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CAF123
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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!