# Coarse graining procedure for the Ising model

To make things more clear, let us see how the coarse graining procedure works for the Ising model. If we call the local magnetization at the -th site and the dimensionality of the system, every "block" will have volume ; we define for every block of the system centered in the coarse grained magnetization as:

We now must express the partition function in terms of , and as we have stated before:

- bulk component
- Suppose that every block of volume is separate from the rest of the system; inside every one of them the magnetization is uniform (since the linear dimension of the blocks is much smaller than the correlation length), so we can use Landau theory for uniform systems. In the case of the Ising model, it led to:

- interaction component
- We now must take into account the fact that adjacent blocks do interact. In particular since as we have stated does not vary much on microscopic scales, the interaction between the blocks must be such that strong variations of magnetization between neighbouring blocks is energetically unfavourable. If we call a vector of magnitude that points from one block to a neighbouring one, the most simple analytic expression that we can guess for such a term can be a harmonic one
^{[1]}:

*real*interaction energy.

Now, since the linear dimension of the blocks is much smaller than the characteristic length of the system we can treat as a continuous variable and thus substitute the sum over with an integral:

*finite*number of nearest neighbours). Therefore:

Keeping in mind that , the interaction term can be rewritten in terms of :

- If the energy of the system has the same structure of the one used in Landau theory

- The term proportional to is completely new but we could have introduced it intuitively to a Landau-like mean field functional, since the introduction of spatial variations in the order parameter has an energetic cost which must depend on how it varies in space, i.e. it depends on the gradient of . In particular, it must involve because of the symmetries of the model: since the system is isotropic and -invariant, we must use combinations of derivatives that are invariant under rotations and parity, and is the simplest of them
^{[2]}.

If there is also an external magnetic field , we must add to the Hamiltonian the term:

*functional derivatives*instead of usual derivatives.

## Saddle point approximation: Landau theory[edit | edit source]

We can now compute , as a first approach, using The saddle point approximation; as we will see this will reproduce a Landau-like mean field theory which will also take into account the presence of inhomogeneities. In particular thanks to the new term involving we will be able to compute the fluctuation correlation function^{[3]} and so also to determine the critical exponents and .

Therefore we approximate with the leading term of the integral, i.e. we must determine the function that maximizes the exponent, namely minimizes:

^{[4]}:

### Correlation function in the saddle point approximation[edit | edit source]

We can now proceed to compute the correlation function within our approximations. In order to do that, we take the (functional) derivative of the state equation with respect to , so that appears:

In case of translationally invariant (i.e. uniform) systems, is constant and equal to the equilibrium values given by the Landau theory for the Ising model; in particular, depending on the sign of there are two possible situations:

- In this case , so the last equation becomes:

- In this case the magnetization is:

We will shortly see that and are the expressions of the correlation length for and , respectively. We can therefore see that in both cases we get:

Thus, for both the cases and the correlation function can be obtained by solving the differential equation:

then transforming both sides of the equation we get:

^{[5]}. From this last equation we can also foresee that when , since we have and so , from which we have that the critical exponent is null (we will see that explicitly once we have computed ). Therefore, renaming we can now determine with the Fourier antitransform:

We see now clearly that the correlation function has indeed an exponential behaviour (as we have stated also in Long range correlations) and that is really the correlation length; furthermore, and from the definition of the exponent we have , so since we indeed have .

Therefore, we have seen that for the Ising model . If we also consider the values of the other critical exponents we see that the upper critical dimension for this model is . In other words, mean field theories are actually good approximations for the Ising model if . We will later see some other confirmations of this fact.

## Gaussian approximation[edit | edit source]

Until now even if we have introduced Ginzburg-Landau theory we are still neglecting the effects of the fluctuations since we are regarding the mean field theory approximation for non-homogeneous systems as a saddle point approximation of a more general theory; in other words, since we are approximating as we are still regarding the magnetization as non fluctuating over the system. In order to include the fluctuations we must do more and go further the simple saddle point approximation.
The simplest way we can include fluctuations in our description is expanding expressed as a functional integral around the stationary solution and keeping only quadratic terms; this means that we are considering fluctuations that follow a normal distribution around the stationary value. The important thing to note, however, is that in this approximation these fluctuations *are independent*, i.e. they do not interact with each other^{[6]}.
As we will see, with this assumption the values of some critical exponents will differ from the "usual" ones predicted by mean field theories.

Let us apply this approximation from easier cases to more complex ones (and finally to the one we are interested in).

