Thermal entanglement properties of small spin clusters

Exchange interactions in spin systems can give rise to quantum entanglement in the ground and thermal states of the systems. We consider a spin tetramer, with spins of magnitude 1 2, in which the spins interact via nearest-neighbor, diagonal, and four-spin interactions of strength J(1), J(2), and K, respectively. The ground- and thermal-state entanglement properties of the tetramer are calculated analytically in the various limiting cases. Both bipartite and multipartite entanglements are considered and a signature of the quantum phase transition (QPT), in terms of the entanglement ratio, is identified. The first-order QPT is accompanied by discontinuities in the nearest-neighbor and diagonal concurrences. The magnetic properties of a S=1/2 antiferromagnetic poly-oxovanadate compound V12 are well explained by tetramers, with J(2)=0, K=0, in which the spins interact via the isotropic Heisenberg exchange interaction Hamiltonian. Treating the magnetic susceptibility chi as an entanglement witness (EW), an estimate of the lower bound of the critical entanglement temperature T-c below which entanglement is present in the experimental compound, is determined. Two other cases considered include the symmetric tetramer - i. e., tetrahedron (J(1)= J(2), K=0)-and the symmetric trimer. In both the cases, there is no entanglement between a pair of spins in the thermal state but multipartite entanglement is present. A second EW based on energy provides an estimate of the entanglement temperature T-E below which the thermal state is definitely entangled. This EW detects bipartite entanglement in the case of the tetramer describing a square of spins (the case of V12) and multipartite entanglement in the cases of the tetrahedron and symmetric trimer.