Entanglement witnesses in the XY chain: Thermal equilibrium and postquench nonequilibrium states

TL;DR

This paper investigates entanglement detection in the XY spin chain using energy-based and negativity-based witnesses, analyzing thermal and postquench states, and exploring effects of phase transitions and finite-size corrections.

Contribution

It introduces and compares energy and negativity-based entanglement witnesses in the XY chain, including analytical determination of entangled regions after quenches.

Findings

Energy-based witnesses detect entanglement below a temperature bound.

Negativity-based witnesses identify entanglement in broader parameter regions.

Postquench states near phase boundaries are often detected as entangled.

Abstract

We use entanglement witnesses to detect entanglement in the XY chain in thermal equilibrium and determine the temperature bound below which the state is detected as entangled. We consider the entanglement witness based on the Hamiltonian. Such a witness detects a state as entangled if its energy is smaller than the energy of separable states. We also consider a family of entanglement witnesses related to the entanglement negativity of the state. We test the witnesses in infinite and finite systems. We study how the temperature bounds obtained are influenced by a quantum phase-transition or a disorder line in the ground state. Very strong finite-size corrections are observed in the ordered phase due to the presence of a quasi-degenerate excitation. We also study the postquench states in the thermodynamic limit after a quench when the parameters of the Hamiltonian are changed suddenly. In the case of the Ising model, we find that the mixed postquench state is detected as entangled by the two methods if the parameters of the Hamiltonian before and after the quench are close to each other. We find that the two witnesses give qualitatively similar results, showing that energy-based entanglement witnesses are efficient in detecting the nearest-neighbor entanglement in spin chains in various circumstances. For other XY models, we find that the negativity based witnesses also detect states in some parameter regions where the energy-based witness does not, in particular, if the quench is performed from the paramagnetic phase to the ferromagnetic phase and vice versa. The domains in parameter space corresponding to postquench states detected as entangled by the energy-based witness have been determined analytically, which stresses further the utility of our method.