General Characterization

What does Renoamalization Methods Do?#

Given a critical system \(\mathcal{S}\) with parameter \(\theta\), when \(\theta\) approaches a critical value \(\theta_c\), methods used when the value of the parameter \(\theta\) is non-critical become incorrect:

Divergence of the correlation length \(\xi\) makes local analysis impossible both in the real space and in the conjugate space.
The divergence of \(\xi\) maeks it impossible to do any analysis which assumes that the system of size \(L\) is homogeneous on a scale \(l<<L\) or describes it as such. For example, mean-field theories fail, in which the interactions between the elementary constituents are replaced by an effective homogeneous exterior influence depending only on the statistical properties of the system.
\(\xi\) coincides with the size of the inhomogeneities of the microscopic distribution, so that its divergence makes it impossible to treat the statistical fluctuations perturbatively starting from a macroscopic description using only statistical averages.
It is impossible to determine the state function in the form of analytic expansions in \(\theta -\theta_c\). The problem of critical divergences becomes crucial in perturbative methods. The convergence of the expansion becomes slower as the critical point \(\theta_c\) is approached.
There is a problem with the inversion of the thermodynamic limit \(N\to \infty\) and the “critical” limit \(\theta\to\theta_c\). We observe \(L_1 = \lim_{\theta\to\theta_c} \lim_{N\to\infty}\) but we know how to compute \(L_2=\lim_{N\to\infty}\lim_{\theta\to\theta_c}\). Because of the critical singularities \(L_1\) and \(L_2\) are different since the exchange of the order of limit is impossible.

Renormalization methods are used to make up for this problem.

The main question relative to critical system is to explain how short-range (spatial or temporal) physical couplings generate a phenomenon perceptible on a large scale. The answer lies in the existence of a collective behavior organized hierarchically from the microscopic up through the observation scales.

The analysis of critical systems, then, must be global and deal not with the microscopic details but with the way in which they cooperate. The different scales must be related to each other in such a way that the expected scale invariances are made explicit. The analysis should also be detached from the specific details of the system in order to give universal results which should be the same for any system for which some set of generic hypotheses is satisfied.

The aims of renormalization methods are thus

prove the existence of scaling laws
give constructive expressions for the critical exponents
show their universal nature and to determine the associated univerality classes

The Method#

The method

Reduce the number (infinite in the thermodynamic limit) of degrees of freedom necessary to describe a critical system on the macroscopic scale. Depending on the scale invariance associated to the divergence of \(\xi\), decompose \(\mathcal{S}\) into nested subsystems which can be deduced from each other by scalings.
Weaken the critical divergences by increasing the resolution (minimal scale) in the neighborhood of a singularity, or bring asymptotic behavior into the field of observation.

The step:

Specify the space \(\Phi\) of structure rules: functions \(\phi\) defined on the phase space \(\mathcal{X}\) of the system \(\mathcal{S}\).
Introducing a renormalization operator \(R\) acting on \(\Phi\). Instead of the state of \(\mathcal{S}\) in \(\mathcal{X}\), study the relationship, described by \(\phi\in\Phi\), between the state of \(\mathcal{S}\) and the given information about \(\mathcal{S}\).
Main idea: simultaneous modification of the number of degree of freedom, reduced by an arbitrary factor \(K>1\) by a decimation operation, and of the structure rule \(\phi\in\Phi\). Renormalization operator \(R_k\), labeled by \(k\) is devised so that the scale invariance of the system is expressed by the invariance of its structure rule \(\phi = R_k\phi\). The fixed points of \(R_k\) are thus typical critical systems.
If \(\phi\) depends on the control parameter \(\theta\) of the critical phenomenon, try in the neighborhood of \(\theta_c\) to move the renormalization to \(\theta\): \(R_k\phi_\theta \approx \phi_{r_k \theta}\). THe fixed point equation becomes \(r_k\theta_c = \theta_c\). The condition that the trajectory of \(\phi_\theta\) under the action of \(R_k\) converge towards a non-trivial fixed point indicates the manner in which we can let \(N\to\infty\) and simultaneously modify \(\theta\) and thus directly gives the \(\theta\)-dependence of the macroscopic quantities in the neighborhood of the critical point.

This is fruitful if we can display a hyperbolic fixed point \(\phi^\ast\) and do the linear and if possible non-linear analysis of \(R_k\) around this fixed point.

The critical exponents are related to the logarithm of the eigenvalues of \(DR_k(\phi^\ast)\).
The set of structure rules converging to \(\phi^\ast\) under the action of \(R_k\) is the assocaited universality class.
\(\phi^\ast\) appears as a universal critical system, a typical representative of the class which reflects properties common to all of its elements.