Subsections

• Universality
• Scaling Invariance

Universality and Scaling Invariance

Universality

We neglected the term dM⁴. If we include it, free energy is no longer a sum of independent terms--fluctuations interact!

$$\int M^4(\mathbf{r})\,d\mathbf{r}= V\sum_{\mathbf{k}_1,\math... ...{k}_3} M_{-\mathbf{k}_1-\mathbf{k}_2-\mathbf{k}_3-\mathbf{k}_4}$$

When fluctuations are small, we can expand this term. When they are not--we are out of luck!

Close enough to critical point only interactions of long wavelength fluctuations determine the properties. But the long wavelength limit is described by Landau Hamiltonian!

In other words, near critical point we start from Landau expression, calculate fluctuations and their interactions--and obtain everything!

Universality:

Close enough to critical point ($\left\lvert T-T_c\right\rvert/T_c\ll\mathsf{Gi}$) the behavior of the system is determined only by the symmetry of order parameter.

In particular, for scalar order parameter all systems can be described by partition function  

$$\Xi = \int\mathcal{D}M(\mathbf{r})\,\exp\left[-\beta\int (aM^2+dM^4-HM)\,d\mathbf{r}\right]$$ (5)

where $\mathcal{D}M(\mathbf{r})$ is integral over all realizations of $M(\mathbf{r})$.

Now we need to calculate (5)--a very tough task!

Scaling Invariance

Fluctuations with the wavelength $\lambda\lesssim\xi$ do not ``feel'' the critical point. The only fluctuations that do, are those with $\lambda\gtrsim\xi$

Scaling Hypothesis:

The fluctuations at all length scales look exactly the same. Change of the length scale can be compensated by the change of the temperature

Renormalization Group Approach:

We change length scale in (5) and require $\Xi$ to be invariant! By this method people obtained approximation for $\Xi$ (and received Nobel Prize).