Phases of fuzzy field theories

Phases of fuzzy field theories

Juraj Tekel

Department of Theoretical PhysicsFaculty of Mathematics, Physics and Informatics

Comenius University, Bratislava

Seminar aus Mathematischer Physik, University of Vienna24.3.2015

Juraj Tekel Phases of fuzzy field theories

Short outline

what are the noncommutative/fuzzy spacesproperties of scalar field theories on these spacesbasics of matrix modelsmatrix model corresponding to fuzzy scalar fieldsresults for the triple point and phase diagramcurrent research, challenges and outlook


Noncommutative spaces


NC spaces

The usual sphere S2 is given by coordinates

xixi = R2 , xixj − xjxi = 0

which generate the algebra of functions

f ={∑(

ak1k2k3

∏xkii

) ∣∣∣xixi = R2}

which is commutative by definition.Using this algebra, we can reconstruct all the information about thesphere.It turns out that any commutative algebra corresponds to somedifferentiable manifold in a similar way.


NC spaces

Noncommutative spaces are (by definition) spaces, which are incorrespondence with noncommutative algebras

[xi, xj ] = iΘij

The choice of Θ determines what the kind of space we have.xi are operators on a Hilbert space. If this space is finite dimensional, wecall the space "fuzzy". It is a finite mode approximation to compactmanifold.


NC spaces

For the fuzzy sphere S2F

xixi = ρ2 , xixj − xjxi = iθεijkxk

This can be realized as a N = 2j + 1 dimensional representation of theSU(2)

xi =2r√N2 − 1

Li , θ =2r√N2 − 1

, ρ2 =4r2

N2 − 1j(j + 1) = r2

The coordinates xi still carry an action of SU(2) and thus the space stillhas the symmetry of the sphere.xi are N ×N matrices and functions on S2

F are combinations of products→ a herminitian matrix M


NC spaces

Hermitian N ×N matrix can be decomposed into

M =

N−1∑l=0

m=l∑m=−l

clmTlm

matrices T lm are called polarization tensors and

[Li, [Li, Tlm]] = l(l + 1)Tlm

Tr (TlmTl′m′) = δll′δmm′

there is an analogy with spherical harmonics on S2

f(φ, θ) =

∞∑l=0

m=l∑m=−l

almYlm(φ, θ)

algebra of functions on S2F is the algebra of functions on S2 with a basis

cut off, such that functions with l > L a L = N − 1The multiplication rule has changed form a commutative pointwisemultiplication to a noncommutative matrix multiplication.Limit of large N reproduces the original sphere S2.


NC spaces

Hermitian N ×N matrix can be decomposed into

M =

N−1∑l=0

m=l∑m=−l

clmTlm

there is an analogy with spherical harmonics on S2

f(φ, θ) =∞∑l=0

m=l∑m=−l

almYlm(φ, θ)

algebra of functions on S2F is the algebra of functions on S2 with a basis

cut off, such that functions with l > L a L = N − 1

The multiplication rule has changed form a commutative pointwisemultiplication to a noncommutative matrix multiplication.Limit of large N reproduces the original sphere S2.


NC spaces

By introducing a cut off on l we introduce a maximal momentum ∼ N ,which in turn means a smallest possible distance ∼ 1/N ≈ θ.Number of extrema of Y l

m is ∼ 2l2, so one extremum corresponds to area∼ 1/l2 and function Y l

m can give us information about distances ∼ 1/l.Points on S2 are encoded in the algebra of functions as sequences ofgaussians converging to the corresponding δ-function. But after the cutoffwe can not construct an arbitrarily narrow gaussian and we can no longerconstruct and arbitrarily precise approximation of a point.The points on S2

F no longer exist.


Fuzzy field theory


Scalar field on S2F

Scalar field is a function on S2F , i.e. an N ×N hermitian matrix.

