Thermalization hierarchy from irreducible degrees of freedom

The decomposition of the Hilbert space of a quantum many-body system into the irreducible representations of its bond and commutant algebras yields a finer structure of dynamically isolated subspaces than the mere decomposition into symmetry sectors. While it has been recognized that subspaces associated with low bond-irrep dimensions $D_{\lambda }$ tend to violate the eigenstate thermalization hypothesis (ETH), here we show that $D_{\lambda }$ controls thermalization continuously across the full spectrum of dynamical subspaces. Using SU(2)-symmetric spin-1/2 chains as a paradigmatic example, we demonstrate that $\mathrm{l}\mathrm{o}\mathrm{g}{D}_{\lambda }$ quantitatively accounts for the average eigenstate entanglement entropy within each sector, establishing a thermalization hierarchy that interpolates from exact quantum many-body scars at $D_{\lambda }=1$ to volume-law ergodic states at large $D_{\lambda }$. To make this concrete, we introduce the notion of irreducible degrees of freedom (IDOF), defined as the number of independently-varying spatial coordinates parametrizing a many-body state within a given bond-algebra sector, which provides a microscopic interpretation of $D_{\lambda }$ and of the resulting thermalization hierarchy. Finally, we show that by selectively breaking symmetries while preserving chosen bond-algebra sectors, one can embed families of nonthermal eigenstates at prescribed entanglement levels into an otherwise ergodic spectrum, generalizing restricted spectrum-generating algebras from towers of individual states to entire dynamical subspaces.