Thermal entanglement properties of small spin clusters

Abstract

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 J₁, J₂, 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 polyoxovanadate compound V12 are well explained by tetramers, with J₂=0, K=0, in which the spins interact via the isotropic Heisenberg exchange interaction Hamiltonian. Treating the magnetic susceptibility ? 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 (J₁=J₂,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.