Hilbert-Space Fragmentation and Constrained Dynamics

Hilbert-space fragmentation occurs when local kinetic rules split a conventional symmetry sector into many dynamically disconnected fragments, often called Krylov sectors in this literature. It is an exact statement about the coordinate-space connectivity generated by a Hamiltonian, not merely slow relaxation or strong initial-state dependence. Thermalization may still occur inside a fragment even though no dynamics connects different fragments.

Required background. Quenches and relaxation supplies the dynamical tests, and the eigenstate thermalization hypothesis supplies the ensemble comparison. Helpful background. Quantum-scar models provide the nearby but distinct case of exceptional eigenstates.

Krylov sectors as a connectivity problem

Choose a local product basis $\{\lvert a\rangle\}$ in a fixed sector of all ordinary global symmetries. Make a graph whose vertices are basis states and whose edges connect pairs with $\langle b\rvert H\lvert a\rangle\ne0$. If $C(a)$ is the connected component containing $\lvert a\rangle$, its coordinate span

$$\mathcal F_a=\operatorname{span}\{\lvert b\rangle:b\in C(a)\}$$

is an invariant fragment. The cyclic Krylov space generated by one basis vector,

$$\mathcal K(H,a) =\operatorname{span}\{\lvert a\rangle,H\lvert a\rangle,H^2\lvert a\rangle,\ldots\},$$

satisfies $\mathcal K(H,a)\subseteq\mathcal F_a$ but need not equal it: destructive interference or spectral degeneracy can prevent powers of $H$ from spanning every coordinate direction in the connected component. Equality requires $\lvert a\rangle$ to be a cyclic vector for that block.

After ordering the basis by connected components, the Hamiltonian block-decomposes as

$$\mathcal H_{Q}=\bigoplus_\alpha\mathcal F_\alpha, \qquad H=\bigoplus_\alpha H_\alpha,$$

within an ordinary charge sector $\mathcal H_Q$. The projectors onto the components commute with $H$, even when they are highly nonlocal in the original degrees of freedom.

Exponentially many components are not by themselves enough to describe the strength of fragmentation. Let $D_Q$ be the dimension of the conventional symmetry sector and $D_{\max}$ its largest fragment. In the standard taxonomy, strong fragmentation has $D_{\max}/D_Q\to0$ with size, whereas weak fragmentation has $D_{\max}/D_Q\to1$ even though many smaller components remain. A limit strictly between zero and one is an intermediate finite-fraction case and should be named explicitly rather than folded into either asymptotic class. The precise limit, boundary conditions, filling, and basis tied to the kinetic constraint must be stated.

This connectivity-based definition and the distinction from ordinary localization are developed by Khemani, Hermele, and Nandkishore 2020.

Charge-and-dipole-conserving motion

A simple one-dimensional bosonic move on an open or infinite chain removes two particles from site $j$ and adds one to each neighbor:

$$H_{\mathrm{pair}} =-t\sum_j \left(b_{j-1}^\dagger b_j^2 b_{j+1}^\dagger+\mathrm{h.c.}\right)+H_{\mathrm{diag}}.$$

It conserves charge $Q=\sum_j n_j$ and dipole moment $P=\sum_j jn_j$, because the change $(+1,-2,+1)$ has both zero total and zero first moment. A periodic chain needs a separately defined modular polarization or must omit wraparound moves; the naive $P$ above is not conserved across that boundary. Even on the open chain, fixed $(Q,P)$ does not guarantee connectivity. Occupation constraints can leave frozen configurations with no allowed move and larger components whose dimensions differ greatly.

This example separates two layers. Charge and dipole conservation are conventional labels visible from local continuity rules. Fragment labels encode the full reachability graph and can be much more numerous. A generalized Gibbs ensemble containing only $Q$ and $P$ averages over components that the dynamics cannot visit.

Dipole-conserving Hamiltonians provide a canonical realization of this finer decomposition Sala et al. 2020.

For a small chain, fragmentation can be established constructively:

1. enumerate all basis states at fixed $Q$ and $P$;
2. add an edge for every nonzero off-diagonal matrix element;
3. find connected components;
4. verify that exact diagonalization is block diagonal in that ordering; and
5. repeat with boundary conditions and size held explicit.

The result should agree whether components are found by graph search or by repeated reachability on the support of basis states. A numerical Krylov iteration from one vector recovers the whole component only after cyclicity has been checked; otherwise it may produce a proper subspace. Comparing these constructions catches missing moves, destructive cancellations, and numerical thresholds mistaken for exact zeros.

Sector-resolved thermalization

An initial state in $\mathcal F_\alpha$ remains there exactly. The appropriate microcanonical comparison is therefore

$$\langle O\rangle_{E,\alpha} =\frac{\operatorname{Tr}\!\left(P_{E,\Delta E}P_\alpha O\right)} {\operatorname{Tr}\!\left(P_{E,\Delta E}P_\alpha\right)},$$

where $P_\alpha$ projects onto the fragment and $P_{E,\Delta E}$ selects an energy window. Eigenstate thermalization can hold inside a large fragment even when expectation values differ between fragments at the same energy and ordinary charges. The observable memory then comes from exact sector selection, not from every eigenstate being nonthermal.

