Hilbert Space Fragmentation in Quantum Systems

• Hilbert space fragmentation is a phenomenon where local constraints partition the full many-body state space into disconnected Krylov sectors.
• It leads to sector-dependent dynamics with limited thermalization, vividly captured in observables like imbalance overshoot and suppressed entanglement growth.
• Induced by conservation laws, kinetic constraints, and lattice geometries, this mechanism provides key insights into ergodicity breaking and potential quantum information applications.

Hilbert space fragmentation is a mechanism in quantum many-body systems where the allowed dynamics, often constrained by local conservation laws or kinetic constraints, partition the full exponentially large Hilbert space into an extensive number of dynamically disconnected Krylov sectors. This phenomenon prevents full thermalization, as each sector evolves independently, limiting equilibration to only a subportion of the many-body state space. The paper of Hilbert space fragmentation is foundational for understanding ergodicity breaking beyond conventional symmetry sectors and has significant implications for both equilibrium and nonequilibrium quantum dynamics.

1. Fundamental Mechanisms and Formalism

Hilbert space fragmentation arises when not all pairs of many-body configuration basis states can be connected via a sequence of allowed local moves, even after factoring out all conventional symmetries (e.g., magnetization, particle number). In models with strong local constraints (e.g., conservation of domain walls, dipole moment, or pairwise occupation), an initial product state typically can only explore a restricted region—the Krylov sector—under the system’s dynamics.

Mathematically, given a parent Hamiltonian $H$, the application of $H$ to a state $|\psi\rangle$ generates a Krylov subspace $$\mathcal{K}_{|\psi\rangle} = \text{span}\{H^n|\psi\rangle\}$$ for $n \geq 0$. The full Hilbert space $\mathcal{H}$ then decomposes as

$\mathcal{H} = \bigoplus_{j} \mathcal{K}_j,$

where the index $j$ labels the Krylov sectors. In fragmented models, the number of sectors grows exponentially with system size and their internal structure is determined by the detailed form of dynamical constraints.

Several paradigmatic spin models, including the XXZ, Heisenberg, and various constrained lattice models (e.g., PXP, folded XXZ, pair-hopping, t-J$_z$), realize this formalism. In the strong fragmentation limit, the Krylov sectors become so small that the typical sector comprises a vanishing fraction of the full Hilbert space.

2. Physical Realizations and Dynamical Constraints

Fragmentation is realized in a variety of physical systems:

• Domain wall and dipole conservation: Models that conserve the number of domain walls or the dipole moment, such as those arising in strongly tilted (Stark) chains or spin-1 dipole-conserving models, lead to strong fragmentation, as only specific arrangements of excitations are accessible without violating the local constraint (Moudgalya et al., 2021, Łydżba et al., 30 Jan 2024).
• Kinetic constraints or projectors: Hamiltonians with terms that permit state changes only when certain local spin arrangements are met (e.g., XNOR, PXP, or folded XXZ models) generate fragmentation, as the imposed constraints restrict allowed sequences of local flips (Vuina et al., 16 Jun 2025, Yang et al., 20 Mar 2024).
• Lattice geometry: Lattices with nontrivial connectivity, such as the Vicsek fractal or higher-dimensional kagome/pyrochlore structures, impose additional geometric obstacles to dynamics. These can further enhance fragmentation by immobilizing defects (e.g., domain walls) in regions where kinetic rules interact nontrivially with the lattice structure (Lee et al., 2020, Harkema et al., 11 Apr 2024).

Table 1: Examples of fragmentation-inducing mechanisms and models

Constraint type	Example model(s)	Reference
Conservation law	Spin-1 dipole or domain wall	(Moudgalya et al., 2021, Łydżba et al., 30 Jan 2024)
Kinetic projector	Folded XXZ, XNOR, PXP	(Vuina et al., 16 Jun 2025, Iadecola, 7 Oct 2025)
Lattice geometry	Vicsek fractal/kagome/pyrochlore	(Lee et al., 2020, Harkema et al., 11 Apr 2024)

3. Characterization and Diagnostic Observables

Fragmentation manifests in both static and dynamic observables:

• Sector-resolved observables: Local order parameters (e.g., sublattice imbalance $I(t)$ or magnetization profiles) often have sector-dependent expectation values. Preparation in different Krylov sectors leads to distinct late-time averages for observables, making these values a diagnostic of sector identity (Vuina et al., 16 Jun 2025). For instance, in the strongly interacting XXZ chain, initial states with different domain wall patterns relax to different minimum imbalance values before returning to zero when sector-mixing becomes possible.
• Entanglement structure: Fragmentation restricts entanglement growth within sectors. For certain initial states, entanglement entropy quickly saturates to values corresponding to random states within the Krylov sector, which can be substantially lower than the ETH (Page) value for the full Hilbert space (Hahn et al., 2021, Zhang et al., 2023).
• Relaxation dynamics and memory: Infinite-temperature autocorrelation or Loschmidt echo dynamics display nonvanishing late-time values, reflecting incomplete ergodization and the “memory” of initial conditions within a sector (Harkema et al., 11 Apr 2024).
• Experimental signatures: In systems with controlled dephasing, the imposed noise mixes states within a Krylov sector but leaves sector boundaries intact. The characteristic “overshoot” in the time evolution of observables like imbalance reflects the sector’s mean observable value and serves as a reliable experimental witness of fragmentation (Vuina et al., 16 Jun 2025).

