Towards a CFT describing the 3D Isingmodel at criticality
A Project Report
submitted by
Sutapa Samanta(PH11C041)
in partial fulfilment of the requirementsfor the award of the degree of
MASTER OF SCIENCE (in PHYSICS)
DEPARTMENT OF PHYSICSINDIAN INSTITUTE OF TECHNOLOGY, MADRAS
CHENNAI-600 036,INDIAMAY 2013
THESIS CERTIFICATE
This is to certify that the thesis titled Towards a CFT describing 3DIsing Model at criticality, submitted by Sutapa Samanta, to the IndianInstitute of Technology, Madras, for the award of the degree of Master inScience, is a bona fide record of the research work done by her under mysupervision. The contents of this thesis, in full or in parts, have not beensubmitted to any other Institute or University for the award of any degreeor diploma.
Prof. Suresh GovindarajanProject GuideProfessorDept. of PhysicsIIT-Madras, 600 036
Place: Chennai
Date: 26th April 2013
ACKNOWLEDGEMENTS
I would like to thank my guide and teacher Prof. Suresh Govindarajanfor guiding me throughout the project in a organized way and teaching mephysics over the past two years. Doing hard work and trying to actuallysolve a problem with pen and paper is learnt from him. His endless energyand patience inspired me a lot.
I would also like to thank Prof. Rajesh Narayanan for helping me withvaluable discussions that really provides me a good understanding of severaltopics related to this project.
i
Contents
ACKNOWLEDGEMENTS i
LIST OF FIGURES iv
1 Introduction 11.1 Organization of the thesis . . . . . . . . . . . . . . . . . . . . 2
2 Conformal Invariance 32.1 Conformal Group . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Conformal invariance . . . . . . . . . . . . . . . . . . . . . . . 82.3 Constraints on two, three and four point functions . . . . . . . 82.4 Conformal partial wave and differential equation in d-dimension 12
2.4.1 Solution in two and four dimensions . . . . . . . . . . . 142.4.2 Solution in three dimension . . . . . . . . . . . . . . . 16
2.5 The three-dimensional Ising model . . . . . . . . . . . . . . . 17
3 The O(N) symmetric universality class 183.1 The O(N) model . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Construction of Hamiltonian . . . . . . . . . . . . . . . 183.1.2 Mean field approximation: . . . . . . . . . . . . . . . . 193.1.3 Continuous symmetry breaking and Goldstone mode . 213.1.4 Effect of fluctuations . . . . . . . . . . . . . . . . . . . 21
3.2 Scaling hypothesis . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Renormalization Group . . . . . . . . . . . . . . . . . . . . . . 243.4 Solution of Gaussian model using RG . . . . . . . . . . . . . . 26
4 Renormalization perturbation to solve Φ4 model 294.1 Perturbative expansion of Φ4 term in the power of u . . . . . 294.2 First order perturbation: . . . . . . . . . . . . . . . . . . . . . 324.3 β-functions at first order . . . . . . . . . . . . . . . . . . . . . 374.4 Second order perturbation theory . . . . . . . . . . . . . . . . 39
ii
4.5 The Wilson-Fisher fixed point: . . . . . . . . . . . . . . . . . . 41
5 The spectrum of the O(N) model 445.1 Character of unitary representation of d = 3 conformal group: 445.2 Primary operators of O(N) Theory . . . . . . . . . . . . . . . 46
Conclusion and way to further research 49
Appendices 50
A Labeling the eigenstates of Conformal Group 50
B Polynomials 52B.1 Associated Legendre polynomial . . . . . . . . . . . . . . . . . 52B.2 Gegenbauer polynomial . . . . . . . . . . . . . . . . . . . . . . 53
iii
List of Figures
4.1 Elements of the diagrammatic representation of perturbation theory 324.2 Diagrammatic representation of Two point function . . . . . . 334.3 Diagrammatic representation of a vertex . . . . . . . . . . . . 334.4 Diagrammatic representation of 〈~m(q1) · ~m(q2) ~m(q3) · ~m(q4)〉 344.5 Diagrammatic representation of 〈~µ(q1) · ~m(q2) ~m(q3) · ~m(q4)〉 344.6 Diagrammatic representation of 〈~µ(q1) · ~µ(q2) ~m(q3) · ~m(q4)〉 354.7 Diagrammatic representation of 〈~µ(q1) · ~m(q2) ~µ(q3) · ~m(q4)〉 354.8 Diagrammatic representation of 〈~µ(q1) · ~µ(q2) ~µ(q3) · ~m(q4)〉 364.9 Diagrammatic representation of 〈~µ(q1) · ~µ(q2) ~µ(q3) · ~µ(q4)〉 . 364.10 Number of terms = 8 . . . . . . . . . . . . . . . . . . . . . . . 404.11 Number of terms = 32 . . . . . . . . . . . . . . . . . . . . . . 404.12 Number of terms=32 . . . . . . . . . . . . . . . . . . . . . . . 40
iv
Chapter 1
Introduction
The 2D Ising model at criticality is well defined as a Conformal Field Theory.Its spectrum and correlation function have been determined. But the Isingmodel in 3D is not known that well. Our aim is to describe the spectrumand correlation function of the 3D Ising model at critical point.
In order to proceed towards the solution of 3D Ising model at criticality,we are assuming that it is invariant under full conformal group at criticality.In this project, we have started with learning the global conformal groupand conformal algebra. Four-point correlation functions are of particularinterest as it can be written as the functions of conformally invariant cross-ratios. We have considered four point function as a eigenfunction of Casimiroperator of Conformal group[2] and obtained the differential equation fromthe corresponding eigenvalue equation by transforming the co-ordinate to thehomogeneous co-ordinate for general d-dimension. Since we are interested in3-dimension, we tried to understand the solution of the differential equationin 3-dimension as obtained in [4].
Universality of critical exponents suggests to generalize all the phasetransition phenomena into one model with N -component vector field in d-dimension, known as O(N) universality class. N = 1 corresponds to uni-versality class of the Ising Model. Critical points are the fixed points underrenormalization group with co-dimension one, that is with only one relevantoperator associated with it. The free O(N) theory known as Gaussian Modelhas one fixed point at d ≥ 4, known as Gaussian Fixed Point. For d < 4,Φ4 interaction becomes relevant and need to do renormalization perturbationto obtain the fixed point for small ǫ = 4 − d. Wilson introduced the idea ofthis kind of expansion which is called the ǫ expansion. Under renormaliza-tion perturbation, the O(N) model in d ≤ 4 flows towards a non-trivial fixedpoint known as Wilson-Fisher fixed point.
The critical O(N) theory is obtained from N free scalars, Φ, perturbed
1
by a potential of the form (Φ ·Φ)2. This perturbation makes the theory flowfrom UV(Gaussian Fixed point) to the IR(Wilson-Fisher fixed point), a newconformal field theory[9]. The UV theory has higher spin symmetry – it hasan infinite number of higher spin currents that are conserved in the large Nlimit. But the IR CFT has spin symmetry broken by 1
Neffect for large N .
The free theory spectrum contains single trace currents which falls into theshort representation of Conformal algebra. The interacting theory includesmulti-trace operators that fall into long representation of Conformal Algebragiving rise to 1
Ncorrections to the conserved currents[10].
1.1 Organization of the thesis
In the second chapter we introduce the Conformal group and Conformalalgebra associated with it. Then we define the correlation functions andobtained conditions for those imposing conformal symmetry. We identifythe four point function as the eigenfunction of Casimir operator of Conformalgroup and obtain the differential equation satisfied by it from the eigenvalueequation, study the solution of those differential equations in some lowerdimensions. Finally introduce the idea of three-dimensional Ising model.
The chapter three is dedicated to solve the O(N) symmetric model with-out any interaction i.e free O(N) theory. First we have constructed theHamiltonian of the theory from the symmetry, applied mean field approx-imation to it and seen how it breaks for physical systems. Then motivatethe idea of scaling hypothesis and Introduce Renormalization Group. As anapplication to it, we have solved the simple Gaussian model using Renormal-ization group.
The fourth chapter is sort of continuation of third chapter. Now we addthe interaction to the free theory known as Φ4 theory. Applied renormaliza-tion group very rigorously to solve that. Here we are able to reach a fixedpoint which determines criticality for dimension d < 4 after doing a secondorder perturbation.
In the fifth chapter we get back the Conformal Group again but this timewe tried relate it to O(N) symmetric theory. We obtain the operator spec-trum for O(N) theory for large N limit after imposing conformal symmetryon it.
At the last chapter we have tried to conclude the work done so far andget a direction to proceed further.
2
Chapter 2
Conformal Invariance
This chapter is a review of conformal group and conformal algebra. Theconformal condition i.e preservation of angle imposes several conditions [1]on co-ordinate transformation. From these conditions we can extract theform of transformations and corresponding generators. Then we look atthe form of correlation functions. As we can construct conformal invariantsusing four point function, we tried to investigate little bit more about it. Wefound out the differential equation satisfied by this four point function ind-dimension and tried to identify the Hypergeometric function as a solutionto this differential equation in different dimensions.
2.1 Conformal Group
Conformal Group in d-dimension
Let us consider a flat space in d-dimension and transformations thereof whichlocally preserve the angle between two lines. Such transformations are calledconformal transformations. Let gµν be the metric tensor in a space-time ofdimension d. By definition a conformal transformation of the coordinates isan invertible mapping x→ x′ which leaves the metric tensor invariant up toa scale.
g′µν(x′) = Λ(x)gµν(x) , (2.1)
where the positive factor Λ(x) is called the scale factor.The set of conformal transformation manifestly forms a group. When
gµν = ηµν , the Poincare group is the subgroup of conformal transformationssince the Poincare group corresponds to special case when Λ(x) = 1. Forx
′µ = xµ+ǫµ(x), a infinitesimal change of co-ordinates corresponds to change
3
in tensor gµν :
g′µν =∂xα
∂x′µ
∂xβ
∂x′νgαβ
= (δαµ − ∂µǫα)(δβν − ∂νǫ
β) gαβ
= gµν − (∂µǫν + ∂νǫµ) +O(ǫ2).
For simplicity we will consider flat space with a constant metric of the formη = diag(−1, . . . ,+1, . . .). Considering terms up to first order in ǫ
ηµν → ηµν − (∂µǫν + ∂νǫµ)
For the transformation parametrized by ǫ to be conformal, we need
∂µǫν + ∂νǫµ = f(x) ηµν , (2.2)
where f(x) is some function of x and can be determined by taking trace inthe both side of the equation with ηµν . We obtain
f(x) =2
d(∂ρǫ
ρ) =2
d(∂ · ǫ) .
Thus the condition for conformality of the co-ordinate transformation can bewritten as
(∂µǫν + ∂νǫµ) =2
d(∂ · ǫ) ηµν . (2.3)
By taking derivatives ∂ρ of the above equation (Eq. (2.3)) and permutingindices we obtain
∂ρ∂µǫν + ∂ρ∂νǫµ =2
d∂ρ(∂ · ǫ) ηµν .
∂ν∂ρǫµ + ∂µ∂ρǫν =2
d∂ρ(∂ · ǫ) ηρµ ,
∂µ∂νǫρ + ∂ν∂µǫρ =2
d∂ρ(∂ · ǫ) ηνρ .
Subtracting the first line from the sum of last two lines we have
2∂µ∂νǫρ =2
d(−ηµν∂ρ + ηρµ∂ν + ηνρ∂µ)(∂.ǫ) (2.4)
Upon contracting with ηµν we obtain
2∂2ǫρ =2(2− d)
d∂ρ(∂ρǫ
ρ) = (2− d)∂ρf . (2.5)
4
Applying ∂ν on the both sides of this equation, we get
2∂ν∂2ǫµ = (2− d)∂µ∂νf . (2.6)
Applying ∂2 on Eq. (2.2)
∂2∂µǫν + ∂2∂νǫµ = ∂2f(x)ηµν .
