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

• 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

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

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

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

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

• 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

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

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…

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)

chiral anti-chiraln=2, m=2 n=3, m=3

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)

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 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.

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 approach

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)

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

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 here…

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