Lecture 6: Simple Real-Space RG Transformations
Naoki KAWASHIMAISSP, U. Tokyo

May 27, 2024

Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 6

May 27, 2024

1 / 15In this lecture, we see
The fixed point of the RG transformation is important inunderstanding our world.
Real-space renormalization group transformation is generally
impossible to carry out in dimension higher than 1. Therefore, it
requires some approximation.
A decimation-based RG can be approximately done by a method
proposed by Migdal and Kadanoff.
Generally, the fixed point of RG transformation (RGT) represents thecritical point.
MKRG produces a non-trivial evaluates of critical exponents.
However, they do not generally agree with the correct values, and it is
not obvious how to systematically improve the approximation.

Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 6

May 27, 2024

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

Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 6

May 27, 2024

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Coarse-graining flow“https://youtu.be/MxRddFrEnPc” by Douglas Ashton“Critical point” = “Fixed point of RG transformation”

Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 6

May 27, 2024

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1D was easy. Can we do the same in 2D case?

Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 6

May 27, 2024

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RG is more tricky for d > 1
Consider, for example, coarse-graining by decimation.
S̃x ≡ Sx

for x ∈ Ω′ ≡ {(2ma, 2na)|m, n = 0, 1, 2, · · · , L/2}

The partial trace can be taken (at least formally) ase−H̃2a (S̃,K̃) ≡Tr{Sx }x∈Ω\Ω′

e−Ha (S,K)

There are paths that connect two remaining spins, say Sr and Sr′(r, r ′ ∈ Ω′ ), through Ω\Ω′ . Tracing out the spins along the path giverise to the long-range interaction between Sr and Sr′ .
There are n-body (n >P2) interactions among the remaining spins,because, for example, S0 (1 + tS0 S1 )(1 + tS0 S2 )(1 + tS0 S3 )(1 + tS0 S4 ) contains the term like t4 S1 S2 S3 S4 .
As a result, unlike the 1D case, the renormalized Hamiltonian is toocomplicated to deal with.
Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 6

May 27, 2024

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d > 1 needs approximation

The renormalized Hamiltonian

is more likethan.

We can carry out the RG mapping only approximately.

Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 6

May 27, 2024

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Migdal-Kadanoff approximation for 2D Ising model

Bunch up two vertical lines.
1 : th K̃ = th2 K)
Partial trace of spins (×) on horizontal bonds. (○2 : Ḱ = 2K̃)Bunch up two horizontal lines. (○Partial trace of intermediate spins (×) on vertical bonds.
Trivial re-scaling (do nothing on Ḱ).

simple Migdal-Kadanoff
2t22′t = th(2 ath(t )) =1 + t4
Naoki KAWASHIMA (ISSP)(t ≡ th K, t′ ≡ th Ḱ)Statistical Mechanics I: Lecture 6May 27, 2024

8 / 15Effect of RGT on correlation function
Consider a decimation-based RG transformation with scaling factor bt′ = Rb (t),e.g., R2 (t) = th(2 ath(t2 )) for MKRG.
For the decimation-based RG (∗) , the renormalized spin is simply theoriginal spin at the corresponding position, i.e., S ′ (r′ ) = S(r)(r′ ≡ r/b). Therefore, the correlation function must satisfy
C(b−1 r|t′ ) ≡ ⟨S ′ (b−1 r)S ′ (0)⟩t′ = ⟨S(r)S(0)⟩t ≡ C(r|t).

(1)

Assuming that C(r|t) decays asymptotically like e−r/ξ(t) , (1) yields
b−1 r/ξ(t′ ) = r/ξ(t). Thus, we obtain
ξ(t′ ) = b−1 ξ(t).

