Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems

Block Spin Transformation of 2D O(N) sigma model,

Toward solving a Millennium Problems

K.R.Ito

Inst. for Fundamental Sciences Setsunan Univ.

2015 Mar 06, Courant Inst.

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Millennium Problem

From HomePage of Clay Institute

1. Construction of 4D YM Field Theory (Jaffe, Witten) 2. Solution of Navier-Stokes Equation (Feffermann) What kind of Analysis do we need in these problems ?

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Millennium Problem

Difficulties in 4D LGT, 2D Sigma and NvS Eq

1. The system is non-linear. Difficult to find linear part (or Gaussian part)

2. There appear relevant terms (increasing coupling constants by naive scaling or by BST)

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Hisroty of 2D Spin System

History of 2D Spin Systems

2D O(N)Spin Model is simple, but hard to analyze.

1. 2D Ising spin, existence of spontaneous magnetization, R.Peierls (1936), L.Onsager (1944)

2. Kosterlitz-Thouless Transition in 2D XY model, J.Fröhlich and T.Spencer (1982)

3. non-existence of phase transition in the Heisenberg model with large N (∼quark confinement in YM₄) (this talk)

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems 2D Sigma model & BST

The Model

The 2D O(N)Heisenberg model,ϕ(x)∈S^N−1:

⟨· · · ⟩= Z

(· · ·)exp[X

n.n.

ϕ_xϕ_y]Y

x

δ(ϕ²(x)−Nβ)d^Nϕ_x

= Z

(· · ·)exp[X

n.n.

ϕ_xϕ_y − g₀ 2N

X

x

³

ϕ²(x)−Nβ

´2

]Y

x∈Λ

d^Nϕ_x

whereϕ(x) = (ϕ₁(x),· · ·, ϕ_N(x))and x,y are lattice points x,y ∈Λ⊂Z². Typical double-well potential¡

ϕ²(x)−Nβ¢2

:

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems 2D Sigma model & BST

Gibbs measure:

⟨f(ϕ)⟩= Z

f(ϕ)exp[−W₀(ϕ)]Y

x

d^Nϕ(x)

W₀= 1

2⟨ϕ,(−∆ +m²₀)ϕ⟩+ g₀

2N⟨:ϕ²:_G,:ϕ²:_G⟩ :ϕ²:_G (x) =

XN i=1

ϕ²_i(x)−NG(0), β=G(0) (−∆)xy =4δ_xy −δ_1,|x−y|, Lattice Laplacian HereG(0) =β meansm₀²∼32e^−4πβ:

G(x) = 1

−∆ +m²₀(x) = Z _π

−π

Z _π

−π

e^ipx m²₀+2P

(1−cos p_i)

Ydp_i 2π

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems 2D Sigma model & BST

Set

G₀(x,y) = 1

−∆ +m²₀(x,y) G_n(x,y) = 1

L⁴ X

ζ,ξ∈∆0

G_n−1(Lx+ζ,Ly+ξ)

ϕ_n(x) = (Cϕ_n−1)(x) = 1 L²

X

ζ∈∆0

ϕ_n−1(Lx+ζ)

C=Block Spin Operator

=Arithmetic average (L⁻²P

) +scaling (Lx →x ):

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems 2D Sigma model & BST

Then

⟨ϕn(x)ϕn(y)⟩=G_n(x,y)

We find matrices A_n+1and Q (by Gaw-Kupi) such that ϕ_n(x) = A_n+1ϕ_n+1

| {z }

averaged spin

(x) + Qξ_n

|{z}

zero−average fluct.

