Lecture 10: Operator Product Expansion
Naoki KAWASHIMAISSP, U. Tokyo

June 17, 2024

Statistical Mechanics I: Lecture 10

June 17, 2024

1 / 18In this lecture, we see ...
Product of two scaling operators can be expanded in terms of scaling
operators. (OPE)Such an expansion determines the RG-flow structure around the fixed
point (will be discussed in the next lecture).
For the Gaussian fixed point, all the coefficients of the OPE can beexactly obtained.

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General Framework
Product of two scaling operators, defined at some distance from eachother, can be expanded as a linear combination of scaling operators.
Considering the 3-point correlators and taking into account of thescaling properties of operators, the general form of the OPE
coefficients can be fixed up to universal constants.

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Product of two is expandable
Previously, we introduced scaling operators {φµ } as something thatspans the space of local operators:!
X∀Q(x)∃qµ Q(x) =qµ φµ (x)(1)µThe RGT’s action on φµ is φ′µ (x′ ) = bxµ φµ (x), which leads to⟨φµ (x)φν (y)⟩ ∼ |x − y|−2xµ
Let us consider the product of two scaling operators, φµ (x)φν (y).
This product must appear to be a “local” operator when we view itfrom a point z far away from x and y (i.e., |z − x| ≫ |y − x|).
Then, we should be able to expand it:

X
x+yα
φµ (x)φν (y) =Cµν(x − y)φα2α(Equality holds only asymptotically.)
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Three-point correlator (1/2)
Let us consider three-point correlation function:Gµνλ (x, y, z) ≡ ⟨φµ (x)φν (y)φλ (z)⟩By applying the RGT and then expanding φµ φν , we haveGµνλ (x′ , y ′ , z ′ ) = bxµ +xν +xλ Gµνλ (x, y, z)

X
x+yα
= bxλ +xµ +xνCµν(x − y)Gαλ,z2α
By reversing the order of the operations, we have ′′Xx+yαGµνλ (x′ , y ′ , z ′ ) =Cµν(x′ − y ′ )Gαλ, z′2αXx+yα,z=Cµν(x′ − y ′ )bxα +xλ Gαλ2αStatistical Mechanics I: Lecture 10

June 17, 2024

(2)

(3)

5 / 18Three-point correlator (2/2)
Comparing (2) and (3), we concluder
α−xµ −xν +xα αCµν (r) = bCµν.
bThis leads to
αα∃cµν Cµν(r) =cαµνrxµ +xν −xλ.

Therefore, we have the operator-product expansion:

φµ (x)φν (y) =Xαcαµν
rxµ +xν −xλφα (x)Statistical Mechanics I: Lecture 10(OPE)

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Universality
By normalizing the scaling operators, φµ , so thatlim|x−y|→∞|x − y|2xµ ⟨φµ (x)φµ (y)⟩ = 1

the OPE coefficient cαµν can be fixed. The resulting coefficients areuniversal numbers.
We assume that thus fixed OPE coefficients cαµν are universal, andcharacterizing property of the fixed-point, together with the scaling
dimensions, xµ . In other words, they do not depend on the details ofthe system, but depend only on the symmetry, the space dimension,etc. (This assumption of universality is similar to the assumption ofvery existence of the fixed-point of the RGT. It is at least supported
by several exactly solvable cases.)Statistical Mechanics I: Lecture 10

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OPE at the Gaussian fixed point
The scaling operators at the Gaussian fixed point can be obtained
through the normal order product: φn ≡ [[ϕn ]].
We can compute exact OPE coefficients of the gaussian fixed point.

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A hint for scaling operators — Wick’s theoremConsider the operator (ϕ(x))2 andits correlation function.
⟨ϕ2 (x)ϕ2 (y)⟩ = ⟨ϕ(x) ϕ(x) ϕ(y) ϕ(y)⟩1234= ⟨ϕ(x) ϕ(x)⟩⟨ϕ(y) ϕ(y)⟩1234+ ⟨ϕ(x) ϕ(y)⟩⟨ϕ(x) ϕ(y)⟩1324+ ⟨ϕ(x) ϕ(y)⟩⟨ϕ(x) ϕ(y)⟩12423= G (0) + 2G2 (r)(4)

where r ≡ |x − y| and
1d−2G(r) ∼ 2x where x ≡r2(x ≡ The scaling dim. of ϕ1 = ϕ)Statistical Mechanics I: Lecture 10

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Scaling operators
What are scaling operators at the Gaussian fixed point?
If the constant term G(0)2 in ⟨ϕ2 (x)ϕ2 (y)⟩ = G(0)2 + 2G(r)2 wereabsent, ϕ2 (x) would be regarded as a scaling operator.
This observation leads us to define
φ2 (x) ≡ ϕ(x)2 − ⟨ϕ(x)2 ⟩,which has the “pure” two-point correlator⟨φ2 (x)φ2 (y)⟩ = ⟨(ϕ2 (x) − G(0))(ϕ2 (y) − G(0))⟩= ⟨ϕ2 (x)ϕ2 (y)⟩ − G(0)2 = 2G(r)2 =2r4x

Therefore, φ2 is the scaling operator with the dimension x2 ≡ 2x.

