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The Landau theory makes a mean-field approximation on the order parameter, which assumes that there are no fluctuations in the value of the order parameter at different sites (neglects the effects of fluctuations).

Then, near the critical point the order parameter is very small, we expand the free energy in powers of the order parameter. I have a question at this moment.

Landau theory makes a mean field approximation near the critical point. However, at the critical point fluctuation may not be small. For example, in the Ising model, near the critical point there are large fluctuations of the magnetic moment between positive and negative values. So making mean field approximation (neglecting fluctuation or correlation) near critical point seems like a contradiction to me.

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So, critical phenomena can't be addressed via naive mean field. BUT, it is the first simple thing you can do to analyze stuff. Plus, if all you want just for the start is a qualitative description of the phase diagram, MFT is the way to go.

Landau's procedure can give a very good qualitative sense of what goes on near criticality. Actually, if you are well above the critical dimension https://en.m.wikipedia.org/wiki/Critical_dimension, MFT gives the right physics (universally speaking)

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As it is well known, the equilibrium state of a physical system can be obtained from the extremisation of thermodynamic potential. In the context of phase transitions, the corresponding thermodynamic potential is the (Gibbs) free energy $G(m)$, where I assume for simplicity that the order parameter is constant in space, and is thus just a number $m$.

The potential $G(m)$ is obtained from the (Helmholtz) free energy $F(h)$ by a Legendre transformation. Here $h$ is the field conjugated to $\phi$, the degrees of freedom the average of which give the order parameter $\langle \phi\rangle =m$. In practice, $F(h)$ is obtained from the partition function (up to $k_BT$ factors) $$ e^{-F(h)}=\int \mathcal D \phi\, e^{-H_{\rm eff}[\phi]+h\int_x \phi(x)}, $$ with $H_{\rm eff}[\phi]$ a (possibly effective) Hamiltonian. Here $\phi$ could be a continuous (vector) field, a quantum field, etc. depending on the context. All these details are hidden in the measure, what $x$ represents and Hamiltonian.

Assuming we have $F(h)$, then $m(h)=\frac{\partial F}{\partial h}$. If we are interested in the $h=0$ case, the physical value of the order parameter is given by $m_0=m(h=0)$, which depends on all the other parameters of the Hamiltonian (which will depend on the temperature, interactions of the degrees of freedom, and so on). Note that $F$ is concave.

The thermodynamic potential $G(m)$ is obtained as $$ G(m)=\sup_h (F(h)+m \,h), $$ which gives a convex function. Now, the physical value of the order parameter is found by minimizing $G(m)$. If $m_0\neq0$, this implies that $G$ is \emph{not} a double well! More on that later.

Right away, we can dissipate one of the misconceptions of the OP. For a second-order phase transition, which happens at small $m$, we only care about the behavior of $G(m)$ at small $m$. Assuming it is analytical, we can expand the \emph{true} $G(m)$, with all fluctuations included, at small $m$. This does make sense. However, in most relevant cases (dimension three and lower, or in the ordered phase), one cannot expand $G(m)$, although this picture is useful.

Of course, the main problem is that it is not possible in general to compute neither $F$ nor $G$. One thus as to do approximations. The most common one is to do a saddle-point (or mean-field) approximation of $F(h)$. Then one directly gets that $G_{MF}(m)=H_{\rm eff}(m)$. (I ignore here gradient terms in the Hamiltonian, since I assumed that $m$ is constant. However, the discussion would not change much if we kept them.)

This construction recovers Landau theory of phase transitions. We recover all the usual physical results from it. As discussed above, for second-order phase transitions, since the transition happens at small $m$, we can expand $G(m)$ around $m=0$ to recover the usual business. But of course, it has some drawbacks: it is not convex if $H_{\rm eff}$ is not --which is usually the case when there is a phase transition-- although this is rarely discussed; it ignores fluctuations, which can dramatically change the picture when included (change of critical exponent, or even the order of the transition, from second-order at mean-field to first-order).

Let me now discuss these two points. I will start by discussing fluctuations at and above $T_c$ (to discuss the ordered phase, we need to take the convexity into consideration in any case). There are two possibilities. Above the upper critical dimension (usually $d=4$), the fluctuations do not change the picture much: while $G(m)$ is different from $G_{MF}(m)$, it is so in minor ways. It is still analytic, only the coefficients are different (meaning for instance that the true critical temperature is not the mean-field one). In particular, the critical exponents are the mean-field ones. Below the upper critical dimensions, things change quite a lot. First, the transition can disappear (below the lower critical dimension). Second, the behavior of the coefficients of the expansion of $G(m)$ will change with different exponents than the mean-field ones.

We can finally address the behavior of $G(m)$ below $T_c$. Because $G$ is convex, there is no double-well behavior. In fact $G(m)$ is defined only for $|m|\geq |m_0|$ (in the thermodynamic limit), and it is thus non-analytic close to $m_0$. (This is true for all dimensions above the lower critical dimension.) As stated above, while Landau theory is unphysical in that case, the picture is nevertheless useful to understand what is going on qualitatively.

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  • $\begingroup$ It is a very nice answer - it certainly deserves an upvote. The only problem might be that it does not answer the OP's question on their level of understanding. But it is up to them to decide. $\endgroup$ Oct 13, 2020 at 15:05
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Yes you are correct the Landau theory ignores any effect due to the fluctuations. But the theory is phenomenological theory used to explain the essence of second order phase transitions. That is the order parameter adopts a nonzero value below a critical temperature as this is favourable due to free energy minimisation conditions. To address the effects due to fluctuation at or near the critical temperature one has to look at a slightly modified version.

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Landau theory breaks down near the critical point - precisely for the reason that you described!

Not also that Landau free energy is a theoretical construction rather than an expansion of the actual free energy that one obtains from the partition function.

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    $\begingroup$ Both sentences are misleading... Landau theory breaks down only for dimensions $d\geq 4$ (marginally in $d=4$). It is valid up to changes of the numerical values of its parameters. Furthermore, it can be constructed explicitly from microscopic models in some cases (for instance, classical spins models, BCS superconductivity, some models of quantum phase transitions, etc.) $\endgroup$
    – Adam
    Oct 12, 2020 at 19:17
  • $\begingroup$ Adam, should not your inequality be inverted? (higher than 4 is safe territory for Landau) $\endgroup$
    – daydreamer
    Oct 12, 2020 at 19:27
  • $\begingroup$ @Adam I don't agree, but if you're sure about this, write your answer. $\endgroup$ Oct 12, 2020 at 19:50
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    $\begingroup$ @daydreamer yes, of course... Vadim: My answer would be pretty much similar to that of daydreamer. But I could expand on the derivation of Landau theory from microscopic models $\endgroup$
    – Adam
    Oct 12, 2020 at 21:49
  • $\begingroup$ Vadim, I think he thought it misleading because your first sentence is dimension-specific: valid only under critical dimension. As of the second affirmation, it is not strictly true: I get that you are ellucidating its aspect as a phenomenological approach, but we could also obtain it from free energy expansion. Why not? $\endgroup$
    – daydreamer
    Oct 12, 2020 at 22:06

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