Derivatives become commutators with generators Li, integrals becometraces and we can write and Euclidean field theory action

S = −1

2Tr ([Li,M ][Li,M ]) +

1

2rTr

(M2)

+ V (M) =

=1

2Tr (M [Li, [Li,M ]]) +

1

2rTr

(M2)

+ V (M)

The theory is given by functional correlation functions

〈F 〉 =

∫dM F [M ]e−S[M ]∫dM e−S[M ]

These can be computed using a noncommutative version of the Feynmannrules with a propagator

〈clmcl′m′〉 =δll′δmm′

l(l + 1) + r

Fuzzy scalar field theory is a matrix model!Juraj Tekel Phases of fuzzy field theories

Scalar field on CP nF

Taking Li to be generators of SU(n+ 1), we can obtain xi’s as coordinateson fuzzy version of CPn, with a corresponding field theory action

S =1

2Tr (M [Li, [Li,M ]]) +

1

2rTr

(M2)

+ V (M)

where i = 1, . . . , (n+ 1)2 − 1, polarization tensors

[Li, [Li, Tlm]] = l(l + n)Tlm

Tr (TlmTl′m′) = δll′δmm′

a more complicated dependence of N on l, but a theory which is still aparticular matrix model.


UV/IR mixing

The trademark property of the noncommutative field theories is theUV/IR mixing, which arises as a consequence of the nonlocality of thetheory.Quanta can not be compressed into arbitrarily small volume and if we tryto do so in one direction, they will stretch in a different direction.Processes at high momenta contribute to processes at low momenta.In theories on noncompact spaces, e.g. Moyal plane or R4, divergence ofnon-planar diagrams at small momentum

Due to no clear separation of scales the theory is not renormalizable.The commutative limit of such noncommutative theory is (very) differentthan the commutative theory we started with.


UV/IR mixing

To obtain a renormalizable theory with a well defined commutative limit,we have to modify the original action such that we remove the UV/IRdivergenceHighly nontrivial, several approaches

B.P. Dolan, D. O’Connor and P. PrešnajderMatrix φ4 models on the fuzzy sphere and their continuum limits[arXiv:/0109084]H. Grosse and R. WulkenhaarRenormalization of φ4 theory on noncommutative R4 in the matrix base[arXiv:0401128]R. Gurau, J. Magnen, V. Rivasseau and A. TanasaA translation-invariant renormalizable non-commutative scalar model[arXiv:0802.0791]


UV/IR mixing

In our setting the UV/IR mixing will arise as an extra phase in the phasediagram, not present in the commutative case.The commutative field can oscillate around zero, in which caseTr (M) = 0 or around some nonzero value Tr (M), which is a minimum ofthe potential.Noncommutative theories will exhibit a third phase, where the field doesnot oscilate around any fixed value. This is a consequence of the nonlocality of the theory.


Random Matrix Models


Some basic results of random matrix theory

The random variable is N ×N hermitian matrix M , expressed as

M =

N−1∑l=1

l∑m=−l

clmTlm

We will expect SU(N) symmetric probability distribution

1

Ze−N

2S[M ] , S[M ] =1

N

[1

2rTr

(M2)

+

N∑n=3

gnTr (Mn)

]The mean value of matrix function f(M) is

〈f〉 =1

Z

∫dM e−N

2S[M ]f(M)

wheredM =

∏i≤j

dReMij

∏i<j

dImMij

What is the distribution of the eigenvalues of the random matrix M?Juraj Tekel Phases of fuzzy field theories


The key is diagonalization, we write M = UΛU−1 for some U ∈ SU(N)and Λ = diag(λ1, . . . , λN ), the integration measure becomes

dM =

(dU

N∏i=1

dλi

)×∏i<j

(λi − λj)2

The problem is now N in an effective potential (eigenvalue repulsion)

Seff =1

N

[1

2r∑i

λ2i + g

∑i

λ4i

]− 2

N2

∑i<j

log(λi − λj)

The eigenvalue distribution

ρ(λ) =1

N

N∑i=1

δ(λ− λi)

will become continuous in the large N limit



In this limit ∑i

g(xi) = N

∫dx g(λ)ρ(λ)

