# Introduction: Ginzburg criterion

As we have seen, the main assumption (and the most important problem) of mean field theories is that the fluctuations of the order parameter are completely neglected in the computation of the partition function; this approximation breaks down in the neighbourhoods of critical points, where as we have seen in Long range correlations the correlation length becomes comparable with the size of the system. What we would now like to do is to include these fluctuations in a mean field theoretical framework; this will lead to the so called Ginzburg-Landau theory.

As a first approach we can try to estimate how big is the error we make in mean field theories neglecting the fluctuations of the order parameter near a critical point, so that we can understand under which conditions mean field theories are actually a good approximations. To make things explicit, let us use the Ising model as a base for our considerations. We have seen in Weiss mean field theory for the Ising model that the Weiss mean field theory for the Ising model is based on the assumption that ${\displaystyle \left\langle S_{i}S_{j}\right\rangle =\left\langle S_{i}\right\rangle \left\langle S_{j}\right\rangle }$, i.e. that the spins are statistically independent; therefore, a possible estimate of the error made with this assumption can be:

${\displaystyle E_{ij}={\frac {|\left\langle S_{i}S_{j}\right\rangle -\left\langle S_{i}\right\rangle \left\langle S_{j}\right\rangle |}{\left\langle S_{i}\right\rangle \left\langle S_{j}\right\rangle }}}$
The numerator of ${\displaystyle E_{ij}}$ is, by definition (see also Long range correlations), the two-point connected correlation function:
${\displaystyle G_{c}(i,j)=\left\langle S_{i}S_{j}\right\rangle -\left\langle S_{i}\right\rangle \left\langle S_{j}\right\rangle }$
If we neglect the fluctuations of the order parameter (which in our case is of course ${\displaystyle \left\langle S_{i}\right\rangle }$) we see that ${\displaystyle G_{c}}$ is constantly zero: therefore, in order to have non-null correlation functions we need that the system exhibits some kind of inhomogeneity, not necessarily due to thermal fluctuations. In fact, the connected correlation function describes not only the spatial extension of the fluctuations of the order parameter, but also the way it varies in space in response to an external inhomogeneous field. Let us see this explicitly. We know that from the partition function of the Ising model in an inhomogeneous external field ${\textstyle {\vec {H}}}$, i.e.:
${\displaystyle Z=\operatorname {Tr} e^{\beta \left(J\sum _{\left\langle ij\right\rangle }S_{i}S_{j}+\sum _{i}H_{i}S_{i}\right)}}$
we have:
${\displaystyle \left\langle S_{i}\right\rangle ={\frac {1}{\beta Z}}{\frac {\partial Z}{\partial H_{i}}}={\frac {1}{\beta }}{\frac {\partial \ln Z}{\partial H_{i}}}}$
Similarly:
${\displaystyle \left\langle S_{i}S_{j}\right\rangle ={\frac {1}{\beta ^{2}Z}}{\frac {\partial ^{2}Z}{\partial H_{i}\partial H_{j}}}}$
and thus:
${\displaystyle G_{c}(i,j)={\frac {1}{\beta ^{2}Z}}{\frac {\partial ^{2}Z}{\partial H_{i}\partial H_{j}}}-{\frac {1}{\beta ^{2}Z^{2}}}{\frac {\partial Z}{\partial H_{i}}}{\frac {\partial Z}{\partial H_{j}}}={\frac {1}{\beta ^{2}}}{\frac {\partial ^{2}\ln Z}{\partial H_{i}\partial H_{j}}}}$
Therefore:
${\displaystyle {\frac {\partial \left\langle S_{i}\right\rangle }{\partial H_{j}}}={\frac {\partial }{\partial H_{j}}}\left({\frac {1}{\beta }}{\frac {\partial \ln Z}{\partial H_{i}}}\right)={\frac {1}{\beta }}{\frac {\partial ^{2}\ln Z}{\partial H_{i}\partial H_{j}}}=\beta G_{c}(i,j)}$
so ${\displaystyle G_{c}(i,j)}$ can indeed be seen as a response function. If we now call:
${\displaystyle M=\sum _{i}\left\langle S_{i}\right\rangle }$
we will have:
${\displaystyle {\frac {\partial M}{\partial H_{j}}}=\sum _{i}{\frac {\partial \left\langle S_{i}\right\rangle }{\partial H_{j}}}=\beta \sum _{i}G_{c}(i,j)}$
If our system is invariant under translations and subject to a uniform field, then:
${\displaystyle {\frac {\partial M}{\partial H}}=\sum _{j}{\frac {\partial M}{\partial H_{j}}}{\frac {\partial H_{j}}{\partial H}}=\beta \sum _{i,j}G_{c}(i,j)}$
and since ${\displaystyle \chi _{T}=\partial M/\partial H}$ we get:
${\displaystyle \chi _{T}=\beta \sum _{i,j}G_{c}(i,j)=\beta \sum _{i,j}\left(\left\langle S_{i}S_{j}\right\rangle -\left\langle S_{i}\right\rangle \left\langle S_{j}\right\rangle \right)}$
which is a version of the fluctuation-dissipation theorem.

