Lecture 9: Perturbative Renormalization Group
Naoki KAWASHIMAISSP, U. Tokyo

June 17, 2019

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 20191 / 18

In this lecture, we see ...
When there is a fixed point and we know its OPE, by a perturbativeargument, we can derive a set of equations describing RG flow aroundit. (Then, we can study the behavior of other fixed points in its
vicinity, as we will discuss in the next lecture.)
We can obtain the renormalized Hamiltonian up to the 2nd order (ormore if we try harder) in the case of GFP, which is the lowest
non-trivial order.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 20192 / 18[9-1] General perturbative RG
We decompose the field operator into the high-frequency component
and the low-frequency component.
Tracing out the high-fruquency component, followed by rescaling,yields the RG flow equations.
In the RGT from the scale a to ab (b = 1 + δ), the product of two
scaling operators within the distance of a, gives rise to new
perturbative terms through OPE, which contributes non-linear terms
in the RG flow equation.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 20193 / 18

Expanding the Hamiltonian around a fixed pointConsider some fixed-point Hamiltonian, Ha∗ , with short-distant cut-off(lattice constant) a, and consider a general Hamiltonian expressed interms of the scaling-operators at Ha∗ :!
X ZHa ≡ Ha∗ + ∆Ha∆Ha ≡gα dx φα (x)αawhere φα is the scaling operator at H∗ with the dimension xα .
φα (x) → φ́α (x́) = Rb φα (x) = b xα φα (x)Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 20194 / 18

RGT to the expansion
Let us carry out the general program of RG: (i) partial trace, and (ii)rescaling.
We introduce the ultra-violet cut-off in the form of the restriction on
the integral region in (i) that no two operators cannot be within themutual distance a.
By the partial trace, we will shift the cut-off length a toá ≡ e λ a ≈ (1 + λ)a.
Then, the partial trace is equivalent to application of the OPE toevery pair of operators that come within the mutual distance of á,and taking the summation with respect to the relative position of thetwo (This yields the factor Vd (ád − ad ) ≈ Vd dλad , where Vd is thevolume of unit sphere.).

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 20195 / 18

The partial trace (0th order term)
By denoting the partial trace by Tr 0 , the perturbative expansion
becomes10 −Ha∗ −∆Ha
20−Ha∗
Tr e= Tr e1 − ∆Ha + (∆Ha ) − · · ·2−H̃ab (φl )≡e∗ byWe define Zh and H̃ab∗∗l(0th order term) = Tr 0 e −Ha (φ) = Zh × e −H̃ab (φ ) .

(1)

where the superscript l in φl is symbolic and reminder of therestriction that two operators cannot come closer than á. (We also∗ will become back to H∗ after the rescaling, because
demand that H̃aba∗
Ha is the fixed-point Hamiltonian.)
Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 20196 / 18

The partial trace (1st order term)e −H̃a = Tr 0∗1e −Ha 1 − ∆Ha + (∆Ha )2 − · · ·2∆Ha ≡Xα!

Zgαdx φα (x)aBecause of the absense of interaction, the 1st order term is easy:X Z∗(1st-order term) = − Tr 0 e −Ha (φ)
gα dx φα (x)
α(∗)

∗

X
l= −Zh e −H̃ab (φ )α= −Zh e∗ (φl )−H̃abZ
gαadx φlα (x)ab∆Hab (φl )(2)In (∗), we have usedhi∗∗l
Tr 0 e −Ha (φ) Q(φ) = Zh e −H̃ab (φ ) Q(φl )Naoki KAWASHIMA (ISSP)

Statistical Machanics IJune 17, 20197 / 18

The partial trace (2nd order term)∗
1e −H̃a = Tr e −Ha 1 − ∆Ha + (∆Ha )2 − · · ·2φh∆Ha ≡Xα!

Zgαdx φα (x)aFor the 2nd-order term, we use OPE:
(2nd-order term)

Z
∗
1X
=gα gβ Tr 0 e −H̃adxdy φα (x)φβ (y)
2aαβ

Z


∗
0−H̃∗aTr edxdy φα (x)φβ (y)Tr 0 e −H̃aaZ
Z=dxdy φlα (x)φlβ (y) +dxdy φα (x)φβ (y)|x−y|>aba<|x−y|<abZZ≈dxdy φlα (x)φlβ (y) +dxdy φα (x)φβ (y)aba<|x−y|<ab“trivial term”

Naoki KAWASHIMA (ISSP)“collision term”

Statistical Machanics I

June 17, 20198 / 18

OPE for the collesion term
For the collision term, we use OPE:
Zdxdy φα (x)φβ (y)a<|x−y|<abZ≈

dxdya<|x−y|<abµZ

dµcαβ

Xdaxα +xβ −xγXφlµ (x)µcαβ

φl (x)
xα +xβ −xγ µaµZX µdyα +yβ −yµ= Vd (b − 1)cαβ adx φlµ (x)dx Vd ((ab) − a )=ababµNaoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 20199 / 18

The 2nd order term
Putting together, the 2nd order term becomes(2nd-order term)= Zh e1−H̃∗abX2(Zgα gβdxdy φlα (x)φlβ (y)abαβZX µdyα +yβ −yµ+Vd (b − 1)cαβ adx φlµ (x)abµ= Zh e−H̃∗ab)

