Complexity of PXP scars revisited

We revisit a quantum quench scenario in which either a scarring or thermalizing initial state evolves under the PXP Hamiltonian. Within this framework, we study the time evolution of spread complexity and related quantities in the Krylov basis. We find that the Lanczos coefficients $b_n$, as functions of the iteration number $n$, exhibit a characteristic arched growth and decay, followed by erratic oscillations which we refer to as buttress. The arched profile predominantly arises from contributions within the quantum many-body scar subspace, while the buttress is linked to thermalization dynamics. To explain this behavior, we utilize the representation theory of $\mathfrak{s}l_3(\mathbb{C})$, allowing us to decompose the PXP Hamiltonian into a linear component and a residual part. The linear term governs the formation and width of the arch, and we observe that that there exists a threshold of arch width which determines whether a given initial state exhibits scarring. Meanwhile, the residual term accounts qualitatively for the emergence of the buttress. We estimate an upper bound for the extent of the buttress using Lucas numbers. Finally, we demonstrate that spread complexity oscillates periodically over time for scarred initial states, whereas such oscillations are suppressed in thermalizing cases.