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

2. Outline Overview of Renormalization Critical Points
Conformal Invariance

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

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

5. 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.

6. Renormalization:QED

7. Renormalization:QCD

8. 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.

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

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

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

12. 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+ =

13. 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.

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

15. 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.

16. 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

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

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

19. 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

20. Renormalization

21. 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

22. 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

23. 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.

24. 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/ v1