So the effective action

Seff =1

N

[1

2r∑i

λ2i + g

∑i

λ4i

]− 2

N2

∑i<j

log(λi − λj)

becomes

Seff =

∫dλ

[1

2rλ2 + gλ4

]ρ(λ)− 2

∫dλ dλ′ log(λ− λ′)ρ(λ)ρ(λ′)

becomes

Seff =

∫dλ

[1

2rλ2 + gλ4

]ρ(λ)− 2


Thanks to the factor N2 the large N average 〈f〉 will be given by theconfiguration of eigenvalues which minimizes this action

〈f〉 =∑i

f(xi) =

∫dλ f(λ)ρ(λ)

Euler-Lagrange equation for ρ is

rλ+ 4gλ3 = 2P

∫dλ′

ρ(λ′)

λ− λ′

which can be solved by complex analysis, with an additional assumptionon the support of the distribution



In this limit ∑i

g(xi) = N

∫dx g(λ)ρ(λ)

So the effective action becomes

Seff =

∫dλ

[1

2rλ2 + gλ4

]ρ(λ)− 2


Thanks to the factor N2 the large N average 〈f〉 will be given by theconfiguration of eigenvalues which minimizes this action

〈f〉 =∑i

f(xi) =

∫dλ f(λ)ρ(λ)

Euler-Lagrange equation for ρ is

rλ+ 4gλ3 = 2P

∫dλ′

ρ(λ′)

λ− λ′

which can be solved by complex analysis, with an additional assumptionon the support of the distribution



Assuming the support to be only a single interval (−a, a), we get thedistribution (normalized to 1, not N)

ρ(λ) =1

π

(1

2r + ga2 + 2gλ2

)√a2 − λ2

a2 =1

6g

(√r2 + 48g − r

)




ρ(λ) =1

π

(1

2r + ga2 + 2gλ2

)√a2 − λ2

a2 =1

6g

(√r2 + 48g − r

)for g = 0 the Wigner semicircle




ρ(λ) =1

π

(1

2r + ga2 + 2gλ2

)√a2 − λ2

a2 =1

6g

(√r2 + 48g − r

)for nonzero g




ρ(λ) =1

π

(1

2r + ga2 + 2gλ2

)√a2 − λ2

a2 =1

6g

(√r2 + 48g − r

)for negative r the potential has two wells




ρ(λ) =1

π

(1

2r + ga2 + 2gλ2

)√a2 − λ2

a2 =1

6g

(√r2 + 48g − r

)for negative r less dense around origin



If we allow negative r, the dip in the potential can be so deep that itsplits the eigenvalues completely.



If we allow negative r, the dip in the potential can be so deep that itsplits the eigenvalues completely.At that point, the formula

ρ(λ) =1

π

(1

2r + ga2 + 2gλ2

)√a2 − λ2

becomes negative and looses the interpretation of a probabilitydistributionThis happens, if

r ≤ −4√g

We need to change the one-cut assumption and look for a distributionsupported on two disjoint intervals (−a,−b) ∪ (b, a) with the solution

ρ(λ) =2g|λ|π

√(a+ λ)(a− λ)(λ+ b)(λ− b)

a2 =|r|4g

+1√g, b2 =

|r|4g− 1√g

We thus obtain the phase diagram.Juraj Tekel Phases of fuzzy field theories

Asymmetric eigenvalue distributions

What if we start with probability distribution, which is not symmetric, eg.

S[M ] =1

N

[1

2rTr

(M2)

+ hTr(M3)

+ gTr(M4)]

we still get the one-cut and the two-cut distributions, but qualitativelydifferent third possibility - an asymmetric one-cut distribution.This happens if one of the wells of the potential is deep enough.


