Lecture 8: Operator Product Expansion
Naoki KAWASHIMAISSP, U. Tokyo

June 10, 2019

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 20191 / 18

In 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.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 20192 / 18[8-1] 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.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 20193 / 18

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)µAt the fixed point, we required that the RGT acts on φµ as
Rb φµ (x) = b xµ φµ (x)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 when viewed from a distant point.)
Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 20194 / 18Three-point correlator (1/2)
Let us consider three-point correlation function:Gµνλ (x, y, z) ≡ hφµ (x)φν (y)φλ (z)i
By applying the RGT and then expanding φµ φν , we haveGµνλ (x́, ý, ź) = b xµ +xν +xλ Gµνλ (x, y, z)Xx+yxλ +xµ +xνα,z=bCµν (x − y)Gαλ2αBy reversing the order of the operations, we haveXx́ + ýαGµνλ (x́, ý, ź) =Cµν (x́ − ý)Gαλ, ź2αXx+yα
xα +xλ=Cµν (x́ − ý)bGαλ,z2α

Naoki KAWASHIMA (ISSP)

Statistical Machanics IJune 10, 2019

(2)

(3)

5 / 18Three-point correlator (2/2)
Comparing (2) and (3), we conclude
r
αCµναCµν(r) = xµ +xν b−xαbThis leads toαα∃cµν Cµν(r ) =αcµνr xµ +xν −xλ.

Therefore, we have the operator-product expansion:
φµ (x)φν (y) =Xα

Naoki KAWASHIMA (ISSP)

αcµνr xµ +xν −xλφα (x)

Statistical Machanics I(OPE)

June 10, 20196 / 18

Universality
By normalizing the scaling operators so thatlim|x−y|→∞|x − y|2xµ hφµ (x)φν (y)i = 1

α can be fixed (and become universal
the OPE coefficient cµνquantities).
α are universal, andWe assume that thus fixed OPE coefficients cµνcharacterizing 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.)

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 20197 / 18[8-2] 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.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 20198 / 18

A hint for scaling operators — Wick’s theorem
Consider the operator (φ(x))2 andits correlation function.
hφ2 (x)φ2 (y)i = hφ(x) φ(x) φ(y) φ(y)i1234= hφ(x) φ(x)ihφ(y) φ(y)i1234+ hφ(x) φ(y)ihφ(x) φ(y)i1324+ hφ(x) φ(y)ihφ(x) φ(y)i (4)1423= G 2 (0) + 2G 2 (r )where r ≡ |x − y| andG (r ) ∼Naoki KAWASHIMA (ISSP)

1r 2x

wherex≡d −22

Statistical Machanics I

June 10, 20199 / 18

Scaling operators
What are scaling operators at the Gaussian fixed point?
If the constant term G (0)2 in
hφ2 (x)φ2 (y)i = G (0)2 + 2G (r )2were absent, φ2 (x) would be regarded as a scaling operator.
This observation leads us to define
φ2 (x) ≡ φ(x)2 − hφ(x)2 i,
which has the two-point correlator
hφ2 (x)φ2 (y)i = h(φ2 (x) − G (0))(φ2 (y) − G (0))i= hφ2 (x)φ2 (y)i − G (0)2 = 2G (r )2 =2r 4x
Therefore, φ2 is the scaling operator with the dimension x2 ≡ 2x.
Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 201910 / 18

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 ]] a scaling operator. For example,

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 201911 / 18

Scaling operators φ2 ≡ [[φ2 ]] and φ3 ≡ [[φ3 ]]For φ2 , as we have seen alreadyφ2 ≡ [[φ2 ]] = φ2 − hφ2 i
For φ3 , from the diagram below, we obtain
φ3 = [[φ3 ]] + 3hφ2 iφTherefore,
φ3 ≡ [[φ3 ]] = φ3 − 3G (0)φNaoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 201912 / 18

Scaling operator φ4 ≡ [[φ4 ]]For φ4 , again from the diagram, we obtain
φ4 = [[φ4 ]] + 6hφ2 i[[φ2 ]] + 3hφ2 i2Therefore,
φ4 = [[φ4 ]] = φ4 − 6G (0)φ2 + 3G (0)2
= φ4 − 6G (0)φ2 − 3G (0)2

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 201913 / 18

Scaling operators of Gaussian fixed-point
To summarize, the scaling operators of Gaussian fixed point are
φn ≡ [[φn ]]For the standard normalization, considerXN!
hφn (x)φn (y)i =hφ(x)φ(y)in = N!G n (r ) = 2nx .
rall connection
patterns

1Therefore, φ̂n ≡ √ φn is the normalized scaling operator.
N!
The scaling dimension is obviously
xn ≡ nx = n(d − 2)/2The scaling operators are orthogonal to each otherδmn
hφ̂m (x)φ̂n (y)i = 2xmr

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 201914 / 18

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

From this diagram, we obtainx+yφ2 (x)φ2 (y) ∼ φ4+ 4hφ(x)φ(y)iφ(x)φ(y) + 2hφ(x)φ(y)i22x+yx+y= φ4+ 4G (r )φ2+ 2G 2 (r ).
22
Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 201915 / 18

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

(m+n)/2  
nx+ykφm (x)φn (y) ∼k!G (r )φm+n−2kk2k=0lXx+ycmnφl=x+x−xmnl|x − y|2Xmkl

l = m n k!
(cmn( k )(k )(k≡ m+n−l))
2Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 201916 / 18

SummaryGenerally, we can expand the product of the two scaling operators interms of scaling operators (OPE), which takes the form
lX
cmnx+yφm (x)φn (y) =φl.
|x − y|xm +xn −xl
2ll
The constants cmn
are universal quantities (provided that the scalingoperators 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−ll
cmn =k!
k≡kk
2

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 201917 / 18

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

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 10, 201918 / 18