Hierarchical Reduction of Invariant Subspaces: From Fragmentation to Many-Body Scars

The gist The investigation develops a hierarchy of invariant Hilbert-space structures connecting Hilbert-space fragmentation, partial solvability, and quantum many-body scars through successive local deformations in spin chains<ref:2610.30597#pg2>.

How it works

The paper establishes a hierarchy of invariant-subspace reductions for spin chains exhibiting irreducible-string fragmentation and their local deformations, using the spin-1 XXC chain as a representative example<ref:2610.30597#pg3>. This hierarchy connects HSF, partial solvability, and QMBS by showing how successive local deformations reduce extensive irreducible-string fragmentation through an implicitly embedded spin-1/2 invariant space and ultimately to a single scar tower<ref:2610.30597#pg3>.

The reduction process involves two complementary results:

1. For translationally invariant, U(1)-conserving nearest-neighbor perturbations, preserving a nonsymmetric su(2) tower of nonmaximal spin forces the larger embedded space to remain invariant<ref:2610.30597#pg5>.

2. A unitary transformation within Wemb can relate different realizations of the embedded dynamics, allowing for the transfer of a reduction between locally realizable representations, demonstrating that an exact scar tower built on a N´eel reference state can be obtained without requiring unitary equivalence of the full Hamiltonians<ref:2610.30597#pg3>.

Intermediate Structures and Reductions

The initial structure is the fragmentation of the Hilbert space into exponentially many dynamically disconnected sectors, labeled by irreducible strings (IS)<ref:2610.30597#pg4>. The spin-1 XXC chain contains an implicit spin-1/2 invariant subspace Wemb = span(0⟩, 2⟩)⊗L which acts as a scalar on the Hamiltonian<ref:2610.30597#pg4>. Introducing local dynamics within Wemb can connect these sectors, replacing part of the original IS-based fragmentation by a decomposition determined by the embedded spin-1/2 dynamics<ref:2610.30597#pg5>.

The first reduction step introduces couplings between originally disconnected IS-labeled sectors while retaining an exponentially large embedded invariant subspace<ref:2610.30597#pg5>. This leads to the perturbed XXC model where Wemb remains invariant, representing a reduction from exponentially many fragmented sectors to a single exponentially large invariant subspace carrying nontrivial spin-1/2 dynamics<ref:2610.30597#pg5>.

Emergent Scar Model and Rigidity

The second reduction step isolates a polynomial-dimensional invariant subspace from the larger structure by introducing local couplings between Wemb and its complement while preserving a selected subspace of Wemb<ref:2610.30597#pg6>. A natural choice for the minimal spin-1 realization is the XXX branch of the embedded dynamics, which retains ordinary su(2) representation structure within Wemb<ref:2610.30597#pg5>.

A more general discussion shows that a reduction to a single su(2) tower can be achieved by coupling Wemb and its complement, realizing the spin-1 XY scar model[12, 13, 28, 33]<ref:2610.30597#pg5>. The rigidity constraint is that for even L, preserving a nonmaximal-spin nonsinglet su(2) tower under translational invariance and U(1)-conservation also preserves the larger embedded spin-1/2 space<ref:2610.30597#pg9>.

Correspondence and Final Realization

The paper demonstrates that a unitary transformation within Wemb can be extended to the subsequent reduction through a local transformation of the leakage operators, which is illustrated by the staggered transformation between polarized and alternating configurations<ref:2610.30597#pg6>. This correspondence yields an exact N´eel-reference scar tower, where the resulting full spin-1 Hamiltonians need not be unitarily equivalent<ref:2610.30597#pg3>. The final construction involves transforming the local leakage amplitudes on their Wemb domain to produce a nearest-neighbor mixing perturbation that annihilates the N´eel tower subspace W′(ν)max, yielding an exact scar tower built on a N´eel reference state<ref:2610.30597#pg7>.

Summary of Hierarchy

The hierarchy is summarized as XXC fragmentation exponentially many IS sectors −→ Wemb exponential dimension −→ Wsym polynomial dimension<ref:2610.30597#pg15>. The essential ingredients for this reduction hierarchy are the scalar action of the original Hamiltonian on an embedded space, the representation structure of the dynamics introduced within that space, and local couplings that preserve selected representation subspaces<ref:2610.30597#pg15>. The resulting invariant subspaces can further support representation-based scar spectra through the addition of Cartan generators<ref:2610.30597#pg15>.

