Circle patterns and critical Ising models Marcin Lis February 18, - - PowerPoint PPT Presentation

Circle patterns and critical Ising models

Marcin Lis February 18, 2019

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Can we compute the critical point of the Ising model?

• For d = 1, βc = ∞ (Ising ’25).
• For d = 2, we have many examples:

◮ square lattice ◮ biperiodic graphs ◮ isoradial graphs ◮ circle patterns ◮ s-embeddings (?)

• For d ≥ 3, the problem seems hopeless.

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Ising model

Let G = (VG, EG) be an infinite non-degenerate planar graph embedded in C. Let G = (VG, EG) be a finite connected subgraph. Let ΩG = {−1, +1}VG be the spin configurations. Let J = (Je)e∈E ∈ (0, ∞)EG be coupling constants. The Ising model on G at inverse temperature β > 0 with free or ‘+’ boundary conditions conditions ✷ ∈ {f, +} is a probability measure on ΩG given by P✷

G,β(σ) =

1 Z✷

G,β

exp

• β
• {u,v}∈EG

J{u,v}σuσv + 1{✷=+}β

• v∈VG

u/ ∈VG

J{u,v}σv

• .

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Critical temperature

The squared average magnetization in G is M+

G = v∈VG σv

2 |VG|2 +

G,β

We define the (magnetic) critical (inverse) temperature according to the behaviour of M: βc = inf

• β > 0 : inf

G⊂G M+ G > 0

• β < βc

β = βc β > βc

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Square lattice

• Kramers–Wannier duality (’41) between high and low-temperature
• expansion. Let EG be the set of even subgraphs of G.

Zf

G,β = Cf G,β

• ω∈EG
• e∈ω

xe Z+

G,β = C+ G∗,β

• ω even
• e∈EG∗

x∗

e ,

where the dual parameters xe = tanh βJe and x∗

e = e−2βJe

are related by x + x∗ + xx∗ = 1.

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Square lattice

• Since Z2 = (Z2)∗, the self dual point x = x∗ satisfying 2x + x2 = 1

should be critical (x = √ 2 − 1).

• A rigorous confirmation of this prediction came with the exact solution
• f Onsager.

Theorem (Onsager ’44)

The isotropic Ising model (J ≡ 1) on the square lattice undergoes a phase transition at βc = −1 2 log( √ 2 − 1).

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Square lattice

• More generally, for the anisotropic model

tanh Jhorizontal = e−2Jvertical yields the critical point.

• Even more generally, one can consider arbitrary biperiodic coupling

constants.

Theorem (Li ’10)

Let J be coupling constants on Z2 invariant under mZ × nZ. Then, βc is determined by the condition that the spectral curve of the corresponding dimer model on the Fisher graph has a real zero on T2.

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Biperiodic graphs

We say that G is biperiodic if it is invariant under a ≃ Z2 action (together with the coupling constants J).

G1

Theorem (Cimasoni & Duminil-Copin ’12)

If G is biperiodic, then βc is the only positive root of an explicit polynomial in (tanh βJe)e∈EG1 .

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Isoradial graphs

We say that G is isoradial (or a rhombic lattice) if every face can be inscribed in a circle of a common radius.

e θe θe∗ e∗

The critical Z-invariant coupling constants introduced by Baxter are given by tanh Je = tan θe 2

• r equivalently

e−2Je = tan θe∗ 2 .

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Isoradial graphs

• Graphs admitting an isoradial embedding were characterized by

Kenyon & Schlenker ’04

• The critical Z-invariant Ising model on isoradial graphs was proved by

Chelkak & Smirnov ’09 to be conformally invariant in the scaling limit.

• Baxter’s Ising model is critical:

Theorem (L. ’13)

If G is isoradial and satisfies the bounded angle property: ∃ε>0 ∀e∈EG ε ≤ θe ≤ π

2 − ε,

then for the self-dual Z-invariant coupling constants, βc = 1.

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Circle patterns

We say that G is a circle pattern if each face (of its dual) can be inscribed in a circle of arbitrary radius with the center of the circle being inside the face.

θ

e

θ−

e

e

The coupling constants are tanh Je =

• tanθ

e

2 tanθ−

e

2 .

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Circle patterns

• One recovers Baxter’s model as a special case.
• These coupling constants were first (and independently) considered on

triangulations by Bonzom & Costantino & Livine ’15 in relation to supersymmetry between the Ising model and spin networks (relevant in LQG).

• The model is critical:

Theorem (L. ’17)

If G is a cirlce patterns satisfiying the bounded angle property, and the bounded radius property: ∃R<∞ ∀v∈VG 1 R ≤ rv ≤ R, then for the coupling constants as above, βc = 1.

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Circle patterns

• dual of a triangulation with acute angles
• circle patterns from circle packings
• the square lattice with stretched/squeezed rows and columns

tanh Ji,i+1 = e−Ji−Ji+1

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The Kac–Ward solution of the Ising model

(uv, vw) u v w Figure: The turning angle from the vector uv to the vector vw.

The Kac–Ward transition matrix ΛG(x) is defined by ΛG(x)−

→ uv,− → wz =

• xuvei∠(uv,vw)/2

if v = w and u = z;

• therwise,

where x = (xe)e∈EG is a complex weight vector.

