---
title: Strong Quantum Fragmentation in Many-Body Physics
url: https://www.emergentmind.com/topics/strong-quantum-fragmentation
type: topic
---

# Strong Quantum Fragmentation in Many-Body Physics

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 [2512.00751]. 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 [1502.04889].

## 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 [1910.06341]. 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 $\mathcal H$ [2604.12907].

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 $N_{\rm irr}(L)\to\infty$ [2604.05218].

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 [2512.00751].

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 [1502.04889].

| Setting | Strong-fragmentation criterion | Representative source |
|---|---|---|
| Hilbert-space fragmentation | $d_{\max}(L)/D(L)\sim e^{-\alpha L}$ | [1910.06341] |
| Generalized-symmetry fragmentation | $\lim_{L\to\infty}\dim\mathcal K_{\max}/\dim\mathcal H=0$ | [2604.12907] |
| Quantum fragmentation after removing EFS | $D_{\max}/D_q\to0$, equivalently $N_{\rm irr}(L)\to\infty$ | [2604.05218] |
| Learnable fragmented algebras | $\sum_\alpha \dim(\mathcal K_\alpha)=\poly(L)$ | [2512.00751] |
| Condensate fragmentation | $F=1-|n_1-n_2|/N\approx1-n_1/N$ | [1502.04889] |

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 [2604.12907].

## 2. Algebraic formulation and entangled-basis structure

A unifying algebraic formulation starts from a family of local Hermitian generators
$$
\{h_1,\dots,h_m\}\subset \mathrm{End}(\mathcal H),\qquad m=\poly(L),
$$
and the associative algebra
$$
\mathcal A=\langle h_1,\dots,h_m\rangle\subset\mathrm{End}(\mathcal H).
$$
Its commutant is
$$
\mathcal C=\{\,X\in\mathrm{End}(\mathcal H):[X,A]=0\ \forall A\in\mathcal A\},
$$
and the von Neumann bicommutant theorem yields the block decomposition
$$
\mathcal H=\bigoplus_{\lambda=1}^{\Lambda}\bigl(\mathcal H_\lambda^{(A)}\otimes \mathcal H_\lambda^{(C)}\bigr).
$$
The subspaces $\{\mathcal H_\lambda^{(A)}\}$ are precisely the Krylov subspaces of the family of Hamiltonians in $\mathcal A$, and for any state $\ket{\psi}$,
$$
\mathrm{Krylov}(\psi)=\mathrm{Span}\{A\ket{\psi}:A\in\mathcal A\}.
$$
In the fragmentation setting, each irreducible component $\mathcal H_\lambda^{(A)}$ is invariant under every $A\in\mathcal A$ [2512.00751].

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 [2305.06918].

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 $\Omega$ is built from tensor products of noncrossing dimer singlets on bonds and entangled frozen segments on complementary intervals. For $N=2$, $\dim\Omega(L)=2^L$, and if a bipartition cuts a dimer or a frozen segment the entanglement entropy is $O(1)$; for a non-separable frozen segment of length $L$ one has
$$
S\le \ln W\lesssim \ln|s|\le \ln L,
$$
so the entangled basis is sub-volume-law while remaining sharply distinct from both product-state fragmentation and generic ergodic eigenbases [2604.06461].

## 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 $t\!-\!J_z$ chain, the entire spin pattern carried by the fermions is an exact constant of motion, producing $2^{N_F}$ disconnected blocks. In the dipole-conserving spin-1 chain $H_3$, the complete set of conserved labels includes $\{Q,P\}$, the two boundary charges, the defect number, the defect charges, and regional dipole moments, together labeling every connected block of $\mathcal H$ [1910.06341].