### Gaussian approximation for one degree of freedom[edit | edit source]

Let us consider a system with a single degree of freedom , and call its Hamiltonian. Supposing that is a minimum for , i.e. , expanding around we get:

### Gaussian approximation for N degrees of freedom[edit | edit source]

This is a simple generalization of the previous case; the Hamiltonian will now be a function of the -component vector , and calling the minimum of , expanding around we get:

### Gaussian approximation for infinite degrees of freedom[edit | edit source]

Let us now move to the really interesting case, i.e. the case of infinite degrees of freedom. In general terms (we shall shortly see this explicitly for the Ising model) we want to compute a partition function of the form:

### Gaussian approximation for the Ising model in Ginzburg-Landau theory[edit | edit source]

Let us now apply what we have just only stated to a concrete case, i.e. the Ginzburg-Landau theory for the Ising model we were considering. In this case:

and so setting :

Let us make some remarks on what happens when we apply Fourier transformations in this case.
If our system is enclosed in a cubic box of volume , we can define the Fourier components of the magnetization as:

Let us now compute the partition function of the system in the simpler case ^{[7]}, so that in the end we can determine the free energy of the system.
In this case so , and therefore substituting:

We can now compute the specific heat of the system, and so determine its critical exponent .
We therefore want to compute:

^{[8]}, since the density of states in Fourier space is high (it is proportional to , see the footnote). Now, using the definition of that we have previously seen we have:

^{[9]}for . We can also wonder what happens for . From what we have stated about the rescaled form of we could think that the integral diverges, but we must also take into account the prefactor , which tends to zero for as the transition is approached (since ). The net result is

*finite*, as could also be argued from the original (unscaled) form of ; in fact if in then (in spherical coordinates) the integrand is proportional to , and since:

Let us now consider the second contribution to .
In particular, as we have done before we rewrite it substituting the sum with an integral and also using the definition of , so that:

*not*itself) in the limit converges if , i.e. ; this means that for and we have that behaves as:

^{[10]}:

### Two-point correlation function in the Gaussian approximation[edit | edit source]

We know that the (simple) correlation function is defined as:

- In this case (which can be re-expressed as ) the two coefficients and are distinct, and in the numerator the double integral factorizes into two integrals of the form:

- In this case (equivalent to ) we can either have so that , or so that .

Let us first consider the case . Using polar coordinates, we define so that the measure in the complex plane becomes:

^{[11]}:

Therefore since the correlation function in Fourier space is non null only when , in general we can write:

## Interaction between fluctuations: expansion to the fourth order[edit | edit source]

We have therefore seen that mean field theories can be improved including the fluctuations of the order parameter around its extremal values; in particular with the Gaussian approximation we have stopped the expansion at the second order and this led to a change in the critical exponent of the specific heat, which now really diverges instead of exhibiting a jump discontinuity as simple mean field theories predict. However, the quartic term that we ignore within the Gaussian approximation (and which basically represent the interactions between the fluctuations) becomes crucial when we approach a critical point. We could thus wonder if the Gaussian approximation can be improved. In particular, reconsidering the expression of with :

### Dimensional analysis of Landau theory[edit | edit source]

We know that the partition function of the system~is:

*dimensionless*

^{[12]}effective Hamiltonian . Now, since is dimensionless all the three integrals that appear in must be so; this means that , and must have precise dimensions. In fact, from the first contribution we have that:

The most standard procedure to apply in this case would be a perturbative method, namely to consider the dimensionless parameter as small, i.e. , and expanding the exponential term containing :

A final remark.
The argument we have shown is not really convincing. In fact, we have only shown that every term of the perturbation theory diverges as ; however, this does not necessarily mean that the whole perturbation series is divergent. For example, consider the exponential series:

- ↑ Sometimes this approximation is called
*elastic free energy*. - ↑ At this point we could wonder why the interaction part of the Hamiltonian does not contain other terms, like : in fact this is in principle perfectly acceptable since is of second order in and is invariant under rotations and parity . However, we have that:
and so when we integrate over , the first term vanishes in the thermodynamic limit supposing that the magnetization or its gradient goes to zero sufficiently rapidly as . Therefore, we are left only with : the two terms are perfectly equivalent.
- ↑ We will do our computations on the Ising model, as usual.
- ↑ This follows from the integral form of . In fact, if in general a functional is of the form:
then from the definition of functional derivative we have:where is an arbitrary function that vanishes on the boundary of integration. Integrating by parts we get:and so finally:
- ↑ In the following, for the sake of simplicity we will indicate the magnitude of a vector simply removing the arrow sign.
- ↑ In solid state physics this assumption is often called
*random phase approximation*, while in field theory*free field approximation*. - ↑ We could have equivalently considered the case , but it is a bit more complicated since and so there is another term that contributes to the free energy. In other words, if then in we are not considering the term with (which is exactly equal to ).
- ↑ The substitution: can be justified as follows:Now, is quantized since is finite and we have periodic boundary conditions, and in particular . Therefore:
- ↑ A little remark: the origin of this divergence does
*not*come from the behaviour of the integral for large wavelengths. In fact, in the original definition of the integral has an upper limit, , so it cannot diverge because of the large behaviour; if it diverges, it must be because of the behaviour for , which corresponds to large wavelengths (this is also why this divergence is sometimes called*infrared divergence*). - ↑ We stress again that the same calculations could have been done in the case , but we have not done so only for the sake of simplicity.
- ↑ In order to compute the integral in the numerator we have used a standard "trick":
- ↑ Remember that has the dimension of the inverse of an energy.