Fragmenting Hilbert space: Polynomial Factorization and Commutant Algebra

Hilbert space fragmentation is characterized by the exponential growth of Krylov subspaces. However, the precise definition of Krylov subspace remains ambiguous. Particularly, we focus on Krylov subspaces defined by commutant algebra and Integer characteristic polynomial factorization. While these two definitions differs significantly, it has been observed in the literature that they coincide in many systems. In this paper, we showed that Krylov subspaces defined by the latter are always finer than the former, provided that each generator of the center of the bond algebra takes rational eigenvalues. We explicitly showed that this condition holds for a wide range of models.