Ground state of the Heisenberg ferromagnet

As a first step in the study of the Heisenberg ferromagnet, we express the dot product of two spin operators, with the help of the spin raising 5+ and lowering 5 operators (for details see Appendix B), as 

When this result is substituted into the Heisenberg spin model, Eq. (7.21), taking into account that raising and lowering operators at different lattice sites commute with each other because they operate on different spins, we find 

There are two configurations that can be constructed with one 5, value lowered by one unit, 

Since the hamiltonian contains a sum over all values of R, R, both 5 (R)5+(R ) and S-(R )S+(R) will appear in it. When applied to the two configurations defined 

We also notice that these states are eigenfunctions of the product of operators 52(R)52(R ) or 52(R )5j(R), with eigenvalue 5(5 — 1). Ignoring all other spins which are fixed to 5j = 5, we can then apply the hamiltonian to the states 5 + ), 5 ), to find that they are both eigenstates with energies  

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