Successive local deformations reduce extensive irreducible-string fragmentation through an implicitly embedded spin-1/2 invariant space. This process ultimately leads to a single scar tower connecting the XXC, perturbed XXC, and spin-1 XY scar models. Local dynamics couples states from distinct frozen irreducible-string sectors via this intermediate embedded space and selects a new decomposition into representations. Smaller invariant spaces are then able to be retained as a result. A key constraint on isolating nonsymmetric towers is also established. Furthermore, a constructive correspondence transferring a reduction between locally realizable representations of the embedded dynamics is demonstrated.

## Unitary Transformation Preserves Reduction Hierarchy in Invariant Subspaces The reduction hierarchy can be transferred through a unitary transformation within an embedded subspace, yielding an exact scar tower built on a Néel reference state without requiring full Hamiltonian equivalence. I. Introduction The boundary between thermalization and exact solvability is more nuanced than simply categorizing quantum systems as either nonintegrable or integrable. Typically, generic interacting systems exhibit thermal behaviour following the eigenstate thermalization hypothesis (ETH), while integrable systems avoid this fate due to their extensive set of conserved quantities.

However, a third possibility exists: constrained dynamics may survive only within a proper invariant subspace of an otherwise nonintegrable Hamiltonian. These partially solvable systems are valuable not just as exceptions to ETH but also for exploring a fundamental question, how much can an invariant Hilbert-space structure be reduced by local deformations. What larger structures must necessarily remain intact to protect a specific nonthermal subspace?

Hilbert-space fragmentation (HSF), isolated solvable sectors, and quantum many-body scars (QMBS) offer distinct perspectives on this challenge. In fragmented systems, the local Hamiltonian decomposes the Hilbert space into numerous dynamically disconnected sectors, often exponentially with system size, that lack conventional global symmetries. Partially solvable systems retain closed dynamics within selected extended sectors while allowing complementary portions of the Hilbert space to become nonintegrable or thermalizing.

QMBS represent an even more refined structure: sparse sets of atypical eigenstates embedded in otherwise thermal spectra can potentially generate coherent revivals from simple initial states. Exact scar towers are frequently stabilised by local annihilation conditions or restricted spectrum-generating algebras; these phenomena therefore define different levels of a hierarchy based on how much invariant Hilbert-space structure persists. This intermediate level isn’t merely a container for scar states, but rather introduces dynamics connecting fragmentation labels and organizing them into new representations.

Dynamics introduced at this stage connect states carrying distinct fragmentation labels and organize them within a novel representation-based decomposition. Connections between HSF and QMBS have been explored extensively, with recent work demonstrating a direct link through partially activating kinetically constrained fragmented structures to generate scarred dynamics. We adopt an alternative approach here: examining how underlying invariant subspaces are successively reduced under local deformations, what structures can be removed while preserving smaller ones?

When does attempt preservation inevitably restore larger configurations. These questions become particularly acute when considering scar towers built on nonsymmetric reference states; their preservation is weaker than that of extended sectors yet locality and symmetry may prevent separating these two conditions. We develop a construction for invariant subspace hierarchies in spin chains exhibiting irreducible-string fragmentation alongside their local deformations.

The spin-1 XXC chain serves as a representative example, with its Hamiltonian acting scalarly upon the exponentially large embedded space Wemb containing states from distinct frozen irreducible-string (IS) sectors. Introducing local dynamics within this space connects those sectors, replacing some original IS-based fragmentation by a decomposition determined by the embedded spin-1/2 dynamics. This initial reduction yields the perturbed XXC model where Wemb remains unchanged; subsequent coupling preserves a polynomial-dimensional representation subspace while breaking invariance of Wemb itself, as seen in the spin-1 XY scar model.

These models therefore provide successive levels for an invariant subspace reduction hierarchy. We establish two complementary results: first, translationally invariant U-conserving nearest neighbour perturbations preserving a nonsymmetric su tower with nonmaximal spin forces preservation of the larger embedded space.

Nonmaximal-spin scar towers are nevertheless possible beyond these constraints, as demonstrated elsewhere. Second, applying a unitary transformation within Wemb can relate different realizations of the embedded dynamics and transfer reductions to smaller subspaces when corresponding leakage operators transform locally; we demonstrate this correspondence through staggered transformations between polarized and alternating configurations yielding an exact Néel reference scar tower without requiring full Hamiltonian equivalence.

The remainder of this paper details our approach, starting by reviewing IS fragmentation in Sec. Section III constructs the partially solvable level via perturbations within that implicit subspace while Sec. IV further reduces it to the scar level establishing rigidity for nonsymmetric su towers. In Sec. V, we explain how a unitary transformation extends reduction using local operator changes illustrated with polarized and alternating IS configurations before carrying these transformations through the hierarchy in Sec. VI obtaining the Néel scar construction; finally, we conclude in Sec. VII providing technical details in appendices.

II. IMPLICIT SPIN-1/2 INVARIANT SUBSPACE OF THE XXC MODEL We begin by considering the open XXC model which provides our fragmented limit case for hierarchical discussion. Beyond its well-known HSF, this model contains an implicit spin-1/2 invariant subspace serving as the foundation of all subsequent constructions. Consider a spin-l/2 chain on (Cl+1)⊗L = span{|0⟩.

Because the local interactions never exchange species in A and B, the Hilbert space fragments according to initial ordering of those species within each set.

This research demonstrated a connection between fragmented quantum systems, partially solvable models, and the emergence of quantum many-body scars. This process ultimately leads to a single scar tower, achieved via unitary transformations within this subspace without requiring full Hamiltonian equivalence; notably, they constructed such a tower based on a Néel reference state. These findings establish relationships in how complex quantum dynamics evolve from fragmented states towards more ordered configurations.

👉 More information
🗞 Hierarchical Reduction of Invariant Subspaces: From Fragmentation to Many-Body Scars
✍️ Chihiro Matsui (The University of Tokyo)
🧠 ArXiv: https://arxiv.org/abs/2610.30597

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