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CAF123
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Consider an Ising model system where the total energy is ##E = −J \sum_{<ij>} S_iS_j ##, ##S_i = \pm 1## and ##< ij >## implies sum over nearest neighbours. For ##J < 0## the ground state of this system at ##T = 0## is antiferromagnetic. (All adjacent spins misaligned so net magnetisation zero and thus 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##. 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. Why is this the case?
I know that ##F## is some polynomial expansion in order parameter ##\psi## and as far as I understand from the Landau theory, it is constructed 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 beinvariant 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 they obtained the expressions 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)##. The case ##\psi = 0## corresponds to the case when ##m_1 = m_2## so that this could be realized in the ground state. I just would like the argument as to why we infer the dependence of ##A## on ##T## and that ##B## is a constant.
Or by using the given expression, 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<Tc we get two minima. Is this correct understanding? Thanks :)
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##. 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. Why is this the case?
I know that ##F## is some polynomial expansion in order parameter ##\psi## and as far as I understand from the Landau theory, it is constructed 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 beinvariant 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 they obtained the expressions 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)##. The case ##\psi = 0## corresponds to the case when ##m_1 = m_2## so that this could be realized in the ground state. I just would like the argument as to why we infer the dependence of ##A## on ##T## and that ##B## is a constant.
Or by using the given expression, 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<Tc we get two minima. Is this correct understanding? Thanks :)