# Entanglement properties of the time periodic Kitaev Chain

## TL;DR

This paper investigates the entanglement properties of a periodically driven Kitaev chain, revealing how topological Majorana modes manifest in the entanglement spectrum and vary over a drive cycle, with distinctions between eigenstates and physical states.

## Contribution

It introduces a detailed analysis of entanglement spectra in Floquet topological phases, highlighting the time-dependent behavior of Majorana modes and their dependence on initial states and driving.

## Key findings

- Majorana zero and pi modes appear in the entanglement spectrum.
- The number of Majorana modes varies over the drive cycle, peaking at a symmetric point.
- Only zero modes persist in the physical state, while pi modes merge with bulk excitations.

## 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 $\pi$ ($\mathbb{Z}_{\pi}$) modes, where the zero modes are present even in the absence of the drive, while the $\pi$ 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 quasi-energy spectrum. The number of Majorana edge modes in the ES of the FGS vary in time from $|\mathbb{Z}_0-\mathbb{Z}_{\pi}|$ to $\mathbb{Z}_0+\mathbb{Z}_{\pi}$ within one drive cycle, with the maximal $\mathbb{Z}_0+\mathbb{Z}_{\pi}$ 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 wavefunction, namely the $\mathbb{Z}_0$ modes, appear in the ES. The $\mathbb{Z}_{\pi}$ 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 $\pi$ modes in the ES are also explained by mapping the parent wavefunction to a Bloch sphere.