In statistical field theory, one usually considers the so-called Landau Hamiltonian: $$\beta H = \int d^{d}x\bigg{[}\frac{t}{2}m^{2}(x) + \alpha m^{4}(x)+\frac{\beta}{2}(\nabla m)^{2}+\cdots+ \vec{h}\cdot \vec{m}(x)\bigg{]}$$ This Hamiltonian seems to be general enough to study Landau's theory on phase transitions. The process of constructing the above Hamiltonian is to coarse-grain the spin system and consider the order parameter (in my case, the magnetization) as a field. I'd like to better understand the motivations of such procedure: why to turn spin systems into fields? Does it improve our results? Does our models become more realistic this way? What kind of objects are we aiming to study with this field theory? And so on.


If we were to consider spins individually, then what we would have is an Ising model; the problem with this is that to get analytic solutions of Ising at 2D or above is really difficult. The benefit of a Landau formulation is that we can just work directly from the symmetries of the system to get a Hamiltonian, which we can then use; for example, this theory assumes a symmetry in spin all up or all down, which is why we exclude the odd powers. This formulation also has the benefit that corrections can quite easily be added by considering higher powers or excluding certain powers. So the overall reason we convert to mean field formulations is really a simplification to make the problem solvable.

The general idea of field theory is that we create a framework to study things, rather than a model for a specific system; for example, Landau theory was originally created for continuous first order transitions. This means any systems that exhibit such a transition can be studied with this theory; the obvious application may be to magnetic materials, but as someone who works in soft matter I have used it in nematic liquid crystal systems and bacterial swarms as well.

This does not mean the model is more accurate than, say, Ising, in any sense; in fact, due to the assumption that we can expand the free energy as a power series, we make the implicit assumption that the free energy is analytic, which may not be the case at the transition point. So it is important to note that field theory may not always work, but generally they can give a good idea as to, say, when a phase transition may occur.

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