New Results on Second-Order Phase Transitions and Conformal

New Results on Second-Order Phase Transitionsand Conformal Field Theories

Zohar Komargodski

Weizmann Institute of Science, Israel

April 26, 2015

Zohar Komargodski New Results on Second-Order Phase Transitions and Conformal Field Theories

We have Ld spins with some nearest-neighbor interaction energyJ > 0 if they are misaligned. So the spins want to be aligned atlow temperatures. The magnetization M is the order parameter.

Alternatively, think of the 2nd order water-vapor transition diagram

At Tc there is a phase transition. Long range correlations develop.Lattice structure becomes irrelevant. Quantities depend singularlyon the temperature and external field. Ginzburg-Landau theory:

∫ddx

(r(∇M)2 + cM2 + λM4 + ...

)The partition function

∫[dM]e−H

encodes all the thermodynamics properties at the phase transition.

Some experimentally interesting quantities are the usualα, β, γ, δ, η, ν exponents:

C ∼ (T − Tc)−α , M ∼ (Tc − T )β , χ ∼ (T − Tc)−γ ,

M ∼ h1/δ , 〈M(~n)M(0)〉 ∼ 1

|~n|d−2+η, ξ ∼ (T − Tc)−ν .

Amazingly, one discovers four relations between these 6 quantities:

α + 2β + γ = 2 ,

γ = β(δ − 1) ,

γ = ν(2− η) ,

νd = 2− α .

The explanation of this miracle is that at Tc the symmetry of thesystem is enhanced.

SO(d)× Rd → SO(d)× Rd ×∆ ,

with∆ : x → λx

and λ ∈ R+.

The dilaton charge ∆ can be diagonalized. If we have a localoperator O in the theory, ∆(O) would uniquely determine itstwo-point correlator

〈O(n)O(0)〉 ∼ 1

n2∆(O).

Local operators could also have spin s, but we suppress it in themeantime.

In the Ising model, two of the infinitely many operators in thetheory are M(x) and ε(x), which are the magnetization and energyoperators. The four miraculous relations among α, β, γ, δ, η, ν canbe simply understood from scale invariance:

α =d −∆ε

d −∆ε,

β =∆M

d −∆ε,

γ =d − 2∆ε

d −∆ε,

δ =d −∆M

η = 2− d + 2∆M ,

d −∆ε.

This was essentially understood more than 70 years ago.

There has been a lot of recent progress based on the observationthat the symmetry is actually bigger!!

SO(d)× Rd → SO(d)× Rd ×∆→ SO(d + 1, 1)

Theories with this big symmetry are called Conformal FieldTheories.

SO(d + 1, 1) is the conformal group that acts on Rd . This is theset of all transformations that preserve orthogonal lines.It consists of

d(d − 1)/2 rotations

d translations

1 dilation (∆)

d special conformal transformations

The last d symmetry generators were not known to Landau (Ithink...)

Essentially all the examples that we know of second order phasetransitions which have

SO(d)× Rd ×∆

have the full SO(d + 1, 1). It has been verified “experimentally” insome examples, it has been proven to be correct theoretically insome general situations etc. So we are quite sure about this.

The d bonus special conformal transformations act on space as

x i → x i − bix2

1− 2b · x + b2x2,

where bi is any vector in Rd .It turns out that these are enough to fix three-point correlationfunctions as follows

〈O1(n′)O2(n)O3(0)〉 =CO1O2O3

n∆2+∆3−∆1(n − n′)∆1+∆2−∆3n′∆1+∆3−∆2

Remember

〈Oi (n)Oj(0)〉 ∼δij

n2∆i (O).

SO(d+1,1) symmetry fixes the two- and three-point functions interms of a collection of numbers

∆i , Cijk

which are called the “CFT data.”

It turns out that ALL the correlation functions are fixed in terms ofthe CFT data.

How can we say anything useful about this collection of numbers ?

{∆i} , {Cijk}

This collection is infinite because there are infinitely manyoperators in every Landau-Ginzburg theory. It is easiest to measurethe relevant, low dimension, operators (such as M, ε), but also theothers exist.

Consider a four-point function with operators at n1, n2, n3, n4. Wecan form conformally invariant ratios:

12n234

, v =n2

13n224

So a four-point function could contain an arbitrary function of u, v

〈O1(n1)O2(n2)O3(n3)O4(n4)〉 ∼ F (u, v) .

We represent the four point function as a sum over infinitely manythree point functions:

~ C12X

CX34 1,2,3,4,X

Therefore,

F (u, v) ∼∑X

C12XCX34G (∆1,2,3,4,X , u, v)

The functions G are partial waves. Thus, once we know the CFTdata, the four-point function can be in principle computed.

