2N~ two-valued Ising spins are arranged sequentially on a circle, so that the last spin is a neighbor of the first. The Hamiltonian is
H=2j=1∑2N~Jjσjσj+1−Bj∑σj
where Jj=Je if j is even and Jj=J0 if j is odd. Assume J0>Je.
There are two options for carrying out a mean field theory for this system. The first option is to note that all spins are equivalent. Hence one can choose an unperturbed system of 2N~ single spins, each acted on by an effective field (to be evaluated variationally). The second option is to recognize that we can choose a pair of spins coupled by J0 (the larger exchange interaction). Each such pair is coupled to two other pairs by the weaker exchange interactions Je. The unperturbed system consists of N~ such pairs.
Carry out each of the mean field theories described above. Discuss the relative merits of these two procedures.