Landau free energy form for antiferromagnetic Ising model

[CAF123]

Homework Statement

The total energy of the Ising model is $E = −J \sum_{<ij>} S_iS_j$, where $S_i = \pm 1$ and $< ij >$ implies sum over nearest neighbours. For $J < 0$ explain why the ground state of this system at $T = 0$ is antiferromagnetic.

Let $m_{1,2}$ be the magnetisations in the two sublattices of an antiferromagnetic Ising model. Define the order parameter $\psi ≡ m_1−m_2$. Argue why the Landau free energy for this system should have the form, $$F = at\psi^2 + b\psi^4 + ...,$$ where $t \equiv (T − T_c)/T_c$ and $a, b > 0$ are constants.

Homework Equations

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F some polynomial expansion in order parameter $\psi$.

The Attempt at a Solution

So as far as I understand from the Landau theory, we construct a function F for the free energy based solely on the symmetries of the system, particularly that obeyed by the order parameter. $m_i \rightarrow -m_i$ can be viewed as a rotation of the system by $\pi$ so F should be invariant under $m_i \rightarrow -m_i$ which is to say $\psi \rightarrow -\psi$ is a symmetry. So we construct $F = c + a(T)\psi^2 + b(T)\psi^4 + ...$, where c is a constant, can be set to 0. I am just a bit confused as to how we solve for a(T) and b(T). Imposing that at equilibrium, $\partial F/\partial \psi = 0$ then this means $$2\psi(a(T) + 2b(T) \psi^2) = 0$$ ie $\psi = 0$ or $\psi^2 = -a(T)/2b(T)$. But how should I progress? The case $\psi = 0$ corresponds to the case when $m_1 = m_2$ so that this could be realized in the ground state. Thanks!

 

[CAF123]

Ok I guess I can work with what is given and try to justify it. At the critical temperature (or temperature at which we reach criticality) the first term vanishes which implies the equilibrium situation is one in which we have $\psi=0.$ Similarly for the case $T>T_c$. (We must choose $\psi=0$ otherwise we get an imaginary solution for $\psi$ which is unphysical.) For $T<T_c$ we get two minima, with $\psi=0$ a maximum. Is this correct understanding? It seems that this is the same analysis for the case $J>0$.