3.2. Mean field theory for coupled lattices Consider a system of Ising spins made of two-dimensional square lattices A and B, coupled together. Let Ja and Jb be the coupling constants between next-neighbour spins in the lattices A and B respectively, and J the coupling constant between the nextneighbour spins of the two lattices (Figure 3.13).

a. Generalize the mean field theory to this system and show that the spontaneous magnetizations of the two lattices satisfy the coupled system of equations Ma = f (JaMa +JMb), Mb = f (JbMb +JMa)

Determine the explicit form of the function f .

b. Estimate the critical temperature Tc in the mean field approximation.

c. Show that, for T > Tc, the magnetic susceptibility is expressed by the ratio of linear and quadratic polynomials in T.

d. Discuss the case Ja = Jb > 0 while J < 0 and study the magnetic susceptibility for these values of the couplings.

e. Discuss the limits J→±∞.