1. Understanding phase transitions and critical phenomena from conformal bootstrap
Yu Nakayama (Kavli IPMU, Caltech)
in collaboration with Tomoki Ohtsuki (Kavli IPMU)

2. Critical point of H2O phase diagram
At T= 647K, P = 22MPa, we have a critical point
Second order phase transition
Critical behavior is universal

3. Universal critical behavior 1
Various thermodynamic quantities show scaling law
The origin of the critical behavior is scale invariance at the critical point as a result of renormalization group flow
At second order phase transition, critical behavior appears in thermodynamic quantities

4. Universal critical behavior 2
Various thermodynamic quantities scale as
The origin of the critical behavior is scale invariance at the critical point (fixed point of RG flow)
One of the greatest challenges to human intellect is to understand the origin of universality, and determine critical exponents
The same critical behavior is seen in 3D Ising model

5. Scaling hypothesis
Assume free energy shows the scaling behavior
Then, scaling relations can be derived
The scaling hypothesis and universality may be understood from the renormalization group (Wilson)
But the scaling hypothesis itself does not explain the value of and
At the critical point, the thermodynamic free energy satisfies the scaling law

6. From scale to conformal hypothesis
At the critical point, the system is not only scale invariant, but is invariant under the enhanced symmetry known as conformal symmetry
The universality of the critical behavior is governed by the conformal symmetry as a result of local renormalization group
The critical exponent may be understood from the hidden conformal symmetry (~ solution of 3d Ising)
There exists hidden enhanced symmetry called conformal invariance

7. Scale vs Conformal invariance
Scale transformation
Conformal transformation

8. Conformal transformation
Scale transformation:
Conformal transformation:
It is not immediately obvious if global scale invariance means conformal invariance (see my review paper arXiv: )  

9. Conformal hypothesis in 3D Ising model
Consistency of 4-point functions in conformal invariant system gives a bound on scaling dimensions of operators (El-Showk et al)
Explains critical exponents (3d Ising model solved)!
Assuming the conformal invariance, critical exponents can be determined from conformal bootstrap

10. O(n) x O(m) symmetric CFTs and critical phenomena

11. O(n)xO(m) Landau-Ginzburg model
Field transforms vector x vector rep under O(n) x O(m) global symmetry
u preserves O(nm) symmetry, but v breaks it
Always exists O(nm) symmetric Heisenberg fixed point with v = 0
d= 3 model appears in effective theories of frustrated spins or chiral transition in QCD
Although we won’t need Hamiltonian (Lagrangian), we start with the concrete model…

12. Frustrated spins in non-collinear order
Effective theory = O(n) x O(m) LG model:
n = components of spin, m = non-collinear dim
1st order phase transition or 2nd order phase transition? Huge debate in experiments
Theoretical controversy as well. Monte Carlo, epsilon expansions, large N expansions, exact RG all disagree which values of n and m, the fixed points exist ( 2nd order phase transition)…
Anti-ferro spins in frustrated lattice (Kawamura)
n=2, m=2
n=3, m=3
chiral
anti-chiral

13. Chiral phase transition in QCD
A long standing debate if the QCD chiral phase transition with Nf=2 massless flavors is 1st order or 2nd order
Lattice simulations are again controversial
Effective theory description is SU(2) x SU(2) x U(1) (= O(4) x O(2)) LG model in d=3
RG computation is also controversial…
SUSY does not help (with many respects…)
What is the order of chiral phase transition in QCD (Pisarsky-Wilczek)

14. Schematic RG picture (Un)stable one is called (anti-)chiral fixed
For sufficiently large n with fixed m, they both exist
Nobody has agreed what happens for smaller n
Multiple fixed points cannot appear in SUSY theories…

15. Why controversial?
Large n (with fixed m) expansion or epsilon expansion are asymptotic
Results depend on how you resum the diverging 5- loop or 6-loop series (need artisan technique. OK for Ising but…)
Exact (or functional) RG directly in d=3 needs “truncation”, which is not a controlled approximation
No SUSY, no large n, no holography. We are talking about real problems.

16. The questions to be answered
To fix the conformal window for O(n) x O(m) symmetric Landau-Ginzburg models in d=3
(Non-)Existence of non-Heisenberg fixed point  determine the order of phase transitions
Compute critical exponents to compare with experiments (or simulations)
Our conformal bootstrap approach is non-perturbative without assuming any Hamiltonian (c.f. “Hamiltonian is dead”)

17. Conformal Bootstrap approach

18. Schematic conformal bootstrap equations
Consider 4pt functions
OPE expansions
I: SS, ST, TS, TT, AS, SA, AA … (S: Singlet, T: Traceless symmetric, A: Anti-symmetric)
Crossing relations
Assume spectra (e.g , )
to see if you can solve the crossing relations (non-trivial due to unitarity )
 convex optimization problem
(but 100 times more complicated than Ising model)

19. Results Begin with O(3) x O(m) with m=15
Can we see Heisenberg/chiral/anti-chiral fixed point?
Each plots need 1~2 weeks computation on our cluster computers
Hypothesis: non-trivial behavior of the bound indicates conformal fixed point

20. Heisenberg fixed point in SS sector
Constraint is same as O(45) (symmetry enhancement)
 “Kink” is Heisenberg fixed point
Consistent but cannot see chiral/anti-chiral

21. Anti-chiral fixed point in TA spin 1 op
We can read spectra at the “kink”
Dimension of SS operator
Seems to agree with large n prediction of anti-chiral fixed point

22. Anti-chiral fixed point in ST spin 0 op
We can read spectral at the “kink(?)”
Dimension of SS operator
Agrees with anti-chiral fixed point?

23. Chiral fixed point in TS spin 0 op
We can read spectral at the “kink”
Dimension of SS operator
Seems to agree with large n prediction of chiral fixed point

24. Finding conformal window n*(m=3)
Change n (with m=3) to see if the kink disappears (suggesting no anti-chiral fixed point!)
n = 6~7 seems the edge of the conformal window?

25. Finding conformal window n*(m=3)
Differentiated plot
Kink disappears for n<6~7!

26. Quick summary for O(n) x O(3)
A single conformal bootstrap equation can detect all Heisenberg/chiral/anti-chiral fixed points in different sectors
Large n (with fixed m) analysis agrees with us
We predict that n= 6~7 is the edge of the conformal window for anti-chiral fixed point in m=3 (e.g. large n expansion n= 7.3, epsilon expansion n = 9.5)
First example of determining conformal window from (numerical) conformal bootstrap

27. Toward O(n) x O(2) under controversies
Situation is much controversial
n > n*~5,6, chiral and anti-chiral exit
n =2,3,4, some say there are (non-perturbative) chiral fixed point (cannot seen in 1/n expansions)
Can we see it?
Found conformal window in spin 1 sector
We have preliminary results on controversial regime, but my collaborator refuses to show them here…

28. Summary and discussions
Conformal hypothesis is very powerful
O(n) x O(m) bootstrap is exciting
Applications to real physics (frustrated spin, QCD…)
Determination of conformal window is now possible!
Theoretical backup needed? Still empirical science.
If you have any models to be studied with conformal bootstrap, let us know