Four interacting spins: addition of angular momenta, spin-spin correlation functions, and entanglement

We study four spins on a ring coupled through competing Heisenberg interactions between nearest neighbors, $J$, and next-nearest neighbors, $J_2\equiv \alpha J>0$. The spectrum is obtained in a simple way by using the rules for addition of 4 angular momenta. This allows us to follow the evolution of the ground state with $\alpha$, characterized by level crossings and by analyses of spin-spin correlation functions. Further insight is obtained by examining the entanglement between different parts of the system: we observe that the entanglement entropy is strongly dependent on how the system is partitioned.