(x)

⟨ϕ_n,G⁻_n¹ϕ_n⟩Λn = ⟨ϕ_n+1,G⁻_n+1¹ ϕ_n+1⟩Λn+1+⟨ξ,Q⁺G⁻_n¹Qξ⟩Λn

whereΛn=L⁻ⁿ∩Λand

CA_n+1=1, CQ=0

A_n+1=G_nC⁺G⁻_n+1¹ :R^Λn+1 →R^Λn Q:R^Λ′n →R^Λn

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems 2D Sigma model & BST

Thus we decompose⟨ϕ,(−∆ +m²₀)ϕ⟩into many Gaussians {z_n}with covariancesΓn=Q⁺G_n⁻¹Q:

⟨ϕ,(−∆ +m²₀)ϕ⟩=⟨ϕ,G⁻₀¹ϕ⟩

=⟨ϕ₁,G⁻₁¹ϕ₁⟩+⟨z₀,Q⁺G⁻₀¹Q

| {z }

Γ⁻₀¹

z₀⟩

=⟨ϕ₂,G⁻₂¹ϕ₂⟩Λ2+⟨z₁,Q⁺G⁻₁¹Q

| {z }

Γ⁻₁¹

z₁⟩Λ1+⟨z₀,Q⁺G₀⁻¹Q

| {z }

Γ⁻₀¹

z₀⟩Λ0

where

Γ⁻_n¹≡Q⁺G⁻_n¹Q⁺>O(1)on QR^Λ′n

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems 2D Sigma model & BST

The zero-ave fluctuationa are of short range:

Qz_n = QΓ^1/2_n ξn=block average zero fluctuations whereξ_nobeysN(1,0)

where

Γ_n(x,y)∼exp[−c|x−y|] This kills long-range spin waves:

Gn(x,y)∼βn−log|x−y|

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems 2D Sigma model & BST

Fluctuations z

= Qξ

influenced by Double-Wells

exp h−g

N((ϕn+1+z_n)²−Nβ_n)² i

with|ϕ²_n+1−Nβ_n+1|=O(1) meanszn is ⊥ϕ_n+1:

-2 -1 0 1 2

-2-1012

0 5 10 15

-2 -1 0 1 2

-2-1012

0 5 10 15

Fluctuationsξ_n(x)are strongly influenced by block spins. I.e., they can live only on the bottom of bottles, andξn(x)propagate along the direction orthogonal toϕ_n+1.

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems 2D Sigma model & BST

BST=Perturbation around the Gaussians, but not in the present case since ϕ

changes:

C leaves the fundamental Gaussian measures invariant.

G_n(x,y) = (CG_n−1C⁺)(x,y)∼G₀(x,y) Can we expect W_nkeeps its main terms invariant under the influence of domain walls ?

W_n(ϕ_n, ψ_n) = 1

2⟨ϕ_n,G⁻_n¹ϕ_n⟩+ g_n

2N⟨ϕ²_n:_Gn, ϕ²_n:_Gn⟩ +correction

Gn(0) = βn∼β₀−const.n

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Mathematical Meanings of RG

D(ϕn) = Large and/or non-smooth configuration ofϕn

= Dw(ϕ_n+1)∪R(zn=Qξn)

= Long Domain Walls + Short Domain walls Domain walls D_w =STRONG SPIN ROTATION REGION

|ϕ_n(x)ϕ_n(y)−Nβ_n|>N^1/2+εexp[(c/10)|x−y|]

∀x ∈Dw,∃y ∈Dw

1/2 is thecentral limit theoremforP

:ξ_i²:. Outside of Dw,

|ϕn(x)ϕn(y)−Nβn|<N^1/2+εexp[(c/10)|x−y|]

∀x ∈D_w^c,∀y ∈D_w^c

Thus ϕn(x)ϕ_n(y) =NG_n(x,y) on(D_w)^c

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Mathematical Meanings of RG

Configuration ofϕn=A_n+1ϕ_n+1+Qzn

small waves Qznon domain-walls (tsunami=A_n+1ϕ_n+1)

-10 -5 0 5 10

-10 -5 0

5 10

-2 0 2

-10 -5 0 5 10

-10 -5 0 5 10

-1.0 -0.5 0.0 0.5 1.0

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Mathematical Meanings of RG

Block Spin=Trimming short waves

-10 -5

0 5

10

-10 -5 0 5 10

-2 -1 0 1 2

Fluctuationsξ_n(x)perpenficular toϕ_n(x)have N−1 degrees of freedom of gaussian fields.

