Super-Exponential Scrambing and Slow Entanglement Growth in the Deep Hilbert Space of All-to-All Systems

Quantum dynamics of spin systems with uniform all-to-all interactions have often been studied in the totally symmetric space (TSS) of maximal total spin. However, the TSS states are atypical in the full many-body Hilbert space. We explore several aspects of the all-to-all quantum dynamics away from the TSS, and reveal surprising features of the "deep Hilbert space'' (DHS). We study the out-of-time order correlator (OTOC) in the infinite-temperature ensemble of the full Hilbert space and show that the OTOC can grow super-exponentially in the large-N limit, due to the fast dynamics in an unbounded phase space. By a similar mechanism, the Krylov complexity grows explosively. We also study the entanglement growth in a quantum quench from a DHS product state, i.e., one of non-aligned spins that resemble the DHS infinite-temperature ensemble with respect to the statistics of the collective spins. Using a field-theoretical method, We exactly calculate the entanglement entropy in the large-N limit. We show that, unlike in TSS, fast OTOC growth does not imply fast entanglement growth in the DHS.