Strong Quantum Fragmentation in Many-Body Physics

• Strong quantum fragmentation is a phenomenon in quantum many-body systems where the Hilbert space splits into exponentially many small, dynamically disconnected Krylov sectors.
• It arises through various mechanisms such as algebraic symmetries, peak-valley rules, and rank deficiencies, leading to non-thermal spectra and anomalous dynamical behavior.
• This fragmented structure underpins efficient learnability in quantum machine learning and offers practical experimental avenues in Rydberg arrays and condensate systems.

Strong quantum fragmentation denotes several distinct but related non-ergodic structures in quantum many-body physics. In the Hilbert-space literature, it refers to a decomposition of the many-body Hilbert space into many dynamically disconnected Krylov sectors, with “strong” distinguishing regimes in which no single sector retains a finite fraction of the available state space, or, in a more restrictive learnability setting, regimes in which the summed dimension of all distinct Krylov subspaces is only polynomial in system size (Mints et al., 30 Nov 2025). In bosonic many-body physics, by contrast, fragmentation refers to macroscopic occupation of more than one natural orbital, and low-density two-dimensional systems with long-range interactions provide a regime explicitly identified as strong quantum fragmentation in that sense (Fischer et al., 2015).

1. Definitions and competing criteria

The literature uses several non-equivalent criteria for “strong” fragmentation because the relevant decomposition depends on the physical setting. In Hilbert-space fragmentation, one begins from invariant Krylov sectors generated by the Hamiltonian or by a bond algebra. One standard criterion is that the largest disconnected block becomes exponentially small relative to the full Hilbert space,

$$d_{\max}(L)\big/D(L)\sim e^{-\alpha L},\qquad \alpha>0,$$

so that almost every symmetry sector splits into exponentially many small, dynamically disconnected pieces (Rakovszky et al., 2019). A closely related formulation requires

$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$

equivalently that no single Krylov sector occupies a finite fraction of $\mathcal H$ (Budde et al., 14 Apr 2026).

In projector-based quantum fragmentation, the definition is refined after separating off the entangled frozen subspace. Writing

$$\mathcal K_q=\bigoplus_{\mu=1}^{N_{\rm irr}(L)}\mathcal K_{q,\mu},\qquad D_q=\dim\mathcal K_q,\qquad D_{\max}=\max_\mu\dim\mathcal K_{q,\mu},$$

strong quantum fragmentation is defined by

$\frac{D_{\max}}{D_q}\xrightarrow[L\to\infty]{}0,$

equivalently $N_{\rm irr}(L)\to\infty$ (Zhou et al., 6 Apr 2026).

A more restrictive criterion appears in the learnability setting for fragmented algebras. If $\{\mathcal K_\alpha\}$ denotes the set of Krylov subspaces, then strong fragmentation is compactly stated as

$\sum_\alpha \dim(\mathcal K_\alpha)=\poly(L),$

so that the total dimension of all distinct Krylov subspaces remains polynomial in system size (Mints et al., 30 Nov 2025).

In condensate physics, the object being fragmented is not Hilbert space but the condensate itself. Diagonalizing the one-body density matrix gives natural-orbital occupations $n_i$, and a useful two-mode fragmentation measure is

$F=1-\frac{|n_1-n_2|}{N}\approx 1-\frac{n_1}{N}.$

Within that framework, low-density two-dimensional gases with unscreened Coulomb-like interactions become strongly fragmented (Fischer et al., 2015).

Setting	Strong-fragmentation criterion	Representative source
Hilbert-space fragmentation	$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$0	(Rakovszky et al., 2019)
Generalized-symmetry fragmentation	$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$1	(Budde et al., 14 Apr 2026)
Quantum fragmentation after removing EFS	$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$2, equivalently $$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$3	(Zhou et al., 6 Apr 2026)
Learnable fragmented algebras	$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$4	(Mints et al., 30 Nov 2025)
Condensate fragmentation	$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$5	(Fischer et al., 2015)

These definitions are not interchangeable. Some quantify the size of the largest connected component, some quantify the multiplicity of irreducible quantum blocks after removing entangled frozen states, and some quantify the total dimension of distinct Krylov sectors. A further caution is explicit in the generalized-symmetry framework: the presence of exponentially many Krylov sectors does not by itself imply ergodicity breaking (Budde et al., 14 Apr 2026).

