Conformal Invariance and Critical Phenomena Jason Therrien April 21, 2009 Solid State II

Outline Overview of Renormalization Critical Points Conformal Invariance

Renormalization We can use Feynman diagrams to represent interactions The vertex is proportional to the coupling constant.

Renormalization But that is only the Classical picture, Quantum effects must be taken into account = + + + ...

Renormalization These Quantum Effects will change the coupling constant. We say that Quantum Effects “Dress” our theory. When we initially write down our theory we must remember that we will have to make quantum corrections to our coupling constants, fields, and the masses present in our theory.

Renormalization:QED

Renormalization:QCD

Asymptotic Freedom Means that as we increase the energy to probe QCD the system becomes decoupled. This is known as Asymptotic Freedom If you reduce the energy at which you probe the system, it becomes very strongly bound. This is seen in Condensed Matter with the Kondo effect.

Renormalization:Mass Quantum Effects will also change the mass of the particles involved in your theory.

Renormalization:Mass 1-Particle Irreducible Diagrams are diagrams which if an internal line is cut stays connected. 1-PI Not 1-PI

Renormalization:Mass 1-PI's are very special diagrams 2 + =

Renormalization:Mass The First Term ~1/(K^2-m^2) The Second goes like We can factor out an overall Then we will have 2 X 2 ( +( ) + ....) 1+ =

Renormalization:Mass This is our old friend the Geometric series Summing we get It is interesting to note that this has done more than just Renormalize the mass.

Criticality We are still some ways away from Condensed Matter Conformal Invariance Scale Invariance

Ferromagnet For a Ferromagnet near the 2nd order phase transition the free energy goes like This is very similar to the Klein-Gordon Equation We can vary G just like an action.

Ferromagnet This leads to a wave equation whose solution is a Yukawa potential Notice that as you increase the mass this becomes an increasingly short range effect

Coherence Length This leads to the definition of the Coherence length This means the critical exponent n=.5

Scale Invariance We have not taken into account any Quantum Effects Wilson-Fisher fixed points Dimensional Regularization

Dimensional Regularization Due to t' Hooft Noticed that if we were only working in lower dimensional systems we wouldn't have some of the problems that we do in Relativistic Field theory. Lets work in 4-e Dimensions Works great for this system

Renormalization

After a lot of math ..... After a lot of math I will not take credit for, the renormalization procedures give us a new critical exponent. For First order, you get .6 According to Peskin & Schroeder further orders give .63 Experimentally it is found to be .625

Summary Quantum Effects must be taken into account Near Critical points all notions of scale is lost Even very crude models can be very powerful

Conformal Invariance Formally a conformal transformation is a transformation that leaves the metric invariant up to a scale change Meaning that the angle between 2 vectors is preserved under this transformation This leads to many usefull things, including a traceless Stress Tensor which gives us theories without scale.

References Peskin & Schroeder's Introduction to Quantum Field Theory Or really any Field theory book worth the paper it is printed on Applied Conformal Field Theory by Paul Ginsparg-arXiv:hep-th/9108028v1