Surprises in the Deep Hilbert Space of all-to-all systems: From super-exponential scrambling to slow entanglement growth

Abstract

The quantum dynamics of spin systems with uniform all-to-all interaction are often studied in the totally symmetric space (TSS) of maximal total spin. However the TSS states are atypical in the full many-body Hilbert space. In this work, 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. We derive a phase-space representation of the DHS OTOC and show that the OTOC can grow super-exponentially in the large $N$ limit, due to the fast dynamics in an unbounded phase space (in finite systems, we observe numerically that the super-exponential growth ends precociously and gives way to a power-law one until saturation). 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, in the DHS, fast OTOC growth does not imply fast entanglement growth, in contrast to the Zurek-Paz relation derived in the TSS.

arXiv comment25 pages, 6 figures; v2: minor changes, updated references, accepted version

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• Dynamics of spin-momentum entanglement from superradiant phase transitions
• Chaos controlled and disorder driven phase transitions induced by breaking permutation symmetry
• Out-of-time-order correlator computation based on discrete truncated Wigner approximation
• Krylov Winding and Emergent Coherence in Operator Growth Dynamics