Combining this with Eq. (2.6), we obtain
2∂ν∂2ǫµ = ∂2f(x)ηµν . (2.7)
Equating the right hand sides (RHS) of Eqs. (2.6) and (2.7), we find
∂2f(x)ηµν = (2− d)∂µ∂νf (2.8)
Upon contracting with ηµν , we obtain
(d− 1)∂2f(x) = 0 . (2.9)
This implies that for d > 1, f(x) is a harmonic function. We will considerthe case d ≥ 3. For d ≥ 3, Eqs. (2.7) and (2.8) imply that
∂µ∂νf = 0 . (2.10)
The solution of this differential equation is given by
f(x) = A+Bµxµ . (2.11)
Thus the function f(x) is linear in co-ordinates and as f(x) = 2d∂ρǫ
ρ, ǫ canbe at most quadratic in co-ordinates. This motivates the following ansatz forǫ:
ǫµ = aµ + bµν xν + cµνρ xνxρ (2.12)
where cµνρ is symmetric in last two indices. We will now study various termsof equation (2.12) separately because the constraints for conformal invariancehave to be independent of position xµ.
1st term: The constant term aµ is not constrained by equation(2.2). It describesinfinitesimal translation x′µ = xµ + aµ, and corresponding generator ismomentum operator, Pµ = −i∂µ.
2nd term: Substituting the linear term in equation (2.2)
bµν + bνµ =2
dbρρηµν .
5
We see that the traceless symmetric part of bµν vanishes and thus bµνis the sum of an antisymmetric part and a pure trace. We thus writebµν = αηµν + mµν , where mµν = −mνµ is the antisymmetric part andαηµν is the pure trace part. The pure trace part αηµν corresponds toinfinitesimal scale transformation:
x′µ = (1 + α)xµ , (2.13)
with generator, D = −ixµ∂µ. The antisymmetric part corresponds toinfinitesimal rotations (and boosts)
x′µ = (δµν +mµν)x
ν , (2.14)
with generator Lµν = i(xµ∂ν − xν∂µ).
3rd term: Substituting the quadratic term in equation (2.4) and using ∂ · ǫ =bµµ + 2cµµρx
ρ, we have
cµνρ = ηµρbν + ηµνbρ − ηνρbµ , (2.15)
where bµ := 1dcσσµ The resulting transformations are called Special
Conformal transformations (SCT) and have the form
x′µ = xµ + 2 (b · x) xµ − x2 bµ ,
and the corresponding generators are given by
Kµ = −i (2xµxν ∂ν − (x · x) ∂µ) .
These transformations are infinitesimal conformal transformation. Howeverin order to determine conformal group, we will need finite conformal trans-formations. The finite conformal transformations corresponding to the aboveare
1. x′µ = xµ + aµ (Translation)
2. x′µ = α xµ (Dilation)
3. x′µ = Mµν xν (Rigid rotation)
4. x′µ = xµ−bµx2
1−2bx+b2x2 (Special conformal transformation)
6
The special conformal transformation can be written as
x′µ
x′ · x′=
xµ
x · x − bµ .
The above relation says that the special conformal transformation can beunderstood as an inversion of xµ, followed by a translation bµ and againfollowed by an inversion.
The generators of above Conformal transformations are:
• Pµ = −i∂µ (Translation)
• D = −ixµ∂µ (Dilation)
• Lµν = i(xµ∂ν − xν∂µ) ( rotation)
• Kµ = −i (2xµxν∂ν − x2∂µ) (SCT)
These obey following algebra:
[D,Pµ] = iPµ
[D,Kµ] = −iKµ
[Kµ, Pµ] = 2i(ηµνD − Lµν)
[Kρ, Lµν ] = i(ηρµKν − ηρνKµ (2.16)
[Pρ, Lµν ] = i(ηρµPν − ηρνPµ)
[Lµν , Lρσ] = i(ηνρLµσ + ηµσLµσ − ηµρLνσ − ηνσLµρ)
We will now see that this algebra can be written in compact form. Let usdefine Jm,n with m,n = −1, 0, . . . , d as follows:
Jµ,ν = Lµ,ν , J−1,µ =1
2(Pµ +Kµ) , J−1,0 = D and J0,µ =
1
2(Pµ −Kµ) .
The Jmn satisfy the following commutation relation:
[Jmn, Jrs] = i(ηmsJnr + ηnrJms − ηmrJns − ηnsJmr) , (2.17)
where η−1,−1 = −1 and η0,0 = 1. Thus for d-dimensional Euclidean space, thed+2 dimensional metric ηmn used above is ηmn = diag(−1, 1, . . . , 1) and thecommutation relation (2.17) is the commutation relation of the Lie algebraso(d + 1, 1). More generally, if the d-dimensional metric ηµν has signature(p, q), then the (d+2) dimensional metric ηmn has signature (p+1, q+1) andthe commutation relations are identical to the Lie algebra so(p+ 1, q + 1).
7
2.2 Conformal invariance
Let us consider conformally invariant functions Γ(N)(x1, . . . , xN ) of N points.We shall now study the constraints on their forms due to conformal invari-ance.
1. Translational invariance implies that Γ(N)(x1, . . . , xN)should be a func-tion of differences of pairs of co-ordinates i.e, (xi−xj). We need N ≥ 2to create such pairs.
2. Rotational invariance implies that Γ(N)(x1, . . . , xN) should be a func-tion of only the absolute value of pairwise co-ordinate differences i.e,|xi − xj |.
3. Scale invariance needs Γ(N)(x1, . . . , xN) to a function to be of ratios
such as|xi−xj |
|xj−xk|. We need N ≥ 3 to construct such ratios.
4. Under special conformal transformation, the distance between two pointstransform as
|x′i − x′
j | =|xi − xj |
(1− 2b.xi + b2x2i )
12 (1− 2b.xj + b2x2
j )12
.
So the function Γ(N)(xi) will be invariant under special conformal trans-formation if it is a function of ratios involving at least four points i.e,N ≥ 4:
|xi − xj ||xk − xl||xi − xk||xj − xl|
. (2.18)
In summary, it is impossible to construct conformally invariant functionsfor N = 2, 3 though there are scale-invariant functions appearing at N =3. The simplest conformally invariant function involves four points and itis necessarily a function of the combinations |x1−x2||x3−x4|
|x1−x3||x2−x4|and |x1−x2||x3−x4|
|x1−x4||x2−x3|.
These combinations are called anharmonic ratios or cross-ratios. ForN > 3, the number of independent cross-rations is N(N − 3)/2.
2.3 Constraints on two, three and four point
functions
A field theory is conformal invariant at classical level requires the actionshould be conformal invariant. This symmetric could be broken by quantumcorrections as quantum field theories introduce dependence on scale. Thus
8
Conformal symmetry at classical level breaks in the quantum level except forcertain value of parameters. We will now study the restrictions on variouscorrelation functions due to conformal invariance.
Irreducible representations of the conformal algebra carry two sets oflabels: one arises from the rotation/boosts and is nothing but the spin ofthe representation and the second arises from scaling and is called the scalingdimension. In quantum field theory, we will consider operators that transformin such irreducible representations. Such operators/fields are called quasi-primary operators/fields. For simplicity, let us consider quasi-primary spinzero fields ϕ1(x1), ϕ2(x2). A quasi primary field ϕj(x) of (scaling) dimension∆j transforms as follows under conformal transformations:
ϕj(x) =
∣
∣
∣
∣
∂x′
∂x
∣
∣
∣
∣
∆j/d
ϕj(x′) (2.19)
under global conformal transformation x → x′. The N -point function of Nquasi-primary fields is defined as in any quantum field theory as the path-integral
〈ϕ1(x1) · · ·ϕN(xN)〉 =1
Z
∫
[dΦ] ϕ1(x1) · · ·ϕN(xN) e−S[Φ] (2.20)
where [dΦ] is the path integral measure for the fields Φ that define the quan-tum field theory and Z is partition function. We will now study the con-straints on two, three and four point functions of quasi-primary fields in aCFT.
Two-point functions:
The quasi-primality of the two fields entering the two-point correlation func-tion transforms as
〈ϕ1(x1)ϕ2(x2)〉 =∣
∣
∣
∣
∂x′
∂x
∣
∣
∣
∣
∆1d∣
∣
∣
∣
∂x′
∂x
∣
∣
∣
∣
∆2d
〈ϕ1(x′1)ϕ2(x
′2)〉 (2.21)
Let us now consider the scale transformation
x→ λ x .
Under this transformation equation (2.21) becomes
〈ϕ1(x1)ϕ2(x2)〉 = (λd)∆1d (λd)
∆2d 〈ϕ1(x
′1)ϕ2(x
′2)〉
〈ϕ1(x1)ϕ2(x2)〉 = λ∆1+∆2〈ϕ1(x′1)ϕ2(x
′2)〉 .
9
Now translational invariance requires 〈ϕ1(x1)ϕ2(x2)〉 = f(x1 − x2). Next,rotational invariance requires 〈ϕ1(x1)ϕ2(x2)〉 = f(|x1 − x2|). Under scaletransformations f(x) = λ∆1+∆2f(λx). Thus f(x) has the form
〈ϕ1(x1)ϕ2(x2)〉 = f(|x1 − x2|) =c12
|x1 − x2|∆1+∆2,
since there is no nontrivial conformally invariant function of two co-ordinates.It remains to study what happens to the two-point function under special
conformal transformations:
x′µ =xµ − bµx2
1− 2bx+ b2x2.
We find that one needs
c12|x1 − x2|∆1+∆2
=1
γ∆1
1 γ∆2
2
c12(γ1γ2)∆1+∆2
2
|x1 − x2|∆1+∆2
where γi = (1 − 2b.xi + b2x2i ). For c12 6= 0, this constraint is satisfied only
if ∆1 = ∆2. Thus we obtain that two-point functions in any conformallyinvariant field theory must have the form:
〈ϕ1(x1)ϕ2(x2)〉 ={
c12|x1−x2|2∆1
, if ∆1 = ∆2 ,
0 , if ∆1 6= ∆2 .(2.22)
Three-point functions:
Covariance under translation, rotation and dilation require three point func-tions to be of the form
〈ϕ1(x1)ϕ2(x2)ϕ3(x3)〉 =c123
xa12x
b23x
c13
,
where c123 is a constant and a + b + c = ∆1 +∆2 +∆3 and xij = |xi − xj|.Under special conformal transformations one has 1
(x′ij)
a =(γiγj)a/2
xaij
. Thus
〈ϕ1(x′1)ϕ2(x
′2)ϕ3(x
′3)〉 =
c123x
′a12x
′b23x
′c13
=c123(γ1γ2)
a/2(γ2γ3)b/2(γ1γ3)
c/2
xa12x
b23x
c13
.
Now,
〈ϕ1(x1)ϕ2(x2)ϕ3(x3)〉 =∣
∣
∣
∣
∂x′
∂x
∣
∣
∣
∣
∆1/d
x=x1
∣
∣
∣
∣
∂x′
∂x
∣
∣
∣
∣
∆2/d
x=x2
∣
∣
∣
∣
∂x′
∂x
∣
∣
∣
∣
∆3/d
x=x3
〈ϕ1(x′1)ϕ2(x
′2)ϕ3(x
′3)〉
⇒ C(abc)123
xa12x
b23x
c13
=C
(abc)123
γ∆1
1 γ∆2
2 γ∆3
3
(γ1γ2)a/2(γ2γ3)
b/2(γ1γ3)c/2
xa12x
b23x
c13
10
⇒ ∆1 =a
2+
c
2, ∆2 =
a
2+
b
2, ∆3 =
b
2+
c
2
⇒ a = ∆1 +∆2 −∆3 , b = ∆2 +∆3 −∆1 , c = ∆3 +∆1 −∆2 .
So correlation function of three quasi-primary field has the form
〈ϕ1(x1)ϕ2(x2)ϕ3(x3)〉 =c123
x∆1+∆2−∆3
12 x∆2+∆3−∆1
23 x∆3+∆1−∆2
13
. (2.23)
-
Four-point functions:
For two and three-point functions, the non-existence of non-trivial confor-mally invariant functions lead to fairly simple expressions involving constantsc12 and c123 and the scaling dimensions of the fields. However, this is nolonger true for four-point functions and we will see the appearance of anundetermined function of cross-ratios defined in Eq. (2.18).
