Exact Solvability and Integrability Signatures in a Periodically Driven Infinite-Range Spin Chain: The Case of Floquet interval $π/2$

We study the signatures of quantum integrability (QI) in a spin chain model, having infinite-range Ising interaction and subjected to a periodic pulse of an external magnetic field. We analyze the unitary operator, its eigensystem, the single-qubit reduced density matrix, and the entanglement dynamics for arbitrary initial state for any $N$. The QI in our model can be identified through key signatures such as the periodicity of entanglement dynamics and the time-evolved unitary operator, and highly degenerated spectra or Poisson statistics. In our previous works, these signatures were observed in the model for parameters $τ=π/4$ and $J=1,1/2$, where we provided exact analytical results up to $12$ qubits and numerically for large $N$ [Phys. Rev. B \textbf{110}, 064313,(2024)}; arXiv:2411.16670 (2024)}]. In this paper, we extend the analysis to $τ=mπ/2$, and arbitrary $J$ and $N$. We show that the signatures of QI persist for the rational $J$, whereas for irrational $J$, these signatures are absent for any $N$. Further, we perform spectral statistics and find that for irrational $J$, as well as for rational $J$ with perturbations, the spacing distributions of eigenvalues follow Poisson statistics. The average adjacent gap ratio is obtained as $\langle r \rangle=0.386$, consistent with Poisson statistics. Additionally, we compute the ratio of eigenstate entanglement entropy to its maximum value ($\langle S \rangle /S_{Max}$) and find that it remains significantly below $1$ in the limit $N\rightarrow \infty$, which further confirms the QI. We discuss some potential experimental realizations of our model.