2. Algebraic formulation and entangled-basis structure

A unifying algebraic formulation starts from a family of local Hermitian generators

$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$6

and the associative algebra

$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$7

Its commutant is

$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$8

and the von Neumann bicommutant theorem yields the block decomposition

$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$9

The subspaces $\mathcal H$0 are precisely the Krylov subspaces of the family of Hamiltonians in $\mathcal H$1, and for any state $\mathcal H$2,

$\mathcal H$3

In the fragmentation setting, each irreducible component $\mathcal H$4 is invariant under every $\mathcal H$5 (Mints et al., 30 Nov 2025).

This algebraic language also clarifies the distinction between classical and quantum fragmentation. When the commutant is Abelian, it may be generated by projectors onto simple product-state configurations; each fragment is then spanned by a product-state basis, and one speaks of classical fragmentation. By contrast, if one cannot find a complete product-state basis in which all projectors in the commutant are simultaneously diagonal, but only an entangled basis, one has quantum fragmentation (Li et al., 2023).

A systematic construction of quantum-fragmented Hamiltonians makes this entangled-basis character explicit. In that construction, the true Krylov-sector structure can be fully resolved only in an entangled basis, and a convenient basis $\mathcal H$6 is built from tensor products of noncrossing dimer singlets on bonds and entangled frozen segments on complementary intervals. For $\mathcal H$7, $\mathcal H$8, and if a bipartition cuts a dimer or a frozen segment the entanglement entropy is $\mathcal H$9; for a non-separable frozen segment of length $$\mathcal K_q=\bigoplus_{\mu=1}^{N_{\rm irr}(L)}\mathcal K_{q,\mu},\qquad D_q=\dim\mathcal K_q,\qquad D_{\max}=\max_\mu\dim\mathcal K_{q,\mu},$$0 one has

$$\mathcal K_q=\bigoplus_{\mu=1}^{N_{\rm irr}(L)}\mathcal K_{q,\mu},\qquad D_q=\dim\mathcal K_q,\qquad D_{\max}=\max_\mu\dim\mathcal K_{q,\mu},$$1

so the entangled basis is sub-volume-law while remaining sharply distinct from both product-state fragmentation and generic ergodic eigenbases (Han et al., 7 Apr 2026).

3. Mechanisms that generate strong quantum fragmentation

Several mechanisms now organize the known examples.

In the framework of statistically localized integrals of motion, a fragmented Hamiltonian admits conserved operators whose eigenvalues label connected components of the Hilbert space. In the $$\mathcal K_q=\bigoplus_{\mu=1}^{N_{\rm irr}(L)}\mathcal K_{q,\mu},\qquad D_q=\dim\mathcal K_q,\qquad D_{\max}=\max_\mu\dim\mathcal K_{q,\mu},$$2 chain, the entire spin pattern carried by the fermions is an exact constant of motion, producing $$\mathcal K_q=\bigoplus_{\mu=1}^{N_{\rm irr}(L)}\mathcal K_{q,\mu},\qquad D_q=\dim\mathcal K_q,\qquad D_{\max}=\max_\mu\dim\mathcal K_{q,\mu},$$3 disconnected blocks. In the dipole-conserving spin-1 chain $$\mathcal K_q=\bigoplus_{\mu=1}^{N_{\rm irr}(L)}\mathcal K_{q,\mu},\qquad D_q=\dim\mathcal K_q,\qquad D_{\max}=\max_\mu\dim\mathcal K_{q,\mu},$$4, the complete set of conserved labels includes $$\mathcal K_q=\bigoplus_{\mu=1}^{N_{\rm irr}(L)}\mathcal K_{q,\mu},\qquad D_q=\dim\mathcal K_q,\qquad D_{\max}=\max_\mu\dim\mathcal K_{q,\mu},$$5, the two boundary charges, the defect number, the defect charges, and regional dipole moments, together labeling every connected block of $$\mathcal K_q=\bigoplus_{\mu=1}^{N_{\rm irr}(L)}\mathcal K_{q,\mu},\qquad D_q=\dim\mathcal K_q,\qquad D_{\max}=\max_\mu\dim\mathcal K_{q,\mu},$$6 (Rakovszky et al., 2019).

