Lecture 6: General Framework of Renormalization
Group — Fixed Points and Scaling Operators
Naoki KAWASHIMAISSP, U. Tokyo

May 27, 2019

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 27, 20191 / 22

In this lecture, we see ...
Having seen a few examples of the real-space RG transformations, weformulate it as a general framework for discussing the phase diagram
and the critical phenomena.
As an exactly-treatable example of the RG framework, we consider
the Gaussian model, which is easy to solve and provides us thestarting point for perturbative renormalization group.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 27, 20192 / 22

[6-1] Fixed-point and scaling operators
As a Gedunkenexperiment, we consider a generic Hamiltonian, and its
exact renormalization group transformation. (As long as it exists, it
doesn’t matter whether or not we can actually compute such things.)
We’ll see that the RGT defines a “RG-flow” in the parameter space,which provides us with a framework of understanding the phase
diagram.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

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Critical point is scale-invariant“https://youtu.be/fi-g2ET97W8” by Douglas Ashton

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

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RG flow“https://youtu.be/MxRddFrEnPc” by Douglas Ashton

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Statistical Machanics I

May 27, 20195 / 22

Generic Hamiltonian
Any Hamiltonian is expressed as an expansion w.r.t. local operators.
XXHa (S|K, L) = −Kα Sα (x)(1)
xαwhere {Sα } spans the space of all local operators, i.e.,!
X∀Q(x) ∃qα Q(x) =qα Sα (x)(2)α

Example: A generic model defined with Ising spins.
K1 = HK2 = Jx
K3 = Jy
K4 = Q
K5 = Q..
.
Naoki KAWASHIMA (ISSP)

S1 (x) = Sx
S2 (x) = Sx Sx+ax
S3 (x) = Sx Sx+ay
S4 (x) = Sx Sx+ax Sx+2ax
S5 (x) = Sx Sx+ax Sx+ay..
.
(a , a , · · · : lattice unit vectors)
xy

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RG flow diagram
The RGTHa (φ, K) → Ha (φ́, Ḱ)
can be regarded as a map fromthe parameter space onto itself
K → Ḱ ≡ Rb KAn RG trajectory is a RGT-invariant curve.
We assume that the trajectory is continuous. (In other words, theRGT is defined for continuous b, such that Rb1 Rb2 = Rb1 b2 .)A trajectry converging to the unstable fixed point (FC ) is called acritical line (LC ). The parameter along it is called irrelevant (uw ).
The parameter along a trajectry emanating from the unstable fixed
point is called relevant. (ut ).
Naoki KAWASHIMA (ISSP)

Statistical Machanics IMay 27, 20197 / 22

Critical properties are controled by unstable fixed-point
RGT with b maps the points
1, 2, · · · , 7 to 10 , 20 , · · · , 70 .
RGT with b 0 > b maps thenarrower region including only
2, 3, 4, 5, 6, to 200 , 300 , · · · , 600 ,distributed in the same range ofut , but closer to the uw = 0 line.
In this way, a narrower region is mapped closer to the uw = 0 line.
So, the critical properties on the t-axis, is identical to the property
around the unstable fixed point (“FC ”) on the uw = 0 line.
The irrelevant fields of our system determine how far we must
approach to the critical point to observe the correct critical behavior.
Applying a small irrelevant field does not qualitatively change thenature of the critical point, while a relevant field does.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

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Expansion around unstable fixed point
Consider the local Hamiltonian Ha (S(x), x) and its fixed point form:Ha∗ (S(x), x) ≡ Ha (S(x), x|K∗ ).

(3)(In what follows, we drop some or all of the parameters, a, x and S(x), and use theabbreviation like H∗ for Ha∗ (S(x), x).)Let us denote the RGT by Rb where b is the renormalization factor.
Then, Rb (H∗ ) = H∗ .
Let us expand the Hamiltonian around this fixed point.
XH = H∗ −hα Sα (x) = H∗ − h · S(4)αwhere hα is the deviation of the parameter Kα from its fixed-pointvalue, i.e., hα ≡ Kα − Kα∗

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Statistical Machanics I

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Linearization of RGT
Now consider the transformation applied to the local Hamiltoniannear the fixed point:Rb (H∗ − h · S(x)) = H∗ − h́ · S(x)To the lowest order, h́ depends linearly on h in the lowest order, i.e., alinear operator Tb exists such thath́ ≈ Tb h.
We assume that Tb is diagonalizable with real eigenvalues.

λ1


λ2
P −1 Tb P =  ≡ Λb..
.