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Normal-ordered operator
The key to finding general scaling operators is to eliminate thediagrams with “internal connections” such as the first term in (4).
Therefore, it would be convenient to introduce a symbol [[· · ·]] asAll terms represented by[[A(x)]] ≡ A(x) − diagrams with internal
connections

The operator thus defined is called “normal-ordered.”
When considering correlations among normal-ordered operators, bydefinition, we can forget about the internal lines. Therefore, 2-point
correlators do not have constant terms, which makes thenormal-ordered operator φn ≡ [[ϕn ]] a scaling operator. For example,Statistical Mechanics I: Lecture 10

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Scaling operators φ1 ≡ [[ϕ]], φ2 ≡ [[ϕ2 ]] and φ3 ≡ [[ϕ3 ]]Obviously, φ1 ≡ [[ϕ]] = ϕ .
For ϕ2 , as we have seen already
ϕ2 = [[ϕ2 ]] + ⟨ϕ2 ⟩ ⇒ φ2 ≡ ϕ2 − ⟨ϕ2 ⟩For ϕ3 , from the diagram below, we obtain
ϕ3 = [[ϕ3 ]] + 3⟨ϕ2 ⟩ϕTherefore,
φ3 ≡ [[ϕ3 ]] = ϕ3 − 3G(0)ϕStatistical Mechanics I: Lecture 10

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Scaling operator φ4 ≡ [[ϕ4 ]]By explicitly writing down the diagrams contributing to ϕ4 ,ϕ4 = [[ϕ4 ]] + 6⟨ϕ2 ⟩[[ϕ2 ]] + 3⟨ϕ2 ⟩2

Therefore,
φ4 = [[ϕ4 ]] = ϕ4 − 6G(0)φ2 − 3G(0)2= ϕ4 − 6G(0)ϕ2 + 3G(0)2

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Scaling operators of Gaussian fixed-point
To summarize, the scaling operators of Gaussian fixed point areφn ≡ [[ϕn ]]For the standard normalization, considerXn!
⟨φn (x)φn (y)⟩ =⟨ϕ(x)ϕ(y)⟩n = n!Gn (r) = 2nx .
rall pairingpatterns

1Therefore, φ̂n ≡ √ φn is the normalized scaling operator.
n!
nThe scaling dimension is obviously xn ≡ nx = (d − 2) .
2Because of the “no internal line” condition, the scaling operators areorthogonal to each other⟨φ̂m (x)φ̂n (y)⟩ =δmn.
r2xm

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Expansion of φ2 (x)φ2 (y)Consider the product of two operators φ2 (x)φ2 (y).

From this diagram, we obtainx+yx+yφ2 (x)φ2 (y) ∼ φ4+ 4G(r)φ2+ 2G2 (r),22or more symbolically, φ2 · φ2 ∼ φ4 + 4φ2 + 2.

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OPE of Gaussian fixed-point
We can generalize the product ϕ2 ϕ2 to general two operators.

min(m,n)  
m nx+ykφm (x)φn (y) ∼k!G (r) φm+n−2k2kkk=0
Xx+yclmnφl=|x − y|xm +xn −xl
2l=|m−n|,|m−n|+2,··· ,m+n  mnm+n−lclmn =k!
k≡
2kkX

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SummaryGenerally, we can expand the product of the two scaling operators interms of scaling operators (OPE), which takes the form


X
clmn
x+y
φm (x)φn (y) =φl.
|x − y|xm +xn −xl2l

The constants clmn are universal quantities (provided that the scaling
operators are properly normalized).
For the Gaussian model, the scaling operator can be explicitly definedby the normal-ordering as φn ≡ [[ϕn ]], and its scaling dimension isxn ≡ nx = n(d − 2)/2.
The OPE at the Gaussian fixed-point is characterized by  mnm+n−lclmn =k!
k≡kk
2

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Exercise 10.1: For the Gaussian model, obtain φ5 in terms of ϕk(k = 1, 2, 3, · · · ), following the same argument as in the lecture.

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