Role of external fields in enhancing long-distance entanglement at finite temperatures

We investigate the end-to-end entanglement of a general XYZ-spin chain at the non-zero temperatures. The entanglement usually vanishes at a certain critical temperature Tc, but external fields can make Tc higher. We obtain a general statement on the increase of the critical temperature Tc by the external fields. We prove that if the two end spins are separated by two spins or more, then the critical temperature cannot be higher than a certain finite temperature T-bar c (Tc≤ T-bar c), that is, the entanglement must vanish above the temperature T-bar c for any values of the external fields. On the other hand, if the two end spins are separated by one spin, then the entanglement maximized by the external fields exhibits a power-law decay of the temperature, being finite at any temperatures. In order to demonstrate the former case, we numerically calculate the temperature T-bar c in XX and XY four-spin chains. We find that the temperature T-bar c shows qualitatively different behavior, depending on the conservation of the angular momentum in the z direction. (paper)