Lecture 4: Introduction to Renormalization Group
Naoki KAWASHIMAISSP, U. Tokyo

May 10, 2019

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 10, 20191 / 23

To begin with ...
There are cases where we can rely on the mean-field theory even forthe critical behavior. (Ginzburg criterion)
In one dimension, we may be able to carry out coarse-graining andobtain correct critical behavior. However, in higher dimensions, similar
approaches would not yield computationally tractable solutions.
Real-space renormalization group transformation can beapproximately carried out (Migdal-Kadanoff RG) and produces a
non-trivial (non-MF) evaluates of critical exponents. However, they
do not generally agree with the correct values, nor they provide a way
to systematically improve the approximation.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 10, 20192 / 23[4-1] 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)

Naoki KAWASHIMA (ISSP)

Statistical Machanics IMay 10, 20193 / 23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  hφr i
Typically, this condition is not satisfied at the scale of latticepconstant, e.g., for the Ising model, hφr i ≈ 0 and δφr = hδφ2r i ≈ 1.
However, the MF description can still be qualitatively correct at larger
length-scales relevant to the critical behavior, i.e., ξ.
1 X
So, we consider the local average of φ, i.e., φ̄R ≡ dφrbr∈Ωb (R)The condition for asymptotic validity of MF is δ φ̄R  hφ̄R i for someb ∼ ξ.

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 10, 20194 / 23

Self-consistency of mean-field approximation (1)ρ|∆t|∼ 2uuξ
For the amplitude of the fluctuation, we have a 2d X2∗ ξh(δ φ̄)2 iMF =hδφr0 δφr i ∼ d (∗ see supplement)bρbr,r0 ∈Ωb (R)
ρ
1ud−4
Thus, the validity condition becomes 2  d−2 ξ 2uξρρξFor d > 4, the condition is asymptotically satisfied as one approaches
the critical point, whereas it is not for d < 4.
For hφ̄i, below Tc , we have hφ̄i2MF ∼ m2 ∼Ginzburg criterion (Upper critical dimension)
The MF approximation to φ4 model cannot be correct asymptoticallybelow 4 dimensions while it can be correct above 4.

Naoki KAWASHIMA (ISSP)

Statistical Machanics IMay 10, 20195 / 23

Supplement: MF estimate of fluctuation (1)In Lecture 3, we sawhδφr0 δφr i ∼1 κ0d−2 −κ0 |r0 −r|e
ρ (κ0 r ) d−12(κ0 ≈√−∆t)from which we obtain a 2d X a d X ρ−1 κ0d−2−κ0 |∆r|h(δ φ̄)2 i =hδφr0 δφr i ∼d−1 e0bb∆r (κ |∆r|) 2r,r0 ∈Ωb (R)Z bZ κ0 b−1 0d−2d−111 1d−1 ρ κ−κ0 r∼ ddr r∼ d 02dx x 2 e −xd−1 eb 0b ρκ 0(κ0 r ) 2( d+1!
x 2(x  1)f (κ0 b)∼ 02 df (x) ∼ρκ b
f∞ (a dimension-less constant) (x  1)Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 10, 20196 / 23

Supplement: MF estimate of fluctuation (2)Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 10, 20197 / 23[4-2] General 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)

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 10, 20198 / 23[4-2] 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 Machanics IMay 10, 20199 / 23

Coarse-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,
Xe −Ha (S|K,L) → e −Hab (S̃|K̃,L) ≡∆(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)

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 10, 201910 / 23

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 − hi=0L−1X

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

S1 ,S3 ,··· ,SL−1e −H2a (S̃|K̃ ,L) =X
S1e K (S0 +S2 )S1X

e K (S2 +S4 )S3 · · ·S3∼ e K̃ S0 S2 e K̃ S2 S4 · · · e K̃ SL−2 S0 ∼ e −H2a (S̃|K̃ ,L)

Naoki KAWASHIMA (ISSP)

Statistical Machanics IX

e K (SL−2 +S0 )SL−1

SL−1(K̃ ≡ ath(th2 K ))May 10, 201911 / 23

Example: Rescaling of Ising chain (b = 2)
Let us use t ≡ e −2K in stead of K for the parameter. Then, the effectof the coarse-graining on t ist̃ =2t.
1 + t2

The rescaling in the present case is simplyŕ ≡ r/2,Śŕ ≡ S̃r ,and t́ ≡ 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 Machanics It́ =2t.
1 + t2May 10, 201912 / 23

Example: Critical exponent ν
From the whole RG procedure, we can deducee −r /ξ(t) ∼ hSr S0 it = hSŕ S0 it́ ∼ e −ŕ /ξ(t́)
Because ŕ = r /2,
ξ(t) = 2ξ(t́)

t́ =2t
1 + t2.

Since t́ ≈ 2t near the fixed-point t = 0,
ξ(t) ≈ 2ξ(2t).
Therefore, for t ≈ 0,
ξ(t) ∼

Naoki KAWASHIMA (ISSP)

1t⇒ν=1(Exact!)