Correlation function of four scalar primaries ϕi of dimension ∆i, which isfixed by conformal invariance have the form
〈ϕ1(x1)ϕ2(x2)ϕ3(x3)ϕ4(x4)〉 =(
x224
x214
)1
2∆12(
x214
x213
)1
2∆34
× g(u, v)
(x212)
12(∆1+∆2)(x2
34)12(∆3+∆4)
, (2.24)
where xij ≡ (xi− xj), ∆ij ≡ (∆i−∆j) and g(u, v) is a conformally invariantfunction of the two independent cross-ratios
u =x212x
234
x213x
224
, v =x214x
223
x213x
224
.
We will now show that functions such as g(u, v) satisfy certain differentialequations whose solutions provide a basis for conformally invariant functions.This is called the conformal partial wave expansion in analogy with the par-tial wave expansion in scattering theory where spherical waves provide thebasis.
11
2.4 Conformal partial wave and differential
equation in d-dimension
The conformal group in d dimensional Euclidean space is isomorphic to thegroup SO(d + 1, 1) and can be realized as linear transformation in d + 2dimensional projective space[2, 3]. The homogeneous co-ordinate ηm on theprojective null cone gmnη
mηn = η2 = 0, with m,n ∈ −1, 0, 1, 2, . . . , d and thed+ 2 dimensional metric gmn = diag(−1, 1, 1, . . . , 1) .
Conformal transformations act on η linearly ηm ∼ ληn and the corre-sponding operators are
Jmn = ηm∂
∂ηn− ηn
∂
∂ηm(2.25)
The four-point correlation function of four spin-zero quasi-primaries in termsof homogeneous coordinates is given by
〈ϕ1(η1)ϕ2(η2)ϕ3(η3)ϕ4(η4)〉 =1
(η1 · η2)12(∆1+∆2)(η3 · η4)
12(∆3+∆4)
×(
η2 · η4η1 · η4
)12∆12(
η1 · η4η1 · η3
)12∆34
F (u, v) (2.26)
where ∆ij = (∆i − ∆j) as before and u, v are two conformally invariantcross-ratios:
u =η1 · η2 η3 · η4η1 · η3 η2 · η4
, v =η1 · η4 η2 · η3η1 · η3 η2 · η4
.
The conformal partial waves G(l)∆ (u, v), that we will define later provide a
basis for the conformal wave expansion
F (u, v) =∑
∆,l
a∆,l G(l)∆ (u, v) . (2.27)
The G(l)∆ (u, v) is the contribution of an operator O
(l)∆ with scale dimension ∆
and spin l and its descendants that appear in the operator product expansionof ϕ1ϕ2 or ϕ3ϕ4. This provides constraints on the spectrum of quasi-primariesin any CFT.
The operator O(l)∆ and also all its descendants are eigenvectors of Casimir
operators formed form the generator Jmn = −Jnm of conformal group SO(d+
1, 1) action on O(l)∆ and which obey the algebra
[Jmn, Jrs] = i(ηmsJnr + ηnrJms − ηmrJns − ηnsJmr) ,
12
with ηmn = diag(−1, 1, . . . , 1).
J2〈ϕ1(η1)ϕ2(η2)ϕ3(η3)ϕ4(η4)〉 = −C∆,l〈ϕ1(η1)ϕ2(η2)ϕ3(η3)ϕ4(η4)〉
where J2 = 12JmnJmn. The Casimir value −C∆,l can be calculated from
group theoretical considerations.
J2 =1
2JmnJmn =
1
2JµνJµν + J−1,0J−1,0 + J−1µJ−1µ + J0µJ0µ
with m,n ∈ −1, 0, 1, 2, 3, 4 and µ, ν ∈ 1, 2, 3, 4. Recall that
Jµ,ν = Lµ,ν , J−1,µ =1
2(Pµ +Kµ) , J−1,0 = D , J0,µ =
1
2(Pµ −Kµ) .
Thus, we see that
J2 =1
2JµνJµν −D2 −KµP
µ − idD .
The quadratic Casimir is determined in terms of the scaling dimension andspin to be[4]
C∆,l = ∆(∆− d) + l(l + d− 2) . (2.28)
The Casimir value for dimension d = 3 can be evaluated easily. In thisdimension the quadratic casimir looks like
J2 =1
2J ijJij −D2 −KiP
i − i3D .
The term 12J ijJij is the quadratic Casimir of SU(2) with eigenvalue l(l+ 1),
where l is the spin of the operator. If ∆ be the scaling dimension, the highestweight state of the operator is given by |l,∆〉. Since P is the raising operator,it gives zero acting on the highest weight state. So the quadratic Casimir ind = 3 acting on the state |l,∆〉 gives (Obtained explicitely in Appendix A)
J2|l,∆〉 =(
l(l + 1) + ∆2 − 3∆)
|l,∆〉= ∆(∆− 3) + l(l + 1) .
Now consider the four-point function in the limit where x1 → x2. Let J :=J1+J2 denote the sum of the so(d+1, 1) generators acting on the co-ordinatesx1 and x2. In this limit, one anticipates that
limx1→x2
ϕ1(x1)ϕ(x2) ∼∑
∆,l
a∆,l |x1 − x2|α∆,l [O∆,l(x2)] , (2.29)
13
[O] denotes the quasi-primary field O and all its descendants. The sumruns over all quasi-primaries that appear in the fusion of ϕ1 and ϕ2. This isusually called the operator product expansion or OPE in short. The essentialequation determining the conformal partial wave associated with [O∆,l] isgiven by the action of J2 = (J1+J2)
2 on the four-point function and isolatingthe contribution of [O∆,l], we get
1
2(J1 + J2)mn(J1 + J2)
mn〈ϕ1(η1)ϕ2(η2)ϕ3(η3)ϕ4(η4)〉= −C∆,l 〈ϕ1(η1)ϕ2(η2)ϕ3(η3)ϕ4(η4)〉 (2.30)
Now using the expression for four point function in terms of homogeneousco-ordinates from (2.26), above equation becomes
1
2(J1 + J2)mn(J1 + J2)
mn 1
(η1 · η2)12(∆1+∆2)(η3 · η4)
12(∆3+∆4)
×(
η2 · η4η1 · η4
)12∆12(
η1 · η4η1 · η3
)12∆34
a∆,l G(l)∆ (u, v) =
− C∆,l1
(η1 · η2)12(∆1+∆2)(η3 · η4)
12(∆3+∆4)
×(
η2 · η4η1 · η4
)12∆12(
η1 · η4η1 · η3
)12∆34
a∆,l G(l)∆ (u, v) . (2.31)
After a long calculation [4], the eigenvalue equation becomes
J2G(l)∆ (u, v)−(∆12−∆34)
(
(1 + u− v)(u∂
∂u+ v
∂
∂v)− (1− u− v)
∂
∂v
)
G(l)∆ (u, v)
− 1
2∆12∆34(1 + u− v)G
(l)∆ = −C∆,lG
(l)∆ (u, v) , (2.32)
where,
1
2J2G(u, v) = −
(
(1− v)2 − u(1 + v)) ∂
∂vv∂
∂vG(u, v)−(1−u+v)u
∂
∂uu∂
∂uG(u, v)
+ 2(1 + u− v)uv∂2
∂u∂vG(u, v) + du
∂
∂uG(u, v) . (2.33)
2.4.1 Solution in two and four dimensions
To determine the partial wave G(l)∆ [3] let us introduce new variables x, z such
thatu = xz, v = (1− x)(1 − z)
14
Manifestly all results are symmetric under x↔ z . Differentials with respectto u and v can now be expressed in terms of derivative w.r.t x and z as
∂
∂u=
1− x
z − x
∂
∂x− 1− z
z − x
∂
∂z
and∂
∂v=
x
z − x
∂
∂x+
z
z − x
∂
∂z.
The eigenvalue equation (2.32) then becomes
2Dε = C∆,lG(l)∆ , (2.34)
where 2Dε is a symmetric differential operator. In terms of new coordinate
2Dε = x2(1−x) ∂2
∂x2+z2(1−z) ∂
2
∂z2+c
(
x∂
∂x+ z
∂
∂z
)
−(a+b+1)
(
x2 ∂
∂x+ z2
∂
∂z
)
− ab(x+ z) + 2εxz
x− z
(
(1− x)∂
∂x− (1− z)
∂
∂z
)
, (2.35)
where the parameters a, b, c, ε takes the values
a = −12∆12, b =
1
2∆12, c = 0, ε =
1
2(d− 2)
The equation (2.35) can be written as
Dε = Dx +Dz + 2εxz
x− z
(
(1− x)∂
∂x− (1− z)
∂
∂z
)
, (2.36)
with
Dx = x
(
x(1 − x)d2
dx2+ (c− (a+ b+ 1)x)
d
dx− ab
)
and
Dz = z
(
z(1− z)d2
dx2+ (c− (a+ b+ 1)z)
d
dz− ab
)
.
This Dx and Dz satisfy the eigenvalue equations
Dxxλ1F (λ1 + a, λ1 + b, 2λ1 + c; x) = λ1(λ1 + c− 1)xλ1F (λ1 + a, λ2 + b, 2λ1 + c; x)
(2.37)
Dzzλ1F (λ2 + a, λ2 + b, 2λ2 + c; z) = λ2(λ2 + c− 1)zλ2F (λ2 + a, λ2 + b, 2λ2 + c; z)
(2.38)
15
where, F (a, b, c; x) is hypergeometric function [5].When ε = 0 (d = 2) the eigenvalue equation (2.36) is the sum of two hyper-geometric operators. So the eigenfunction of D0 is then
xλ1zλ2F (λ1+a, λ1+ b, 2λ1+ c; x)F (λ2+a, λ2+ b, 2λ2+ c; z)+x↔ z (2.39)
with the eigenvalue λ1(λ1 + c− 1) + λ2(λ2 + c− 1).For ε = 2(d = 4) the solution of equation (2.36) can be obtained in terms
of the solution of D0 using the relation
D2xz
x− z=
xz
x− z(D0 + c− 2) . (2.40)
2.4.2 Solution in three dimension
To determine the Conformal Partial wave G∆,l(u, v) in dimension d = 3, wewill use the power series expansion of G∆,l(u, v) for general d [6]. But nowwe will consider only the conformal blocks of four identical scalars, so that∆12 = ∆34 = 0. For scalar exchange l = 0, conformal blocks have doublepower series representation([2], Eq.(2.32)):
G∆,0(u, v) = u∆/2∞∑
m,n=0
[(∆/2)m(∆/2)m+n]2
m!n!(∆ + 1− d2)m(∆)2m+n
um(1− v)n ,
where (x)n is the Pochhammer symbol. For exchange operators of non-zero spin, the conformal partial waves can be computed by various recursionrelations. The recursion relations in transformed co-ordinates as obtained in[4] involves taking derivatives of G∆,l(x, z), which is not very easy to performnumerically. But along x = z line the terms involving derivatives drop outand recursion relations becomes simple [6]:
(l + d− 3)(2∆ + 2− d)G∆,l(z)
= (d− 2)(δ + l − 1)G∆,l−2(z) +2− z
2z(2l +D − 4)(∆− d+ 2)G∆+1,l−1(z)
− ∆(2l + d− 4)(∆ + 2− d)(∆ + 3− d)(∆− l − d+ 4)2
16(∆ + 1− d2)(∆− d
2+ 2)(l −∆+ d− 5)(l −∆+ d− 3)
G∆+2,l−2(z).