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Mathematical Meanings of RG

RG=Contraction Map on Banach Space H

Namely we consider of Flow of SpaceKnof Spin Configurations

K1⊃ K2⊃ · · · ⊃ Kn

Kn=smoothly propagating spin waves on the surfaces of balls 1. no domain walls

|ϕ_n(x)ϕ_n(y)−Nβ_n|<N^1/2+εexp[(c/10)|x−y|]

∀x,y ∈K

2. |ϕ_n(x)²−Nβ_n|<N^1/2+ε 3. |∇ϕn(x)|<N^1/2+ε

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Mathematical Meanings of RG

Serious difficulty is

ϕn(x) =Anϕ_n+1+Qξ(x)∼ϕ_n+1([x/L])+Qξ(x) Namelyϕn(x),|x|<L/2 contain L²ofϕn+1([x/L]). Thus

X

x

ϕ²_n(x)∼L²X

x

ϕ²_n+1(x) X

x

(:ϕ²_n:_Gn (x))²∼L²X

x

(:ϕ²_n+1(x) :_Gn+1)² ϕ⁴term increasesexponentiallyin n, i.e. relevant term.

BUT THIS DOES NOT HAPPEN.

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Main Theorem on RG flow

Theorem on the RG flow

The main part of W_nis represented by 3 terms and 4 parametersβn, gn,γnand m_n², :

Wn(ϕn, ψn) = 1

2⟨ϕn,G⁻_n¹ϕn+ g_n

2N⟨:ϕ²_n:_Gn,:ϕ²_n:_Gn⟩ +1

2γn< ϕ²_n,E^⊥G_n⁻¹E^⊥ϕ²_n>

where

1. G⁻_n¹=−∆ +m_n², m²_n=L²ⁿm₀² 2. γn= (Nβn)⁻¹.

3. gn→g^∗ =O(1)>0 (convergetnt to the fixed point) 4. E^⊥=projection toN(C) ={f;Cf =0}

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Main Theorem on RG flow

I the first two terms = marginal (main term)

I the last term is irrelevant. it fades e away.

I (:ϕ²_n :)²is relevant butgnconverges to a constant in the scaling region

The flow is described by three parameters

m²_n=L²ⁿm₀²∼exp[−4πβ+2n log L]→O(1), β_n =β−const.n→O(1)

γn=O((βnN)⁻¹) gn=O(1)

All this means is thatsystem goes to the single-well potential, and thenabsence of phase transitions follows.

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Proof of the Main Theorem

Sketch of the Proof

Main Ideas and Theorems:

Setϕ_n =A_n+1ϕ_n+1+z_n, z_n=Qξ_nso that

< ϕn,G⁻_n¹ϕn> = < ϕ_n+1,G⁻_n+1¹ ϕ_n+1>+< ξn,Γ⁻_n¹ξn>, Γ⁻_n¹ = Q⁺G_n⁻¹∼Q⁺(−Λ)Q>O(1)

:ϕ²_n(x) :_Gn = :ϕ²_n+1(x) :_Gn+1 +q(x) q(x) = 2ϕ_n+1(x)z_n(x)+ :z(x)²_n:Γn

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Proof of the Main Theorem

Calculate the distribution function of q(ξ)_x:

P(φ_n+1,p) = Z

exp

"

√i N

X

x

(p−q(ξ))xλx

#

dµ(ξ)Y dλx

= exp

· i

√N⟨(p−q(ξ)), λ⟩

¸

dµ(ξ)Y dλ_x dµ(ξ) = exp[−⟨ξ,Γ⁻_n¹ξ⟩]Y

dξx

the distribution function of q(ξ) =2ϕ_n+1(x)z_n(x)+ :z(x)²_n:_Γn with respect to dµ(ξ)

