Discrete parafermions and Coulomb gas in the square-lattice O ( n ) - PowerPoint PPT Presentation

Discrete parafermions and Coulomb gas in the square-lattice O ( n ) model Yacine Ikhlef Section Math´ ematiques, Gen` eve Thursday 27th May 2010 Ascona

This talk is based on : ◮ YI, J. Cardy, J. Phys. A 42 , 102001 (2009) M. Rajabpour, J. Cardy, J. Phys. A 42 , 14703 (2007) V. Riva, J. Cardy, J. Stat. Mech P12001 (2006) ◮ S. Smirnov, ICM vol. II, 1421 (2006) ◮ S.0. Warnaar, M.T. Batchelor and B. Nienhuis, J. Phys. A 25 , 3077 (1992) H.W.J. Bl¨ ote, B. Nienhuis, J. Phys. A 22 , 1415 (1989)

Plan The square-lattice O ( n ) model Integrable models on a regular rhombic lattice Discrete parafermions

The square-lattice O ( n ) model

1. The model ◮ Partition function Π( G ) = n # loops ( G ) � � ω ( G , j ) , Z = Π( G ) site j subgraph G ◮ Local Boltzmann weights t w 1 w 2 u 1 u 2 v

2. Yang-Baxter Equations (YBE) ◮ Plaquette diagram (“ˇ R -matrix”) � � λ := t ( λ ) + u 1 ( λ ) + + . . . + w 2 ( λ ) ◮ Yang-Baxter Equations λ ′ λ = λ ′′ λ ′′ λ ′ λ with λ ′′ = λ − λ ′ − 3( π − θ ) 4 ◮ Commutation of transfer matrices λ ′ λ ′ λ ′ λ ′ λ λ λ λ λ ′′ λ ′′ = λ ′ λ ′ λ ′ λ ′ λ λ λ λ

3. Solution of the Yang-Baxter Equations [Nienhuis] n = − 2 cos 2 θ − cos 2 λ + sin 5 θ 2 − sin 3 θ 2 − sin θ t ( λ ) = 2 � 3 θ − π � u 1 ( λ ) = 2 sin θ cos − λ 4 � 3 θ − π � u 2 ( λ ) = 2 sin θ cos + λ 4 � � cos 2 λ + sin 3 θ v ( λ ) = − 2 � � cos( θ − 2 λ ) + sin θ w 1 ( λ ) = − 2 � cos( θ + 2 λ ) + sin θ � w 2 ( λ ) = − . 2

4. Three physical regimes [Nienhuis et al. ] ◮ Regime I : 0 < θ < π ◮ Central charge : c eff = 1 − 6(1 − g ) 2 g = 2 θ , g π � 2 √ g ± m √ g � ◮ Conformal dimensions : h , ¯ h = 1 e , e , m ∈ Z 4 Simple Coulomb gas (= compactified GFF)

4. Three physical regimes [Nienhuis et al. ] ◮ Regime I : 0 < θ < π ◮ Central charge : c eff = 1 − 6(1 − g ) 2 g = 2 θ , g π � 2 √ g ± m √ g � ◮ Conformal dimensions : h , ¯ h = 1 e , e , m ∈ Z 4 Simple Coulomb gas (= compactified GFF) ◮ Regime II : − π < θ < − π 3 2 − 6(1 / 2 − 2 g ) 2 ◮ Central charge : c eff = 3 g = π + θ , 2 π g ◮ Conformal dimensions : � 2 2 � ( e / √ 2 g + m √ 2 g ) , ( e / √ 2 g + m √ 2 g ) + 1 h ∈ , e ≡ m [2] 8 8 2 2 h = ( e / √ 2 g + m √ 2 g ) + 1 16 , e ≡ m + 1 [2] 8 Coulomb gas + Ising

