Exact Solvability Of Entanglement For Arbitrary Initial State in an Infinite-Range Floquet System

Sharma and Bhosale [\href{https://journals.aps.org/prb/abstract/10.1103/PhysRevB.109.014412}{Phys. Rev. B \textbf{109}, 014412 (2024)}; \href{https://journals.aps.org/prb/abstract/10.1103/PhysRevB.110.064313}{Phys. Rev. B \textbf{110}, 064313,(2024)}] recently introduced an $N$-spin Floquet model with infinite-range Ising interactions. There, we have shown that the model exhibits the signatures of quantum integrability for specific parameter values $J=1,1/2$ and $\tau=\pi/4$. We have found analytically the eigensystem and the time evolution of the unitary operator for finite values of $N$ up to $12$ qubits. We have calculated the reduced density matrix, its eigensystem, time-evolved linear entropy, and the time-evolved concurrence for the initial states $\ket{0,0}$ and $\ket{\pi/2,-\pi/2}$. For the general case $N>12$, we have provided sufficient numerical evidences for the signatures of quantum integrability, such as the degenerate spectrum, the exact periodic nature of entanglement dynamics, and the time-evolved unitary operator. In this paper, we have extended these calculations to arbitrary initial state $\ket{\theta_0,\phi_0}$, such that $\theta_0 \in [0,\pi]$ and $\phi_0 \in [-\pi,\pi]$. Along with that, we have analytically calculated the expression for the average linear entropy for arbitrary initial states. We numerically find that the average value of time-evolved concurrence for arbitrary initial states decreases with $N$, implying the multipartite nature of entanglement. We numerically show that the values $\langle S\rangle/S_{Max} \rightarrow 1$ for Ising strength ($J\neq1,1/2$), while for $J=1$ and $1/2$, it deviates from $1$ for arbitrary initial states even though the thermodynamic limit does not exist in our model. This deviation is shown to be a signature of integrability in earlier studies where the thermodynamic limit exist.