Fock-space delocalization and its control in scarred-vs-ergodic dynamics

Introduction

Quantum many‑body scarred dynamics—most famously observed in Rydberg‑atom arrays—exhibit striking revivals where the system repeatedly returns to its initial state despite the underlying complexity of the Hilbert space. A central question is how far the wave function spreads across the allowed configurations between these revivals. In the recent work by Francesco Perciavalle, the author addresses this by measuring the fraction of the Fock space that becomes occupied during PXP dynamics starting from two canonical initial states: the polarized state (which thermalizes) and the Néel state (which displays scarred dynamics). The analysis uncovers contrasting delocalization behaviors and shows how a weak quasiperiodic field can tune the explored fraction, leading to a non‑monotonic dependence on field strength.

Measuring Fock‑space delocalization

The PXP model imposes a kinetic constraint that forbids adjacent excitations, dramatically reducing the size of the accessible configuration space compared with an unconstrained spin‑½ chain. To quantify delocalization, the author defines the occupied fraction as the ratio of distinct basis configurations visited by the evolving wave function to the total number of allowed configurations in the thermodynamic limit. This metric is evaluated numerically for system sizes up to the computationally feasible limit, and extrapolated to infer scaling trends.

Polarized versus Néel initial states

• Polarized state: Starting from a fully polarized product state, the dynamics quickly spreads over a large portion of the allowed configurations. The occupied fraction approaches a finite value close to the anticoncentration limit expected for a random pure state, but chiral symmetry of the PXP Hamiltonian caps this value at roughly one‑third of the full space.
• Néel state: Despite the presence of pronounced revivals, the Néel‑initiated dynamics also explores exponentially many configurations. However, the occupied fraction shrinks to zero as the system size grows, indicating that the scarred trajectory occupies a vanishingly small portion of the Hilbert space in the thermodynamic limit.

Controlling delocalization with a quasiperiodic field

Introducing a weak quasiperiodic potential modifies the kinetic constraints without destroying the underlying scar structure. The study finds two regimes:

1. Weak field – For the polarized state, lifting the chiral symmetry allows the occupied fraction to rise to about one‑half of the configuration space, a substantial increase over the symmetry‑restricted value. The Néel state, however, requires a stronger field before its occupied fraction shows a comparable rise.
2. Strong field – As the field strength continues to increase, both initial states experience a reduction in the occupied fraction. This non‑monotonic behavior signals that stronger quasiperiodic modulation suppresses the ability of the system to explore new configurations, even though the dynamics does not necessarily become localized in the conventional sense.

Implications and outlook

The contrasting delocalization patterns highlight a nuanced picture of scarred dynamics: revivals do not imply confinement to a few configurations, yet the scarred trajectory remains a vanishing subset of the full Hilbert space. The ability to tune the explored fraction with external fields offers a practical handle for experimental platforms seeking to manipulate ergodic versus non‑ergodic behavior. Moreover, the observed non‑monotonic dependence on field strength suggests that delocalization and localization are not simply opposite ends of a single parameter axis but can coexist in a richer phase diagram.

Practical takeaways

• Experimental relevance: Quasiperiodic fields can be implemented in Rydberg‑atom or trapped‑ion setups to adjust the degree of configuration space exploration, potentially optimizing coherence times for scarred dynamics.
• Theoretical tools: The occupied‑fraction metric provides a clear, quantitative way to compare scarred and thermalizing dynamics across models with kinetic constraints.
• Future directions: Extending the analysis to larger system sizes, different constraints, or incorporating disorder could further clarify the relationship between scarred revivals, delocalization, and many‑body localization phenomena.

In summary, Perciavalle’s work offers a systematic framework for assessing how many‑body wave functions traverse their allowed configuration space, revealing that scarred dynamics, while exhibiting strong revivals, still spread over exponentially many states—but only a vanishing fraction of the total— and that external quasiperiodic fields provide a versatile knob to control this delocalization.