Statistical Machanics IMay 10, 201913 / 23

Can we do the same in 2D case? (1)Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 10, 201914 / 23

Can we do the same in 2D case? (2)Coarse-graining by decimation.
S̃r ≡ Srfor r ∈ Ω0 ≡ {(2ma, 2na)|m, n = 0, 1, 2, · · · , L/2}

The partial trace can be taken (at least formally)e −H̃2a (S̃,K̃ ) ≡Tr{Sr }r∈Ω\Ω0e −Ha (S,K )In general, unlike the 1D case, H̃2a contains terms other than thetwo-body nearest-neighbor interactions. For example, it contains thelong-range interaction −Krr0 Sr0 Sr where |r − r0 | > a, as well as
many-body interactions such as −Kr1 r2 r3 r4 Sr1 Sr2 Sr3 Sr4 .
Therefore, it is not feasible to study such a model (unless we usemachines).

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 10, 201915 / 23

Can we do the same in 2D case? (3)The renormalized Hamiltonian

is more like

Naoki KAWASHIMA (ISSP)

than

Statistical Machanics I.

May 10, 201916 / 23

[4-3] Migdal-Kadanoff approximation for 2D Ising model

Bunch up two vertical lines.
Take the partial trace of intermediate spins (×). (Step 1 )th K̃ = th2 K
Bunch up two horizontal lines. (Step 2 )Ḱ = 2K̃
Take the partial trace of intermediate spins (×).

simple Migdal-Kadanofft́ =Naoki KAWASHIMA (ISSP)

2t 2
1 + t4(t ≡ th K , t́ ≡ th Ḱ )Statistical Machanics I

May 10, 201917 / 23

RG fixed point and 1/ν (general argument)
Suppose some RG transformation (RGT) yieldst́ = Rb (t)22t(b: the rescaling factor, e.g., R2 (t) = 1+t4)

We define the RG fixed-point tc by tc = Rb (tc ).
Then, the ‘deviation’ from the fixed-point changes by RGT asδt → δ t́ = t́ − tc = Rb (t) − Rb (tc ) ≈ λδt(λ ≡ Rb0 (tc ))The correlation length after RGT must be smaller than the original byfactor b. So, we obtain ξ(λδt) ≈ b −1 ξ(δt), which leads toξ(δt) ∝ (δt)−ν where λ−ν = b −1

Naoki KAWASHIMA (ISSP)

Statistical Machanics Iorν≡log b.
log λMay 10, 2019(1)18 / 23

RG fixed point and 1/ν (numerical estimates)
For the Migdal-Kadanoff RGT for 2D Ising model, we have2tc2
= tc → tc = 0.54368 · · ·1 + tc4√(cf: tcexact = 2 − 1 = 0.4142 · · · )R2 (tc ) =With some arithmatics, we can get
2(1 − tc )≈ 1.676tc→ yt ≡ 1/ν ≈ log 1.676/ log 2 ≈ 0.747

R20 (tc ) =(cf: ytexact = 1, ytmean field = 2)
Not bad, but ad-hoc (not obvious how to improve).

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 10, 201919 / 23

An improvement of MKRG
Consider the MKRG step in which b lines, instead of 2, are bunchedup at a time. The resulting RG transformation will beK̃ = bK and th Ḱ = thb K̃ ,orth Ḱ = thb (bK )(The order of bunching and tracing was changed.)“bunching-up” two lines to one might be too crude. It may becomeless harmful if we bunch-up as small number of lines as possible.
For b = 1 + λ (λ  1), defining t ≡ th K , we obtaint́ = Rb (t) = t + λ(1 − t 2 ) ath t + λt log t,dRb= (1 − t 2 ) ath t + t log t ≡ f (t)d log tNaoki KAWASHIMA (ISSP)

Statistical Machanics IorMay 10, 201920 / 23

Infinitesimal RG (general argument)
In general, suppose some RG transformation with continuous scaling
factor b = e λ yieldsdR(t)= f (t).
λ→0 dλlim

Obviously, the fixed-point is determined by tc = f (tc ).
Starting from the previously obtained expression for 1/ν, we get blog dRdt t
1
1 dR1+λcyt = ==−1νlog bλdt 
d R1+λ − td R1+λ − R1d===f (t)
dtλdt
λdttc

Naoki KAWASHIMA (ISSP)

Statistical Machanics IMay 10, 201921 / 23

Infinitesimal MKRG (numerical estimates)
From tc = f (tc ) = (1 − tc2 ) ath tc + tc log tc ,, we obtain√
tc = 2 − 1 = tcexact !
As for yt , we have
yt = f 0 (tc ) = 0.7535 · · · ,slightly closer to ytexact = 1 than the simple MKRG with b = 2.

Better, but still ad-hoc (not obvious how to further improve).

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 10, 201922 / 23

Exercise
By solving the 1D Ising model, compute the correlation function
G (r ) ≡ hSr S0 i and the correlation length ξ. Verify ξ ∝ t −1 .
hint:
.  hSr S0 i = Tr T L−r σT r σ
Tr T L
where0TS 0 S ≡ e KS S (2 × 2 matrix)

1 0σ=0 −1

Naoki KAWASHIMA (ISSP)

Statistical Machanics I

May 10, 201923 / 23