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 $T_c$, but external fields can make $T_c$ higher. We obtain a general statement on the increase of the critical temperature $T_c$ by the external fields. We prove that if the two end spins are separated by two spins or more, the critical temperature cannot be higher than a certain finite temperature $bar{T}_c$ ($T_c le bar{T}_c$), that is, the entanglement must vanish above the temperature $bar{T}_c$ for any values of the external fields. On the other hand, if the two end spins are separated by one spin, 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 $bar{T}_c$ in $XX$ and XY four-spin chains. We find that the temperature $bar{T}_c$ shows qualitatively different behavior, depending on the conservation of the angular momentum in the $z$ direction.