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

Circle patterns and critical Ising models Marcin Lis February 18, 2019 1 / 13

Can we compute the critical point of the Ising model? • For d = 1, β c = ∞ (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. 2 / 13

Ising model Let G = ( V G , E G ) be an infinite non-degenerate planar graph embedded in C . Let G = ( V G , E G ) be a finite connected subgraph. Let Ω G = {− 1 , + 1 } V G be the spin configurations . Let J = ( J e ) e ∈ E ∈ ( 0 , ∞ ) E G 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 Ω G given by 1 � � � � P ✷ G ,β ( σ ) = exp β J { u , v } σ u σ v + 1 { ✷ =+ } β J { u , v } σ v . Z ✷ G ,β { u , v }∈ E G v ∈ V G u / ∈ V G 3 / 13

Critical temperature The squared average magnetization in G is � + � 2 �� � v ∈ V G σ v M + G = | V G | 2 G ,β We define the (magnetic) critical (inverse) temperature according to the behaviour of M : � � G ⊂ G M + β c = inf β > 0 : inf G > 0 β < β c β = β c β > β c 4 / 13

Square lattice • Kramers–Wannier duality (’41) between high and low-temperature expansion. Let E G be the set of even subgraphs of G . Z f G ,β = C f � � Z + G ,β = C + � � x ∗ x e e , G ,β G ∗ ,β e ∈ ω ω even ω ∈E G e ∈E G ∗ where the dual parameters x ∗ e = e − 2 β J e x e = tanh β J e and are related by x + x ∗ + xx ∗ = 1. 5 / 13

Square lattice • Since Z 2 = ( Z 2 ) ∗ , the self dual point x = x ∗ satisfying 2 x + x 2 = 1 √ should be critical ( x = 2 − 1). • A rigorous confirmation of this prediction came with the exact solution of Onsager. Theorem (Onsager ’44) The isotropic Ising model (J ≡ 1 ) on the square lattice undergoes a phase transition at √ β c = − 1 2 log ( 2 − 1 ) . 5 / 13

Square lattice • More generally, for the anisotropic model tanh J horizontal = e − 2 J vertical yields the critical point. • Even more generally, one can consider arbitrary biperiodic coupling constants. Theorem (Li ’10) Let J be coupling constants on Z 2 invariant under m Z × n Z . Then, β c is determined by the condition that the spectral curve of the corresponding dimer model on the Fisher graph has a real zero on T 2 . 5 / 13

Biperiodic graphs We say that G is biperiodic if it is invariant under a ≃ Z 2 action (together with the coupling constants J ). G 1 Theorem (Cimasoni & Duminil-Copin ’12) If G is biperiodic, then β c is the only positive root of an explicit polynomial in ( tanh β J e ) e ∈ E G 1 . 6 / 13

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 . θ e e ∗ θ e ∗ e The critical Z-invariant coupling constants introduced by Baxter are given by tanh J e = tan θ e e − 2 J e = tan θ e ∗ 2 . or equivalently 2 7 / 13

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 ∈ E G ε ≤ θ e ≤ π 2 − ε, then for the self-dual Z-invariant coupling constants, β c = 1 . 7 / 13

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 e θ − � e The coupling constants are � tan θ � 2 tan θ − � e e tanh J e = 2 . 8 / 13

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 ∈ V G 1 R ≤ r v ≤ R , then for the coupling constants as above, β c = 1 . 8 / 13

Circle patterns • dual of a triangulation with acute angles • circle patterns from circle packings • the square lattice with stretched/squeezed rows and columns tanh J i , i + 1 = e − J i − J i + 1 8 / 13

The Kac–Ward solution of the Ising model w ( uv, vw ) v u Figure: The turning angle from the vector uv to the vector vw . The Kac–Ward transition matrix Λ G ( x ) is defined by � x uv e i ∠ ( uv , vw ) / 2 if v = w and u � = z ; Λ G ( x ) − wz = → uv , − → otherwise , 0 where x = ( x e ) e ∈ E G is a complex weight vector. 9 / 13

The Kac–Ward solution of the Ising model The Kac–Ward matrix T G is defined by T G ( x ) = Id − Λ G ( x ) , and T G is its restriction to � E G . Theorem (Kac & Ward ’52) 1 � � 2 T G ( x ) = det x e ω ∈E G e ∈ ω Corollary Setting x e = tanh β J e yields Z f e = e − 2 β J e yields Z + G ,β , and x ∗ G ∗ ,β . 9 / 13