4. Three physical regimes [Nienhuis et al. ] ◮ Regime I : 0 < θ < π ◮ Central charge : c eff = 1 − 6(1 − g ) 2 g = 2 θ , g π � 2 √ g ± m √ g � ◮ Conformal dimensions : h , ¯ h = 1 e , e , m ∈ Z 4 Simple Coulomb gas (= compactified GFF) ◮ Regime II : − π < θ < − π 3 2 − 6(1 / 2 − 2 g ) 2 ◮ Central charge : c eff = 3 g = π + θ , 2 π g ◮ Conformal dimensions : � 2 2 � ( e / √ 2 g + m √ 2 g ) , ( e / √ 2 g + m √ 2 g ) + 1 h ∈ , e ≡ m [2] 8 8 2 2 h = ( e / √ 2 g + m √ 2 g ) + 1 16 , e ≡ m + 1 [2] 8 Coulomb gas + Ising ◮ Regime III : − π 3 < θ < 0 ◮ Coupling of CG and Ising ? ◮ Full low-energy spectrum is not known

Integrable models on a regular rhombic lattice

5. Transfer matrix for rhombic lattice ◮ One-row transfer matrix (periodic transverse BCs) 1 α L ◮ Scaling limit T L ( α ) ∼ const L exp( − sin α H ) exp( i cos α P ) ◮ Conformal invariance H = 2 π P = 2 π L 0 − c L ( L 0 + ¯ L ( L 0 − ¯ 12) , L 0 ) , where L n , ¯ L n are Virasoro generators ◮ CFT prediction for eigenvalues of T L ( α ) − log Λ L ( α ) ≃ Lf ∞ − 2 π h − c h − c � ie i α � � − ie − i α � �� ¯ L 24 24

6. Relation between α and λ ◮ YBE − → − → asymptotics of Λ L Bethe Ansatz ◮ Result − log Λ ≃ Lf ∞ + 2 π h − c h − c � e i ρ ( θ ) λ � � + e − i ρ ( θ ) λ � �� ¯ L 24 24 For example, in regime I, ρ ( θ ) = 2 π/ (3 π − 3 θ ). ◮ Simple relation α = π | λ | < π 2 − ρ ( θ ) λ , 2 ρ

Discrete parafermions

7. Discretely holomorphic functions ◮ Morera’s theorem (in the continuum) : � If F is continuous and ∀ C closed circuit, C F ( z ) dz = 0, then F is holomorphic. ◮ Discrete version : Let F be defined on the edges of the lattice L . We say that F is discretely holomorphic on L iff, for every plaquette P with corners { z i } : � z i + z j � � ( z i − z j ) F = 0 . 2 � i j �∈ ∂ P

8. Definition of the discrete parafermion [Smirnov, Cardy-Rajabpour-Riva-YI] ◮ Introduce a pair of defects at 0 and z , and let ψ s ( z ) = 1 � Π( G ) e − isW ( z ) Z G | [0 and z carry defects] ◮ Example configuration ( W ( z ) = − π ) z 0

9. Discrete holomorphicity equations ◮ Around a plaquette : � � ψ ( z ) � δ z = 0. ◮ Linear equations on the Boltzmann weights   t . . A ( θ, s , α )  = 0   .  w 2 ◮ Singularity condition : det A ( θ, s , α ) = 0 ⇔ s = 3 θ − π 2 π ◮ Solution for Boltzmann weights = solution of YBE ! One recovers the relation λ ↔ α

10. Relation to SLE [Smirnov] ◮ Prove (or assume) convergence of � ψ s � to an analytic function ◮ In the upper half plane H , � ψ s � H solves a boundary value problem : Arg � ψ s � H = π s 2 sgn ( z ) for real z ⇒ � ψ s � H = const z s ◮ If g t is the conformal map g t : H \ γ t → H t / g t ) s is a martingale then ( g ′ ◮ Consequence : the driving function W t is Brownian W t = √ κ B t , with s = 6 − κ 2 κ

Discussion and Conclusion ◮ The square O ( n ) model is integrable, with three physical regimes ◮ In regimes I and II, effective degrees of freedom were determined by Bethe Ansatz + asymptotic calculation ◮ Regime III is analytically and numerically harder ◮ We found a discrete parafermion for all regimes ◮ Smirnov’s argument ( modulo convergence) connects the model to SLE : In regimes II and III, how to describe fermions in the SLE formalism ?

Thank you for your attention !