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).(adsbygoogle = window.adsbygoogle || []).push({});

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 realised 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 :)

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Construction of free energy using Landau theory

Loading...

Similar Threads for Construction free energy |
---|

A Particle in a 2d "corridor" |

I What does this equation for a free particle mean? |

A Can disjoint states be relevant for the same quantum system? |

I Atom construction |

A Construction of wavepackets |

**Physics Forums | Science Articles, Homework Help, Discussion**