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

TL;DR

This paper analytically and numerically investigates the entanglement dynamics of an exactly solvable infinite-range Floquet spin model for arbitrary initial states, revealing signatures of quantum integrability and multipartite entanglement behavior.

Contribution

It extends previous work by deriving analytical expressions for entanglement measures for any initial state and provides numerical evidence of integrability signatures beyond small system sizes.

Findings

Exact analytical expressions for entanglement dynamics for arbitrary initial states.

Average linear entropy approaches maximum value as system size increases.

Deviations in concurrence indicate integrability for specific parameter values.

Abstract

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.