The Kac–Ward solution of the Ising model g ↓ � � e → The fermionic observable F G ( x ) of Smirnov is 1 � e − i 2 α ( γ ω ) � F G ( x ) � g = δ � g + x e . e ,� e ,� Z f G ,β e ∈ ω ω ∈E G ( � e ,� g ) Theorem (L. ’13, Cimasoni ’13) F G ( x ) = T G ( x ) − 1 9 / 13

Proof of criticality for circle patterns ( I ) To get vanishing of magnetization for β < 1, we use an argument based on the local geometry to prove that for every β < 1, there exists C β > 0 such that for all G ⊂ G , and u , v ∈ V G , � σ u σ v � f G ,β ≤ e − C β d ( u , v ) . This, together with the modified Simon–Lieb inequality yields � 1 � M + G = O . | V G | ( 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 ∈ V G , and all β > 1, G ,β ≥ β s − 1 � σ v � + β s + 1 . This part uses an inequality due to Duminil-Copin and Tassion. 10 / 13

Proof of criticality for circle patterns • High temperature expansion of the 2-point function: v u Let E G ( u , v ) be the set of subgraphs that have odd degree at u and v and even degree everywhere else. We have 1 � σ u σ v � f � � G ,β = x e . Z f G ,β e ∈ ω ω ∈E G ( u , v ) 10 / 13

Proof of criticality for circle patterns • Correlations and fermionic observables are comparable : v v τ u u Lemma (L. ’17) For every v , u ∈ V G , there exists a signed weight x τ such that g | ≤ � σ u σ v � f g ∈ Out v | F G ( x τ ) � g ∈ Out v | F G ( x ) � G ,β ≤ deg ( u ) deg ( v ) g | . max max e ,� e ,� � e ∈ In u ,� � e ∈ In u ,� 10 / 13

Vanishing of magnetization for β < 1 • Let x β = tanh β J , where J is as above, and let � · � = � · � l 2 ( � E G ) . E G ) → l 2 ( � Lemma (L. ’13) For every choice of signs τ and every β < 1 , we have � Λ( x τ β ) � < 1 . Proof 1. Consider the involution J : C � E G → C � E G induced by � e → − � e . 2. ˜ Λ = J Λ is block-diagonal with self-adjoint blocks (˜ Λ v ) v ∈ V G of size deg ( v ) . 3. ρ (˜ Λ v ) = � ˜ Λ v � < 1. ✷ 10 / 13

Vanishing of magnetization for β < 1 • We have � σ u σ v � f g ∈ Out v | F G ( x τ ) � G ,β ≤ deg ( u ) deg ( v ) max g | e ,� � e ∈ In u ,� g ∈ Out v | T − 1 G ( x τ ) � ≤ C g | max e ,� � e ∈ In u ,� g ∈ Out v | ( Id − Λ G ( x τ )) − 1 = C g | max � e ,� � e ∈ In u ,� ≤ C ′ � Λ G ( x τ ) � d ( u , v ) 1 − � Λ G ( x τ ) � . • We finish by using the modified Simon-Lieb inequality. ✷ 10 / 13

Nonzero magnetization for β > 1 • For S ⊂ V G , v ∈ S , let � � � σ v σ u � f � � ϕ S ,β ( v ) = tanh β J uw S ,β . u ∈ S w / ∈ S Lemma (Duminil-Copin & Tassion ’16) For any finite G = ( V G , E G ) , any v ∈ V G , and β > 0 , d d β � σ v � + 1 − ( � σ v � + G ,β ≥ 1 � G ,β ) 2 � ϕ S ,β ( v ) β inf . S ∋ v S ⊂ V G 10 / 13

Nonzero magnetization for β > 1 • Let x 1 = tanh J . We show that T G ( x 1 ) has a nontrivial kernel . Lemma (L. ’17) Define ρ : � � E G → ( 0 , ∞ ) by ρ � e = ρ − � e = | e ∗ | . Then T G ( x 1 ) = 0 . • Note that by the bounded angle and radius property e ∈ � e ∈ � r := inf { ρ � e : � E G } > 0 R := sup { ρ � e : � E G } < ∞ . and 10 / 13

Nonzero magnetization for β > 1 • Define G to be the graph induced by S ⊂ V G together with the external edges ∂ G pointing outside the boundary. • Let ζ = T G ( � 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. uv in the bulk of ¯ • We have for any directed edge � e = � G , r ≤ ρ � e T − 1 � � = G ( � x ) ζ � e � T − 1 = G ( � x ) � g ρ � e ,� g � g ∈ ∂ G � � σ v σ w � f ≤ G , 1 ρ � g � g =( w , z ) ∈ ∂ G ≤ CR ϕ S , 1 ( v ) . ✷ 10 / 13