Participation entropy and Krylov dimension can diagnose restricted exploration, but finite-time confinement is weaker than exact fragmentation. In a tilted lattice, an effective dipole-conserving Hamiltonian may emerge only perturbatively; higher-order processes eventually connect components. The resulting prefragmented regime has a long lifetime controlled by the tilt or constraint-breaking coupling rather than an exact block decomposition of the microscopic Hamiltonian.

Distinguishing nearby mechanisms

Fragmentation differs from disorder localization because it can occur in a clean translation-invariant model and is defined by exact zeros in the connectivity graph. It differs from many-body scarring because scarring singles out atypical eigenstates inside an otherwise connected block, whereas fragmentation partitions the entire basis into invariant blocks. It differs from an ordinary symmetry decomposition because two states can share every declared local or global charge and still belong to different fragments.

For a systematic comparison with scars and other ETH exceptions, see Moudgalya, Bernevig, and Regnault 2022.

A perturbation $\lambda V$ that violates the kinetic constraint supplies the decisive test. Exact fragment projectors cease to commute with $H+\lambda V$, and memory should decay on a scale that changes systematically with $\lambda$. A lifetime insensitive to that perturbation calls for another explanation, such as disorder, energetic suppression, or measurement limits.

Experiments in tilted Hubbard systems and programmable processors have observed strong initial-state dependence and restricted participation compatible with fragmentation regimes. A 2025 superconducting-processor study compared states with the same energy and conventional quantum numbers, measured participation entropy, and followed systems up to 24 qubits Wang et al. 2025. Such evidence probes finite devices and effective Hamiltonians; exact asymptotic fragmentation still depends on the controlled microscopic constraints and their corrections.

Evidence status

This assessment was checked through 10 August 2026. Exact fragmentation is established in multiple models, and finite-platform evidence for restricted Krylov dynamics is substantial. Stability under generic perturbations, thermodynamic classification near an exactly fragmented point, and the relation to Stark-localized regimes remain model-dependent research questions. Updates belong in the Quantum Matter and Emergence Research dossier.

Exercises

1. Conservation by a local move. Verify explicitly that the occupation change $(+1,-2,+1)$ on sites $(j-1,j,j+1)$ conserves $Q$ and $P$.

Solution

The charge change is $1-2+1=0$. The dipole change is $(j-1)-2j+(j+1)=0$. Thus every allowed off-diagonal move remains within fixed $(Q,P)$, although those two numbers need not specify a connected component.

2. Weakly broken fragments. Let $P_\alpha$ project onto a fragment of $H_0$, and take $H=H_0+\lambda V$. Show that the initial rate of change of its weight is zero for a state inside the fragment, while leakage probability begins at order $\lambda^2t^2$.

Solution

For $p_\alpha(t)=\langle P_\alpha\rangle$, $\dot p_\alpha(0)=i\lambda\langle[V,P_\alpha]\rangle$. If $P_\alpha\lvert\psi_0\rangle=\lvert\psi_0\rangle$, the two terms have equal expectation value, so the derivative vanishes. Expanding the amplitude into the complementary sector gives $(1-P_\alpha)e^{-iHt}\lvert\psi_0\rangle=-i\lambda t(1-P_\alpha)V\lvert\psi_0\rangle+O(t^2)$, hence leakage probability is $\lambda^2t^2\lVert(1-P_\alpha)V\lvert\psi_0\rangle\rVert^2+\cdots$.

References

• Khemani, Vedika, Michael Hermele, and Rahul Nandkishore. “Localization from Hilbert Space Shattering: From Theory to Physical Realizations.” Physical Review B 101, 174204 (2020). DOI.
• Moudgalya, Sanjay, B. Andrei Bernevig, and Nicolas Regnault. “Quantum Many-Body Scars and Hilbert Space Fragmentation: A Review of Exact Results.” Reports on Progress in Physics 85, 086501 (2022). DOI.
• Sala, Pablo, Tibor Rakovszky, Ruben Verresen, Michael Knap, and Frank Pollmann. “Ergodicity Breaking Arising from Hilbert Space Fragmentation in Dipole-Conserving Hamiltonians.” Physical Review X 10, 011047 (2020). DOI.
• Wang, Yong-Yi, Yun-Hao Shi, Zheng-Hang Sun, Chi-Tong Chen, Zheng-An Wang, Kui Zhao, Hao-Tian Liu, Wei-Guo Ma, Ziting Wang, Hao Li, Jia-Chi Zhang, Yu Liu, Cheng-Lin Deng, Tian-Ming Li, Yang He, Zheng-He Liu, Zhen-Yu Peng, Xiaohui Song, Guangming Xue, Haifeng Yu, Kaixuan Huang, Zhongcheng Xiang, Dongning Zheng, Kai Xu, and Heng Fan. “Exploring Hilbert-Space Fragmentation on a Superconducting Processor.” PRX Quantum 6, 010325 (2025). DOI.