Entanglement transitions in a periodically driven non-Hermitian Ising chain

We study entanglement transitions in a periodically driven Ising chain in the presence of an imaginary transverse field $\gamma$ as a function of drive frequency $\omega_D$. In the high drive amplitude and frequency regime, we find a critical value $\gamma=\gamma_c$ below which the steady state half-chain entanglement entropy, $S_{L/2}$, scales with chain length $L$ as $S_{L/2} \sim \ln L$; in contrast, for $\gamma>\gamma_c$, it becomes independent of $L$. In the small $\gamma$ limit, we compute the coefficient, $\alpha$, of the $\ln L$ term analytically using a Floquet perturbation theory and trace its origin to the presence of Fisher-Hartwig jump singularities in the correlation function of the driven chain. We also study the frequency dependence of $\gamma_c$ and show that $\gamma_c \to 0$ at special drive frequencies; at these frequencies, which we analytically compute, $S_{L/2}$ remain independent of $L$ for all $\gamma$. This behavior can be traced to an approximate emergent symmetry of the Floquet Hamiltonian at these drive frequencies which we identify. Finally, we discus the behavior of the driven system at low and intermediate drive frequencies. Our analysis shows the presence of volume law behavior of the entanglement in this regime $S_{\ell} \sim \ell$ for small subsystem length $\ell \le \ell^{\ast}(\omega_D)$. We identify $\ell^{\ast}(\omega_D)$ and tie its existence to the effective long-range nature of the Floquet Hamiltonian of the driven chain for small subsystem size. We discuss the applicability of our results to other integrable non-hermitian models.