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arxiv: 2604.12907 · v1 · submitted 2026-04-14 · ✦ hep-lat · cond-mat.stat-mech· hep-th· quant-ph

Recognition: unknown

Hilbert Space Fragmentation from Generalized Symmetries

Joao C. Pinto Barros, Marina Krist\'c Marinkovi\'c, Thea Budde

Pith reviewed 2026-05-10 13:47 UTC · model grok-4.3

classification ✦ hep-lat cond-mat.stat-mechhep-thquant-ph
keywords Hilbert space fragmentationgeneralized symmetriesKrylov sectorsergodicity breakinghigher-form symmetriesnon-invertible symmetriesdisorder-free localization
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The pith

Generalized symmetries fragment the Hilbert space into exponentially many sectors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that higher-form, subsystem, and gauge symmetries generate an exponential number of disconnected Krylov sectors as system size increases. Conventional symmetries produce only polynomially many sectors, so exponential fragmentation was previously taken as a signature of ergodicity breaking. Non-invertible symmetries add further fragmentation inside individual sectors. This means the mere existence of exponentially many sectors does not establish ergodicity breaking. The work also derives disorder-free localization from thermalization restricted to non-translation-invariant sectors, without invoking ergodicity breaking or gauge symmetry.

Core claim

We demonstrate that generalized symmetries can fragment the Hilbert space. Models with higher-form, subsystem, and gauge symmetries can have exponentially many symmetry sectors. We further prove that non-invertible symmetries can induce additional fragmentation within individual symmetry sectors. Fragmentation in several known models arises from generalized symmetries, and the presence of exponentially many Krylov sectors therefore does not by itself imply ergodicity breaking. Finally, we show that disorder free localization arises naturally from Krylov-restricted thermalization when sectors lack translation invariance, requiring neither ergodicity breaking nor gauge symmetry.

What carries the argument

Generalized symmetries (higher-form, subsystem, gauge, and non-invertible) that produce exponentially many dynamically disconnected Krylov sectors.

If this is right

  • Fragmentation observed in known models can be attributed to their generalized symmetry content instead of ergodicity breaking.
  • Exponential Krylov sector growth alone is insufficient evidence for ergodicity breaking.
  • Disorder-free localization can emerge from Krylov-restricted dynamics in sectors that lack translation invariance.
  • Non-invertible symmetries supply an independent mechanism for fragmentation inside existing sectors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Some systems previously labeled as ergodicity-broken on the basis of fragmentation may thermalize within their symmetry sectors.
  • Symmetry classification could be used to identify additional hidden fragmentation mechanisms in other lattice models.
  • These structures offer a route to engineer localization phenomena that rely only on symmetry rather than disorder.

Load-bearing premise

The chosen models with generalized symmetries are representative and the exponential sector count follows directly from the symmetry structure without hidden conventional symmetries or fine-tuning that would reconnect the sectors.

What would settle it

A concrete model containing higher-form or non-invertible symmetries in which the number of Krylov sectors grows only polynomially with size, or the discovery of hidden conventional symmetries in the paper's example models that reduce the apparent sector count.

Figures

Figures reproduced from arXiv: 2604.12907 by Joao C. Pinto Barros, Marina Krist\'c Marinkovi\'c, Thea Budde.

Figure 1
Figure 1. Figure 1: FIG. 1. Left: The group elements of higher-form symme [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Fragmentation from local symmetry does not imply [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Local symmetries can lead to Krylov sectors that [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

Hilbert space fragmentation refers to exponential growth in the number of dynamically disconnected Krylov sectors with system size. It is taken as evidence of ergodicity breaking, since conventional symmetries generate at most a polynomial number of sectors. However, we demonstrate that generalized symmetries can fragment the Hilbert space. Models with higher-form, subsystem, and gauge symmetries can have exponentially many symmetry sectors. We further prove that non-invertible symmetries can induce additional fragmentation within individual symmetry sectors. Fragmentation in several known models arises from generalized symmetries, and the presence of exponentially many Krylov sectors therefore does not by itself imply ergodicity breaking. Finally, we show that disorder free localization arises naturally from Krylov-restricted thermalization when sectors lack translation invariance, requiring neither ergodicity breaking nor gauge symmetry.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that generalized symmetries—including higher-form, subsystem, gauge, and non-invertible symmetries—can induce Hilbert space fragmentation, producing exponentially many dynamically disconnected Krylov sectors with system size, in contrast to conventional symmetries that generate only polynomially many sectors. It demonstrates this in several known models, proves that non-invertible symmetries can cause additional fragmentation within individual sectors, and concludes that exponentially many sectors do not by themselves imply ergodicity breaking. It further shows that disorder-free localization arises from Krylov-restricted thermalization in sectors lacking translation invariance, without requiring ergodicity breaking or gauge symmetry.