(2)(∗) Essentially the same argument can be made for more general RG, but there is no need for

additional argument about the nature of the renormalized correlation function/length for thedecimation-based RG as can be seen below.
Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 6

May 27, 2024

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RG fixed point and yt = 1/ν
The RG fixed-point tc is defined by tc = Rb (tc ).
The RGT amplifies the ‘deviation’ from the fixed-point as
δt → δt′ = t′ − tc = Rb (tc + δt) − tc ≈ Rb′ (tc )δt
Therefore, (2) means ξ(tc + Rb′ δt) ≈ b−1 ξ(tc + δt).
Since the exponent ν is defined by ξ ∝ (δt)−ν ,(Rb′ δt)−ν = b−1 (δt)−ν
→→Rb′−ν= b−1

log Rb′ (tc )1yt ≡ =
νlog b(3)

Derivatives of RG transformation are critical exponents.

Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 6

May 27, 2024

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RG fixed point and yt = 1/ν (numerical estimates)
For the Migdal-Kadanoff RGT for 2D Ising model, we have
2t2c
R2 (tc ) == tc → tc = 0.54368 · · ·1 + t4c√(cf: texact=2 − 1 = 0.4142 · · · )cWith some arithmetic, we can get
2(1 − tc )≈ 1.676tc→ yt ≡ 1/ν ≈ log 1.676/ log 2 ≈ 0.747

R2′ (tc ) =(cf: ytexact = 1, ytmean field = 2)

Not bad, but ad-hoc (not obvious how to improve).

Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 6

May 27, 2024

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Infinitesimal MKRG
The MKRG mapping with the scaling factor b can be summarized ast′ = R2 (t) ≡ th(2 ath(t2 )).
This can be generalized formally to general integer b > 1 ast′ = Rb (t) ≡ th(b ath(tb )).
Although the corresponding operation cannot be defined for
non-integer b, let us assume that it is still meaningful.
After all, “bunching-up” two lines to one by one step might be toocrude. It may become less harmful if we bunch-up as small number oflines as possible, i.e., taking b = 1 + λ where 0 < λ ≪ 1Does this infinitesimal RG still yield sensible results?

Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 6May 27, 2024

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Infinitesimal RG (general argument)
In general, suppose some RG transformation t′ = Rb (t) with
continuous scaling factor b = 1 + λ.
Using the notation f˙ ≡ ∂f /∂b, the critical point is determined by
tc = R1+λ (tc ) = R1 (tc ) + λṘ1 (tc ) ⇒ Ṙ1 (tc ) = 0.
(Ṙ1 (t) is called “beta function” and the symbol β(t) is often used.)Using R1 (t) = t (therefore R1′ (t) = 1, and R1′′ (t) = 0), in the lowest
order in λ, the scaling dimension yt can be obtained by′log R1+λ(tc (1 + λ))(See Eq.(3))
yt (1 + λ) =log(1 + λ)1′′′′≈ log R1 (tc (1)) + R1 (tc (1))∆tc + Ṙ1 (tc (1))λ ≈ Ṙ1′ (tc (1)).
λṘ1 (tc ) = 0 and yt = Ṙ1′ (tc )Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 6

May 27, 2024

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Infinitesimal MKRG (numerical estimates)
For b = 1 + λ (λ ≪ 1), defining t ≡ th K, we obtaint′ = Rb (t) = th(b ath tb ) ≈ t + λρ(t)ρ(t) ≡∂Rb (t)= (1 − t2 ) ath t + t log t∂b b→1

The critical point t = tc is determined by ρ(tc ) = 0, which has thesolution√tc = 2 − 1 (Exactly agrees with the correct value!)
As for yt , we have
yt = ρ′ (tc ) = 2 +√√2 log( 2 − 1) = 0.753549 · · · ,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 Mechanics I: Lecture 6May 27, 202414 / 15

Exercise 6.1: Try the idea of MKRG (i.e., bunching up and trace overintermediate spins) on the Ising model in higher dimensions (d > 2).

Naoki KAWASHIMA (ISSP)

Statistical Mechanics I: Lecture 6

May 27, 2024

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