Editing Renormalization group (section)

==Block spin==
This section introduces pedagogically a picture of RG which may be easiest to grasp: the block spin RG, devised by [[Leo P. Kadanoff]] in 1966.<ref name=Kadanoff/>

Consider a 2D solid, a set of atoms in a perfect square array, as depicted in the figure.

[[File:Rgkadanoff.png|180px]]

Assume that atoms interact among themselves only with their nearest neighbours, and that the system is at a given temperature {{mvar|T}}. The strength of their interaction is quantified by a certain [[coupling constant|coupling]] {{mvar|J}}. The physics of the system will be described by a certain formula, say the Hamiltonian {{math|''H''(''T'', ''J'')}}.

Now proceed to divide the solid into '''blocks''' of 2×2 squares; we attempt to describe the system in terms of '''block variables''', i.e., variables which describe the average behavior of the block. Further assume that, by some lucky coincidence, the physics of block variables is described by a ''formula of the same kind'', but with '''different''' values for {{mvar|T}} and {{mvar|J}}: {{math|''H''(''{{prime|T}}'', ''{{prime|J}}'')}}. (This isn't exactly true, in general, but it is often a good first approximation.)

Perhaps, the initial problem was too hard to solve, since there were too many atoms. Now, in the '''renormalized''' problem we have only one fourth of them. But why stop now? Another iteration of the same kind leads to {{math|''H''(''T"'', ''J"'')}}, and only one sixteenth of the atoms. We are increasing the '''observation scale''' with each RG step.

Of course, the best idea is to iterate until there is only one very big block. Since the number of atoms in any real sample of material is very large, this is more or less equivalent to finding the ''long range'' behavior of the RG transformation which took {{math|(''T'',''J'') → (''{{prime|T}}'',''{{prime|J}}'')}} and {{math|(''{{prime|T}}'', ''{{prime|J}}'') → (''T"'', ''J"'')}}. Often, when iterated many times, this RG transformation leads to a certain number of '''fixed points'''.

To be more concrete, consider a [[magnetic]] system (e.g., the [[Ising model]]), in which the {{mvar|J}} coupling denotes the trend of neighbor [[Spin (physics)|spin]]s to be aligned. The configuration of the system is the result of the tradeoff between the ordering {{mvar|J}} term and the disordering effect of temperature.

For many models of this kind there are three fixed points: 
# {{math|1=''T'' = 0}} and {{math|''J'' → ∞}}. This means that, at the largest size, temperature becomes unimportant, i.e., the disordering factor vanishes. Thus, in large scales, the system appears to be ordered. We are in a [[ferromagnetic]] phase.
# {{math|''T'' → ∞}} and {{math|''J'' → 0}}. Exactly the opposite; here, temperature dominates, and the system is disordered at large scales.
# A nontrivial point between them, {{math|1=''T'' = ''T''<sub>''c''</sub>}} and {{math|1=''J'' = ''J''<sub>''c''</sub>}}. In this point, changing the scale does not change the physics, because the system is in a [[fractal]] state. It corresponds to the [[Curie point|Curie]] [[phase transition]], and is also called a [[critical point (thermodynamics)|critical point]].

So, if we are given a certain material with given values of {{mvar|T}} and {{mvar|J}}, all we have to do in order to find out the large-scale behaviour of the system is to iterate the pair until we find the corresponding fixed point.