Dynamical witnesses and universal behavior across chaos and non-ergodicity in the tilted Bose-Hubbard model

Technical Summary: Dynamical Witnesses and Universal Behavior Across Chaos and Non-Ergodicity in the Tilted Bose–Hubbard Model

Problem Statement
While the static spectral properties of the chaotic phase in the tilted Bose–Hubbard model (TBH) have been well characterized through level statistics and multifractal analysis, the dynamical signatures of the transition between quantum chaos and regularity (integrability) remain less explored. Specifically, there is a need to understand how the breakdown of ergodicity, as the system moves from the chaotic phase toward integrable limits (Wannier-Stark Localization and Hard-Core Bosons), manifests in the real-time dynamics of experimentally accessible observables. This work addresses the gap between static spectral diagnostics and the time-evolution of physical quantities in the TBH.

Methodology
The authors numerically simulate non-equilibrium quantum quench dynamics in a one-dimensional TBH model with $N$ bosons on $M=N$ sites under open boundary conditions. The Hamiltonian is governed by tunneling amplitude $J$ (set as the energy unit), on-site interaction $U$, and linear tilt $D$.

The study investigates two parameter paths:

1. Tilt-dominated path: Increasing $D$ at fixed $U$, tracing the transition from the chaotic Bose-Hubbard regime to the integrable Wannier-Stark Localization (WSL) limit.
2. Interaction-dominated path: Increasing $U$ at fixed $D$, tracing the transition from the WSL regime through chaos to the integrable Hard-Core Boson (HCB) limit.

The analysis employs three primary dynamical observables, averaged over ensembles of initial Fock product states (with on-site occupation $n_i \le 3$):

• Survival Probability ($SP(t)$): Analyzed for the presence of a "correlation hole" (a dip in the fidelity before reaching the inverse participation ratio plateau), which serves as a signature of long-range spectral correlations.
• Single-site Entanglement Entropy ($S$): The average entropy between a single site and the rest of the chain, used to track thermalization and relaxation values.
• Half-chain Imbalance ($I$): Defined as the normalized difference in particle number between the left and right halves of the chain, tracking memory of the initial state.

Static properties (mean level-spacing ratio $\langle r \rangle$, participation ratio, and eigenstate entanglement entropy) are also computed to benchmark the dynamical results against established spectral indicators.

Key Results

1. Hierarchy of Sensitivity: The three observables exhibit a clear hierarchy in their sensitivity to the chaos-regular transition.

   • Survival Probability: Emerges as the most robust indicator. In the chaotic regime, $SP(t)$ displays a distinct correlation hole (dip-ramp-plateau structure). As the system approaches integrability (either via strong tilt or strong interaction), this correlation hole vanishes, and the dynamics are dominated by initial fluctuations or Bloch oscillations (in the tilt-dominated case).
   • Imbalance: Exhibits a pronounced distinction between regimes. In the chaotic phase, it relaxes rapidly to near zero. In the regular WSL regime, it remains close to its initial value (freezing), showing deep oscillations.
   • Entanglement Entropy: Shows a smoother transition. While the relaxation value decreases as the system moves toward regularity, the change is less abrupt compared to the other two observables.

2. Universal Scaling: A critical finding is the existence of universal behavior across different system sizes ($N=M=7$ to $10$).

   • When the depth of the correlation hole is scaled by the Hilbert space dimension (specifically the GOE reference value $D_{GOE}$), and the relaxation values of entanglement entropy and imbalance are scaled by their respective theoretical limits (Page value and initial imbalance), the curves for all system sizes collapse onto a single universal curve.
   • This scaling allows for a size-independent characterization of the transition, with the maximum chaos (deepest correlation hole, highest entropy) occurring around $U/(NJ) \approx 0.15$ for the interaction path.

3. Asymmetry in Transition: The transition from chaos to regularity is asymmetric. Chaos is rapidly suppressed as the tilt $D$ increases at fixed $U$, whereas it remains more robust when the interaction $U$ is increased at fixed $D$. This asymmetry is reflected in both static spectral properties and dynamical responses.

4. Dynamical Signatures of Limits:

   • WSL Regime ($D \gg U$): Characterized by Bloch oscillations, the absence of a correlation hole, reduced long-time entanglement, and a "frozen" imbalance.
   • HCB Regime ($U \gg D$): Characterized by the absence of Bloch oscillations but the persistence of regular dynamics signatures: slow relaxation, lower equilibrium entanglement, and strong memory retention in the imbalance.

Significance and Claims
The paper claims to provide a complementary and experimentally actionable perspective on the phase diagram of the TBH, moving beyond static spectral measures to real-time dynamics. The primary contribution is the identification of the survival probability, specifically the depth of its correlation hole, as the sharpest and most direct dynamical witness of the transition between chaos and regularity.

While entanglement entropy and imbalance are also effective indicators, the authors note that the survival probability offers a more sensitive probe of the loss of spectral rigidity. Furthermore, the demonstration of universal scaling collapses suggests that the chaos-regular crossover in the TBH can be described by universal laws independent of system size and particle number. The work fills a gap in understanding how ergodicity breakdown manifests dynamically, offering specific observables that can be directly probed in cold-atom experiments.