Lecture 11: Perturbative Renormalization Group
Naoki KAWASHIMAISSP, U. Tokyo

June 24, 2024

Statistical Mechanics I: Lecture 11

June 24, 2024

1 / 18In this lecture, we see ...
When there is a fixed point for which we know its OPE, we canderive, by a perturbative argument, a set of equations describing RG
flow around it. (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.

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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 + λ), 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.

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Expanding the Hamiltonian around a fixed point
Consider 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∗ + Va
Va ≡gα dx ϕα (x)αawhere ϕα is the scaling operator at H∗ with the dimension xα .
ϕα (x) → ϕ′α (x′ ) = Rb ϕα (x) = bxα ϕα (x)Statistical Mechanics I: Lecture 11June 24, 2024

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Outline of RGT for the expansion
We carry out the general RGT program: partial trace and rescaling.
We introduce the ultra-violet cut-off, a, which means two conditions:(a) When we expand e−Va (ϕ) , the spatial integration likeZ
dx1 dx2 · · · dxn ϕα1 (x1 )ϕα1 (x2 ) · · · ϕα1 (xn )ais restricted in the region where no two xi and xj are closer than a,and (b) ϕα contains only low-frequency component with k < 1/a.
The partial trace will shift the cut-off a to a′ ≡ eλ a ≈ (1 + λ)a.
The OPE is applied to every pair of operators that come within themutual distance of a′ , and taking the summation with respect to therelative position of the two (This yields the factor Ωd (a′ d − ad )≈ Ωd dλad , where Ωd is the volume of unit sphere.).

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5 / 18The partial trace
We decompose the field operator as ϕ = ϕℓ + ϕs whre ϕℓ and ϕs are
the long wave-length (k < 1/a′ ) and the short (1/a′ < k < 1/a)wave-length components of ϕ, respectively.
In what follows, Ha (≡ Ha∗ + Va ) is the perturbed Hamiltonian, Ha∗ isthe fixed point Hamiltonian, Zs is the short wave-length contribution
to the partition function, H̃a′ is the coarse-grained (but not yetre-scaled) perturbed Hamiltonian, and H̃a∗ is the coarse-grained
fixed-point Hamiltonian. More specifically,∗∗ℓ

Zs e−H̃a′ (ϕ ) ≡ Tr e−Ha (ϕ) ,(1)

{ϕs }

ℓ

Zs e−H̃a′ (ϕ ) ≡ Tr e−Ha (ϕ)
{ϕs }

1
∗
= Tr e−Ha (ϕ) 1 − Va (ϕ) + (Va (ϕ))2 − · · ·2{ϕs }

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The short wave-length average and the 1st order term
The partial trace over ϕs goes like
1 2= Tr e
1 − Va (ϕ) + Va (ϕ) + · · ·Tr e2{ϕs }{ϕs }
∗ (ϕℓ )
1−H̃a2′= Zs e1 − ⟨Va (ϕ)⟩s +(2)V (ϕ) s + · · ·2 a
∗ (ϕ)−V (ϕ)
−Haa

∗ (ϕ)
−Ha

where the short w.l. average is defined as∗∗⟨· · · ⟩s ≡ Tr e−Ha (ϕ) · · · / Tr e−Ha (ϕ) .
{ϕs }

{ϕs }

The first order term is simplyZZXXgα ⟨ϕα (x)⟩s =gα ϕℓa (x) = Va′ (ϕℓ )⟨Va (ϕ)⟩s = dxdxaa′ααStatistical Mechanics I: Lecture 11

June 24, 2024

7 / 18The 2nd order term
We can split the double integration into 2 parts:
ZX
2⟨Va (ϕ)⟩s = dxdy
gα gβ ⟨ϕα (x)ϕβ (y)⟩sa=Xα,βZgα gβα,β!

Z+a′a<|x−y|<a′dxdy⟨ϕα (x)ϕβ (y)⟩s(3)The first term is simply (Va′ (ϕℓ ))2 :ZXgα gβdxdy⟨ϕα (x)ϕβ (y)⟩sa′α,β=Xα,βZgα gβa′dxdy ϕℓα (x)ϕℓβ (y) = (Va′ (ϕℓ ))2

