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

Abstract

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.