Exact Solvability and Integrability Signatures in a Periodically Driven Infinite-Range $p$-Spin Kicked Top Model
We study the signatures of quantum integrability (QI) in a $p$-spin kicked top model with infinite-range Ising interactions, subjected to periodic pulses of an external magnetic field, with the precession angle fixed at $α=mπ$. We analytically compute the unitary operator, its eigensystem, time evolution, single-qubit reduced density matrix, and entanglement dynamics for arbitrary initial states. We show that the spectrum of the unitary operator exhibits pronounced clustering of eigenphases in some regions, leading to strongly non-uniform spectral features with $p$. The entanglement measures and unitary operator dynamics exhibit periodic behavior, while the eigenvalue spectrum shows strong degeneracies when the kicking strength $k'$ is a rational multiple of $π$ for even $p$. For odd $p$, both the periodic behavior and spectral degeneracies persist for all values of $k'$. This periodic behavior extends to the strong-kicking regime ($k'=J j^{p-1}π$), where for rational values of $J$ the dynamics remain periodic. For any $k'$ and $p$ the spectral statistics shows Poissonian behavior and the average adjacent gap ratio $\langle r \rangle = 2\ln 2 - 1$, consistent with integrable behaviour. The spectral form factor further supports this nature. Additionally, we compute the ratio of average eigenstate entanglement entropy to its maximum value ($\langle S \rangle /S_{Max}$) and find that it remains significantly far below from $1$ in the limit $N\rightarrow \infty$. These results collectively establish robust signatures of QI across all $k', p,$ and system sizes. We also discuss possible experimental realizations of the model.