Fragmented eigenstate thermalization versus robust integrability in long-range models

Abstract

Understanding the stability of integrability in many-body quantum systems is key to controlling dynamics and predicting thermalization. While the breakdown of integrability in short-range interacting systems is well understood, the role of long-range couplings -- ubiquitous and experimentally realizable -- remains unclear. We show that in fully connected models, integrability is either robust or extremely fragile, depending on whether perturbations are non-extensive, extensive one-body, or extensive two-body. In contrast to finite short-range systems, where any of these perturbations can induce chaos at finite strength, in fully connected finite models, chaos is triggered by extensive two-body perturbations and even at infinitesimal strength. Chaos develops within energy bands defined by symmetries, leading to a fragmented realization of the eigenstate thermalization hypothesis and clarifying how microcanonical shells can be constructed in such systems. We also introduce a general symmetry-based framework that explains the stability of integrability.

Figures (10)

• Figure 1: In the all-to-all coupling limit ($\alpha=0$), (a) the density of states (DOS) is split into energy bands when the system ($L=8$) is subjected to (b) perturbations from any of the three categories. However, the (c) level statistics within the most populated energy band reveal that only class (iii) perturbations induce many-body quantum chaos, as indicated by the Wigner-Dyson level spacing distribution for system ($L=14,~J=1,~h=1$). The representative examples from each class are: (i) $\delta \, \sigma_{L/2}^z$, (ii) $\sum_i^L h_i \sigma_{i}^z$ with $h_i \in [-\delta,\delta]$, (iii) $\delta \sum_{i=1}^{L-1} \hat{\sigma}_i^x \hat{\sigma}_{i+1}^x$, where $\delta =10^{-4}$. Parity and inversion symmetries were taken into account accordingly.
• Figure 2: Verification of the eigenstate thermalization hypothesis (ETH) for Hamiltonian \ref{['eq:LRTFIM']} with $\alpha=10^{-4}$, $L=14$. The top panels show results for the full energy spectrum, with the vertical red line marking the most populated band analyzed in the bottom panels. (a) and (d): Density of states, (b) and (e): Entanglement entropy, (c) and (f): Eigenstate expectation values of $\hat{S}_z$, and (g): Off-diagonal elements of $\hat{S}_z$ for 200 eigenstates in the middle of the selected energy band. (f) and (g): Diagonal and off-diagonal ETH, respectively, are satisfied. Parity and inversion symmetries are taken into account.
• Figure S3: For the Hamiltonian given in Eq. (1) of the main text with $\alpha=0$, this figure demonstrates that introducing a small but finite $\alpha$ acts as a perturbation, since the ratio of the norms can be fitted to the form $\varepsilon = 0.49\, \alpha^{1.0}$.
• Figure S4: For the fully connected limit of the long-range transverse field Ising model with the Hamiltonian given in Eq. (2) of the main text and $L = 8$ spins, panel (a) shows the density of states (DOS), revealing 25 distinct energy bands. Panel (b) confirms that $\hat{S}^2$ is diagonal in the eigenstates of the Hamiltonian in Eq. (2) of the main text. Panel (c) shows that each band is uniquely identified by its corresponding total spin quantum number $s$.
• Figure S5: For the Hamiltonian at $\alpha=0$ in Eq. (2) of the main text, we present the maximally populated $s_\mathrm{max}$-sector for increasing $L$. Even for numerically accessible system sizes, we find variations of $s_\mathrm{max}$ with $L$.