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Proof of the Main Theorem

Thorem 1:

P(p, φ) = exp[− 1 4N⟨p, 1

Mp⟩] M = Γ^◦_n²+ 2

N(ϕnϕn)◦Γn

= Γ^◦_n²+2β_nΓ_n+ :ϕ_nϕ_n:

| {z } domain wall term

◦Γ_n/N

where

(Γn)(x,y) = (QG_n⁻¹Q⁺)(x,y)∼exp[−|x−y|] ((ϕϕ)◦Γ)(x,y) = (ϕ(x)ϕ(y))Γ(x,y)∼NG(x,y)Γ(x,y)

spec M = { |{z}κ0 O(1)>0

, κ1,· · · , κ_L²₋₁

| {z }

O(βn)

}

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Proof of the Main Theorem

Corol.2: Assume

|:ϕn(x)ϕn(y) :◦Γn(x,y)|<N^1/2+ε×1 Then

⟨p, 1

Mp⟩ = X blocks:U⊂Λn

⟨p_U, Ã

1 κ0

P₀+X

i

1 κi

P_i

! p_U⟩

∼ X

blocks:U⊂Λn

⟨p_U, µ 1

κ₀P₀

¶ p_U⟩

= X

blocks:U⊂Λn

1

κ₀(P₀p_U)² where

P₀ = projection to block-wise cobstant functions P_i = projection to zero-average functions

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Proof of the Main Theorem

Definition of Domain Wall

Domain walls are paved set such that

|ϕ_n(x)ϕ_n(y)−Nβ_n|>N^1/2+εexp[(c/10)|x−y|]

∀x ∈Dw,∃y ∈Dw

1/2 is thecentral limit theoremforP

:ξ_i²:. Outside of Dw,

|ϕn(x)ϕn(y)−Nβn|<N^1/2+εexp[(c/10)|x−y|]

∀x ∈D_w^c,∀y ∈D_w^c

Thus ϕn(x)ϕ_n(y) =NG_n(x,y) on(D_w)^c

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Proof of the Main Theorem

Theorem 2

Domain Wall region D_w has high energy:

Z

exp[−1

2⟨φn,G₀⁻¹φn⟩Dw]dµ(ξ)<exp[−N^2ε|D_w|]

Outside of D_w, we can replaceφφby NG_n, and we have a Gaussian integral over p.

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Proof of the Main Theorem

We integrate overξunder the influence of long spin wave by p variables. Using:φ²_n:²= (:φ²_n+1: +p)², we replaceξ⁴by p²: Theorem 3

Z

exp[−g_n

2N⟨:φ²_n:,:φ²_n:⟩+ (...)]dµ(ξ)

= Z

exp[−g_n

2N⟨:φ²_n+1: +p,:φ²_n+1: +p⟩]P(p, φ)Y dp P(p) = exp[− 1

4N⟨p,M⁻¹p⟩]

= exp[− 1 4N⟨p,

µ1 κ0

P₀

¶

p⟩] =exp[− 1 4Nκ₀

X(P₀p_U)²]

where P₀p is block-wise constant spins (block spin type.)

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Proof of the Main Theorem

Final Step:

Put g_n 2N

X

x

(:φ²_n+1: +p)²+ 1 2N

X

x

(P₀p)²

= g_n 2N

X

x

h

(P₀(:φ²_n+1: +p))²+ ((1−P₀)(:φ²_n+1: +p))² i

+ 1 2N

X

x

(P₀p)²

Integrate over P₀p and(1−P₀)p apply steepest descent + perturbation. Since P(p) =P_n(p)is a gaussian for all n, Theorem 4gnconverges in the scaling region: gn→g^∗

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Block Spin Transformation of 2D O(N) sigma model, Toward solving a Millennium Problems Greetings

This completes the proof. Thank you very much for your attension and patience!