Naoki KAWASHIMA (ISSP)

Statistical Machanics IMay 27, 201910 / 22

Scaling fields and scaling operators
By definingu ≡ P −1 h, and φ ≡ P T S
we obtainu · φ = (P −1 h)T (P T S) = hT (P −1 )T P T S = h · S.
In addition, u transforms asú ≡ Rb u = P −1 h́ = P −1 Tb h = P −1 Tb Pu = Λb u,
namely, úµ = b yµ uµwith

yµ ≡ logb λµ .

uµ = “scaling field”,

φµ = “scaling operator”,yµ > 0 → uµ is relevant
yµ = “scaling eigenvalue”
yµ < 0 → uµ is irrelevant

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Statistical Machanics IMay 27, 201911 / 22

Scaling dimensions
We have seen that we can formulate the RGT for a general
Hamiltonian expanded around a fixed pointH(φ) = H∗ (φ) − u · φ,asH́(φ́) ≡ Rb H(φ) = H∗ (φ́) −Xb yµ uµ φ́µ .

µThe scaling property of φµ is determined by yµ through the condition
ZZdxuµ (x)φµ (x) = dx́úµ (x́)φ́µ (x́) + (short length-scale term)with x́ = b −1 x and úµ = b yµ uµ . Namely,φ́µ (x́) ≈ b xµ φµ (x)with xµ = d − yµwhich is called “scaling dimension” of the scaling operator φµ .
Naoki KAWASHIMA (ISSP)

Statistical Machanics IMay 27, 201912 / 22

Scaling form of correlation functions
For correlation function in the long-length scale, we haveGµ (|x́ − ý|, Ḱ) = hφ́µ (x́)φ́µ (ý)iH(φ́,Ḱ)≈ b 2xµ hφµ (x)φµ (y)iH(φ,K) = b 2xµ Gµ (|x − y|, K),r 1, Ḱ
orGµ (r , K) ≈ 2xµ Gµbb
Let us consider the case where b is large enough that all irrelevant
field in Ḱ are regarded as zero.
When we have only one non-zero relevant field, say t,r
1Gµ (r , t) ≈ 2xµ Gµ, b yt t .
bb
By choosing b = r , we obtain r 1(gµ (x) ≡ Gµ (1, x yt ))Gµ (r , t) ≈ 2xµ gµ −1/ytrt
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Statistical Machanics I

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Critical exponents ν and η
Let us consider what we can deduce from the scaling form r 
1Gµ (r , t) ≈ 2xµ gµ −1/y .
trtFirst, by comparing it with the defining equation of the correlation
length, Gµ (r , t) ∝ r −ω e −r /ξ(t) we can derive
ξ(t) ∝ t− y1t⇒

ν=1.
yt

Second, by taking the limit t → 0,Gµ (r , t = 0) ≈1gµ (0)r 2xµ(5)which meansd − 2 + ηµ = 2xµ

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Statistical Machanics IMay 27, 201914 / 22

Order parameters and critical exponent β
Consider the expectation value of a scaling field φµmµ (u) ≡ hφµ (x)iu ≈ hb −xµ φ́µ (x́)iu = b −xµ mµ (ú).
It follows that mµ (0) = 0 if xµ 6= 0, which we assume below.
Suppose that spontaneous “magnetization” exists (i.e., hφµ i > 0)slightly away from the critical point. When we have only one non-zerorelevant field t ≡ uν ,mµ (t) ≈ b −xµ mµ (b yt t).
By choosing b = (t/t0 )−1/yt , with t0 being any constant, we obtain
xµmµ (t) ∝ t yt ,Thus, the critical exponent β is related to the scaling dimensions, i.e.,xµβ= .
yt

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 27, 201915 / 22[6-2] Gaussian model and Gaussian fixed pointConsider the Gaussian model:Z LHa (φ|ρ, t) ≡d d x ρ(∇φx )2 + tφ2x − hφxaZ π/a=π/Lddk(ρk 2 + t)φ2k − hφ0 .
(2π)d(∗ The lower-bound of the integrals symbolically specifies the short-range cutoff.)

