Mean field theory for Ising model and Weiss approximation.

For this exercise we use the Weiss Molecular Field approximation for the Ising model in 3 dimensions. Here the interaction between neighbouring spins is replaced by an interaction of the averaged field over all the spins: the Weiss Molecular Field, which we name m.

The Hamiltonian is given by H = −Jzm ∑si - B ∑si, i = 1,,...,N, si = ±1

where z is the number of nearest neighbors, e.g for z = 6 we would have a cube lattice. In addition we have the self-consistency equation <si > = m

a) Calculate the canonical partition function from the Hamiltonian. Derive <si> from it. Prove that the corresponding self-consistency equation for m is given by
m = tanh (βB + βJzm)

b) Let B = 0. Using a graph, determine the solution for m in dependence on zβJ. Discuss the stability of the solution(s). Argue that a phase transition between one phase with and one phase without spontaneous magnetization takes place and determine its critical temperature Tc as a function of J.

c) Let B = 0. If the temperature is near the critical point then there is a small magnetization. Develop the self-consistency equation as a Taylor Series in m to the third order term and derive from this an expression for m(T,J)

d) The susceptibility is defined by χT = N (∂m/∂B)T. Calculate the susceptibility for B = 0. For temperatures slightly under the critical temperature (T < Tc),
we have χT is proportional to (T − Tc)^(−γ), where γ is a critical exponent. Determine γ for the three-dimensional Ising model in the molecular field approximation.

e) Explain why the Molecular Field Theory cannot be applied in 1 dimension

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