(2.41)
Now, the conformal waves can be computed using this recursion relation interms of spin-0 and spin-1 partial waves. Again spin 0 and 1 partial wavesalong x = z lines can be expressed in terms of generalized hypergeometric
16
functions (Appendix B of [6] ):
G∆,0(z) =
(
z2
1− z
)∆/2
3F2
(
∆
2,∆
2,∆
2− (
d
2− 1);
∆ + 1
2,∆− (
d
2− 1);
z2
4(1− z)
)
,
(2.42)
G∆,1(z) =2− z
2z
(
z2
1− z
)∆+1
2
3F2
(
∆+ 1
2,∆+ 1
2,∆+ 1
2− (
d
2− 1);
∆
2+ 1,∆− (
d
2− 1);
z2
4(1− z)
)
.
(2.43)
For the x 6= z line, instead of using partial wave at generic values of z,we can Taylor-expand the conformal bootstrap condition around the pointx = z = 1/2 along both along and transverse to the x = z line. In [6] thegeneral recursion relation is obtained for small value of (x− z).
2.5 The three-dimensional Ising model
One of the most interesting phenomena in the physics of solid state is fer-romagnetism. In some metals, a finite fraction of the atoms becomes spon-taneously polarized in some direction, giving rise to a macroscopic magneticfield below certain temperature known as Curie temperature. The IsingModel is an attempt to simulates a ‘domain’in a ferromagnetic substance.
In 3d Ising Model the system considered is an array of fixed points calledlattice sites that form a 3-dimensional periodic lattice.The operator associ-ated with each lattice site are, a spin variable σ and the product of twoneighboring spin of lattice ǫ.
It is convenient to consider the continuum limit of the lattice and σ, ǫare the local operators of lowest dimension associated with the theory. AtT = Tc, the theory has scale invariance and each operator is characterizedby its scaling dimension ∆ and spin l [6]. We will assume that the 3d Isingcritical point is invariant under full Conformal group. Conformal invariancehas been used in studies of critical O(N) models in large N -limit. ThoughIsing Model is the N = 1 case of O(N) symmetric universality class, we willstudy the operator spectrum and correlation function of an O(N) model asit has higher spin symmetry for large N and hence they are easy to calculate.
The operators σ and ǫ in 3d Ising model can be identified as the operatorsΦ and Φ · Φ of O(N) model respectively.
17
Chapter 3
The O(N ) symmetricuniversality class
The study of phase transition is related to finding of the origin of varioussingularities in the free energy and its descendants and characterizing them[8]. Depending on the order of phase transition, different order derivativeof free energy shows discontinuity near the critical point. The singular be-haviour in the vicinity of a critical point is characterized by a set of criticalexponents. These exponents describe the non-analyticity of various thermo-dynamic functions. Experimental observations indicate that these exponentsare quite universal i.e independent of the material under investigation andnature of phase transition. To explain this universality behaviour we con-sider a system with order parameter Φ(x) which has N -components livingin a d-dimensional space. We construct the Hamiltonian, obtain the exactsolution in d ≥ 4 . For d ≤ 4, fluctuations become important and we needto add Φ4 term perturbatively to the Gaussian model and use the renormal-ization procedure to obtain a non-trivial fixed point known as Wilson-Fisherfixed point.
3.1 The O(N) model
3.1.1 Construction of Hamiltonian
Let Φ(x) = (φ1, φ2, . . . , φN)T ∈ R
N be the order parameter of a system un-dergoing phase transition. The order parameter is a thermodynamic functionwhich has a non-zero expectation value below the critical temperature andzero above that. It has N -components in a d-dimensional space with coordi-nates x ≡ (x1, x2, . . . , xd) ∈ R
d. The Hamiltonian describing the model can
18
be constructed from the principles of locality ,uniformity and symmetry. Letthe Hamiltonian is
βH =
∫
ddx H[Φ(x),x], (3.1)
where H is the Hamiltonian density.
• Uniformity in space implies that we can get rid of the explicit xdependence of H. We generalize Eq. (3.1) by including the gradient offields as
βH =
∫
ddx H[Φ(x),▽Φ(x),▽2Φ(x), . . .] . (3.2)
• Symmetry: The Hamiltonian must be invariant under O(N) trans-formations.
H[R · Φ(x)] = H[Φ(x)] ,where R ∈ O(N). Linear terms in Φ are not consistent with symme-tries. Then the full Hamiltonian consistent with the symmetry is of theform
βH =
∫
ddx
[
t
2(Φ · Φ) + u (Φ · Φ)2 + K
2(∇Φ · ∇Φ) + . . .− h · Φ
]
,
(3.3)where the term −h · Φ is the interaction with the external field.
The Hamiltonian (3.3) is known as Landau-Ginzburg Hamiltonian.The O(N) symmetric Hamiltonian i.e Hamiltonian of Gaussian modelkeeps only quadratic term of expansion and is given by
βH =
∫
ddx
[
t
2(Φ · Φ) + u (Φ · Φ)2 + K
2(∇Φ · ∇Φ)− h · Φ
]
. (3.4)
• Stability requires the coefficient of Φ4 which is u here should be pos-itive to ensure that the probability is not divergent for infinitely largevalue of Φ.
3.1.2 Mean field approximation:
Various thermodynamic quantities can now be obtained from the partitionfunction
Z =
∫
D[Φ] exp−βH [Φ(x)]
=
∫
D[Φ] exp−∫
ddx[
t2(Φ · Φ) + u (Φ · Φ)2 + K
2(∇Φ · ∇Φ)− h · Φ
]
.
(3.5)
19
The free energy is defined by F = − lnZV
. The mean field approximation orsaddle point approximation is given by the condition
δF [Φ(x)]
δΦ(x)
∣
∣
∣
∣
φ0
= 0 .
The extrema occur for constant (independent of x) Φ. We choose Φ = φ0e1,where e1 is a unit vector (parallel to the external field h) in N -dimensionalparameter space. This leads to the following mean field equation for thevalue of φ0.
tφ20 + 4uφ3
0 − h = 0 (3.6)
For a magnetic system we can identify t as the reduced temperature i.et = a(T − Tc), where a is some constant and h is the external magneticfield. Now we can quantify the singular behaviour predicted by saddle pointapproximation of free energy.
F0 ∼[
t2(Φ · Φ) + u (Φ · Φ)2 − h · Φ
]
∣
∣
∣
Φ=φ0e1(3.7)
1. For no external interaction i.e h = 0, we obtain from (3.6)
tφ20 + 4uφ3
0 = 0
φ0 =
√
− t
4u(3.8)
φ0 =
{√
− a4u(Tc − T )
12 if t < 0;
0 if t > 0.
Again,
φ0 ∼ (Tc − T )β ⇒ β =1
2.
Along the critical isotherm(t = 0; h 6= 0), the mean field equationbecomes
4uφ30 − h = 0
φ0 =
(
h
4u
)1/3
∼ h1/δ
δ = 3 (3.9)
2. From the equation(3.6), we obtained
χ−1 =∂φ0
∂h
∣
∣
∣
∣
h=0
= t + 12uφ20 =
{
t if t > 0, h = 0;−2t if t < 0, h = 0.
(3.10)
χ± ∼ A±|t|−γ± =⇒ γ+ = γ− = +1
20
3. Heat capacity at zero external field:
C(h = 0) = −T ∂2F
∂T 2≈ −Tc
∂2
∂t2(kBTcβF ) ∼ V kBa
2T 2c×{
0 if t > 0;18u
if t < 0.
C ∼ t−α =⇒ α = 0
3.1.3 Continuous symmetry breaking and Goldstone
mode
The Hamiltonian density of our model is given by
H = (▽Φ)2 +t
2Φ2 + uΦ4
For stability, u must be positive. Let us assume t is negative(T −Tc < 0) andt = −2a2u. Ground state can be obtained by minimizing the potential,V =uΦ4 − 2a2uΦ2. For V to be minimum
V ′ = 4uΦ3 − 4a2uΦ = 0
⇒ Φ2 − a2 = 0 ⇒ φ = |a|eiθ(x)
If we choose any one of the above minima, the symmetry has spontaneouslybroken. So order phase(T < Tc) corresponds to the spontaneously brokencontinuous symmetry. The fluctuations added to θ leads to massless excita-tions(i.e does not cost any energy), known as Goldstone mode.
3.1.4 Effect of fluctuations
Now we shall examine the effect of fluctuations to the mean field configurationfor zero external field.
Φ(x) = [φ0 + φl(x)]e1 +N∑
α=2
φt,α(x)eα, (3.11)
where, φl and φt refer to longitudinal and transverse fluctuations respectively.
(▽Φ)2 = (▽φl)2 + (▽φt)
2
Φ2 = φ20 + 2φ0φl + φ2
l + φ2t
Φ4 = φ40 + 4φ3
0φl + 6φ20φ
2l + 2φ2
0φ2t +O(φ3
l , φ3t )
21
Now the Hamiltonian is given by
βH = V (t
2φ20 + uφ4
0)
+
∫
ddx
[
K
2(▽φl)
2 +2tφ0 + 8uφ3
0
2φl +
t + 12uφ20
2φ2l
]
+
∫
ddx
[
K
2(▽φt)
2 +t+ 4uφ2
0
2φ2t
]
+O(φ3l , φ
3t ), (3.12)
where, V is the volume of the system. For no external interaction we havethe mean field equation tφ0 + 4uφ3
0 = 0. Therefore
βH = βHsp +
∫
ddx
[
K
2(▽φl)
2 +t+ 12uφ2
0
2φ2l
]
+
∫
ddx
[
K
2(▽φt)
2 +t + 4uφ2
0
2φ2t
]
(3.13)
Let us defineK
ξ2l≡ t+ 12uφ2
0 =
{
t if t > 0 ;−2t if t < 0 .
K
ξ2t≡ t + 12uφ2
0 =
{
t if t > 0 ;0 if t < 0 .
Here we have used equation(3.10) for the 1st case and equation (3.8) for the2nd. ξl and ξt are the longitudinal and transverse correlation length.
βH = βHsp +1
2
∫
ddx
[
K
2(▽φl)
2 +K
ξ2lφ2l
]
+1
2
∫
ddx
[
K
2(▽φt)
2 +K
ξ2lφ2t
]
(3.14)
Now each Gaussian kernel is diagonalized by the Fourier Transform
φ(q) =
∫
ddx√V
exp(−iq · x)φ(x) ,
with eigenvaluesK(q) = K(q2 + ξ−2) .
So partition function is
Z = e−βHsp [detK(q)]−12 . (3.15)
22
Now,
ln detK =∑
q
lnK(q)
= V
∫
ddq
(2π)dln[K(q2 + ξ−2)] .
The free energy is given by
βf = − lnZV
= Fsp +1
2
∫
ddq
(2π)dln[K(q2 + ξ−2
l )]
+N − 1
2
∫
ddq
(2π)dln[K(q2 + ξ−2
t )] . (3.16)
Using the dependence of the correlation length on reduced temperature, thesingular part of heat capacity is obtained as
Csingular ∝ −∂2(βf)
∂t2=
{
0 + N2
∫
ddq(2π)d
1(Kq2+t)2
if t > 0;18u
+ 2∫
ddq(2π)d
1(Kq2−2t)2
if t < 0.
So correction term to the heat capacity is
CF ∝1
K2
∫
ddq
(2π)d1
(q2 + ξ−2)2.
This integral has dimension, (length)4−d. Now we use the fact that for d > 4,the integral diverges at large q and is dominated by upper cutoff Λ ∼ 1
a; where
a is the lattice spacing. For d < 4, the integral converges in both limits. Itcan be made dimensionless by rescaling q by ξ−1 and hence proportional toξ4−d.
For dimension d > 4, the fluctuation correction to the heat capacity addeda constant term in background but for d < 4, the fluctuation correction leadsto divergence of CF . So mean field approximation is no longer reliable belowthe dimension d = 4, which is called upper critical dimension.
Although we obtained this dimension by looking at the fluctuations cor-rections to the heat capacity, we would have reached the same conclusion inexamining the singular part of any other quantity.
