Lecture 5: Introduction to Renormalization Group
Naoki KAWASHIMAISSP, U. Tokyo

May 20, 2024

Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 5

May 20, 2024

1 / 15In this lecture we see ...
There are cases where we can rely on the mean-field theory even forthe critical behavior. (Ginzburg criterion)
However, in low dimensions including d = 3, the mean-field theory is
not self-consistent concerning the critical phenomena.
We can define the renormalization group (RG) transformation, and if
we can calculate its result, we would be able to discuss the criticalproperties of the system.
For 1D Ising model, we can carry out the RG transformation, whichyields correct critical behavior.

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When can MF be valid? — Ginzburg criterionFirst, we will elucidate the meaning of the asymptotic validity anddraw a general criterion.
Then, we will check whether the mean-field theory satisfies thecriterion in a self-consistent way.
We will find that it is indeed self-consistent in some cases, but not ingeneral. (Ginzburg criterion)

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Asymptotic validity of MF approximation
Consider a system just below the critical temperature, where there isa finite but small spontaneous magnetization.
The mean-field (MF) description should be valid when the relative
fluctuation is negligible, i.e., δϕr ≪ ⟨ϕr ⟩Typically, this condition is not satisfied at the scale of latticepconstant, e.g., for the Ising model, ⟨ϕr ⟩ ≈ 0 and δϕr = ⟨δϕ2r ⟩ ≈ 1.
However, the MF description may still be qualitatively correct atlarger length-scales b, greater than the lattice constant, a, and smaller
than the correlation length, ξ, relevant to the critical behavior.
PSo, we consider the local average of ϕ, i.e., ϕ̄R ≡ b1d r∈Ωb (R) ϕr
over the cluster of size b.
The condition for asymptotic validity of MF is that, for any λ(0 < λ < 1), for clusters of size b ≡ λξ, the ratio δ ϕ̄R /⟨ϕ̄R ⟩converges to zero as we approach Tc from below (ξ → ∞).
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Statistical Mechanics I: Lecture 5May 20, 2024

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Self-consistency of mean-field approximation|∆t|ρ∼ 2uuξFor the amplitude of the fluctuation, we have
 a 2dXA(λ)
2⟨(δ ϕ̄) ⟩MF =⟨δϕr′ δϕr ⟩ ≲ d−2 (∗ see supplement)bρξ′For ⟨ϕ̄⟩, below Tc , we have ⟨ϕ̄⟩2MF ∼ m2 ∼r,r ∈Ωb (R)where A depends on ξ only through λ ≡ b/ξ.
It follows that, for any λ, the ratio⟨δ ϕ̄2 ⟩MF /⟨ϕ̄⟩2MF ∼ A(λ)uρ−2 ξ 4−dalways converges to 0 if d > 4, and diverges if d < 4 as ξ → ∞.

Ginzburg criterion (Upper critical dimension)
The MF approximation to ϕ4 model is asymptotically correct if d > 4, andinvalid if d < 4.
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Supplement: MF estimate of fluctuation
In Lecture 3, we saw
1 κ′d−2 −κ′ |r′ −r|⟨δϕr′ δϕr ⟩ ∼e,ρ (κ′ r) d−121 √κ′ = ≈ −∆tξfrom which we obtain a d X ρ−1 κ′d−2
 a 2dX
2−κ′ |∆r|⟨(δ ϕ̄) ⟩ =⟨δϕr′ δϕr ⟩ ∼d−1 e′bb∆r (κ |∆r|) 2r,r ′ ∈Ωb (R)ZZ κ′ b−1 κ′d−2d−1
ρ
111 b′−κ r∼ d ′2
dr rd−1
dx x 2 e−x∼ dd−1 eb 0b ρκ 0(κ′ r) 2!
( d+1′x 2(x ≪ 1)
f (κ b)f (x) ∼
∼ ′2 dρκ b
f∞ (a dimension-less constant) (x ≫ 1)κ′2−d f (κ′ b)
bA(λ)
f (λ)
∼× ′ d = d−2 .
A(λ) ≡ d , λ ≡ρξ(κ b)ρξλNaoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 5

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Supplement: MF estimate of fluctuation

Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 5

May 20, 2024

7 / 15General renormalization group (RG) transformation

In the derivation of the Ginzburg criterion, we introduced thecoarse-graining transformation as a Gedankenexperiment.
The RG transformation consists of two steps: (i) coarse-graining and(ii) rescaling. Schematically,(i)(ii)
Ha (S | K, L) −−→ Hab (S̃ |K̃, L) −−→ Ha (Ś |Ḱ, b−1 L)

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General Renormalization Group Transformation

In the coarse-graining step, we define coarse-grained field and carryout the configuration-space summation of the partition function, withthe constraint imposed by the coarse-grained fields.
In the rescaling step, we redefine the length-scale and the field
variables by multiplying them with scaling factors so that the effective
Hamiltonian may be the same form as the original one.

Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 5

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9 / 15Coarse-graining
In the coarse-graining step of the RG procedure, we first define“coarse-grained field”, S̃R , which is defined in terms of Sr in theneighborhood of R, i.e., S̃R = Σ({Sr }r∈Ωb (R) ), with some function
Σ(· · · ). More formally,X−Ha (S|K,L)−Hab (S̃|K̃,L)e→e≡∆(S̃ |Σ(S))e−Ha (S|K,L) ,S

where K is a set of parameters such as K ≡ (β, H).

Example 1 (Ising chain with b = 3)
Σ(S1 , S2 , S3 ) = S2(Simple decimation)

Σ(S1 , S2 , S3 ) = (S1 + S2 + S3 )/3(Local Average)

Σ(S1 , S2 , S3 ) = sign(S1 + S2 + S3 )(Majority rule)

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Example: Coarse-graining of Ising chain (b = 2)
Consider the Ising model of size L ≡ 2g in one dimension.
Ha (S|K, L) = −K

L−1X

Si Si+1 − h

i=0

L−1X

(K ≡ (K, h))

Sii=0For even L, let us adopt the decimation for the coarse-graining:(for i = 0, 2, 4, · · · , L − 2)X
−H2a (S̃|K̃,L)=e−Ha (S|K,L) . For h = 0 we haveThen, eS̃i = Si

S1 ,S3 ,··· ,SL−1e−H2a (S̃|K̃,L)=Xe

K(S0 +S2 )S1S1X

eK(S2 +S4 )S3···S3eK(SL−2 +S0 )SL−1SL−1∼ eK̃S0 S2 eK̃S2 S4 · · · eK̃SL−2 S0 ∼ e−H2a (S̃|K̃,L)

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X(th K̃ ≡ (th K)2 )Statistical Mechanics I: Lecture 5

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Example: Rescaling of Ising chain (b = 2)
Let us use t ≡ th K instead of K for the parameter. Then, the effectof the coarse-graining on t ist̃ = t2 .
The rescaling in the present case is simplyŕ ≡ r/2,Śŕ ≡ S̃r ,andt́ ≡ t̃.

Together with the coarse-graining, we obtain the whole RG
transformation,
RG

Ha (S|t, L) −−→ Ha (Ś|t́, L/2),
b=2

Naoki KAWASHIMA (ISSP)

with

Statistical Mechanics I: Lecture 5t́ = t2 .

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Example: Critical exponent νFrom the whole RG procedure, we can deducee−r/ξ(t) ∼ ⟨Sr S0 ⟩t = ⟨Sŕ S0 ⟩t́ ∼ e−ŕ/ξ(t́)
Because ŕ = r/2,t́ = t2 .

ξ(t) = 2ξ(t́)

Since t́ = t2 , if we define g ≡ − log t, the correlation length as afunction of g would satisfy
ξ(g) = 2ξ(2g).
From this, we can obtain ξ(g) upto a constant factor,
ξ(g) ∼Naoki KAWASHIMA (ISSP)

1g⇒ν=1(Exact!)

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Exercise 5.1: By solving the 1D Ising model, compute the correlation
function G(r) ≡ ⟨Sr S0 ⟩ and the correlation length ξ. Verify ξ ∝ g −1
where g ≡ − log th K. (Hint: The correlation function can be expressedas⟨Sr S0 ⟩ = Tr T L−r σT r σ Tr T L′

where T is a 2 × 2 matrix definedas TS′ ,S ≡ eKS S and σ is another
1 02 × 2 matrix defined as σ ≡.)0 −1
The matrices T and σ can be diagonalized as
 K
ee−K
2 ch K
0T = −K=UU,02 sh KeeKNaoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 5
σ=U
0 1U,
1 0May 20, 2024

14 / 15
1 1. Therefore, the correlation function, C(r) ≡ ⟨Sr S0 ⟩, of
1 −1a periodic system of length L can be computed aswhere U ≡ √12C(r) =tr + tL−r(2 ch K)L−r (2 sh K)r + (2 sh K)L−r (2 ch K)r=(2 ch K)L + (2 sh K)L
1 + tLwith t ≡ th K. Therefore, in the limit r ≪ L, the correlation function behaves
like C(r) = tr . From this, we obtain e−1/ξ = t, or ξ = 1/ log(1/t) = 1/g.
This is identical to what we obtained from the coarse-graining of the 1D Ising
chain.

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