Why is it interesting?

As condensed matter physicists were just discovering, when materials that are completely different at the microscopic level are tuned to the critical points at which they undergo phase transitions, they suddenly exhibit the same behaviors and can be described by the exact same handful of numbers. Heat iron to the critical temperature where it ceases to be magnetized, for instance, and the correlations between its atoms are defined by the same “critical exponents” that characterize water at the critical point where its liquid and vapor phases meet. These critical exponents are clearly independent of either material’s microscopic details, arising instead from something that both systems, and others in their “universality class,” have in common. Polyakov and other researchers wanted to find the universal laws connecting these systems. “And the goal, the holy grail of all that, was these numbers,” he said: Researchers wished to be able to calculate the critical exponents from scratch.

Physicists Uncover Geometric ‘Theory Space’

If one is to pick out a single feature that epitomizes the power and successes of RG theory, one can but endorse Gallavotti and Benfatto when they say "it has to be stressed that the possibility of nonclassical critical indices (i.e., of nonzero anomaly $\eta$) is probably the most important achievement of the renormalization group."

Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao

Layman

For a nice introduction in laymen terms see: Physicists Uncover Geometric ‘Theory Space’

Student

The best explanation can be found in Critical point phenomena: universal physics at large length scales by Bruce, A.; Wallace, D. published in the book The new physics, edited by P. Davies

Researcher

What's the connection between critical exponents and conformal invariance?

What materials at critical points have in common, Polyakov realized, is their symmetries: the set of geometric transformations that leave these systems unchanged. He conjectured that critical materials respect a group of symmetries called “conformal symmetries,” including, most importantly, scale symmetry. Zoom in or out on, say, iron at its critical point, and you always see the same pattern: Patches of atoms oriented with north pointing up are surrounded by patches of atoms pointing downward; these in turn are inside larger patches of up-facing atoms, and so on at all scales of magnification. Scale symmetry means there are no absolute notions of “near” and “far” in conformal systems; if you flip one of the iron atoms, the effect is felt everywhere. “The whole thing organizes as some very strongly correlated medium,” Polyakov explained.

Physicists Uncover Geometric ‘Theory Space’

For a nice discussion with simple formulas, see page 97 in the Conceptual Foundations of Quantum Field Theory, Edited by Cao

Common Question 2

Examples

Example1

Example2:

History