Entanglement in SU(2)-invariant quantum spin systems

We analyze the entanglement of SU(2)-invariant density matrices of two spins $S_1,$ $S_2$ using the Peres-Horodecki criterion. Such density matrices arise from thermal equilibrium states of isotropic-spin systems. The partial transpose of such a state has the same multiplet structure and degeneracies as the original matrix with the eigenvalue of largest multiplicity being non-negative. The case $S_1=S,$ $S_2=1/2\mathrm{}$ can be solved completely and is discussed in detail with respect to isotropic Heisenberg spin models. Moreover, in this case the Peres-Horodecki criterion turns out to be a sufficient condition for nonseparability. We also characterize SU(2)-invariant states of two spins of length 1.