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

Abstract

Quantum chaos in isolated quantum systems is intimately linked to thermalization and the rapid relaxation of observables. Although the spectral properties of the chaotic phase in the tilted Bose-Hubbard model have been well characterized, the corresponding dynamical signatures across the transition to regularity remain less explored . In this work, we investigate this transition by analyzing the time evolution of the survival probability, the single-site entanglement entropy, and the half-chain imbalance. Our results reveal a clear hierarchy in the sensitivity of these observables: the relaxation value of the entanglement entropy varies smoothly as a function of the Hamiltonian parameters across the chaos-regular transition, while the imbalance exhibits a more pronounced distinction. Most notably, the survival probability emerges as the most robust indicator of the transition between chaos and regularity. When appropriately scaled, all three observables converge onto a common behavior as a function of the Hamiltonian parameters for different numbers of sites and bosons,enabling a universal characterization of the transition between chaotic and regular dynamics.

Figures (11)

• Figure 1: Absolute value of the difference between the mean level-spacing ratio $\braket{r}$ of the eigenspectrum of the tilted Bose-Hubbard model and the chaotic GOE prediction, shown as a function of the interaction $U/J$ and the tilt for a system with $N=M=8$. The red lines indicate the parameter paths explored in the numerical simulations. The acronyms indicate the following limits: WSL (the integrable Wannier-Stark Localization ), HCB (the integrable Hardcore Bosons), and BH (the chaotic Bose-Hubbard model at $D=0$).
• Figure 2: Left: $PR/\mathcal{D}_{GOE}$ of the eigenstates in the Fock basis for $N=8$. Right: scaling analysis averaging over $80\%$ of the eigenstates in the center of the spectrum. In the upper panel, interaction is fixed at $U=0.5J$ and $D$ varied; in the lower panel, tilt is fixed at $D=0.5J$ and $U$ varied.
• Figure 3: Single-site entanglement entropy of eigenstates for $N=8$. Panels (a)--(c) show $S^{(i)}$ grouped by sites (from edge to middle), revealing how spatial position and energy density shape the entanglement landscape; panel (d) displays the site average $\overline{S}$. The black dashed line is the Page value. Parameters are chosen in the chaotic window, where most mid-spectrum eigenstates are near-thermal yet site-dependent deviations persist.
• Figure 4: Left: $\overline{S}$ of the eigenstates in the Fock basis for $N=8$, the black dashed line is the Page value $\mathcal{S_P}$. Right: $\overline{S}/\mathcal{S_P}$ scaling analysis averaging over $80\%$ of the eigenstates in the center of the spectrum . In the upper panel, interaction is fixed at $U=0.5J$; in the lower panel, tilt is fixed at $D=0.5J$ Single-site entanglement entropy of eigenstates for $N=8$.
• Figure 5: Half-chain imbalance for eigenstates for different tilt strengths at fixed interaction $U=0.5J$ (left) and for varying interaction strength at fixed tilt $D=0.5J$ (right). As $D/J$ increases, eigenstates cluster near specific imbalance values, forming a ladder-like structure characteristic of Wannier-Stark Localization tendencies.