1(int)(∆Hab (φl ))2 − ∆Hab2



where1(int)∆Hab ≡ −2

Naoki KAWASHIMA (ISSP)

XαβµZX µyα +yβ −yµgα gβ Vd (b − 1)cαβ adx φlµ (x)dµ

Statistical Machanics IabJune 17, 201910 / 18

Summary of partial trace
Finally, the partial trace results in∗∗l(int)ll

Tr 0 e −Ha (φ)−∆Ha (φ) ≈ Zh e −H̃ab (φ )−∆Hab (φ )−∆Hab (φ )Therefore, our Hamiltonian after the partial trace is(int)∗H̃ab (φl ) = H̃ab(φl ) + ∆Hab (φl ) + ∆Hab (φl )X Z∗= H̃ab(φl ) +gµdx φlµ (x)µab1Xµ yα +yβ −yµgα gβ Vd (b d − 1)cαβa2µαβX Z∗g̃µdx φlµ (x)= H̃ab(φl ) +Z−µwhere g̃µ ≡ gµ −dx φlµ (x)abab1Xµ yα +yβ −yµgα gβ Vd (b d − 1)cαβ
a2µαβ

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 201911 / 18

Rescaling
By x́ ≡ b −1 x and φ́µ (x́) ≡ b xµ φlµ (x) ,X Z
∗H́a (φ́) = Ha (φ́) +g̃µ dx́ b yµ φ́µ (x́)
a µX
1µcαβgα gβ Vd (b d − 1)ayα +yβ −yµ  .
⇒ ǵµ = byµ g̃µ = byµ gµ −2αβ

By absorbing the factor d2 Vd ayµ in the definition of gµ and ǵµ ,,dX µ(b−1)
cαβ gα gβǵµ = b yµ × gµ −dαβ

By rewriting this equation using λ ≡ log b, we finally obtain
X µdgµ= yµ gµ −cαβ gα gβ + O(g 3 )dλαβ

Naoki KAWASHIMA (ISSP)

Statistical Machanics IJune 17, 201912 / 18

[9-2] Perturbative RG around GFPThe criticality of the Ising model in d > 4 is controled by theGaussian fixed-point, though the critical behavior is modified by thedangerously irrelevant field.
For d < 4, the Gaussian fixed-point is not stable w.r.t. the scaling
operator φ4 . This motivates us to look for another fixed point by
examining the perturbative RG flow around the Gaussian fixed point.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 201913 / 18

Critical property of the Ising model above 4-dimensions
Consider the φ4 model.
Z
H = dx |∇φ|2 + tφ2 + uφ4 − hφ
Let us consider the φ2 and φ4 terms as the perturbation to theGaussian fixed point (GFP). Then, it is natural to express theHamiltonian in terms of scaling operators at the GFP.
Z
H = dx |∇φ|2 + tφ2 + uφ4 − hφ
The scaling eigenvalues for these terms arex2 = 2x = d − 2 ⇒ y2 = d − x2 = 2x4 = 4x = 2(d − 2) ⇒ y4 = d − x4 = 4 − d.
Since φ4 is irrelevant if d > 4, the critical behavior of the φ4 model(and therefore the Ising model as well) is described by the GFP.
Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 201914 / 18

Dangerous irrelevant operator for d > 4
According to the general argument (see Lecture 7), the spontaneous
magnetization should have the singularity likem ∝ L−d+yh = L−xh ∝ (t− y1 −xht)

d−2=t 4 .

(wrong)However, we saw that the mean-field theory correctly describes thecritical behavior for d > 4 (Ginzburg criterion), which means that
1m ∝ t 2 . (correct)
This apparent contradiction comes from the nature of the irrelevant
field u. Specifically, since the φ4 model at or below the critical point(t ≤ 0) is not well-defined when u = 0, we cannot simply put u = 0
in the scaling form as we did in the general argument.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 201915 / 18

Perturbative RG around GFPWe have derived the general RG flow
equation around a fixed-point.
X µdgµ= yµ gµ −cαβ gα gβdλ(3)αβIf we apply this to GFP, we immediately
notice that, for d > 4, there is only one
relevant field t, implying that the GFP is thecontroling fixed point.
Even below four dimensions, we may be able to obtain a new fixed
point from (3) if it is near the GFP.
In other words, we may try to find gµ that makes the r.h.s. of (3)zero and deduce its properties from (3). (Next lexture)

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 201916 / 18

Summary
We have derived a set of equations describing RG flow around a given
fixed point.
We can obtain the renormalized Hamiltonian up to the 2nd order (ormore if we try harder) in the case of GFP, which is the lowest
non-trivial order.
Above four dimensions, the critical point is controled by the Gaussian
fixed point.
However, the dangerously irrelevant field, u, modifies the criticalbeheviors to mean-field like.
Below four dimensions, the critical point is not controled by theGaussian fixed point because u becomes relevant.
We may be able to find the “true” fixed point by analyzing the RG
flow equation. (Next lecture)

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 201917 / 18

Exercise
We saw an apparent contradiction between the general scalingargument and the mean-field behaviors expected from the Ginzburg
criterion. Think of a scaling form of the singular part of the freeenergy that obeys the scaling properties expected from the generalargument, and, at the same time, produces the correct mean-fieldcritical behaviors.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

June 17, 201918 / 18