A second mechanism is the peak-valley rule for one-dimensional integer-spin chains. In the dual height representation
$$
h_{j+\frac12}=\sum_{k=0}^{j}F_k^z+h_{-\frac12},
$$
a local $q$-body operator is required to preserve both the local maximum and the local minimum of the partial sums
$$
\max\{0,F_i^z,F_i^z+F_{i+1}^z,\dots\},\qquad
\min\{0,F_i^z,F_i^z+F_{i+1}^z,\dots\}.
$$
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
$$
H_2=\tfrac12\sum_n F_n^+F_{n+1}^-\bigl[1-e^{i\pi(F_n^z+F_{n+1}^z)}\bigr]+\mathrm{h.c.},
$$
and
$$
H_3=\sum_n F_n^+(F_{n+1}^-)^2F_{n+2}^++\mathrm{h.c.},
$$
satisfy the PV condition and strongly fragment [2604.23659].

A third mechanism derives fragmentation from generalized symmetries. If a translation-invariant local Hamiltonian admits an operator $U$ with $t(L)\ge aL^f$ disjointly supported translates, is diagonal in the product basis, and has $q\ge2$ distinct eigenvalues, then the number of symmetry sectors obeys
$$
N_{\rm sym}(L)\ge q^{t(L)}\ge \exp\!\bigl(a\ln(q)L^f\bigr),
$$
and consequently
$$
N_{\rm Krylov}(L)\ge \exp(cL^f),\qquad c=a\ln q.
$$
Gauge, higher-form, subsystem, and non-invertible symmetries all fit into this mechanism. The three-dimensional $U(1)$ quantum-link model and the PXP chain on the full Hilbert space are explicit examples [2604.12907].

A fourth mechanism is rank deficiency of local Hamiltonian terms. For local couplings
$$
h_i=\sum_{w,w'\in R_\alpha}g^{w,w'}\,|w\rangle\langle w'|_{i,\dots,i+\ell-1},
$$
quantum fragmentation requires $\mathrm{rank}\,g<|R_\alpha|$. The resulting local null directions generate an entangled-frozen subspace
$$
\mathcal K_{\rm EFS}^{(\lambda)}=\bigcap_i \ker\!\bigl(h_i\big|_{\mathcal K^{(\lambda)}}\bigr)
=\ker\!\bigl(H\big|_{\mathcal K^{(\lambda)}}\bigr),
$$
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 ${\rm TL}_{L-1}(3)$ and decomposes as
$$
\mathcal K_q=\bigoplus_j \Delta_j\otimes\mathbb C^{m_j},
$$
with $N_{\rm irr}(L)=4,10,16,39,69,\dots$ for $L=5,6,7,8,9,\dots$, forcing $D_{\max}/D_q\to0$ and hence strong fragmentation [2604.05218].

## 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 $(m,\ell)$ one constructs exact eigenstates
$$
|E_{S=0}\rangle
=|\psi_m^\ell\rangle\otimes|\underbrace{\circ\circ\cdots\circ}_{r}\rangle\otimes|\psi_R\rangle,
$$
and the total number of such zero-entanglement states satisfies
$$
\mathcal N_{S=0}\propto \exp(cL).
$$
The largest thermal Krylov block shows GOE level statistics, but the dynamics from a domain-wall initial state displays three regimes in
$$
R(t)=\sqrt{\sum_{i>N_p}(i-N_p)^2\langle n_i(t)\rangle}:
$$
ballistic growth at early times, a long superdiffusive plateau $R(t)\sim t^{0.74}$, and a late logarithmic regime $R(t)\sim A\ln t$ [2210.15607].

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 $F_{N_p+2}\sim \varphi^{N_p}$ and is equivalent to the one-dimensional PXP chain with open boundaries; the “junction of two wires” has
$$
\mathcal D_{\rm junc}(N_p)=F_{x+1}(F_x+F_{x+3})\sim \varphi^{N_p-1}.
$$
Both show GOE level statistics inside a fragment, while special initial product states exhibit periodic revivals with periods $T\approx5$ or $T\approx4$ [2208.13800].

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
$$
r_n=\frac{\min(\delta_n,\delta_{n+1})}{\max(\delta_n,\delta_{n+1})},\qquad
\delta_n=E_{n+1}-E_n,
$$
the distribution approaches
$$
P_{\rm Poisson}(r)=\frac{2}{(1+r)^2},\qquad 0\le r\le1,
$$
because the spectrum is a superposition of an increasing number of independent GOE blocks [2604.05218].