A second mechanism is the peak-valley rule for one-dimensional integer-spin chains. In the dual height representation

$$\mathcal K_q=\bigoplus_{\mu=1}^{N_{\rm irr}(L)}\mathcal K_{q,\mu},\qquad D_q=\dim\mathcal K_q,\qquad D_{\max}=\max_\mu\dim\mathcal K_{q,\mu},$$7

a local $$\mathcal K_q=\bigoplus_{\mu=1}^{N_{\rm irr}(L)}\mathcal K_{q,\mu},\qquad D_q=\dim\mathcal K_q,\qquad D_{\max}=\max_\mu\dim\mathcal K_{q,\mu},$$8-body operator is required to preserve both the local maximum and the local minimum of the partial sums

$$\mathcal K_q=\bigoplus_{\mu=1}^{N_{\rm irr}(L)}\mathcal K_{q,\mu},\qquad D_q=\dim\mathcal K_q,\qquad D_{\max}=\max_\mu\dim\mathcal K_{q,\mu},$$9

When this holds, every regional peak height and valley depth becomes an emergent invariant. Product states are then tagged by alternating sequences of peak and valley labels, and the Hilbert space decomposes into exponentially many autonomously evolving Krylov sectors. The spin-1 models

$\frac{D_{\max}}{D_q}\xrightarrow[L\to\infty]{}0,$0

and

$\frac{D_{\max}}{D_q}\xrightarrow[L\to\infty]{}0,$1

satisfy the PV condition and strongly fragment (Fu et al., 26 Apr 2026).

A third mechanism derives fragmentation from generalized symmetries. If a translation-invariant local Hamiltonian admits an operator $\frac{D_{\max}}{D_q}\xrightarrow[L\to\infty]{}0,$2 with $\frac{D_{\max}}{D_q}\xrightarrow[L\to\infty]{}0,$3 disjointly supported translates, is diagonal in the product basis, and has $\frac{D_{\max}}{D_q}\xrightarrow[L\to\infty]{}0,$4 distinct eigenvalues, then the number of symmetry sectors obeys

$\frac{D_{\max}}{D_q}\xrightarrow[L\to\infty]{}0,$5

and consequently

$\frac{D_{\max}}{D_q}\xrightarrow[L\to\infty]{}0,$6

Gauge, higher-form, subsystem, and non-invertible symmetries all fit into this mechanism. The three-dimensional $\frac{D_{\max}}{D_q}\xrightarrow[L\to\infty]{}0,$7 quantum-link model and the PXP chain on the full Hilbert space are explicit examples (Budde et al., 14 Apr 2026).

A fourth mechanism is rank deficiency of local Hamiltonian terms. For local couplings

$\frac{D_{\max}}{D_q}\xrightarrow[L\to\infty]{}0,$8

quantum fragmentation requires $\frac{D_{\max}}{D_q}\xrightarrow[L\to\infty]{}0,$9. The resulting local null directions generate an entangled-frozen subspace

$N_{\rm irr}(L)\to\infty$0

which splits a mobile classical Krylov sector into a mobile quantum sector and an entangled frozen sector. In the Temperley-Lieb model, the mobile quantum space carries a faithful representation of $N_{\rm irr}(L)\to\infty$1 and decomposes as

$N_{\rm irr}(L)\to\infty$2

with $N_{\rm irr}(L)\to\infty$3 for $N_{\rm irr}(L)\to\infty$4, forcing $N_{\rm irr}(L)\to\infty$5 and hence strong fragmentation (Zhou et al., 6 Apr 2026).

4. Spectral, entanglement, and dynamical signatures

Strong quantum fragmentation leaves distinctive spectral and dynamical traces. In particle-conserving East models, an exponentially large number of disconnected Krylov sectors coexists with exponentially many exact product-like eigenstates with zero entanglement across one or more spatial cuts. For admissible $N_{\rm irr}(L)\to\infty$6 one constructs exact eigenstates

$N_{\rm irr}(L)\to\infty$7

and the total number of such zero-entanglement states satisfies

$N_{\rm irr}(L)\to\infty$8

The largest thermal Krylov block shows GOE level statistics, but the dynamics from a domain-wall initial state displays three regimes in

$N_{\rm irr}(L)\to\infty$9

ballistic growth at early times, a long superdiffusive plateau $\{\mathcal K_\alpha\}$0, and a late logarithmic regime $\{\mathcal K_\alpha\}$1 (Brighi et al., 2022).

