Order Parameter and Mean Values?

Question

I am confused about mean values and order parameters in specifically Ginzburg-Landau Theory. From what I have read$^1$ the order parameter is in general given by: $$\phi=\left<m\right>_\beta\tag{1}$$ where $m$ is some quantity which is not invariant under the spontaneous symmetry we are looking to break.

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Example: For the phase transition in Liquid-Crystals we often take $Q=\left<n_in_j-\delta_{ij}/3\right>_\beta$ as the order parameter.

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The partition function is then given by: $$Z=\int \mathcal{D} \phi e^{-\beta F[\phi]} \tag{2}$$ where $F[\phi]$ is the free energy.

However:

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(A) On One Hand

It is said that the order parameter $\phi$ goes to zero in the unordered phase and non-zero in the ordered phase. This indicates that the average in (1) is needed - else this statement would depend on the specific configuration which due to thermal fluctuations changes.

(B) On the Other Hand

Putting expression (1) into (2) makes little sense - you are effectively taking a thermal average twice.

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Thus my question is as follows: Am I missing something which means that (A) and (B) are actually compatible with one another. Or is it the case that in (2) we actually have an unthermal averaged version of the order parameter - if this is the case when we say 'order parameter' is it common to mean the averaged or unaveraged version.

_{$^1$ Source not publicly available - I will try and look for one.}

Answer 1

The OP is confusing two very different quantities, the fluctuating field $m$, which is integrated over (that is, we are averaging over all possible configuration of $m$), and the average of this field $\phi = \langle m\rangle$.

Obviously, these two quantities cannot be the same thing, and we can very well have $\phi=0$ when doing the average of all possible $m$.

For a given Hamiltonian $H$, the partition function is $$Z=\int \mathcal D m\, e^{-\beta H[m]},$$ and the order parameter is $$ \phi =\frac1Z\int \mathcal D m\, m\,e^{-\beta H[m]}. $$

The reason why the OP is confused is that at the mean-field level, people tend to say that $m$ is the order parameter. The reasoning is the following. Let's call $m_0$ the configuration of the field $m$ such that $\frac{\delta H}{\delta m}|_{m_0}=0$. Then a saddle-point approximation on $Z$ gives $$ Z_{MF}=e^{-\beta H[m_0]}, $$ and thus a Free energy $F_{MF}=-T\ln Z_{MF}=H[m_0]$. And we also have within this approximation $\phi_{MF}=m_0$.

Note that it is not the free energy that has been minimized to find $m_0$, but $H$, even though at this level of approximation they are closely related. In fact, $F$ never depends functionally on the order parameter, but depends on its conjugated field $h$ (a magnetic field for instance, so that $\beta H\to \beta H-h.m$).

If one want to talk about a function that depends on the order parameter $\phi$, it is better to introduce the Legendre transform of $F$ with respect to $h$, call it $G$, that is $$ G[\phi]=F[h]+h.\phi. $$ If we were to compute exactly $Z[h]$ (and thus $F[h]$), and compute exactly $G[\phi]$, then $G[\phi]$ would be the correct functional to minimize in order to find the equilibrium value of the order parameter.

At the mean-field level, however, one finds $G_{MF}[\phi]=H[m=\phi]$, which is obviously minimum for $\phi=\phi_{MF}=m_0$.

Unfortunately, in the literature, all these are often mixed together, especially if one discusses the mean-field physics. However, as shown here, all these quantities are conceptually very different.

Answer 2

Lets focus on Ising model for simplicity.

In (2), $\phi$ is in the sense: "Pick a microscopic configuration of spins, and calculate the average magnetization" and then the "effective free energy" $F$ encapsulates the information about the counting of how many microscopic configurations have the same average magnetization. Explicitly:

$Z=\sum _{\phi}\sum _{<s>=\phi }e^{-\beta E(s)}\equiv \sum _{\phi}e^{-\beta F(\phi )}$

Where s denotes a microscopic configuration. (so indeed, the $\phi$ in (2) is not thermal averaged, but averaged magnetization over a microscopic configuration)

Fluctuations are taken into account by calculating $<s>$ over a blocsk of some size instead of the whole lattice, and summing over configuration with some given value $\phi(x)$ for those averages. The partition function will then be given by a functional integral:

$Z=\int D\phi (x)e^{-\beta F(\phi (x))}$

One could reconstruct the "thermodynamic" magnetization m by doing a saddle point approximation to the partition function, and find a minimum of $F$, which will be the most probable magnetization. When the saddle point approx is valid and we can neglect fluctuations, one could say that the system has magnetization $m$.

I recommend David Tong's Lectures on Statistical Field Theory chapter 1 for a discussion on this issue. (available online)

Answer 3

Oder parameter is a mean value of something that something changes from system to system. Now the question is it is a mean value but respect to what?

This is an important question, order parameter is calculated by using mean field approximation, when you do a mean field approximation you dont take blindly a mean value of your something over all possible states. you choose specific states when you are taking the mean value that is why mean field approximation is rather an art than a straightforward method.

so let's give an example. the electron operator for in momentum representation is given as $$\rho (q)=\sum_{k}c{\dagger}_{k}c_{k+q}$$ if you take a thermal average of this you will find zero. so we would find that crystals does not exist.

but this is not true crystals do exist, there is something deeper is going on, and it is spontaneous symmetry breaking. when you take the thermal average you average over all possible locations that crystal exist. how ever this is not intuitively true. you would need an energy to move a Crystal from one position to another position so each configuration at different positions of the crystal has an energy barrier between them. and the crystal spontaneously chooses one of these positions. and you have to average over only those states.

this is in general a deep concept, and this answer is mostly based on the book of bruss section 4.4.