3.2 Scaling hypothesis
The power law behaviour near the critical point suggests that, if a criticalsystem is blown by a factor λ, apart from a multiplicative factor λp, the
23
resulting system remains statistically similar to the original one.[8]
f(λx) = λpf(x) ,
where, p is called the scale dimension of the quantity f(x). The scalinghypothesis results a number of exponent identities. These are
α + 2β + γ = 2 Rushbrooke’s identity
β + γ = βδ Widom’sidentity
2− α = dν Josephson’s identity
The Landau-Ginzburg probability was constructed on the basis of local sym-metries such as rotation invariance. If we could add to the constraints arisesdue to dilation symmetry, the resulting probability would describe the criti-cal point.Unfortunately, it is not possible to directly see how such a requirement con-straints the effective Hamiltonian. We shall follow a less direct route of fol-lowing the effects of the dilation operation on effective energy: The renor-malization procedure.
3.3 Renormalization Group
Renormalization procedure is a three step procedure([8]). They are the fol-lowing:
1. Coarse grain: In the 1st step we increase the minimum length scaleby a factor b (b > 1) such that
Φi(x) =1
bd
∫
ddx′Φi(x′) .
2. Rescale: Then we decrease all the length scale by the same factor bby setting
xnew =xold
b.
3. Renormalize: In the last step we renormalize the system by dividingit by a factor ζ
Φnew(xnew) =1
ζbd
∫
ddx′Φ(x′) .
24
After renormalization process, the renormalization parameters are the func-tions of original parameters. If there is any point in the parameter space,which remains invariant under renormalization, known as fixed point.
RbS∗ = S∗, where S∗ is the fixed point.
Now we can study the stability of fixed point by applying the RG in theneighbourhood of the fixed point S∗. Under Renormalization Group, a pointS∗ + δS is transformed as
S∗α + δS ′
α = S∗α + (RL
b )βαδSβ + . . . ,
where, (RLb )
βα = ∂S′
α
∂Sβ
∣
∣
∣
s∗.
Let Oi are the eigenvectors of matrix (RLb )
βα and λi(b) are the correspond-
ing eigenvalues. From the closure property of group, we can write
RLb R
Lb′Oi = λi(b)λi(b
′)Oi = RLbb′Oi = λi(bb
′)Oi . (3.17)
Renormalization group should also satisfy the condition
λ(1) = 1. (3.18)
Equation (3.17) and (3.18) imply, λi(b) = byi .
The eigenvectors Oi are called scaling direction associate with the fixedpoint S∗ and yi are the corresponding anomalous dimensions. Any point inthe neighbourhood of the fixed point can be written as
S = S∗ +∑
i
giOi .
Under RG, the point becomes
S ′ = S∗ +∑
i
gibyiOi .
Depending on the value of yi, we can classify the eigenoperators as following:
• If yi > 0; gi increases under scaling, and Oi is called relevant operator.
• If yi < 0; gi decreases under scaling, and Oi is called irrelevant operator.
• If yi = 0; gi remains invariant under scaling, and Oi is called marginaloperator.
For the fixed point to be the critical point of a system, it must be associatedwith only one relevant operator i.e it should have co-dimension one.
25
3.4 Solution of Gaussian model using RG
The Gaussian model Hamiltonian can be obtained by keeping only the quadraticterm in Landau-Ginzburg Hamiltonian(3.3) and is given by
βH =
∫
ddx
[
K
2(▽Φ)2 +
t
2Φ2 − h · Φ
]
,
and corresponding partition function is
Z =
∫
D[Φ] exp−∫
ddx
[
K
2(▽Φ)2 +
t
2Φ2 − h · Φ
]
. (3.19)
This model is well defined only for t > 0, which ensure that the partitionfunction function remains finite for infinite large value of order parameter.Therenormalization of Gaussian model is performed in terms of Fourier modes.The Fourier modes can be achieved by setting
Φ(q) =
∫
ddxeiq·xΦ(x) ;
Φ(x) =
∫
ddx
(2π)de−iq·xΦ(q) .
(3.20)
The largest q are limited by the lattice spacing and confined to a BrillouinZone. We approximate the Brillouin Zone as a hypersphere of radius Λ. TheHamiltonian interms of Fourier modes is given by
βH =
∫
ddx
(2π)d
[
K
2q2 +
t
2
]
|Φ(q)|2 + h · Φ(q = 0) ,
and corresponding partition function
Z =
∫
D[Φ] exp−∫ Λ
0
ddq
(2π)d1
2(Kq2 + t)|Φ(q)|2 + h · Φ(0) . (3.21)
Now we will follow the three steps of Renormalization Group to evaluate theabove partition function(3.21)
1. Coarse Grain: In the 1st step we increase the minimum length scaleby a factor b. That means we eliminate all the fluctuations in the lengthscale a < x < ba which is equivalent to remove the fluctuations withΛb< q < Λ in momentum space. We thus break up the set of order
parameters into two subsets.
Φ(q) =
{
~µ(q) for Λb< q < Λ;
~m(q) for 0 < q < Λb.
26
The partition function can now be written as
Z =
∫
D[~µ(q)]∫
D[~m(q)]e−βH[~µ,~m]
=
∫
D[~m(q)] exp
[
−∫ Λ
b
0
ddq
(2π)d1
2(Kq2 + t)|m(q)|2 + h ·m(0)
]
×∫
D[~µ(q)] exp[
−∫ Λ
Λb
ddq
(2π)d1
2(Kq2 + t)|µ(q)|2
]
=
∫
D[~m(q)] exp
[
−∫ Λ
b
0
ddq
(2π)d1
2(Kq2 + t)|m(q)|2 + h ·m(0)
]
× exp
[
−NV
2
∫ Λ
Λb
ddq
(2π)dln(kq2 + t)
]
. (3.22)
The µ integral in line 2 is just the Gaussian integral and can be eval-uated using the formulae for Gaussian integral(Section 3.5 of [8]) andcontributes just a phase factor. Let us denote it by e−V δfb(t), where
δfb(t) =N2
∫ ΛΛb
ddq(2π)d
ln(kq2+ t).The partition function is now almost the
same as that we have begin with except the phase factor and the limitof the integral. Now we go to the 2nd step to make it similar.
2. Rescale: Rescaling in real space x′ = x/b is equivalant to rescaleq′ = bq in momentum space. Now the rescaled partition function isgiven by
Z = e−V δfb(t)
∫
D[~m(q′)] exp
[
−∫ Λ
0
ddq′
(2π)db−d
2(Kb−2q′2 + t)|m(q′)|2 + h ·m(0)
]
.
3. Renormalize: In the last step we renormalize the Fourier modes ac-cording to ~m′(q′) = ~m(q′)
z. Then the partition function becomes
Z = e−V δfb(t)
∫
D[~m′(q′)] exp
[
−∫ Λ
0
ddq′
(2π)db−d
2z2(Kb−2q′2 + t)|m′(q′)|2 + zh ·m′(0)
]
,
which is statistically self similar to the original partition function with therenormalized parameters
t′ = b−dz2t , (3.23)
h′ = zh , (3.24)
K ′ = z2b−d−2K . (3.25)
27
From the above relations, it can be seen that the singular point t = h = 0is mapped to itself as expected. To make fluctuations scale invariant at thispoint we must ensure that the remaining Hamiltonian stays fixed. This canbe achieved by the choice of z = b1+
d2 which set K ′ = K.
Away from criticality, the two relevant directions now scale as
t′ = b2t ⇒ yt = 2 ;
h′ = b1+d2h ⇒ yh = 1 +
d
2.
The fixed point Hamiltonian in real space is given by
βH∗ =K
2
∫
ddx(▽Φ)2 .
If a general power of Φ, added a small perturbation to βH∗, the scalingdirection to that Perturbation can be written as(see section 4.7 in[8])
yn = n− d(n
2− 1)
.
The first tow term can be identified as the scaling direction corresponding toh and t respectively as obtained from the Gaussian model calculation
y1 = 1 +d
2= yh ,
y2 = 2 = yt .
The next most important operator in the system consistant with the sym-metry is y4 = 4 − d, which is irrelevant for d > 4 but relevant for d < 4.y6 = 6 − 2d is relevant only for d < 3. Majority of operators becomesirrelevant at Gaussian fixed point for d > 2.
28
Chapter 4
Renormalization perturbationto solve Φ4 model
In the previous section, we have solved the Gaussian model using renormal-ization group and arrived at the fixed point known as Gaussian fixed pointwhich explains the critical exponents well above the upper critical dimen-sion, d > 4.But this is no longer valid for for d < 4 where most of the termsremains irrelevant but the uΦ4 term now becomes relevant. Let us add thisterm perturbatively to the Gaussian Model. The normal perturbation leadsthe series to be divergent (Section 5.4 of [8]) close to critical point and hencecannot determine the critical exponents at d < 4.
K.G. Wilson showed that it is possible to combine perturbative and renor-malization group approaches into systematic method for calculating criticalexponents.
4.1 Perturbative expansion of Φ4 term in the
power of u
Gaussian Hamiltonian for zero external interaction is given by
βH0 =
∫
ddx
[
K
2(▽Φ)2 +
t
2Φ2
]
. (4.1)
Perturbation added to this is
βHint = u
∫
ddx Φ4(x) .
The total Hamiltonian is
βH =
∫
ddx
[
K
2(▽Φ)2 +
t
2Φ2 + uΦ4(x)
]
. (4.2)
29
As in case of Gaussian model here also we work in terms of Fourier modes. Interms of Fourier Modes the Hamiltonian can be obtained using the FourierTransform formula(3.20) as
βH =
∫
ddq
(2π)d1
2(Kq2 + t)|Φ(q)|2
+ u
∫
ddx
∫
ddq1ddq2d
dq3ddq4
(2π)4de−ix.(
∑4i=1 qi)φα(q1)φα(q2)φβ(q3)φβ(q4)
=
∫
ddq
(2π)d1
2(Kq2 + t)|Φ(q)|2
+ u
∫
ddq1ddq2d
dq3ddq4
(2π)4d(2π)dδd
(
4∑
i=1
qi
)
φα(q1)φα(q2)φβ(q3)φβ(q4)
=
∫
ddq
(2π)d1
2(Kq2 + t)|Φ(q)|2
+ u
∫
ddq1ddq2d
dq3
(2π)3dφα(q1)φα(q2)φβ(q3)φβ(−q1 − q2 − q3) .
The partition function can now be written as
Z =
∫
D[Φ(q)] exp[
−∫ Λ
0
ddq
(2π)d1
2(Kq2 + t)|Φ(q)|2 − U [Φ(q)]
]
, (4.3)
Where,
U [Φ(q)] = u
∫
ddq1ddq2d
dq3ddq4
(2π)4d(2π)dδd
(
4∑
i=1
qi
)
φα(q1)φα(q2)φβ(q3)φβ(q4).
Now let us apply the Renormalization procedure to the above partition func-tion (4.3).
1. Coarse grain: In the first step we eliminate the small wavelengthfluctuations by suddividing the order parameter into to subsets as wedid in Gaussian model
Φ(q) =
{
~µ(q) for Λb< q < Λ;
~m(q) for 0 < q < Λb.
30
The partition function can now be written as
Z =
∫
D[~µ(q)]∫
D[~m(q)]e−βH[~µ,~m]
=
∫
D[~µ(q)]∫
D[~m(q)] exp
[
−∫ Λ
b
0
ddq
(2π)d1
2(Kq2 + t)(|m(q)|2 + |µ(q)|2)
]
=
∫
D[~m(q)] exp
[
−∫ Λ
b
0
ddq
(2π)d1
2(Kq2 + t)|m(q)|2
]
×∫
D[~µ(q)] exp[
−∫ Λ
Λb
ddq
(2π)d1
2(Kq2 + t)|µ(q)|2
]
exp [−U [Φ(q)]]
=
∫
D[~m(q)] exp
[
−∫ Λ
b
0
ddq
(2π)d1
2(Kq2 + t)|m(q)|2
]
×∫
D[~µ(q)] exp[
−∫ Λ
Λb
ddq
(2π)d1
2(Kq2 + t)|µ(q)|2
]
×∫
D[~µ(q)]e−U exp[
−∫ Λ
Λb
ddq(2π)d
12(Kq2 + t)|µ(q)|2
]
∫
D[~µ(q)] exp[
−∫ Λ
Λ
b
ddq(2π)d
12(Kq2 + t)|µ(q)|2
]
=
∫
D[~m(q)] exp
[
−∫ Λ
b
0
ddq
(2π)d1
2(Kq2 + t)|m(q)|2
]
× exp
[
−NV
2
∫ Λ
Λb
ddq
(2π)dln(kq2 + t)
]
〈e−U [~µ,~m]〉µ .