Two-dimensional ring-exchange models provide a geometrical realization through “quantum drums.” Starting from a product state, the fluctuating plaquettes organize into disjoint connected regions whose spectra depend only on the plaquettes belonging to the drum. The quasi-one-dimensional “wire” drum has fragment dimension $\{\mathcal K_\alpha\}$2 and is equivalent to the one-dimensional PXP chain with open boundaries; the “junction of two wires” has

$\{\mathcal K_\alpha\}$3

Both show GOE level statistics inside a fragment, while special initial product states exhibit periodic revivals with periods $\{\mathcal K_\alpha\}$4 or $\{\mathcal K_\alpha\}$5 (Chattopadhyay et al., 2022).

In the rank-deficient projector setting, strong fragmentation produces Poissonian unresolved level statistics even when each irreducible block is individually GOE. For adjacent gap ratios

$\{\mathcal K_\alpha\}$6

the distribution approaches

$\{\mathcal K_\alpha\}$7

because the spectrum is a superposition of an increasing number of independent GOE blocks (Zhou et al., 6 Apr 2026).

The extended quantum breakdown model gives a distinct entangled-basis realization. At zero spin field and in the $\{\mathcal K_\alpha\}$8 sector, one can construct exponentially many one-dimensional Krylov sectors with

$\{\mathcal K_\alpha\}$9

while numerics up to $\sum_\alpha \dim(\mathcal K_\alpha)=\poly(L),$0 give a largest Krylov dimension

$\sum_\alpha \dim(\mathcal K_\alpha)=\poly(L),$1

The long-time bipartite entropy stays well below the Page value in the strongly fragmented regime; a uniform spin field merges most Krylov subspaces and restores chaos, while strong randomness yields GOE or GUE to Poisson crossovers and logarithmic-in-time entanglement growth in the Poisson regime (Chen et al., 2024).

Floquet settings support analogous behavior. In a periodically kicked XXZ chain with

$\sum_\alpha \dim(\mathcal K_\alpha)=\poly(L),$2

strong ZZ interactions produce approximate conservation of the absolute magnetization $\sum_\alpha \dim(\mathcal K_\alpha)=\poly(L),$3 and the domain-wall number

$\sum_\alpha \dim(\mathcal K_\alpha)=\poly(L),$4

Defining

$\sum_\alpha \dim(\mathcal K_\alpha)=\poly(L),$5

strong fragmentation is diagnosed by $\sum_\alpha \dim(\mathcal K_\alpha)=\poly(L),$6. The fragmented Floquet spectrum develops $\sum_\alpha \dim(\mathcal K_\alpha)=\poly(L),$7-pairs, subharmonic response, and beating dynamics; numerically, the discrete-time-crystal lifetime is independent of the driving frequency, scales as $\sum_\alpha \dim(\mathcal K_\alpha)=\poly(L),$8 with $\sum_\alpha \dim(\mathcal K_\alpha)=\poly(L),$9, and grows exponentially with system size (Tang et al., 16 Dec 2025).

5. Efficient learnability and the Schur transform

A recent development ties strong fragmentation directly to trainable quantum machine learning. For fragmented algebras generated by $n_i$0, one chooses a Schur basis $n_i$1 adapted to the block decomposition

$n_i$2

and defines the Schur transform $n_i$3 by

$n_i$4

where the ancilla of dimension $n_i$5 stores the computational-basis label $n_i$6 and forgets $n_i$7. In this setting, strong fragmentation means that the summed dimension of the distinct Krylov subspaces is polynomial,

$n_i$8

and this condition makes the Schur transform efficiently learnable by a quantum neural network (Mints et al., 30 Nov 2025).