Fuzzy field theory matrix model


As we have seen, action of the fuzzy CPn is given by

S =1

N

[1

2Tr (M [Li, [Li,M ]]) +

1

2rTr(M2)

+ gTr(M4)]

= Skin[M ] +1

N

[1

2rTr(M2)

+ gTr(M4)]

and the correlation functions are computed as

〈F 〉 =

∫dM F [M ]e−N

2S[M ]∫dM e−N2S[M ]

=→ dλF (λ)ρ(λ)

the same kind of integrals as for a matrix model before


A clear problem is that the diagonalization trick no longer works

dM e−N2S[M ] = dΛ ∆2 e−NTr( 1

2 rM2+gM4)dU e−N

2Skin[U†ΛU ]

As long as the function F is invariant, we can write∫dU e−N

2Skin[U†ΛU ] = e−N2Seff [Λ]

and the model becomes a quartic model with new terms

S[Λ] = Seff [Λ] +1

N

[1

2r∑i

λ2i + g

∑i

λ4i

]− 2

N2

∑i<j

log(λi − λj)

the new terms change the EOM for the eigenvalue distribution


Numerical results

dM integral can be treated numerically and computed by Monte-Carlo.Most recently by B. Ydri [arXiv:1401.1529]The following picture from F. Garcia Flores, X. Martin and D. O’Connor[arXiv:0903.1986]


Numerical results


Perturbative calculation

One possible analytic treatment due to D. O’Connor and Ch. Sämann[arXiv:0706.2493,arXiv:1003.4683]Expand the exponential in powers of kinetic term

e−aTr(M [Li,[Li,M ]]) = 1−aTr (M [Li, [Li,M ]])+1

2a2 [Tr (M [Li, [Li,M ]])]

2+. . .

and perform the dU integral explicitly.For the fuzzy sphere, the result is (up to third order in a)

Skin = a1

2

(c2 − c21

)− a2 1

24

(c2 − c21

)2+ a3 1

432

(c3 − 3c1c2 + 2c31

)2where

cn = Tr (Mn)

Note, that we obtain a multi-trace model, since the effective actioncontains powers of traces of M .We will see how odd multi-traces generate asymmetric solutions.The terms become very nasty and difficult to handle beyond the thirdorder.


Bootstrap

Most recently, Ch. Sämann [arXiv:1412.6255] introduced a method usinga version of Schwinger-Dyson equations to compute the effective action upto fourth order in a, with and extra term (in the symmetric c1 = 0 case)

a4 1

2880m4

2 − a4 1

3456

(m4 − 2m2

2

)2From here on, we rescale the matrix M and couplings r, g with N in sucha way, that all the terms in the effective action contribute in the large Nlimit, i.e. are of order N2.


Different Bootstrap

Finaly, A. Polychronakos [arXiv:1306.6645] used the following method tocompute a part of the effective action non-perturbatively.It starts from previous result, that the free theory g = 0 eigenvaluedistribution is still a semicircle, with a rescaled radius. The effectiveaction can thus be split into two parts

Seff =1

2F (m2) +R[mn − tn]

where mn are the symmetrized moments of the distribution

mn = Tr(M − 1

NTr (M)

)n

and tn are the (symmetrized) moments of the semicircle distribution

t2n = Cnmn2

and odd vanish.


Computation of F (t)

We will concentrate on F part, i.e. we work with the action

S[Λ] =1

2F

(∑i

λ2i

)+

1

N

[1

2r∑i

λ2i + g

∑i

λ4i

]− 2

N2

∑i<j

log(λi − λj)

=1

2F (c2) +

1

N

[1

2rc2 + gc4

]− 2

N2

∑i<j

log(λi − λj) (1)

we have assumed a symmetric distribution c1 = 0, keeping this in mindlater.The corresponding EOM for the minimizing eigenvalue distribution is

1

2F ′(c2)

(dc2dλi

)+

1

2r

(dc2dλi

)+ g

(dc4dλi

)= [F ′(c2) + r]λi + 4gλ3

i

= 2∑j

1

λi − λj



[F ′(c2) + r]λi + 4gλ3i = 2

∑j

1

λi − λj

We see, that the multi-trace term containing c2 acts as an effective mass,which depends on the second moment of the distribution. In the sameway, terms containing c1 and c3 will introduce terms into the effectivepotential that break the symmetry.One can use this equation to derive the equation for the radius of theg = 0 semicircle. Then, we need to make sure that the resultingdistribution yields a correct second moment.We get two equations that determine the F (t) function completely.



This approach is applicable for a more general theory, where

Skin[M ] =1

2Tr (MKM)

for which the kinetic action is diagonal, i.e.