Let us now try to understand when the error ${\displaystyle E_{ij}}$ done in mean field theories is negligible. Now, in general terms if we formulate a mean field theory for a system we will make the error ${\displaystyle E_{ij}}$ in the region where correlations are relevant, namely if ${\displaystyle |{\vec {r}}|}$ is the distance between two points of the system the error is made for ${\displaystyle |{\vec {r}}|\leq \xi }$, with ${\displaystyle \xi }$ the correlation length. Supposing ${\displaystyle T, so that the order parameter ${\displaystyle \eta }$ is non null, then:

${\displaystyle E_{\text{TOT}}={\frac {\int _{|{\vec {r}}|\leq \xi }G_{c}({\vec {r}})d^{d}{\vec {r}}}{\int _{|{\vec {r}}|\leq \xi }\eta ^{2}({\vec {r}})d^{d}{\vec {r}}}}}$
where we have called ${\displaystyle d}$ the dimensionality of our system. Therefore, our mean field theory will be a good approximation if ${\displaystyle E_{\text{TOT}}\ll 1}$, i.e.:
${\displaystyle {\frac {\int _{|{\vec {r}}|\leq \xi }G_{c}({\vec {r}})d^{d}{\vec {r}}}{\int _{|{\vec {r}}|\leq \xi }\eta ^{2}({\vec {r}})d^{d}{\vec {r}}}}\ll 1}$
known as Ginzburg criterion. In order to express it in a useful fashion, let us write it in terms of critical exponents; using also the version we have just found of the fluctuation-dissipation theorem we get (supposing our system is continuous)[1]:
${\displaystyle \int _{|{\vec {r}}|\leq \xi }G_{c}({\vec {r}})d^{d}{\vec {r}}\sim k_{B}T_{c}\chi _{T}\sim t^{-\gamma }\quad \qquad \int _{|{\vec {r}}|\leq \xi }\eta ^{2}({\vec {r}})d^{d}{\vec {r}}\sim \xi ^{d}|t|^{2\beta }\sim t^{2\beta -\nu d}}$
where we have used the definitions of critical exponents (see Critical exponents and universality). Therefore, the Ginzburg criterion can be reformulated as:
${\displaystyle E_{\text{TOT}}\sim t^{-\gamma +\nu d-2\beta }\ll 1}$
and in the limit ${\displaystyle t\to 0}$ this is possible only if ${\displaystyle -\gamma +\nu d-2\beta >0}$, i.e.:
${\displaystyle d>{\frac {2\beta +\gamma }{\nu }}:=d_{c}}$
This means that Ginzburg criterion allows us to determine the upper critical dimension${\displaystyle d_{c}}$ of a system, namely the dimension above which mean field theories are good approximations[2]; if ${\displaystyle d fluctuations become too relevant and mean field theories don't work. Let us note that since it depends on the critical exponents, the upper critical dimension ${\displaystyle d_{c}}$ ultimately depends on the universality class of the system considered; furthermore, in order to actually be able to compute ${\displaystyle d_{c}}$ we must generalize Landau theory to systems with spatial inhomogeneities so that we are able to compute the critical exponent ${\displaystyle \nu }$.

1. For the origin of the first equation, see what we have stated above and also in Long range correlations.
2. This also mean that above the upper critical dimension the critical exponents determined with mean field theories are exact, or at least in good agreement with experiments.