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 ≡ α J > 0. From the pedagogical point of view, dealing with few spins illustrates how to add more than two angular momenta in a systematic way. The spectrum is obtained by using the rules for addition of four angular momenta, which allows us to follow the evolution of the ground state with α, characterized by level crossings and by spin–spin correlation functions. The reduced number of spins also allows us to illustrate how to quantify bipartite entanglement. Editor's note: Entanglement is often an elusive property for students. This paper, appropriate for advanced quantum mechanics class, shows on a practical example how to calculate entanglement. It considers four spins-1/2 that are evenly distributed on a ring and coupled to one another though competing nearest- and next-nearest-neighbor interactions. After determining the eigenstates and their energies, which is in itself a nice example of addition of more than two angular momenta, one can determine the entanglement of two subsystems. Spoiler: it depends on how you partition the system.