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 $\frac{1}{2}$, 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 $S=\frac{1}{2}$ antiferromagnetic polyoxovanadate 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.

• Received 10 March 2005

DOI:https://doi.org/10.1103/PhysRevA.72.022314