Entanglement in (1/2, S) mixed-spin antiferromagnetic XY dimer and trimer systems

In this manuscript, the entanglement in antiferromagnetic $(1/2,S)$ mixed-spin XX and XY dimer and trimer systems are investigated by using the concept of negativity. For the $(1/2,S)$ XX dimer models, when the external magnetic field and temperature are zero, the negativity values of the half-integer $S$ systems are the same, and that of the integer $S$ systems are the same. When the system is in the magnetic field, we obtain the threshold magnetic field above which the ground-state entanglement vanishes. For the (1/2,2) mixed-spin XY dimer system, when the temperature approaches zero, the values of thermal entanglement can correspond well to that of mixed-state one. For the (1/2,3/2) mixed-spin trimer system, the critical temperature is independent of the external magnetic field, but it is higher in the ferromagnetic case than in the antiferromagnetic case. In an extremely low temperature, entanglement forms a stable plateau within a certain magnetic field range, but when the magnetic field exceeds a critical value, it vanishes completely without the uniaxial single-ion anisotropy. Particularly, because the trimer system exhibits a certain frustration in the antiferromagnetic case, whether there is a magnetic field or not, entanglement between spins 1/2 is almost zero at any temperature.