The training set is

$n_i$9

with

$F=1-\frac{|n_1-n_2|}{N}\approx 1-\frac{n_1}{N}.$0

The variational circuit has layered form

$F=1-\frac{|n_1-n_2|}{N}\approx 1-\frac{n_1}{N}.$1

where

$F=1-\frac{|n_1-n_2|}{N}\approx 1-\frac{n_1}{N}.$2

The times $F=1-\frac{|n_1-n_2|}{N}\approx 1-\frac{n_1}{N}.$3 are drawn uniformly in $F=1-\frac{|n_1-n_2|}{N}\approx 1-\frac{n_1}{N}.$4, and only the real parameters $F=1-\frac{|n_1-n_2|}{N}\approx 1-\frac{n_1}{N}.$5 are trained. The loss function is

$F=1-\frac{|n_1-n_2|}{N}\approx 1-\frac{n_1}{N}.$6

with gradients estimated by parameter-shift or linear-combination-unitaries methods.

The trainability result has two parts. First, under the hypothesis that $F=1-\frac{|n_1-n_2|}{N}\approx 1-\frac{n_1}{N}.$7 satisfies the full Eigenstate Thermalization Hypothesis in each irreducible sector, the expected squared gradient obeys

$F=1-\frac{|n_1-n_2|}{N}\approx 1-\frac{n_1}{N}.$8

eliminating barren plateaus. Second, the Hessian rank is bounded by

$F=1-\frac{|n_1-n_2|}{N}\approx 1-\frac{n_1}{N}.$9

so once the number of parameters exceeds this polynomial bound, the Hessian is rank-deficient at all points, and the standard overparameterization argument excludes isolated poor local minima. The resulting setting provides a rare example of a physically motivated quantum learning task with no known dequantization, because the algebra defining the fragmentation is not known a priori and its high-order products need not remain sparse (Mints et al., 30 Nov 2025).

6. Quantum information, open systems, and experimental outlook

Strong fragmentation has begun to function as a resource rather than merely a diagnostic. In fragmentation-protected metrology, strong Ising interactions in

$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$00

with $$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$01, produce an emergent fragmented structure in which a chosen set of probe spins is effectively decoupled from a frozen ancilla region. Using an $$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$02-spin GHZ state in the $$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$03 basis as the probe,

$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$04

so the Heisenberg limit is achieved, while the approximate protection persists up to a fragmentation timescale

$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$05

(Yoshinaga et al., 2022).

Open-system generalizations show that the fate of fragmentation depends sharply on the bath. In the Temperley-Lieb chain, dephasing with $$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$06 reduces quantum fragmentation to classical fragmentation and yields separable stationary states, whereas a fragmentation-preserving bath with $$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$07 retains the full non-Abelian commutant. The steady state then has block form

$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$08

retains coherent memory of the initial state through the matrices $$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$09, and exhibits nonzero logarithmic negativity (Li et al., 2023).

A broader operator-algebraic version appears in Pauli-Lindblad models. There the Lindbladian

$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$10

induces operator-space fragmentation of $$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$11. The bond algebra of superoperators and its commutant give a decomposition

$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$12

and frustration graphs determine effective non-Hermitian Hamiltonians inside fragments. This framework yields symmetry-enriched quantum chaos, free-fermion solvable integrable fragments, and dissipation-driven exceptional points, while also suggesting protected logical subspaces and dissipative codes (Paszko et al., 19 Jun 2025).

Experimental realizations are correspondingly diverse. In one-dimensional Rydberg arrays, the large-detuning Rydberg Ising model maps by a Schrieffer-Wolff transformation to a generalized folded XXZ model with conserved Rydberg number and dimer number, and hence exponentially many disconnected Krylov subspaces. The effective hopping scales

$$\lim_{L\to\infty}\frac{\dim\mathcal K_{\max}}{\dim\mathcal H}=0,$$13

produce distinct magnon and hole timescales, allowing continuous tuning from an integrable regime to a Krylov-restricted thermal phase and then to a statistical bubble localization regime. Atomic position disorder further induces a symmetry-selective many-body-localization transition in the hole sector, and the entire structure is visible in quench dynamics of the imbalance and local densities (Yang et al., 2024).

Taken together, these developments show that strong quantum fragmentation is not a single phenomenon but a family of rigorously structured decompositions: product-basis and entangled-basis, closed-system and open-system, static and Floquet, algebraic and geometrical. Its central common feature is the emergence of many dynamically disconnected sectors beyond conventional symmetry resolution, with consequences ranging from anomalous transport and nonthermal spectra to efficient learnability of Schur transforms and protected quantum sensing.