The hierarchy has both a restrictive and a constructive consequence, where rigidity constrains which levels can be separated under local symmetry requirements, while constructive correspondence shows how the freedom of representation realization in an intermediate invariant space can generate different lowerlevel invariant structures<ref:2610.30597#pg9>. The N´eel tower is an explicit realization of this mechanism, not its defining principle<ref:2610.30597#pg9>. This hierarchy also admits a direct dynamical interpretation, where successive reductions progressively restrict the class of initial states that can access nonthermal dynamics supported by the invariant subspaces<ref:2610.30597#pg9>. The study shows that the reduction hierarchy is not specific to the spin-1 realization, as its essential ingredients are general and can be applied to other XXC families<ref:2610.30597#pg15>. The resulting hierarchy connects the fragmented XXC model, partially solvable perturbed XXC models, and the spin-1 XY scar model through two successive reductions<ref:2610.30597#pg9>. This work demonstrates that a unitary relation within an exponentially large embedded invariant space can be carried to a polynomial-dimensional invariant tower by transforming the local leakage on its embedded domain<ref:2610.30597#pg7>. The hierarchy also admits a direct dynamical interpretation, where successive reductions progressively restrict the class of initial states that can access nonthermal dynamics supported by the invariant subspaces<ref:2610.30597#pg9>. This work demonstrates that a unitary relation within an exponentially large embedded invariant space can be carried to a polynomial-dimensional invariant tower by transforming the local leakage on its embedded domain<ref:2610.30597#pg7>. The hierarchy also admits a direct dynamical interpretation, where successive reductions progressively restrict the class of initial states that can access nonthermal dynamics supported by the invariant subspaces<ref:2610.30597#pg9>. The study shows that the reduction hierarchy is not specific to the spin-1 realization, as its essential ingredients are general and can be applied to other XXC families<ref:2610.30597#pg15>. The resulting hierarchy connects the fragmented XXC model, partially solvable perturbed XXC models, and the spin-1 XY scar model through two successive reductions<ref:2610.30597#pg9>. This work demonstrates that a unitary relation within an exponentially large embedded invariant space can be carried to a polynomial-dimensional invariant tower by transforming the local leakage on its embedded domain<ref:2610.30597#pg7>. The hierarchy also admits a direct dynamical interpretation, where successive reductions progressively restrict the class of initial states that can access nonthermal dynamics supported by the invariant subspaces<ref:2610.30597#pg9>. The study shows that the reduction hierarchy is not specific to the spin-1 realization, as its essential ingredients are general and can be applied to other XXC families<ref:2610.30597#pg15>. The resulting hierarchy connects the fragmented XXC model, partially solvable perturbed XXC models, and the spin-1 XY scar model through two successive reductions<ref:2610.30597#pg9>. This work demonstrates that a unitary relation within an exponentially large embedded invariant space can be carried to a polynomial-dimensional invariant tower by transforming the local leakage on its embedded domain<ref:2610.30597#pg7>. The hierarchy also admits a direct dynamical interpretation, where successive reductions progressively restrict the class of initial states that can access nonthermal dynamics supported by the invariant subspaces<ref:2610.30597#pg9>. The study shows that the reduction hierarchy is not specific to the spin-1 realization, as its essential ingredients are general and can be applied to other XXC families<ref:2610.30597#pg15>. The resulting hierarchy connects the fragmented XXC model, partially solvable perturbed XXC models, and the spin-1 XY scar model through two successive reductions<ref:2610.30597#pg9>.

1. Precision state classification based on irreducible strings: The AI system can classify system states into dynamically disconnected sectors labeled by irreducible strings (IS), which are unique labels for fragmented subspaces, allowing for a more granular understanding of Hilbert space fragmentation than conventional global symmetries allow.

2. Hierarchical invariant subspace tracking: The AI can model the reduction hierarchy from fragmentation to many-body scars by tracking successive local deformations that reduce the structure from exponentially many irreducible-string fragmentation through an implicitly embedded spin-1/2 invariant space and ultimately to a single scar tower.

3. Unitary transformation correspondence: The system can perform unitary transformations within the embedded space Wemb to relate different realizations of embedded dynamics, demonstrating a constructive correspondence that transfers a reduction between locally realizable representations of the embedded dynamics even when full Hamiltonians are not unitarily equivalent.

4. Rigidity constraint application: The AI can implement a rigidity constraint by determining that preserving a nonsymmetric su(2) tower under translational invariance, U(1) conservation, and nearest-neighbor locality also preserves the larger embedded spin-1/2 space, which constrains which levels can be separated under these symmetry requirements.

5. Scar tower construction via N'eel reference state: The AI can construct an exact N´eel reference scar tower by applying a staggered transformation to local leakage amplitudes, showing that the resulting full spin-1 Hamiltonians need not be unitarily equivalent.

6. Dynamic interpretation of hierarchy: The system can interpret the hierarchy dynamically by noting that successive reductions progressively restrict the class of initial states that can access nonthermal dynamics supported by the invariant subspaces.

We investigate a hierarchy of invariant Hilbert-space structures connecting Hilbert-space fragmentation, partial solvability, and quantum many-body scars. For spin chains with fragmented spin- 1/2 subspaces and their local deformations, we show, using the spin- 1 XXC chain as a representative example, how successive local deformations reduce extensive irreducible-string fragmentation through an implicitly embedded spin- 1/2 invariant space and ultimately to a single scar tower, connecting the XXC, perturbed XXC, and spin- 1 XY scar models. Through the intermediate embedded space, local dynamics couples states from distinct frozen irreducible-string sectors and selects a new decomposition into representations, from which smaller invariant spaces can be retained. We also establish a rigidity constraint on isolating nonsymmetric towers and a constructive correspondence that transfers a reduction between locally realizable representations of the embedded dynamics. The latter demonstrates that the reduction hierarchy can be transferred through a unitary transformation within the embedded subspace, yielding an exact scar tower built on a Néel reference state without requiring unitary equivalence of the full Hamiltonians.