The extended quantum breakdown model gives a distinct entangled-basis realization. At zero spin field and in the $Q=1$ sector, one can construct exponentially many one-dimensional Krylov sectors with
$$
D(K^{(1)})=2^{M-1},
$$
while numerics up to $M=11$ give a largest Krylov dimension
$$
|K^{\max}|=4\lfloor M/2\rfloor.
$$
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 [2401.16480].

Floquet settings support analogous behavior. In a periodically kicked XXZ chain with
$$
U_F=U_2U_1,\qquad
U_1=\exp\!\bigl[-i(\pi/2-\epsilon)\sum_j\sigma_j^x\bigr],\qquad
U_2=\exp[-iTH_{XYZ}],
$$
strong ZZ interactions produce approximate conservation of the absolute magnetization $Q=|M|$ and the domain-wall number
$$
D=\sum_{j=1}^{L-1}\frac{1-\sigma_j^z\sigma_{j+1}^z}{2}.
$$
Defining
$$
R_q=\frac{\max_p\dim(\mathcal H_q^p)}{\dim(\mathcal H_q)},
$$
strong fragmentation is diagnosed by $R_q\to0$. The fragmented Floquet spectrum develops $\pi$-pairs, subharmonic response, and beating dynamics; numerically, the discrete-time-crystal lifetime is independent of the driving frequency, scales as $\tau/T\sim (V/J)^\mu$ with $\mu\approx5$, and grows exponentially with system size [2512.14182].

## 5. Efficient learnability and the Schur transform

A recent development ties strong fragmentation directly to trainable quantum machine learning. For fragmented algebras generated by $\mathcal A=\langle h_1,\dots,h_m\rangle$, one chooses a Schur basis $\{|\lambda,q_\lambda,p_\lambda\rangle\}$ adapted to the block decomposition
$$
\mathcal H=\bigoplus_\lambda \mathcal H_\lambda^{(A)}\otimes \mathcal H_\lambda^{(C)},
$$
and defines the Schur transform $U_{\rm Schur}$ by
$$
U_{\rm Schur}\,|\lambda,q_\lambda,p_\lambda\rangle\otimes|0\rangle
=
|\lambda,q_\lambda\rangle\otimes|\mathrm{garbage}(p_\lambda)\rangle,
$$
where the ancilla of dimension $N_a=2^{n_a}$ stores the computational-basis label $(\lambda,q_\lambda)$ and forgets $p_\lambda$. In this setting, strong fragmentation means that the summed dimension of the distinct Krylov subspaces is polynomial,
$$
\sum_\alpha \dim(\mathcal K_\alpha)=\poly(L),
$$
and this condition makes the Schur transform efficiently learnable by a quantum neural network [2512.00751].

The training set is
$$
\mathcal D=\Bigl\{\bigl(|\lambda,q_\lambda,p_\lambda\rangle\otimes|0\rangle,\;O_{\lambda,q_\lambda}\bigr)\Bigr\},
\qquad
O_{\lambda,q_\lambda}=-\mathrm{Id}_{H'}\otimes|\lambda,q_\lambda\rangle\!\langle\lambda,q_\lambda|,
$$
with
$$
H'=\bigoplus_{\lambda=1}^{\Lambda'}\mathcal H_\lambda^{(A)}\otimes\mathcal H_\lambda^{(C)}.
$$
The variational circuit has layered form
$$
U(\boldsymbol\theta)
=
e^{+Ht''}
\prod_{i=1}^{p}\Bigl[e^{-Ht_i}e^{A\theta_i}e^{+Ht_i}\Bigr]
e^{-Ht'},
$$
where
$$
H=
\Bigl(\sum_{i=1}^{m}c_i h_i\Bigr)\otimes
\Bigl(\sum_{j=1}^{n_a}c'_{j,x}\sigma_j^x+c'_{j,y}\sigma_j^y+c'_{j,z}\sigma_j^z\Bigr),
\qquad
A\in\mathcal A\otimes \mathrm{Id}_{n_a}.
$$
The times $t_i,t',t''$ are drawn uniformly in $[0,T]$, and only the real parameters $\theta_1,\dots,\theta_p$ are trained. The loss function is
$$
\ell(\boldsymbol\theta)
=
\frac1M\sum_{x=1}^M
\langle x|\,U(\boldsymbol\theta)\,O_x\,U(\boldsymbol\theta)^\dagger|x\rangle,
$$
with gradients estimated by parameter-shift or linear-combination-unitaries methods.

