Entanglement properties of the time-periodic Kitaev chain

Abstract

The entanglement properties of the time-periodic Kitaev chain with nearest-neighbor and next-nearest-neighbor hopping is studied. The cases of the exact eigenstate of the time-periodic Hamiltonian, referred to as the Floquet ground state (FGS), as well as a physical state obtained from time evolving an initial state unitarily under the influence of the time-periodic drive are explored. Topological phases are characterized by different numbers of Majorana zero (${\mathbb{Z}}_{0}$) and $\ensuremath{\pi}$ (${\mathbb{Z}}_{\ensuremath{\pi}}$) modes, where the zero modes are present even in the absence of the drive, while the $\ensuremath{\pi}$ modes arise due to resonant driving. The entanglement spectrum (ES) of the FGS as well as the physical state show topological Majorana modes whose number is different from that of the quasienergy spectrum. The number of Majorana edge modes in the ES of the FGS vary in time from $|{\mathbb{Z}}_{0}\ensuremath{-}{\mathbb{Z}}_{\ensuremath{\pi}}|$ to ${\mathbb{Z}}_{0}+{\mathbb{Z}}_{\ensuremath{\pi}}$ within one drive cycle, with the maximal ${\mathbb{Z}}_{0}+{\mathbb{Z}}_{\ensuremath{\pi}}$ modes appearing at a special time-reversal-symmetric point of the cycle. For the physical state, on the other hand, only the modes inherited from the initial wave function, namely the ${\mathbb{Z}}_{0}$ modes, appear in the ES. The ${\mathbb{Z}}_{\ensuremath{\pi}}$ modes are absent in the physical state as they merge with the bulk excitations that are simultaneously created due to resonant driving. The topological properties of the Majorana zero and $\ensuremath{\pi}$ modes in the ES are also explained by mapping the parent wave function to a Bloch sphere.