The dynamics is in saying that we can make the decomposition intwo different ways. And we get (roughly speaking)

C12XCX34G (∆1,2,3,4,X , u, v) =∑X

C13XCX24G (∆1,3,2,4,X , v , u)

This equation is supposed to determine/constrain the allowed ∆i

and Cijk that can furnish legal conformal theories.

This is extremely surprising:

Maybe we could classify all the possible conformal theories by justsolving self-consistency algebraic equations.

For the mathematically oriented: these equations are similar to theequations of associative rings. It is also very similar in sprit to theclassification of Lie Algebras.

Additional Applications of Conformal Field Theories:

Quantum phase transitions

Fixed points of the Renormalization Group Flow.

FreeTheory

Therefore, if we had a better idea about what the equations

imply, that would be useful in many branches of physics. Moreambitiously, we could hope to classify all the solutions!

Recently, there has been dramatic progress on this problem bothfrom the analytic and numeric points of view.

I will quote, without proofs, three very general analytic results.They are experimentally and numerically testable.

Result I: Additivity of the Spectrum

If we have (∆1, s1) and (∆2, s2) in the spectrum, then there areoperators (∆i , si ) which have ∆i − si arbitrarily close to∆1 − s1 + ∆2 − s2.

Typically, to find operators with ∆i − si ∼ ∆1 − s1 + ∆2 − s2 wewould need to take large ∆i , si .

This leads to a rather peculiar spectrum

log(s)

(In the figure, τ ≡ ∆− s.)

Result II: Convexity

∆i − si approaches the limiting value ∆1 − s1 + ∆2 − s2 in aconvex manner.

These already lead to nontrivial results for the 3d Ising model.Since the spin field has ∆(M) = 0.518..., there needs to be afamily of operators with ∆− s approaching 1.037... from below, ina convex fashion:

This is beautifully verified by the measurement of the spin-4operator dimension. Also for spin 6. The other predictions awaitconfirmation.

Unfortunately there are not that many general analytic constraintson Conformal Field Theories at the moment.

However, there has been rapid and impressive progress onnumerical constraints. In fact, for some problems, the newmethods outdo 4 decades of attempts to measure these criticalexponents or compute them using the ε-expansion!

(El-Showk et al.)

So for example we know ∆(σ) = 0.518154(15)...

Monte Carlo relies on some microscopic realization of theConformal Field Theory (e.g. with spin degrees of freedom)and there are no rigorous bounds on the errors. The newmethod only uses abstract algebraic properties of the theory(no microscopic realization is invoked) and there are rigorousresults on its exponential convergence rate.

The ε-expansion is a computation done near d = 4extrapolated to d = 3. One needs to make assumptions tobound the errors and a microscopic realization is needed.Most of the interesting theories cannot be approached by anε-expansion.

One starts from the algebraic equations

The equations are coupled quadratic equations for the C ’s and∆’s. Way too hard even if the equations are truncated.

One declares that one is only interested in theories that, forexample, have their lowest and next to lowest operator at somedimensions ∆1,∆2. Taking derivatives of the equations w.r.t. u, vand using properties of the partial waves G , one is sometimes ledto a mathematical contradiction. Then one scans these equationsover all ∆1,∆2 until no contradiction arises. This leads to boundson ∆1,∆2.

The combinations of derivatives that one needs to take in order toarrive at a contradiction is a nontrivial problem. The computer justscans over all the possibilities.

It is actually not completely clear why this procedure had to work,but it clearly does.

Some philosophical remarks: In quantum gravity one does not havelocal degrees of freedom, so there are “fewer” measurablequantities. For example, in asymptotically flat space-time we havethe S-matrix

Quantum gravity in AdSd+1 is described by a boundary ConformalField Theory. The S-matrix of the quantum theory of gravity isbasically the set of correlation functions of a d-dimensional CFT.So the program of classifying CFTs or learning exact things abouttheir properties can shed light on quantum gravity in AdS space.

Conclusions

Conformal Field Theories are abundant in physics.

The are determined by an intricate self-consistency condition.We still don’t know much about the general consequences ofthis self-consistency condition.

There are some results though on monotonicity, convexity,and additivity of the spectrum of dimensions.

Extremely powerful numerical techniques were recentlyintroduced. Not unlikely that in the near future would be ableto say very precise things about strongly coupled models thatappear in Condensed Matter systems (e.g. Herbertsmithite) oreven determine the conformal window of QuantumChromodynamics.

New Results on Second-Order Phase Transitions and Conformal

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