The Coarse grained Hamiltonian is given by
βH = V δf 0b +
∫ Λ
b
0
ddq
(2π)d1
2(Kq2 + t)|m(q)|2 − ln〈e−U [~µ,~m]〉µ . (4.4)
This can be evaluated perturbatively as
ln〈e−U [~µ,~m]〉µ = −〈U〉µ +1
2(〈U2〉µ − 〈U〉2µ) + . . . .
31
4.2 First order perturbation:
1st order perturbation term is
〈U [~m, ~µ]〉µ = u
∫
ddq1ddq2d
dq3ddq4
(2π)4d(2π)dδd
(
4∑
i=1
qi
)
〈(~m(q1)+~µ(q1))·(~m(q2)+~µ(q2)) (~m(q3)+~µ(q3))·(~m(q4)+~µ(q4))〉µ .(4.5)
To evaluate this integral, we use the Wick’s Theorem which says
〈l∏
i=1
mi〉 ={
0 for l odd;sum over all pairwise contractions for l even.
For example:
〈mimjmkml〉 = 〈mimj〉〈mkml〉+ 〈mimk〉〈mjml〉+ 〈miml〉〈mjmk〉 .
Diagramatic representation:
To calculate the l-point expectation value at the p-th order in u, therules are the following:
(a) l-external points are labeled by (qi, αi) corresponding to the re-quired correlation function, as shown in4.1. Then p vertices withfour leg each, labeled by external momenta and indices , e.g.(k1, i), (k2, i), (k3, j), (k4, j) are drawn. since four legs are notequivalent, the four point vertex is indicated by two solid branchesjoined by a dotted line.
k1, i
k2, jk4, j
k3, j
(qi, αi)
Figure 4.1: Elements of the diagrammatic representation of perturbationtheory
32
(b) Algebraic values of graphs:
i. A line joining a pair of points represents a two point average4.2 and corresponding algebraic value is
δα1α2(2π)d
δd(q1 + q2)
(t+Kq2)
(q1, α1) (q2, α2)
Figure 4.2: Diagrammatic representation of Two point function
ii. A vertex as shown in 4.3 has algebraic value
u(2π)dδd(k1 + k2 + k3 + k4)
k1, i
k2, jk4, j
k3, j
Figure 4.3: Diagrammatic representation of a vertex
(c) While calculating cumulants, only fully connected diagrams with-out any adjoint pieces need to be included.
The perturbation 1st order in u is
〈U [~m, ~µ]〉µ = u
∫
ddq1ddq2d
dq3ddq4
(2π)4d(2π)dδd
(
4∑
i=1
qi
)
×〈(~m(q1)+~µ(q1))·(~m(q2)+~µ(q2)) (~m(q3)+~µ(q3))·(~m(q4)+~µ(q4))〉µ .
There are total 16 terms that contribute to the expansion of the aboveproduct. Since expectation value is taken over short wavelength fluc-tuations i.e ~µ(q), only contractions with µ will appear.
33
(a) The quantity 〈~m(q1) · ~m(q2) ~m(q3) · ~m(q4)〉 can be representeddiagrammatically as 4.4. Number of such term is one and algebricvalue due to such representation is
u
∫
ddq1ddq2d
dq3ddq4
(2π)4d(2π)dδd
(
4∑
i=1
qi
)
mα(q1)mα(q2)mβ(q3)mβ(q4)
= U [~m] .
1
24
3
Figure 4.4: Diagrammatic representation of 〈~m(q1) · ~m(q2) ~m(q3) · ~m(q4)〉
(b) The terms like 〈~µ(q1) · ~m(q2) ~m(q3) · ~m(q4)〉 has diagrammaticrepresentation as shown in figure 4.5. Number of such terms isfour. Since there is odd number of µ(q), average is zero (fromWick’s theorem). So algebraic value of these terms is zero.
q1, i
q2, i q3, j
q4, j
Figure 4.5: Diagrammatic representation of 〈~µ(q1) · ~m(q2) ~m(q3) · ~m(q4)〉
(c) The quantity of the form 〈~µ(q1) · ~µ(q2) ~m(q3) · ~m(q4)〉 can bediagrammatically represented by 4.6. Number of such terms are2. These terms have one contraction and evaluates to
− 2u
∫
ddq1ddq2d
dq3ddq4
(2π)4d(2π)dδd
(
4∑
i=1
qi
)
δii(2π)dδd(q1 + q2)
t+Kq21~m(q3) · ~m(q4)
= −2Nu
∫ Λb
0
ddq
(2π)d|m(q)|2
∫ Λ
Λb
ddk
(2π)d1
t+Kk2,
34
where the dummy variables q3 and q1 are replaced by q and krespectively.
q1, i
q2, iq3, j
q4, j
Figure 4.6: Diagrammatic representation of 〈~µ(q1) · ~µ(q2) ~m(q3) · ~m(q4)〉
(d) The number of terms of type 〈~µ(q1) · ~m(q2) ~µ(q3) · ~m(q4)〉 is four.Diagrammatically it can be represented as following 4.7. Theseterms also have one contractions but as δjj term does not appear,The algebraic value is given by
−4u∫ Λ
b
0
ddq
(2π)d|m(q)|2
∫ Λ
Λ
b
ddk
(2π)d1
t+Kk2
q1, iq3, j
q4, jq2, i
Figure 4.7: Diagrammatic representation of 〈~µ(q1) · ~m(q2) ~µ(q3) · ~m(q4)〉
(e) The terms 〈~µ(q1)·~µ(q2) ~µ(q3)·~m(q4)〉 corresponds to the algebraicvalue zero as there are odd number µ in the expression. Numberof such terms are four. Diagrammatic representation is given by4.8.
(f) The terms 〈~µ(q1) · ~µ(q2) ~µ(q3) · ~µ(q4)〉 are diagrammatically rep-resented by 4.9. Number of such term is one. Since these variouscontractions of four µ’s are completely independent of m-values,we denote the sum of these terms by, uV δf 1
b .
35
q1, iq3, j
q2, i
q4, j
Figure 4.8: Diagrammatic representation of 〈~µ(q1) · ~µ(q2) ~µ(q3) · ~m(q4)〉q1, i q3, j
q4, jq2, i
+
Figure 4.9: Diagrammatic representation of 〈~µ(q1) · ~µ(q2) ~µ(q3) · ~µ(q4)〉
Collecting all the terms, the coarse grained Hamiltonian is now givenby
βH = V (δf 0b + δf 1
b ) +
∫ Λb
0
ddq
(2π)d
[
1
2(t+Kq2) + (4Nu+ 8u)
∫ Λ
Λb
ddk
(2π)d1
t +Kk2
]
|m(q)|2
+ u
∫ Λb
0
ddq1ddq2d
dq3
(2π)3d~m(q1)~m(q2)~m(q3)~m(−q1 − q2 − q3)
= V (δf 0b + δf 1
b ) +
∫ Λb
0
ddq
(2π)d1
2(t+Kq2)|m(q)|2
+ u
∫ Λb
0
ddq1ddq2d
dq3
(2π)3d~m(q1)~m(q2)~m(q3)~m(−q1 − q2 − q3),
(4.6)
where,
t = t+ 4u(N + 2)
∫ Λ
Λb
ddk
(2π)d1
t+Kk2.
2. Rescale: We rescale the coarse grained Hamiltonian by setting q =
36
b−1q′. Then the rescaled Hamiltonian becomes
βH [~m(q′)] = V (δf 0b + δf 1
b ) +
∫ Λ
0
ddq′
(2π)db−d1
2(t+Kb−2q′2)|m(q′)|2
+ u
∫ Λ
0
ddq′1d
dq′2d
dq′3
(2π)3db−3d ~m(q′
1)~m(q′2)~m(q′
3)~m(−q′1 − q′
2 − q′3) .
(4.7)
3. Renormalize: Now renormalization can be done by setting, ~m = z ~m′.The renormalized Hamiltonian is
βH ′[~m′(q′)] = V (δf 0b + δf 1
b )+
∫ Λ
0
ddq′
(2π)dz2b−d 1
2(t+Kb−2q′2)|m′(q′)|2
+ u
∫ Λ
0
ddq′1d
dq′2d
dq′3
(2π)3db−3dz4 ~m(q′
1)~m(q′2)~m(q′
3)~m(−q′1 − q′
2 − q′3) .
(4.8)
The renormalized parameters are ,
t′ = b−dz2t
K ′ = b−d−2z2K
u′ = b−3dz4u (4.9)
Fixed point is given by, t∗ = u∗ = 0 similar to the Gaussian Model. To makerest of the Hamiltonian invariant at the fixed point, we set z = b1+
d2 such
that K ′ = K. The recursion relations for t and u in the vicinity of the fixedpoint are given by
t′ = b2
[
t + 4u(N + 2)
∫ Λ
Λb
ddk
(2π)d1
t +Kk2
]
;
u′ = b4−du .
(4.10)
4.3 β-functions at first order
Let us convert these recursion relations into continuous differential flow equa-tion. Let b = el ≃ (1 + δl) for an infinitesimal δl. Then
t′ ≡ t(b) = t(1δl) = t + δldt
dl+O(δl2) ;
u′ ≡ u(b) = u(1δl) = u+ δldt
dl+O(δl2).
(4.11)
37
Expanding equation(4.11) to the order of δl, we have
t + δldt
dl= (1 + 2δl)
(
t+ 4u(N + 2)Sd
(2π)d1
t+KΛ2Λdδl
)
;
u+ δldu
dl= (1 + (4− d)δl) u .
(4.12)
where Sd ≡ total solid angle in d-dimension. So we obtain β-functions as:
βt ≡dt
d(lnb)= 2t+
4u(N + 2)KdΛd
t +KΛ2,
βu ≡du
d(lnb)= (4− d)u ,
(4.13)
where Kd =Sd
(2π)d. The recursion relations can be linearized in the vicinity of
the trivial fixed point, t∗ = u∗ = 0 by setting, t = t∗ + δt and u = u∗ + δu
d
dl
(
δtδu
)
=
(
2 4(N+2)KdΛd−2
K
0 4− d
)(
δtδu
)
.
Eigenvalues of this matrix is given by
yt = 2 ;
yu = 4− d .
These give us the anomalous dimension corresponding to the directions of tand u respectively. These results are identical with those of Gaussian modelwith only difference is that this time dimension yu = 4− d is associated withthe direction
t = −u4(N + 2)KdΛd−2
K,
instead of t = 0.For d > 4, the Gaussian fixed point is associated with only one relevant
operator and hence can correctly describe phase transition.For d < 4, there are two relevant directions. So for d < 4 we need to find
another non-trivial fixed point with co-dimension one which can describe thephase transitions.
Unfortunately, in the 1st order there is no other fixed point. But wecan go to next order of the perturbation series. We can anticipate that therecursion relations to the next order are modified to
dt
dl= 2t+
4u(N + 2)KdΛd
t+KΛ2− Au2 , (4.14)
du
dl= (4− d)u− Bu2 , (4.15)
38
where A,B are positive. For a systematic perturbation theory we need tokeep the parameter u small. From the above expressions we can see thatthere is now a new fixed point at u∗ = 4−d
Bfor d < 4. This new fixed point
can be explored systematically only for small ǫ = 4− d.
4.4 Second order perturbation theory
The Hamiltonian, 2nd order in u, after coarse-graining can be written as
βH [m] = V δf 0b+
∫ Λb
0
ddq
(2π)d1
2(t+Kq2)|m(q)|2+〈U〉µ−
1
2
(
〈U2〉µ − 〈U〉2µ)
+(O)(U3) .