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The “collision” term
To conform the new cutoff a′ , the OPE must be applied to thesecond term in (3) representing operators too close to each other:ZXgα gβdxdy ⟨ϕα (x)ϕβ (y)⟩sa<|x−y|<a′α,β=XZ
gα gβα,β=X

dxdya<|x−y|<a′′ dµdgα gβ Ωd ((a ) − a )α,βX

Zdxa′cµαβ

ϕℓµx+x−xαµβ|x − y|Xµx+y2cµαβϕℓµ (x)x+x−xαµβaZX X µ=λdx
cαβ gα gβ (Ωd dayα +yβ −yµ ) ϕℓµ (x)a′µα,β

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9 / 18The final form of the 2nd order term
Putting together, the 2nd order term in (3) becomes⟨(Va (ϕ))2 ⟩s= Zs e∗−H̃a′X µ(Va′ (ϕ )) + λcαβ gα gβ (Ωd dayα +yβ −yµ )ℓ2Zdx ϕℓµ (x)a′µ,α,β∗(int) ℓ−H̃aℓ 2′= Zs e(Va′ (ϕ )) − 2Va′ (ϕ )Z
λ X X µ(int) ℓyα +yβ −yµ Va′ (ϕ ) ≡ −cαβ gα gβ (Ωd da)dx ϕℓµ (x).
2 µa′α,βThus, the expansion (2) becomes
Tr e{ϕs }∗ (ϕ)−V (ϕ)−Haa= Zs e

∗
−H̃a′



1(int)
1 − Va′ + (Va′ )2 − Va′ + · · ·2

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Summary of partial trace
Finally, the partial trace results in∗∗ℓ(int)ℓTr e−Ha (ϕ)−Va (ϕ) ≈ Zs e−H̃a′ (ϕ )−Va′ (ϕ )−Va′(ϕℓ ){ϕs }Therefore, our Hamiltonian after the partial trace is(int)H̃a′ (ϕℓ ) = H̃a∗′ (ϕℓ ) + Va′ (ϕℓ ) + Va′ (ϕℓ )X Z∗ℓ= H̃a′ (ϕ ) +gµdx ϕℓµ (x)a′µλX µ−cαβ gα gβ dΩd ayα +yβ −yµ2µαβ
X Z∗ℓg̃µdx ϕℓµ (x)= H̃a′ (ϕ ) +Za′dx ϕℓµ (x)a′µλX µwhere g̃µ ≡ gµ −cαβ gα gβ dΩd ayα +yβ −yµ2µαβ

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Rescaling and RG flow equation
By re-scaling ( x′ ≡ b−1 x and ϕ′µ (x′ ) ≡ bxµ ϕℓµ (x) ),X Z′′ℓ∗ ′Ha (ϕ ) = H̃a′ (ϕ ) = Ha (ϕ ) +g̃µ dx′ byµ ϕ′µ (x′ )µ⇒ gµ′ = byµ g̃µ = byµ gµ −aλX µcαβ gα gβ dΩd ayα +yβ −yµ  .
2αβBy absorbing the factor d2 Ωd ayµ in the definition of gµ and gµ′ ,X µcαβ gα gβ gµ′ = (1 + λ)yµ × gµ − λαβX µdgµ= yµ gµ −cαβ gα gβ + O(g 3 )dλαβ

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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.

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Critical property of the Ising model above 4-dimensions
Consider the ϕ4 model, with ϕ2 and ϕ4 terms. From the viewpoint of
the perturbative RG around GFP, it is convenient to use the scaling
fields ϕ2 and ϕ4 , instead of ϕ2 and ϕ4 :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 at
t = 0 (and therefore the Ising model at T = Tc as well) is describedby the GFP.

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Dangerous irrelevant operator for d > 4
According to the general argument (see Lecture 7), the spontaneous
magnetization should scale likem∼L−d+yh=L−xh∼txhytd−2=t 4 .

(wrong)However, we saw that the mean-field theory correctly describes thecritical behavior for d > 4 (Ginzburg criterion), which means that1m ∼ t2 .

(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.

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Perturbative RG around GFPWe have derived the general RG flow
equation around a fixed-point.
X µdgµ= y µ gµ −cαβ gα gβdλ(4)αβIf we apply this to GFP, we immediately
notice that, in d > 4, there is no relevant
field other than t, implying that the GFP
governs the critical phenomena of the ϕ4model.
Even below four dimensions, we may be able to obtain a new fixed
point from (4) if it is near the GFP.
In other words, we may try to find gµ that makes the r.h.s. of (4)zero and deduce its properties from (4). (Next lexture)

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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)

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Exercise 11.1: We saw an apparent contradiction between the generalscaling argument and the mean-field behaviors expected from theGinzburg 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-field criticalbehaviors.

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