Hilbert Space Fragmentation and Commutant Algebras

We study the phenomenon of Hilbert space fragmentation in isolated Hamiltonian and Floquet quantum systems using language commutant algebras, algebra all operators that commute with each term or gate circuit. provide a precise definition this formalism as case where dimension grows exponentially system size. Fragmentation can hence be distinguished from conventional symmetries such $U(1)$ $SU(2)$, polynomially Further, also helps distinguish between "classical" "quantum" fragmentation, former refers to product state basis. explicitly construct several exhibiting classical including $t-J_z$ model spin-1 dipole-conserving model, we illustrate connection previously-studied "Statistically Localized Integrals Motion" (SLIOMs). revisit Temperley-Lieb spin chains, biquadratic chain widely studied literature, show they exhibit fragmentation. Finally, contribution full Mazur bounds various cases. In fragmented systems, use expressions for analytically obtain new improved autocorrelation functions local agree previous numerical results. addition, are able rigorously localization on-site operator model.