Magnetic Entanglement in Spin-1/2 XX Chains

TL;DR

This paper analytically and numerically investigates magnetic entanglement between farther neighbor spins in a 1D spin-1/2 XY chain, revealing how entanglement propagates with temperature and magnetic field.

Contribution

It provides the first analytical results for entanglement between second, third, and fourth neighbor spins in the XY model, extending understanding beyond nearest neighbors.

Findings

Entanglement exists between farther neighbors above a critical magnetic field.

Entanglement persists up to a critical temperature, which depends on the magnetic field.

Entanglement spreads stepwise to farther neighbors as temperature decreases.

Abstract

In the study of entanglement in a spin chain, people often consider the nearest-neighbor spins. The motivation is the prevailing role of the short range interactions in creating quantum correlation between the 1st neighbor (1N) spins. Here, we address the same question between farther neighbor spins. We consider the one-dimensional (1D) spin-1/2 XY model in a magnetic field. Using the fermionization approach, we diagonalize the Hamiltonian of the system. Then, we provide the analytical results for entanglement between the 2nd, 3rd and 4th neighbor (denoted as 2N, 3N, and 4N respectively) spins. We find a magnetic entanglement that starts from a critical entangled-field ($h_c^{E}$) at zero temperature. The critical entangled-field depends on the distance between the spins. In addition to the analytical results, the mentioned phenomenon is confirmed by the numerical Lanczos calculations. By adding the temperature to the model, the magnetic entanglement remains stable up to a critical temperature, $T_c$. Our results show that entanglement spreads step by step to farther neighbors in the spin chain by reducing temperature. At first, the 1N spins are entangled and then further neighbors will be entangled respectively. $T_c$ depends on the value of the magnetic field and will be maximized at the quantum critical field.