Scale invariance and Conformal invariance

Understanding phase transitions
and critical phenomena from
conformal bootstrap
Yu Nakayama (Kavli IPMU, Caltech)
in collaboration with Tomoki Ohtsuki (Kavli IPMU)
Critical point of H2O phase diagram
• At T= 647K, P = 22MPa, we have a critical point
• Second order phase transition
• Critical behavior is universal
Universal critical behavior 1
At second order phase transition, critical behavior
appears in thermodynamic quantities
• 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
Universal critical behavior 2
The same critical behavior is seen in 3D Ising model
• 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
Scaling hypothesis
At the critical point, the thermodynamic free energy
satisfies the scaling law
• 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
From scale to conformal hypothesis
There exists hidden enhanced symmetry called
conformal invariance
• 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)
Scale vs Conformal invariance
Scale transformation
Conformal transformation
Conformal transformation
• Scale transformation:
• Conformal transformation:
It is not immediately obvious if global scale invariance means
conformal invariance (see my review paper arXiv:1302.0884)
Conformal hypothesis in 3D Ising model
Assuming the conformal invariance, critical exponents
can be determined from conformal bootstrap
• 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)!
O(n) x O(m) symmetric
CFTs and critical
O(n)xO(m) Landau-Ginzburg model
Although we won’t need Hamiltonian (Lagrangian),
we start with the concrete 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
Frustrated spins in non-collinear order
Anti-ferro spins in frustrated lattice (Kawamura)
n=3, m=3
n=2, m=2
• 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)…
Chiral phase transition in QCD
What is the order of 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…)
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…
Why controversial?
• Large n (with fixed m) expansion or epsilon
expansion are asymptotic
• Results depend on how you resum the diverging 5loop 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
• No SUSY, no large n, no holography. We are talking
about real problems.
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”)
Conformal Bootstrap
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)
• Begin with O(3) x O(m) with m=15
• Can we see Heisenberg/chiral/anti-chiral fixed
• Each plots need 1~2 weeks computation on our
cluster computers
• Hypothesis: non-trivial behavior of the bound
indicates conformal fixed point
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
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
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?
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
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?
Finding conformal window n*(m=3)
• Differentiated plot
• Kink disappears for n<6~7!
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
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
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

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