To calculate(
〈U2〉µ − 〈U〉2µ)
we need to consider all possible decompositionsof two U ’s into ~m and ~µ. In the previous section we have seen that U intosix type of terms. So in 2nd order in U , we should have 6 × 6 = 36 type ofterms. But most of them vanish due to several factors like conservation ofmomentum, parity etc. The terms which are non-zero are the following:
1. The term as shown in figure 4.10 leads to the algebraic value
8× u2
2
∫ Λb
0
ddq1ddq2d
dq3ddq4
(2π)4d
∫ Λ
Λb
ddk1ddk2d
dk′1d
dk′2
(2π)4d(2π)dδd(q1 + q2 + k1 + k2)
× (2π)dδd(k′1 + k′
2 + q3 + q4)δjj′(2π)
dδd(k1 + k′1)
t+Kk′21
× δjj′(2π)dδd(k2 + k′
2)
t+Kk′22
~m(q1) · ~m(q2) ~m(q3) · ~m(q4)
= 4Nu2
∫ Λ
b
0
ddq1ddq2d
dq3ddq4
(2π)4d(2π)dδd
(
4∑
i=1
qi
)
~m(q1) · ~m(q2) ~m(q3) · ~m(q4)
×∫ Λ
Λb
ddk
(2π)d1
(t +Kk2)(t +K(q1 + q2 − k)2).
Number of such terms is 8.
2. The 1-loop term as shown in figure 4.11 leads to algebraic value:
32× u2
2
∫ Λ
b
0
ddq1ddq2d
dq3ddq4
(2π)4d(2π)dδd
(
4∑
i=1
qi
)
~m(q1) · ~m(q2) ~m(q3) · ~m(q4)
×∫ Λ
Λb
ddk
(2π)d1
(t +Kk2)(t +K(q1 + q2 − k)2).
Number of such terms is 32.
39
q2
q1k2, j
k1, jk′1, j′
k′2, j′
q3
q4
Figure 4.10: Number of terms = 8
q2
q1 k2, j
k1, jk′1, j′
k′2, i′
q4
q3
Figure 4.11: Number of terms = 32
3. The diagram given in 4.12 leads to
32× u2
2
∫ Λb
0
ddq1ddq2d
dq3ddq4
(2π)4d(2π)dδd
(
4∑
i=1
qi
)
~m(q1) · ~m(q2) ~m(q3) · ~m(q4)
×∫ Λ
Λ
b
ddk
(2π)d1
(t +Kk2)(t +K(q1 + q2 − k)2).
q2
q1
k1, j
k2, j k′2, i′
k′1, i′ q4
q3
Figure 4.12: Number of terms=32
4. There are another non-zero terms at the order of u2 which containcontractions of all µ’s, and hence independent of m. We denote such aterm by, u2V δf 2
b .
5. At this order some terms appear which contain two loop integrationsand we shall denote net effect of such terms by A.
40
6. Some of the terms also become proportional to m2, m6. This indicatesthat the parameter space is not closed at this order . But we will ignoresuch terms at this order.
Now the integral,∫ Λ
Λ
b
ddk(2π)d
1(t+Kk2)(t+K(q1+q2−k)2)
can be expanded as
f(q1+q2) =
∫ Λ
Λb
ddk
(2π)d1
(t+Kk2)2
[
1− 2Kk.(q1 + q2)−K(q1 + q2)2
t+Kk2+ . . .
]
Collecting all the terms, the coarse grained Hamiltonian at the order of u2
can be written as
βH = V (δf 0b + δf 1
b + δf 2b ) +
∫ Λ
b
0
ddq
(2π)d|m(q)|2
[
t+Kq2
2+ 2u(N + 2)
×∫ Λ
Λb
ddk
(2π)d1
(t+Kk2)− u2
2A(t,K, q2)
]
+
∫ Λb
0
ddq1ddq2d
dq3
(2π)3d~m(q1) · ~m(q2) ~m(q3) · ~m(−q1 − q2 − q3)
×[
u− u2
2(8N + 64)
∫ Λ
Λb
ddk
(2π)d1
(t+Kk2)2+O(u2q2)
]
+O(u2, m6, q2, . . .) +O(u3). (4.16)
4.5 The Wilson-Fisher fixed point:
Ignoring all other parameters those have appeared in the process of renormal-ization group, we can express the parameters of the coarse-grained Hamilto-nian as
t = t + 4u(N + 2)
∫ Λ
Λb
ddk
(2π)d1
t +Kk2− u2A(0) ,
K = K − u2A′(0) ,
u = u− u2
2(8N + 64)
∫ Λ
Λb
ddk
(2π)d1
(t+Kk2)2,
(4.17)
where A(0) and A′(0) are the 1st two terms in the expansion of A(t,K, q).
41
Rescale and Renormalize:
Now we rescale the Hamiltonian by, q = b−1q′ and renormalize it by ~m =z ~m′. The renormalized parameters are given by
K ′ = b−d−2z2K ;
t′ = b−dz2t ;
u′ = b−3dz4u .
(4.18)
As we did before, here also we choose the renormalization parameter z suchthat K ′ = K. That can be obtained by setting
z2 =bd+2
1− u2A′(0)K
= bd+2(1 +O(u2)) .
The value of z depends on the position of fixed point u∗. But as u∗ is of theorder of ǫ, value of z does not change .
z ≈ b1+d2 ,
t′ = b2t = b2
[
t + 4(N + 2)u
∫ Λ
Λb
ddk
(2π)d1
t+Kk2− u2A(0)
]
;
u′ = b4−du = u− 4(N + 8)u2
∫ Λ
Λb
ddk
(2π)d1
t +Kk2.
(4.19)
To calculate the β-functions at this order, let us 1st convert the above equa-tions (4.19) into continuous differential flow equation by setting b = el andexpanding upto the order of δl
b = el ≈ 1 + δl ,
t+ δldt
dl= (1 + 2δl)
[
t + 4(N + 2)uSd
(2π)dΛd
t +KΛ2δl − A(0)u2
]
;
u+ δldu
dl= (1 + (4− d)δl)
[
u− 4(N + 8)u2 Sd
(2π)dΛd
(t+KΛ2)2δl
]
.
(4.20)
Comparing the coefficient of δl from both sides in the above equation(4.20),we obtain the β-functions as
βt =dt
d(lnb)= 2t+
4u(N + 2)KdΛd
t +KΛ2− A(0)u2 ,
βu =du
d(lnb)= (4− d)u− 4(N + 8)KdΛ
d
(t+KΛ2)2u2,
(4.21)
42
where Kd =Sd
(2π)d.
At the fixed point, dtdl
= 0; dudl
= 0. Imposing these conditions we obtaina new fixed point
u∗ =(t∗ +KΛ2)2
4(N + 8)KdΛdǫ ,
t∗ = −2u∗(N + 2)KdΛ
d
t∗ +KΛ2,
(4.22)
where ǫ = 4 − d. Solving above equation pair(4.22) at the order of ǫ, thefixed point becomes
u∗ =K2
4(N + 8)K4
ǫ+O(ǫ2) ;
t∗ = −(N + 2)K4Λ4
KΛ2× K2
2(N + 8)K4ǫ+O(ǫ2)
= − N + 2
2(N + 8)KΛ2ǫ+O(ǫ2) .
(4.23)
Linearizing the recursion relation(4.21) in the vicinity of the fixed point as,t = t∗ + δt and u = u∗ + δu, we have
d
dl
(
δtδu
)
=
(
2− 4(N+2)KdΛd
(t+KΛ2)2u∗ − A′u∗2 4(N+2)KdΛ
d
t+KΛ2 − 2Au∗
8(N+8)KdΛd
(t+KΛ2)3u∗2 ǫ− 8(N+8)KdΛ
d
(t+KΛ2)2u∗
)
(
δtδu
)
=
(
2− 4(N+2)KdΛd
(t+KΛ2)2K2ǫ
4(N+8)K4. . .
O(ǫ2) ǫ− 8(N+8)KdΛd
(t+KΛ2)2K2ǫ
4(N+8)K4
)
.
(4.24)The recursion relations are expressed in terms of ǫ(= 4−d) up to 1st order init(4.24). Since the off-diagonal matrix elements are of the order of ǫ2, can beconsidered zero at the order of ǫ and the diagonal elements can be consideredas the eigenvalues
yt = 2− N + 2
N + 8ǫ+O(ǫ2) ; (4.25)
yu = −ǫ = −(4 − d) . (4.26)
For d < 4, yu < 0. Thus the new fixed point has only one relevant operatorcorresponding to the mass term and hence has co-dimension one and can beused to describe the critical exponents at this point.
This new fixed point is known as Wilson-Fisher fixed point.
43
Chapter 5
The spectrum of the O(N )model
In the previous chapters we have come across various fixed points associatedwith the O(N) model. Now we want to impose the bound to the opera-tors at critical points. An operator at critical point is labeled by its scalingdimension ∆ and spin s. We will see various restrictions imposed by Con-formal symmetry on these labels and relations among them at various fixedpoints. First we want to know the spectrum of the Conformal group ind = 3. Character of representation of a group contains all the informationabout the spectrum. So we first go on calculating the Character of Unitaryrepresentation of Conformal group for different spin.
5.1 Character of unitary representation of d =
3 conformal group:
Unitary representations of the conformal group are labeled by the spin s andscaling dimension ∆ of their primary states. Character of such representationis given by
G(x, µ) = Trx∆µJ3 . (5.1)
The SU(2) character of spin s is given by
χs(µ) = TrµJ3 =
s∑
j=−s
µj .
Let us consider the state |δ, j〉, which is eigenstate of the operator x∆µJ3 witheigenvalue xδµj. the whole set of multiplate is obtained by acting the raising
44
operators P3, P± for each level. Tracing out the whole spectrum we obtainedthe character of the representation as
G∆,s = xδχs(µ)(1 + x+ x2 + . . . )(1 + µx+ µ2x2 + . . . )(1 + µ−1x+ µ−2x2 + . . . )
=xδχs(µ)
(1− x)(1− µx)(1− µ−1x).
Characters χ(x, µ) of short representation of the conformal algebra are ob-tained by evaluating the character and then subtracting out the character ofthe null states.
Null states can be obtained either by method given in [10] or by findingout the smallest element of the inner product matrix constructed by actingraising operators on the state |δ, j〉. This also gives the conditions betweenscaling dimension ∆ and spin s for the representation to be short. We haveused the second method and obtained the null state at first level. Accordingto [10] there is no other null in this representation.
• For s ≥ 1, Condition for unitarity is δ ≥ j+1 and Null state correspondsto δ = s+ 2, j = s− 1.
• For s = 12, Condition for unitarity is δ ≥ 1 and Null state corresponds
to δ = 2, j = 12.
• For s = 0, Condition for unitarity is δ ≥ 12and it has a level 2 Null
state corresponds to, δ = 52, j = 0.
When the equality conditions of the ‘Conditions of Unitarity’ satisfies, therepresentation is known as ‘short representation’ of conformal algebra. Nowthe Characters of short representations of conformal algebra can be writtenas
• For s ≥ 1
χs+1,s(x, µ) = Gs+1,s(x, µ)−Gs+2,s−1(x, µ) ; (5.2)
• For s = 12
χ1, 12(x, µ) = G1, 1
2(x, µ)−G2, 1
2(x, µ) ; (5.3)
• For s = 0χ 1
2,0(x, µ) = G 1
2,0(x, µ)−G 5
2,0(x, µ) . (5.4)
45
5.2 Primary operators of O(N) Theory
In the O(N) vector model there exists a set of traceless totally symmetrictensor currents js constructed from the bilinear field operators.They are con-served for free theory in UV whereas in the IR fixed point theory, the currentsare conserved only to leading order in 1/N . The single trace operators satisfythe condition, ∆ = s+ 1.