KTlm = K(l)Tlm



The final result is

F ′(t) + t =N2

f(t)

wheref(z) =

∑l

dim(l)

K(l) + z

For the fuzzy sphere, we can compute F (t) in closed form

F (t) = log

(t

1− e−t

)For a general CPn, f(z) is given in terms of hypergeometric function andthe equation is transcendental. We are left only with a perturbativecalculation

F (t) = A1t+A2t2 +A3t

3 + . . .



We can compare the coefficients An with the perturbative result

Skin = a1

2

(c2 − c21

)− a2 1

24

(c2 − c21

)2+ a4 1

2880

(c4 − 2c22

)2without too much surprise they do agree.Also for CP 2 and CP 3, which involve also a a3 term.


Phase transition

We can now compute the phase transition line. The two conditions wehave are

vanishing of the distribution → F ′(t) + r = −4√g

consistency on the second moment → t =1√g

This givesr(g) = −4

√g − F ′(1/√g)

For the fuzzy sphere this becomes

r(g) = −5√g − 1

1− e1/√g

For higher CPn only a perturbative expansion in powers of 1/√g.


Phase transition

The series fromr(g) = −4

√g − F ′(1/√g)

is asymptotic for small g. We could use Borel summation to resum theseries, but not enough terms are known for practical purposes.To obtain numerical results, we use Pade approximation by a rationalfunction.But this teaches us a lesson. The phase transition lines and otherexpressions we are going to encounter are not analytic around g = 0 andwe need to be very careful when interpreting the results of perturbativecomputations. For example for the above formula

r(g) = −4√g − 1

2+

1

12√g− 1

720g3/2+

1

30240g5/2+ . . .


Asymmetric distributions

To introduce asymmetry, we recall that we should have

F (t)→ F (c2 − c21)

yielding an equation for the distribution

F ′(c2 − c21)(λi − c1) + rλi + 4gλ3i = 2

∑j

1

λi − λj

We can think of this as an effective potential

V (λ) = −[F ′(c2 − c21)c1

]λ+

1

2

[F ′(c2 − c21) + r

]λ2 + gλ4

i.e. the advertised asymmetry. Note that for symmetric distributions theasymmetric term vanishes, so the one-cut/two-cut phase transition is notmodified.We now look for the solution in the form of an asymmetric one cut,supported on one interval (b, a). We again need to impose self consistencyequations on the first and second moments of the distribution.


Asymmetric distributions

We write the interval as (D −√δ,D +

√δ). The solution then is

ρ(λ) =

[4D2g + 2δg + r + F ′

(δ

4+δ3g

16− 9D2δ4g2

16

)+ 4Dgλ+ 4gλ2

]×

× 1

2π

√δ − (D − λ)2

The conditions on D, δ can be solved only perturbatively.To find the domain of existence of the asymmetric one cut solution, weseek the values of parameters when this distribution becomes negative.


Asymmetric phase transition

We can do this perturbatively

r = −2√

15√g +

4A1

5+

(135A2

1 + 848A2

1500√

15

)1√g

+

+

(2025A3

1 + 10170A1A2 + 11236A3

562500

)1

g+ . . .

where Ai’s fromF (t) = A1t+A2t

2 +A3t3 + . . .

Note, that we again obtain a series in 1/√g, but this time the series is

convergent.



We compute the intersection of the two curves

rC = −0.73 , gC = 0.022

The numerical result is




rC = −0.73 , gC = 0.022


[arXiv:0601012] , rC = −0.8± 0.08 , gC = 0.036± 0.013




rC = −0.73 , gC = 0.022


[arXiv:0601012] , rC = −0.8± 0.08 , gC = 0.036± 0.013

[arXiv:0903.1986] , rC = −2.3± 0.2 , gC = 0.13± 0.005




rC = −0.73 , gC = 0.022


[arXiv:0601012] , rC = −0.8± 0.08 , gC = 0.036± 0.013

[arXiv:0903.1986] , rC = −2.3± 0.2 , gC = 0.13± 0.005

[arXiv:1401.1529] , rC = −2.49± 0.07 , gC = 0.145± 0.025

What is the source of discrepancy?


