0 preserves the quantum fragmentation structure and produces a steady state with coherent memory of the initial condition and finite logarithmic negativity (Li et al., 2023). This shows that fragmentation can stabilize highly entangled stationary states, but only when the dissipation respects the non-Abelian conserved structure.

Non-Hermitian many-body systems exhibit a different extension. In interacting Hatano-Nelson and non-Hermitian SSH chains, strong fragmentation in the large-interaction limit constrains hopping so that each fragment admits a similarity transformation to a Hermitian form. Together with suitable global symmetries, this can enforce a real spectrum above a finite critical interaction strength and produces a many-body exceptional point detectable through the relaxation of a local equal-time correlation function (Ghosh et al., 2023).

The relation between HSF and many-body localization remains nuanced. A decimation study of the random-field Heisenberg chain interprets strong-disorder MBL geometrically as a shattering of the Hilbert-space graph into disconnected resonant clusters, thereby linking multifractality to a fragmentation-like structure in configuration space (Pietracaprina et al., 2019). This is not the same mechanism as clean constrained fragmentation, but it suggests that “Hilbert-space shattering” can be a broader geometric motif across disorder-free and disordered settings.

A final conceptual frontier concerns what fragmentation alone implies about ergodicity. One line of work argues that generalized symmetries can already generate exponentially many sectors, so exponential sector counting should not automatically be equated with intrinsic ergodicity breaking (Budde et al., 14 Apr 2026). Another line emphasizes algebraic and dynamical consequences beyond mere counting, including improved Mazur bounds, coherent-memory effects, logical-qubit encoding, and sector-restricted thermalization (Moudgalya et al., 2021, Iadecola, 7 Oct 2025). Taken together, these developments suggest that the modern theory of Hilbert-space fragmentation is no longer limited to a catalog of disconnected blocks: it is an active framework for understanding constrained dynamics, symmetry organization, transport anomalies, and nonthermal many-body phenomena across closed, open, Hermitian, and non-Hermitian quantum systems.