Broken unitary picture of dynamics in quantum many-body scars

Quantum many-body scars (QMBSs) are a novel paradigm for the violation of the eigenstate thermalization hypothesis -- Hamiltonians of these systems exhibit mid-spectrum eigenstates that are equidistant in energy and which possess low entanglement and evade thermalization for long times. We present a novel approach for understanding the anomalous dynamical behavior in these systems. Specifically, we postulate that QMBS Hamiltonians $H$ can generically be partitioned into a set of terms $O_a$ which do not commute over the entire Hilbert space, but commute to all orders within the subspace of scar states. All states in the scar subspace thus evolve according to a `broken unitary' $U_s(t) = \prod_a e^{-iO_at}$, where $H = \sum_a O_a$; alternatively, the scar subspace may be viewed as being spawned by common eigenstates of all $O_a$. While the scar Hamiltonian $H$ may be non-integrable, operators $O_a$ usually exhibit a trivial spectrum with equidistant eigenergies which gives rise to revivals at an appropriate period. Two classes of scar models emerge in this picture -- those with a finite set of $O_a$, as pertaining to, for instance, scars in the AKLT model, and those with an extensive set of such operators, such as eta pairing scar states in the Hubbard model. Besides discussing how well known scar models fit into the above picture, we use this dynamical decoupling to generalize conditions under which scar subspaces can be engendered in the two settings, generalizing constraints satisfied by quasiparticles in the Matrix Product State (MPS) construction of scar eigenstates, and the Shiraishi-Mori approach. We also discuss connections between this approach and others in the literature.