Some topics of current research


Fourth order contribution

We can use Sämann’s result for the effective action for the computation.

Skin =a1

2

(c2 − c21

)− a2 1

24

(c2 − c21

)2+ a3 1

216

(c3 − 3c1c2 + 2c31

)2+

+ a4 1

2880(c2 − c21)4 − a4 1

3456

((c4 − 4c− 1c3 + c2c

31 − 3c− 14)− 2(c2 − c1)2

)2The main distinction is the presence of c24 term, which acts as an effectivecoupling.The properties of the asymmetric transition line are not altereddramatically. But the symmetric transition line gets shifted quitedrastically, moreover loosing Borel summability! Also Pade approximationis quite involved. The overall effect is to shift the triple point to evenlower values of r and g, further away from the numerical result.


Fourth order contribution

The fourth order effective coupling has another interesting feature. It canbe expressed as

geff = g + 2F1

(c40 − 2c220

)with F1 negative. So for distributions that give negative second term, onecan have geff > 0 even for g < 0.Such distributions are "spread out" more than semicircle, so for anegative reff , we might have a region of parameter space where we obtaina stable solution even for negative coupling.Two caveats

seems that this is not the case for fuzzy sphere,might be only a perturbative effect and the full effective action mightremove this effect.


Free energies

The sole existence of the asymmetric one-cut solution does not guarantee,that the solution is going to be realized. Even with no kinetic term,asymmetric solution exists as long as

r ≤ −2√

15√g

but is never realized since it has higher free energy as the double-cut.


Free energies

The sole existence of the asymmetric one-cut solution does not guarantee,that the solution is going to be realized. Even with no kinetic term,asymmetric solution exists as long as

r ≤ −2√

15√g

but is never realized since it has higher free energy as the double-cut.The result is still interesting, because it gives a top bound on the triplepoint, but for a complete result we should compute the free energy of thedistributions.Or do something else.


Outlook


What is next?

Complete treatment of the effective action.What is the non-perturbatve formula for the kinetic term effective action?Can it be computed?

Different sources of asymmetryLimit CPn → R2n.Phase diagram of the theories with no UV/IR mixing.Phase diagram nonenormalizable theories.Theory on R× S2

F .Theory on S4.


What is next?

Complete treatment of the effective action.Different sources of asymmetryContributions of nontrivial configurations to the saddle point, instantons.

Limit CPn → R2n.Phase diagram of the theories with no UV/IR mixing.Phase diagram nonenormalizable theories.Theory on R× S2

F .Theory on S4.


What is next?

Complete treatment of the effective action.Different sources of asymmetryLimit CPn → R2n.Phase diagram in the non-compact case, phase structure of the theory onnon-commutative plane, etc.

Phase diagram of the theories with no UV/IR mixing.Phase diagram nonenormalizable theories.Theory on R× S2

F .Theory on S4.


What is next?

Complete treatment of the effective action.Different sources of asymmetryLimit CPn → R2n.Phase diagram of the theories with no UV/IR mixing.Does the non-uniform order phase disapear from the diagram? What isthe mechanism? What are the lessons to be learned

Phase diagram nonenormalizable theories.Theory on R× S2

F .Theory on S4.


What is next?

Complete treatment of the effective action.Different sources of asymmetryLimit CPn → R2n.Phase diagram of the theories with no UV/IR mixing.Phase diagram nonenormalizable theories.Is there anything we can learn from the phase structure of theories, whichare not renormalizable even at the commutative level, eg. φ6 on R4 or φ4

on R6.

Theory on R× S2F .

Theory on S4.


What is next?

Complete treatment of the effective action.Different sources of asymmetryLimit CPn → R2n.Phase diagram of the theories with no UV/IR mixing.Phase diagram nonenormalizable theories.Theory on R× S2

F .Theory with a commutative time. There are similar perturbative andnumerical results available to compare with.

Theory on S4.


What is next?


F .Theory on S4.There is a realization of the theory on fuzzy S4 that is suitable for ourapproach.


What is next?


F .Theory on S4.


Thanks for your attention!


Phases of fuzzy field theories

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