Table 2: Observable signatures of fragmentation

Observable	Sector dependence	Fragmentation regime
Imbalance $I(t)$	Mean $I_{ss}$ per sector	Strong sector differentiation (Vuina et al., 16 Jun 2025)
Entanglement $S$	Saturates to sector Page	Sub-sector ETH, not full ETH (Hahn et al., 2021)
Autocorrelations	Mazur bound remains finite	Persistent memory (Harkema et al., 11 Apr 2024)

4. Impact of Controlled Dephasing and Experimental Probes

Unitary dynamics within a Krylov sector may remain nonergodic or slow, complicating direct experimental access to the statistical properties of the sector. Controlled dephasing provides a method to accelerate intrasection mixing without destroying the fragmentation structure itself (Vuina et al., 16 Jun 2025). Dephasing in the computational basis efficiently randomizes intra-sector coherences while preserving the block-diagonal sector structure, enabling measurements of sector-averaged observables on experimentally achievable timescales.

By preparing specific initial configurations and introducing dephasing noise via Lindblad jump operators (e.g., $l_j = \sqrt{\gamma} S_j^z$), one can directly observe the predicted minimum overshoot in observables such as sublattice imbalance, as well as confirm sector-specific relaxation dynamics. These effects are robust to experimental imperfections, as the measurement protocol does not depend on perfect unitary evolution.

5. Mathematical and Analytical Frameworks

The classification and analysis of fragmentation employ several mathematical tools:

• Commutant algebra: Fragmentation can be formally identified by examining the commutant (set of all operators commuting with the local terms of the Hamiltonian). If the commutant grows exponentially with system size, the system is fragmented (Moudgalya et al., 2021). This approach distinguishes “classical” fragmentation—visible in the product state basis—from “quantum” fragmentation that arises only in an entangled basis (e.g., Temperley-Lieb spin chains).
• Krylov sector combinatorics: The dimension and observable averages of each Krylov sector can be computed combinatorially, especially in integrable or strongly constrained models. This allows for analytical predictions of sector-dependent observables such as imbalance or entanglement entropy (Vuina et al., 16 Jun 2025).
• Data compression and singular value decomposition: Finding local integrals of motion (LIOMs) or quantifying the surviving conserved quantities in fragmented models can be mapped to data compression problems via singular value decomposition on the set of time-averaged local operators (Łydżba et al., 30 Jan 2024). Operators with singular value $\lambda=1$ correspond to exactly conserved quantities within fragmented sectors.

6. Implications for Nonequilibrium Dynamics and Quantum Information

Fragmentation alters the paradigm of thermalization and ergodicity in quantum matter. Observables and memory of initial state are sector-dependent, and the presence of exponentially many sectors provides a natural quantum information encoding. Specific sector-dependent features include:

• Sector-resolved ETH: The eigenstate thermalization hypothesis can be restored within individual large Krylov sectors, leading to “restricted” ETH even when full ergodicity fails (Hahn et al., 2021).
• Memory retention and nonthermal dynamics: Fragmented models exhibit nonzero late-time autocorrelations and persistent patterns, thus resisting decoherence and thermalization across the full system (Harkema et al., 11 Apr 2024).
• Experimental accessibility: Observable signatures such as imbalance overshoot, sector-limited entanglement scaling, finite Mazur bounds, and persistent or slow relaxation are experimentally testable in ultracold atoms, trapped ions, and Rydberg platforms, especially with recent proposals for dephasing-assisted protocols (Vuina et al., 16 Jun 2025, Yang et al., 20 Mar 2024).
• Quantum information applications: The emergent structure of the sector space can be leveraged to encode robust logical qubits protected by dynamical constraints and symmetries (Iadecola, 7 Oct 2025). In systems with symmetry-enriched fragmentation, the algebra of sector projectors and symmetry operators leads to exponentially many logical qubits encoded in degenerate eigenstate pairs.

7. Outlook and Future Directions

Future research on Hilbert space fragmentation is likely to pursue:

• Floquet and open-system engineering: Control of fragmentation via periodic driving (Floquet engineering) or dissipation, enabling dynamically tunable fragmentation structures and access to new dynamical phases (Zhang et al., 2023, Li et al., 2023).
• Classification of fragmentation strength: Systematic understanding and measurement of strong versus weak fragmentation, including scaling of largest sector dimension and its implications for observable phenomena (Francica et al., 2023, Harkema et al., 11 Apr 2024).
• Generalization to higher dimensions and complex lattices: Extension of fragmentation theory to 2D, fractal, or nontrivial topological lattices, with a view toward enriched quantum information storage and protected subspaces (Lee et al., 2020, Nicolau et al., 2023, Harkema et al., 11 Apr 2024).
• Interaction with symmetries and disorder: Analysis of the interplay between fragmentation, global and subsystem symmetries, and random disorder, as well as the emergence of new selection rules and localized regimes (e.g., symmetry-selective many-body localization) (Iadecola, 7 Oct 2025, Yang et al., 20 Mar 2024).
• Application to quantum algorithms and memories: Exploration of fragmentation-induced degenerate manifolds for robust encoding of logical qubits and potential for quantum error suppression (Iadecola, 7 Oct 2025).

The theoretical and experimental investigation of Hilbert space fragmentation continues to reveal new classes of ergodicity breaking, memory effects, and protected quantum structures, uniting themes from nonergodic dynamics, symmetry theory, and quantum information science.