Signatures of Integrability and Exactly Solvable Dynamics in an Infinite-Range Many-Body Floquet Spin System

In a recent work Sharma and Bhosale [Phys. Rev. B, 109, 014412 (2024)], $N$-spin Floquet model having infinite range Ising interaction was introduced. In this paper, we generalized the strength of interaction to $J$, such that $J=1$ case reduces to the aforementioned work. We show that for $J=1/2$ the model still exhibits integrability for an even number of qubits only. We analytically solve the cases of $6$, $8$, $10$, and $12$ qubits, finding its eigensystem, dynamics of entanglement for various initial states, and the unitary evolution operator. These quantities exhibit the signature of quantum integrability (QI). For the general case of even-$N > 12$ qubits, we conjuncture the presence of QI using the numerical evidences such as spectrum degeneracy, and the exact periodic nature of both the entanglement dynamics and the time-evolved unitary operator. We numerically show the absence of QI for odd $N$ by observing a violation of the signatures of QI. We analytically and numerically find that the maximum value of time-evolved concurrence ($C_{\mbox{max}}$) decreases with $N$, indicating the multipartite nature of entanglement. Possible experiments to verify our results are discussed.