The thermal pairwise entanglement (TE) of the $S=1/2$ $XY$ chain in a transverse magnetic field is exactly resolved by means of the Jordan-Wigner transformation in the thermodynamic limit $N\to \infty$. It is found that the TE vanishes at a fixed point with temperature $T_c\simeq 0.4843J$, which is independent of the magnetic field. A thermal quantity is proposed to witness the entangled state. Furthermore, the TE of the $S=1/2$ antiferromagnetic-ferromagnetic (AF-F) Heisenberg chain is studied by the transfer-matrix renormalization-group method. The TEs of the spins coupled by AF and F interactions are found to behave distinctively. The vanishing temperature of the field-induced TE of the spins coupled by F interactions is observed dependent on the magnetic field. The results are further confirmed and analyzed within a mean-field framework.

• Received 16 April 2009

DOI:https://doi.org/10.1103/PhysRevA.80.012323