The bilinear operator Φ · Φ has scaling dimension 1 in free theory whereas it becomes 2 in IR. Spin s current should content exactly s number ofderivatives and s derivatives can give rise to at most s free traceless vectorindices. So an operator constructed with s number of derivatives and onebilinear field operator has spin= s and scaling dimension s+1 in free theoryand ∆ > s + 1 for IR fixed point,which allows multi trace operators andhence 1/N corrections to come into the picture. The currents generally havethe form[11]
js =
s∑
k=0
askΦ←−∂ ⊗k−→∂ ⊗s−kΦ− traces, (5.5)
where,
ask =1
2(−1)k
(
sk
)
(d2− 1)s
(d2− 1)k(
d2− 1)s−k
.
And (a)n = Γ(a+n)Γ(a)
is the Pochhammer symbol. All the odd spin currentsvanish due to symmetrization and even spin currents survive. To make theoperator totally symmetric and traceless , we can define a ‘current polyno-mial’as
js[y] =
s∑
k=0
askΦ(y ·←−∂ )k(y · −→∂ )s−kΦ− traces. (5.6)
The current operator can be reobtained by differentiation with respect to y
js =1
s!(∂y)
sjs[y] . (5.7)
As obtained in [11], subtraction of traces for a symmetric tensor of type1s!(t.y)s can be obtained from the formula
1
s!(t · y)s − traces =
|t|s|y|s2s(d
2− 1)s
Cd2−1
s
(
t · y|t||y|
)
, (5.8)
where Cλs are the Gegenbauer polynomials. Few expressions of Gegenbauer
polynomials relevant for our calculation has been listed in Appendix B.2.
46
Now let us write equation (5.6) as
js[y] ∼∑
k
askgk1g
(s−k)2 − traces,
where, g1 = y · ←−∂ and g2 = y · −→∂ . The sum is given by (as obtained in [11])
s∑
k=0
askgk1g
(s−k)2 = (−1) d
2−2Γ(
d2− 1)
2(g1+g2)
d2−2+s(−g1g2)1−
d4P
2− d2
d2−2+s
(
g2 − g1g2 + g1
)
,
(5.9)where Pm
l (x) is the associated Legendre polynomial. We have listed out someexpressions required for our calculation in B.1. We have obtained js[y] andhence js for d = 3 and spin s = 0, 1, 2, 3, 4.
• s = 0, d = 3: Using form of P1
2
− 12
from appendix B.1 we obtain the
s∑
k=0
askgk1g
(s−k)2 = (−1)− 1
41
2. (5.10)
Ignoring the factor (−1)− 14 , from equation (5.6), the current polynomial
can be written as
j0[y] =1
2(Φ · Φ).
Hence the spin zero current, from equation(5.7) is j0 = 12(Φ · Φ).
• s = 1, d = 3: Using the expression P1212
(z) =√
2πz(1 − z2)−
14 , the
quantity of (5.9) can be obtained as
s∑
k=0
askgk1g
(s−k)2 = (−1)− 1
2
√π
2(−g1g2)
1
4 (g1 + g2)1
2P121
2
(
g2 − g1g2 + g1
)
= (−1)− 141
2(g2 − g1).
From equation (5.6), the current polynomial can be written as
j(1)[y] =1
2Φ(y · −→∂ − y · ←−∂ )Φ.
and hence the spin zero current, from equation(5.7) is
j(1)i =
1
2Φ(−→∂ i −
←−∂ i)Φ = 0.
So the spin one current for real field is zero as expected. Similarly allspin odd current will become zero due to symmetrization.
47
• s = 2, d = 3: The quantity of (5.9) in this case is
s∑
k=0
askgk1g
(s−k)2 = (−1)− 1
41
2[g22 + g21 − 6g1g2] .
Current polynomial for spin-2 current is then given by
j(2)[y] = Φ
{
1
2[(y · −→∂ )2 + (y · ←−∂ )2 − 6(y · −→∂ )(y · ←−∂ )]
−16[y2(−→∂ )2 + y2(
←−∂ )2 − 6y2(
−→∂ · ←−∂ )]
}
Φ .
(5.11)
Spin-2 current is then give by
j(2)ij = Φ
{
1
2[−→∂ i
−→∂ j +
←−∂ i
←−∂ j − 3
−→∂ i
←−∂ j − 3
←−∂ i
−→∂ j]
−16δij [−→∂2 +
←−∂2 − 6
←→∂2 ]
}
Φ , (5.12)
where−→∂2 :=
−→∂ · −→∂ ,
←−∂2 :=
←−∂ · ←−∂ and
←→∂2 :=
←−∂ · −→∂ .
• s = 4, d = 3: In this case,
s∑
k=0
askgk1g
(s−k)2 = (−1)− 1
4 [2g41 + 2g42 + 35g21g22 − 6g1g
32 − 6g31g2].
Current polynomial for spin-4 current is then given by
j(4)[y] = Φ{
[2(y · ←−∂ )4 + 2(y · −→∂ )4 + 35(y · ←−∂ )2(y · −→∂ )2 − 6(y · ←−∂ )3(y.−→∂ )
− 6(y · ←−∂ )(y · −→∂ )3] +6
7[2 · y2←−∂ 2(y · ←−∂ )2 + 2(y · −→∂ )2y2
−→∂ 2
+ 35y2←→∂2 (y · ←−∂ )(y · −→∂ )− 6y2
←−∂ 2(y · ←−∂ )(y · −→∂ )− 6y2
−→∂ 2(y · ←−∂ )(y · −→∂ )]
− 8
35[2y46y2
←−∂ 4 + 2y4
−→∂ 4 + 35y4
←→∂4 − 6y4
←→∂2←−∂ 2
−6y4←→∂2−→∂ 2]
}
Φ .
(5.13)The spin four current can be obtained by differentiation it four timesw.r.t y. The full expression is not reproduced here as it is a fairly longone.
48
Conclusion and way to furtherresearch
We aimed at determining the spectrum and correlation functions of 3d isingmodel. In order to do that, we considered O(N) symmetric model whoseN = 1 case corresponds to the Ising model. We have identified the 3d Isingcritical point as the Wilson-Fisher fixed point in IR CFT of the O(N) model.In dimension d = 3 we have obtained the unitarity bound on spin and scalingdimension imposed by Conformal symmetry and observe that free theorycontents only single trace operators and fall into the short representationof the theory. The interacting theory mixes up the single trace and multitrace operators and gives rise to anomalous dimension. We have obtainthe expression for current polynomial [11] for d-dimension of a free O(N)theory of real fields and from that explicitly calculated the currents for fewlower spins for d = 3. These currents are all symmetric traceless and henceconstitute a higher spin symmetry.
Our idea is to now consider the interaction term which will leads toslightly broken higher spin symmetry by a factor 1/N , for large N and thenderive the spin-s current for such theory which will contain single trace as wellas double trace operator. What we are aiming to do next are the following:
• Obtain the currents for Φ4 theory with broken symmetry
• Obtain the four point function with 1/N correction
• Modify the theory with large N symmetry to match with our case ofN = 1
49
Appendix A
Labeling the eigenstates ofConformal Group
The generators of the Conformal algebra can be split into three sub-algebras:H , E+, E−. H is the Cartan subalgebra, E+ consists of generalized raisingoperators and E− consists of generalized lowering operators. In our case(conformal theory in d = 3) elements of such sets are the following:
• H = {J3, D}, where J3 is the z-component of angular momentum andD is the dilation operator.
• E+={J+, Pi}, where J+ is the raising operator of angular momentumalgebra and Pi is the translation operator which raise the scaling di-mension by unity.
• E−={J−, Ki}, where J− is the lowering operator of angular momentumalgebra and Ki is the special conformal transformation operator whichlower the scaling dimension by unity.
A highest weight state is annihilated by operators in E+. This state is labeledby corresponding eigenvalues of Cartan generators. We have J3 and D as theelements of Cartan subalgebra and hence the highest weight state is labeledby their eigenvalues l and ∆ respectively. So the highest weight state is |∆, l〉satisfies the following conditions:
J+|∆, l〉 = Pi|∆, l〉 = 0.
J3|∆, l〉 = l|∆, l〉;D|∆, l〉 = −i∆|∆, l〉.
Now any Casimir operator can be written as the combinations of its Cartanand raising, lowering segments. Any raising operator acting on it gives zero
50
and the commutator of raising and lowering operators again an element ofCartan and hence eigenvalue of Casimir operator can be written in terms ofthe Cartan eigenvalues, in this case ∆ and s. For the 3d conformal algebra,the quadratic Casimir in terms of Cartan and raising and lowering operatorsis
C2 =1
2[J+J− + J−J+] + J2
3 +1
2[PiK
i +KiPi]−D2
=1
2[J+J− + J−J+] + J2
3 +1
2δij[PiKj +KiPj ]−D2 (A.1)
And hence the eigenvalue can be obtained as
C2 |∆, l〉 =(
1
2[J+J− + J−J+] + J2
3 +1
2δij[PiKj +KiPj ]−D2
)
|∆, l〉
=
(
1
2[J+, J−] + J2
3 +1
2δij [Pi, Kj]−D2
)
|∆, l〉
=(
J3 + J23 − 3iD −D2
)
|∆, l〉=(
l + l2 − 3∆ +∆2)
|∆, l〉. (A.2)
We thus obtain
C2 |∆, l〉 =(
l(l + 1) + ∆(∆− 3))
|∆, l〉 .
51
Appendix B
Polynomials
B.1 Associated Legendre polynomial
Here we will give few expressions of Associated Legendre polynomial fordifferent fractional parameter values:
P12
− 12
(z) =
√
2
π(1− z2)−
14 (B.1)
P1212
(z) =
√
2
πz(1− z2)−
14 (B.2)
P123
2
(z) =
√
2
π
2z2 − 1
(1− z2)14
(B.3)
P1252
(z) =
√
2
π
4z3 − 3z
(1− z2)14
(B.4)
P1272
(z) =
√
2
π
1− 8z2 + 8z4
(1− z2)14
(B.5)
52
B.2 Gegenbauer polynomial
Here we will give few explicit expressions of Gegenbauer polynomial for dif-ferent parameter values:
C12
0 (x) = 1 (B.6)
C12
1 (x) = x (B.7)
C12
2 (x) =3x2
2− 1
2(B.8)
C12
3 (x) =5x2
2− 3x
2(B.9)
C12
4 (x) =35x4
8− 15x2
4+
3
8(B.10)
53
Bibliography
[1] Phlippe Di Francesco, Pierre Mathieu, David Senechal, Conformal FieldTheory, Springer(1997).
[2] F.A. Dolan and H. Osborn, Conformal Four Point Functions and the Op-erator Product Expansion, Nucl. Phys., B599, (2001)459-496; ArXiv:hep-th/0011040.
[3] F.A. Dolan and H. Osborn, Conformal Partial Waves and the Opera-tor Product Expansion, Nucl. Phys., B678, (2004) 491-507; ArXiv:hep-th/0309180.
[4] F.A. Dolan and H. Osborn, Conformal Partial Waves:Further mathemat-ical results, ArXiv:hep-th/1108.6194.
[5] R.J. Muirhead, Systems of Partial Differential Equations for Hyperge-ometric Functions of Matrix Argument, The Annals of MathematicalStatistics, 41(3) (1970), 991-1001.
[6] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin,A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap,ArXiv:hep-th/1203.6064v3.
[7] A. Pelissetto and E. Vicari, Critical Phenomena and Renormalizationgroup theory, Phys. Rept. 368(2002) 549-727, arXiv:cond-mat/0012164.
[8] Mehran Kardar, Statistical physics of fields, Cambridge University Press,2007.
[9] J. Maldacena and A. Zhiboedov, Constraining Conformal Field Theorieswith a Slightly Broken higher Spin Symmetry, ArXiv:hep-th/1204.3882v1.
[10] S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S.R. Wadia, X.Yin, Chern-Simons Theory with Vector Fermion Matter, Arxiv:hep-th/1110.4386
54
[11] H. Holzler, AdS/CFT Holography of the O(N) symmetric φ4 Vectormodel, PhD Thesis, University of Gottingen, 2007
[12] S. Minwalla, Restrictions imposed by Superconformal Invariance OnQuantum Field Theories, ArXiv:hep-th/9712074v2
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