We will apply the RG transformation:
Partial Trace: Ha (φ|ρ, t, h) → Hba (φ̃|ρ̃, t̃, h̃)Rescaling: Hba (φ̃|ρ̃, t̃, h̃) → Ha (φ́|ρ́, t́, h́)φ0k0 = b −y φ̃k (k0 ≡ bk)

Naoki KAWASHIMA (ISSP)

Statistical Machanics IMay 27, 201916 / 22Partial trace of short-range fluctuation(Partial trace) Ha (φ|ρ, t, h) → Hab (φ̃|ρ̃, t̃, h̃)
Since each wave-number component is independent from the others,the summation over φk for |k| > π/2a results simply in a
multiplicative constant:
ZR π/a d d k−(ρk 2 +t )φ2k +hφ0−Hba (φ̃|ρ̃,t̃,h̃)e≡ d{φk }|k|> baπ e π/L (2π)dR π/ba− π/Lordd k(ρk 2 +t )φ2k +hφ0(2π)d∼e,Z π/ba dd kHba (φ̃|ρ̃, t̃) =ρk 2 + t φ2k − hφ0 .
d(2π)
π/L

In short, the partial trace amounts toπ, (ρ̃, t̃, h̃) = (ρ, t, h).
φ̃k = φkfor |k| <baNaoki KAWASHIMA (ISSP)

Statistical Machanics IMay 27, 201917 / 22

Rescaling(Rescaling) Hba (φ̃|ρ̃, t̃) → Ha (φ́|ρ́, t́)φ́ḱ = b −yh φ̃k (ḱ ≡ bk)

Z π/a

d d ḱ −dbρb −2 k 02 + t b 2yh φ́2ḱ − hφ0dbπ/L (2π)
Z π/a dd ḱ −(d+2)+2yh  22=bρḱ+bt
φ́2ḱ − b yh hφ́0dbπ/L (2π)

Ha (φ́|ρ́, t́, h́) =The exponent yh should be chosen so that ρ is unchanged by the RG
transformation. Namely, yh = (d + 2)/2.
Then,Z π/a d d ḱ
2Ha (φ́|ρ́, t́, h́) =ρḱ+t́φ́2ḱ − h́φ́0dbπ/L (2π)
with t́ ≡ b 2 t, and h́ ≡ b yh h.
Naoki KAWASHIMA (ISSP)

Statistical Machanics IMay 27, 201918 / 22

RG transformation of the Gaussian model
To summarize,
By RG transformation,Z π/a dd kHa (φ|ρ, t, h) =
ρk 2 + t φ2ḱ − hφ0
dbπ/L (2π)
is transformed into
Z π/aHa (φ́|ρ́, t́, h́) =d d ḱ  2
ρḱ+t́
φ́2ḱ − h́φ́0d(2π)
bπ/L

with
ḱ = bk,φ́ḱ = b −yh φ̃k ,t́ = b yt t̃,h́ = b yh h(6)withyt ≡ 2Naoki KAWASHIMA (ISSP)

and yh ≡d +2.
2

Statistical Machanics I(7)May 27, 201919 / 22

RGT on φx
While in [6-1] we saw xµ = d − yµ in general, its direct derivation in
the case of Gaussian model clarifies the meaning of RGT.
Considering the Fourier components of φ́x́ ,φ́x́ = Ĺ−dπ/aXπ/abe i ḱx́ φ́ḱ = b d L−dXe ikx b −y φkkḱ
π/ab

X= b d−y L−de ikx φk = b x [φx ]k< abπk∗−dHere, [φx ]k<k ∗ ≡ LkXe ikx φk is something one obtains afterkfiltering out the short wave-length part (k > k ∗ ) from φx . Therefore,
φx and [φx ]k<k ∗ are identical in the renormalized description.
Naoki KAWASHIMA (ISSP)

Statistical Machanics IMay 27, 201920 / 22

Implication of RGT
In [6-1], we saw, in general,1yt
d − 2 + ηµ = 2xµν=For the Gaussian model, we have derived
yt = 2and yh =d +22Therefore, for the gaussian model
ν=12

Naoki KAWASHIMA (ISSP)

and η = 0.

Statistical Machanics I

May 27, 201921 / 22

Homework (Submit your report on one of the following)
By an argument similar to the one resulting in βµ = xµ /yt , show thatthe critical exponent γµ that describes the temperature-dependenceof the susceptibility, χµ ≡ ∂hφµ (x)i/∂uµ ∝ t −γµ , is related to the2yµ − dyµ − xµ=.
scaling dimensions/eigenvalues as γµ =yt
yt
Consider a system for which the susceptibility χµ diverges as one
approaches the critical point keeping the condition uµ = 0. Does
application of infinitesimal field uµ qualitatively change the criticalproperties? Can we say the opposite, i.e., that the field does notessentially change the nature of the transition whenever χµ < ∞?
In the rescaling of the Gaussian model, we fixed yh so that the ρwould not change. In principle, we should be able to obtain some
RGT by fixing other parameters in stead of ρ. What would we haveobtained, for example, if we had fixed t rather than ρ?

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 27, 201922 / 22