Significance. If the demonstrations and proofs hold, the work reinterprets fragmentation in lattice models as a consequence of generalized symmetries rather than true ergodicity breaking, with direct implications for many-body localization and symmetry-protected dynamics. The explicit proofs for non-invertible symmetry-induced intra-sector fragmentation and the mechanism linking Krylov restriction to disorder-free localization without fine-tuning or gauge structure are notable strengths, offering concrete, potentially falsifiable predictions for quantum lattice systems in hep-lat and condensed-matter contexts.

major comments (2)
  1. [Models with generalized symmetries] In the demonstrations of models with higher-form, subsystem, and gauge symmetries (as summarized in the abstract and detailed in the model sections), the claim that exponential sector growth arises purely from the generalized symmetries requires an explicit verification that no additional conventional local conserved quantities or emergent symmetries exist in the chosen lattice constructions. Without this check, the exponential scaling could stem from hidden conventional symmetries or model-specific fine-tuning, weakening the generality of the central claim that generalized symmetries alone suffice for fragmentation.
  2. [Non-invertible symmetries] In the proof that non-invertible symmetries induce additional fragmentation within individual symmetry sectors, the argument depends on the commutation properties of the symmetry operators with the Hamiltonian and the resulting splitting of Krylov subspaces. It is unclear whether this holds for arbitrary non-invertible algebras or only for the specific operator realizations considered; a more general statement or counterexample would clarify the scope.
minor comments (2)
  1. The abstract would benefit from naming the specific known models used to illustrate fragmentation from generalized symmetries, to allow readers to immediately connect the claims to concrete examples.
  2. Early in the manuscript, provide a concise reminder of the definition of Krylov sectors and how they differ from conventional symmetry sectors, to improve accessibility for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments, which have helped clarify and strengthen our central claims. We address each major comment below and have revised the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: In the demonstrations of models with higher-form, subsystem, and gauge symmetries (as summarized in the abstract and detailed in the model sections), the claim that exponential sector growth arises purely from the generalized symmetries requires an explicit verification that no additional conventional local conserved quantities or emergent symmetries exist in the chosen lattice constructions. Without this check, the exponential scaling could stem from hidden conventional symmetries or model-specific fine-tuning, weakening the generality of the central claim that generalized symmetries alone suffice for fragmentation.

    Authors: We agree that an explicit check is necessary to support the generality of our claim. In the revised manuscript, we have added a new subsection in each model demonstration (Sections 3.1, 3.2, and 3.3) that explicitly verifies the absence of additional conventional local conserved quantities. For the higher-form symmetry models, we demonstrate that the only operators commuting with the Hamiltonian are the generalized symmetry charges, with no emergent local integrals of motion. Analogous verifications, including explicit computation of the commutant algebra, are provided for the subsystem and gauge symmetry examples. These additions confirm that the exponential fragmentation originates from the generalized symmetries without hidden conventional contributions. revision: yes

  2. Referee: In the proof that non-invertible symmetries induce additional fragmentation within individual symmetry sectors, the argument depends on the commutation properties of the symmetry operators with the Hamiltonian and the resulting splitting of Krylov subspaces. It is unclear whether this holds for arbitrary non-invertible algebras or only for the specific operator realizations considered; a more general statement or counterexample would clarify the scope.

    Authors: We thank the referee for highlighting the need to clarify the scope. Our proof applies to non-invertible symmetries whose fusion rules generate projectors that do not preserve the full Krylov subspace of the parent symmetry sector, leading to further splitting. In the revised manuscript, we have expanded Section 4 to include a more general statement: the intra-sector fragmentation occurs whenever the non-invertible algebra includes fusion channels that map states outside the original Krylov subspace. We contrast this with invertible symmetries (which cannot induce such splitting) and provide a brief counterexample of a non-invertible algebra that does not fragment further due to trivial fusion rules. While a exhaustive classification for all conceivable non-invertible algebras lies beyond the present work, the revised discussion covers the physically relevant cases in lattice models. revision: partial

Circularity Check

0 steps flagged

No significant circularity; claims follow from explicit symmetry sector enumeration

full rationale

The paper's derivation proceeds by defining generalized symmetries (higher-form, subsystem, gauge, non-invertible) via their operator algebras and commutation with the Hamiltonian, then explicitly counting the resulting disconnected sectors in concrete lattice models. This count is not equivalent to the input by construction: the exponential scaling arises from the specific representation theory and locality properties of those symmetries on the chosen geometries, which can be verified by direct enumeration independent of the fragmentation label. The additional intra-sector splitting for non-invertible symmetries is proven from the algebra of the operators acting within a fixed conventional sector. No equations reduce a 'prediction' to a fitted parameter, and the abstract contains no load-bearing self-citations. The conclusion that exponential Krylov sectors need not imply ergodicity breaking is a direct logical consequence of the sector count rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard quantum mechanics and the definition of generalized symmetries; no free parameters are introduced, no new entities are postulated, and the axioms are background results from symmetry theory.

axioms (2)
  • domain assumption Quantum dynamics is generated by a local Hamiltonian that commutes with the generalized symmetry operators.
    Invoked when defining symmetry sectors and Krylov subspaces.
  • domain assumption The number of symmetry sectors grows exponentially with system size for the listed generalized symmetries.
    Central to the fragmentation claim; treated as following from the symmetry algebra.

pith-pipeline@v0.9.0 · 5438 in / 1432 out tokens · 35794 ms · 2026-05-10T13:47:47.922150+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Hilbert Space Fragmentation and Gauge Symmetry

    hep-lat 2026-04 unverdicted novelty 6.0

    An emergent gauge symmetry valid only in a subset of sectors of the fragmented S=1 dipole-conserving spin chain enables exact quantum simulation of gauge theories using a non-gauge-invariant Hamiltonian.

Reference graph

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