The trainability result has two parts. First, under the hypothesis that $H$ satisfies the full Eigenstate Thermalization Hypothesis in each irreducible sector, the expected squared gradient obeys
$$
\mathbb E_{\boldsymbol\theta}\|\nabla\ell(\boldsymbol\theta)\|^2
=
\sum_{\lambda=1}^{\Lambda'}
\frac{2M_\lambda(N_a-1)\mathrm{Tr}(A_\lambda^2)p}
{M^2N_a^4N_\lambda^2}
=\Omega\!\bigl(1/\poly(L)\bigr),
$$
eliminating barren plateaus. Second, the Hessian rank is bounded by
$$
\mathrm{rank}\,[\mathrm{Hess}(\boldsymbol\theta)]
\le
2\sum_{\lambda=1}^{\Lambda'} M_\lambda (N_\lambda^*)^3
=
O\!\bigl(MN_a^3N^3\bigr),
$$
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 [2512.00751].

## 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
$$
H=J\sum_{\langle ij\rangle}Z_iZ_j+h\sum_i X_i,
$$
with $J\gg h$, produce an emergent fragmented structure in which a chosen set of probe spins is effectively decoupled from a frozen ancilla region. Using an $N_P$-spin GHZ state in the $X$ basis as the probe,
$$
F_Q=4N_P^2\tau^2,
$$
so the Heisenberg limit is achieved, while the approximate protection persists up to a fragmentation timescale
$$
\tau_{\rm frag}\sim \frac{J}{h^2}
$$
[2211.09567].

Open-system generalizations show that the fate of fragmentation depends sharply on the bath. In the Temperley-Lieb chain, dephasing with $L_j^{\rm deph}=S_j^z$ reduces quantum fragmentation to classical fragmentation and yields separable stationary states, whereas a fragmentation-preserving bath with $L_j^{\rm frag}=e_{j,j+1}$ retains the full non-Abelian commutant. The steady state then has block form
$$
\rho_\infty=\bigoplus_\lambda\Bigl[M^{(\lambda)}\otimes(\mathbb I_{D_\lambda}/D_\lambda)\Bigr],
$$
retains coherent memory of the initial state through the matrices $M^{(\lambda)}$, and exhibits nonzero logarithmic negativity [2305.06918].

A broader operator-algebraic version appears in Pauli-Lindblad models. There the Lindbladian
$$
\frac{d\rho}{dt}
=
-i[H,\rho]
+\sum_\alpha\Bigl(L_\alpha\rho L_\alpha^\dagger-\tfrac12\{L_\alpha^\dagger L_\alpha,\rho\}\Bigr)
$$
induces operator-space fragmentation of $\mathcal B(\mathcal H)$. The bond algebra of superoperators and its commutant give a decomposition
$$
\mathcal B(\mathcal H)=\bigoplus_\lambda\bigl(\mathcal B_\lambda^{\mathcal C}\otimes \mathcal B_\lambda^{\mathcal A}\bigr),
$$
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 [2506.16518].

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
$$
J_P=\frac{\Omega^2V}{4\delta(\delta+V)},\qquad
J_Q=\frac{\Omega^2V}{4(\delta+V)(\delta+2V)},
$$
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 [2403.13790].

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.

Source: https://www.emergentmind.com/topics/strong-quantum-fragmentation