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The Kac–Ward solution of the Ising model

The Kac–Ward matrix TG is defined by TG(x) = Id − ΛG(x), and TG is its restriction to EG.

Theorem (Kac & Ward ’52)

det

1 2 TG(x) =

• ω∈EG
• e∈ω

xe

Corollary

Setting xe = tanh βJe yields Zf

G,β, and x∗ e = e−2βJe yields Z+ G∗,β.

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The Kac–Ward solution of the Ising model

→

• e
• g ↓

The fermionic observable FG(x) of Smirnov is FG(x)

e, g = δ e, g +

1 Zf

G,β

• ω∈EG(

e, g)

e− i

2 α(γω)

e∈ω

xe.

Theorem (L. ’13, Cimasoni ’13)

FG(x) = TG(x)−1

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Proof of criticality for circle patterns

(I) To get vanishing of magnetization for β < 1, we use an argument based

• n the local geometry to prove that for every β < 1, there exists Cβ > 0

such that for all G ⊂ G, and u, v ∈ VG, σuσvf

G,β ≤ e−Cβd(u,v).

This, together with the modified Simon–Lieb inequality yields M+

G = O

1 |VG|

• .

(II) To get nonzero magnetization for β > 1, we use an argument based on the global geometry to prove that there exists s > 0 such that for all G ⊂ G, v ∈ VG, and all β > 1, σv+

G,β ≥ βs − 1

βs + 1. This part uses an inequality due to Duminil-Copin and Tassion.

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Proof of criticality for circle patterns

• High temperature expansion of the 2-point function:

u v Let EG(u, v) be the set of subgraphs that have odd degree at u and v and even degree everywhere else. We have σuσvf

G,β =

1 Zf

G,β

• ω∈EG(u,v)
• e∈ω

xe.

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Proof of criticality for circle patterns

• Correlations and fermionic observables are comparable:

u v τ u v

Lemma (L. ’17)

For every v, u ∈ VG, there exists a signed weight xτ such that max

• e∈Inu,

g∈Outv|FG(x) e, g| ≤ σuσvf G,β ≤ deg(u)deg(v)

max

• e∈Inu,

g∈Outv|FG(xτ) e, g|.

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Vanishing of magnetization for β < 1

• Let xβ = tanh βJ, where J is as above, and let · = · l2(

EG)→l2( EG).

Lemma (L. ’13)

For every choice of signs τ and every β < 1, we have Λ(xτ

β) < 1.

Proof

• 1. Consider the involution J : C

EG → C EG induced by

e → − e.

• 2. ˜

Λ = JΛ is block-diagonal with self-adjoint blocks (˜ Λv)v∈VG of size deg(v).

• 3. ρ(˜

Λv) = ˜ Λv < 1. ✷

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Vanishing of magnetization for β < 1

• We have

σuσvf

G,β ≤ deg(u)deg(v)

max

• e∈Inu,

g∈Outv|FG(xτ) e, g|

≤ C max

• e∈Inu,

g∈Outv|T−1 G (xτ) e, g|

= C max

• e∈Inu,

g∈Outv|(Id − ΛG(xτ))−1

• e,

g|

≤ C′ ΛG(xτ)d(u,v) 1 − ΛG(xτ) .

• We finish by using the modified Simon-Lieb inequality. ✷

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Nonzero magnetization for β > 1

• For S ⊂ VG, v ∈ S, let

ϕS,β(v) =

• u∈S
• w/

∈S

tanh

• βJuw
• σvσuf

S,β.

Lemma (Duminil-Copin & Tassion ’16)

For any finite G = (VG, EG), any v ∈ VG, and β > 0, d dβ σv+

G,β ≥ 1 β inf S∋v S⊂VG

ϕS,β(v)

• 1 − (σv+

G,β)2

.

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Nonzero magnetization for β > 1

• Let x1 = tanh J. We show that TG(x1) has a nontrivial kernel.

Lemma (L. ’17)

Define ρ : EG → (0, ∞) by ρ

e = ρ− e =

• |e∗|. Then

TG(x1) = 0.

• Note that by the bounded angle and radius property

r := inf{ρ

e :

e ∈ EG} > 0 and R := sup{ρ

e :

e ∈ EG} < ∞.

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Nonzero magnetization for β > 1

• Define G to be the graph induced by S ⊂ VG together with the external

edges ∂G pointing outside the boundary.

• Let ζ = TG(

x)ρ, and note that, by the definition of the Kac–Ward matrix, ζ

e = ρ e for all

e ∈ ∂G, and ζ

e = 0 otherwise.

• We have for any directed edge

e = uv in the bulk of ¯ G, r ≤ ρ

e

=

• T−1

G (

x)ζ

• e

=

• g∈∂G

T−1

G (

x)

e, gρ g

≤

• g=(w,z)∈∂G

σvσwf

G,1ρ g

≤ CRϕS,1(v). ✷

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S-embeddings

Following Chelkak (’17), we say that (G, G∗) is an s-embedding if every quadrangle whose diagonals are e, e∗ is tangential. The coupling constants are expressed in terms of the local geometry of G.

• One recovers circle patterns as a special case.

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Conjecture

The critical Ising model on circle patterns is conformally invariant in the scaling limit. Supporting fact: The fermionic observable satisfies a form of discrete holomorphicity.

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Thank you for your attention!

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