Correlation functions in loop models Yacine Ikhlef LPTHE, Universit - PowerPoint PPT Presentation

Correlation functions in loop models Yacine Ikhlef LPTHE, Universit´ e Paris-6/CNRS collaborators: B. Estienne, J. Jacobsen, M. Picco, R. Santachiara, J. Viti June 2014 Giens

Outline 1. Introduction 2. Computation of OPE constants 3. Signature of Logarithmic CFT

1. Introduction

A simple problem in the O( n ) loop model ◮ The lattice model: → W ( C ) = K # edges ( C ) n # loops ( C ) C = − � CFT with c = 1 − 6(1 − g ) 2 ◮ Scaling limit ( K > K c ): g n = − 2 cos π g , 0 < g < 1 ◮ Three-point connectivity � r 2 r 1 � � r 3 C P ( � r 1 ,� r 2 ,� r 3 ) = r 3 | ) 2 h ( | � r 1 − � r 2 || � r 2 − � r 3 || � r 1 − �

The compact boson CFT [Nienhuis, Di Francesco-Saleur-Zuber, Alcaraz ’80s] ◮ Free-field action: � A [Φ] = g d 2 r ( ∇ Φ) 2 , Φ ≡ Φ + 2 π , Φ = ϕ ( z ) + ϕ (¯ z ) 4 π ◮ Vertex operators: V α, ¯ α ( z , ¯ z ) = exp[ i αϕ ( z ) + i ¯ αϕ (¯ z )] α rs = (1 − r ) √ g − (1 − s ) ( r , s ) ∈ Z 2 ◮ Lattice of charges: 2 √ g , 2 ◮ Spectrum of the O( n ) model: 1. Electric operators V 1 k = V α 1 k ,α 1 k , k ∈ Z ( V = 1 , ǫ . . . ) 2. Watermelon operators W me = V α me ,α − m , e , m = 1 , 2 , 3 . . . e ∈ Z / m W me = “(2 m -leg defect) × exp( ie Φ)”

What is the operator algebra of the O( n ) model? ◮ Operator Product Expansion (OPE): � z − ¯ h a − ¯ h b +¯ ab z − h a − h b + h c ¯ h c O c (0)+ . . . C c O a ( z , ¯ z ) O b (0) ∼ z → 0 c ◮ Defines an operator algebra � C c O a × O b = ab O c c ◮ Questions ◮ Fusion rules and C c ab for the V 1 k ’s and W me ’s ? ◮ ⇒ consistent CFT with generic c < 1? ◮ Role of non-unitarity / indecomposability?

Other CFTs with infinite operator algebras ◮ Liouville CFT ◮ Spectrum = scalar vertex ops. V α,α ◮ OPE constants by conformal bootstrap [Dorn-Otto,Zamolodchikov-Zamolodchikov,Teschner, 90’s] ◮ Analytic continuation to c < 1: “time-like Liouville” ◮ OPE coefficient C σσσ for percolation+FK connectivity [Delfino-Picco-Santachiara-Viti, ’12-’13] ◮ Logarithms in 4-pt functions [Santachiara-Viti, ’12-’13] ◮ Boundary O( n )/Potts models ◮ Spectrum = chiral vertex op. V 1 k ( z ) ◮ Fusion rules V 1 m × V 1 n → V 1 , | m − n | +1 + · · · + V 1 , m + n − 1 ◮ At c = 0 (polymers, perco.), T is a null-vector ( � TT � = 0) ⇒ Logarithmic CFT [ Gurarie, Rozansky-Saleur, Ludwig . . . ’90-’00s] � λ � 1 ◮ Exact Jordan cells in the lattice transfer matrix 0 λ [ Read, Saleur, Jacobsen, Pearce, Rasmussen, Zuber . . . ’00s]

2. Computation of OPE constants

OPE constants from four-point functions (1/2) Conformal bootstrap program ◮ Correlation function C ( z , ¯ z ) = �O 1 (0) O 2 ( z , ¯ z ) O 3 (1) O 4 ( ∞ ) � � p ( z | ¯ h 1 , . . . , ¯ = X p ¯ p F p ( z | h 1 , . . . , h 4 ) F ¯ h 4 ) p , ¯ p ◮ Bases of conformal blocks V 1 (0) V 4 ( ∞ ) V 1 (0) V 4 ( ∞ ) ← → V p V q V 2 ( z ) V 3 (1) V 2 ( z ) V 3 (1) � F p ( z | h 1 , . . . , h 4 ) F q ( z | h 1 , . . . , h 4 ) � β pq � Change of basis: F p ( z ) = F q ( z ) q

OPE constants from four-point functions (2/2) � β pq ¯ ◮ Monodromy invar. ⇒ ∀ q � = ¯ q β ¯ q X p ¯ p = 0 ( M ) p ¯ p , ¯ p ◮ Computing steps: 1. Matrix elements β pq are known, solve ( M ) for X p ¯ p 2. Interpret X p ¯ p as C ( O 1 , O 2 , O p ¯ p ) × C ( O p ¯ p , O 3 , O 4 ) 3. Extract ratios of the form C ( O 1 , O 2 , O p ¯ p ) / C ( O 1 , O 2 , O p ′ ¯ p ′ ) ◮ Variants ◮ Minimal models: [Dotsenko-Fateev ’85] ◮ F p = Q p H dw j . . . j =1 ◮ β pq obtained by contour deformation ◮ Liouville: [Teschner ’95] ◮ Set O 4 = V 12 or O 4 = V 21 , then F p = hypergeometric ◮ Obtain recursion relations C ( α 1 , α 2 , α + b − 1 / 2) C ( α 1 , α 2 , α + b / 2) C ( α 1 , α 2 , α − b / 2) = f , C ( α 1 , α 2 , α − b − 1 / 2) = g ◮ Unique solution C L ( α 1 , α 2 , α 3 ) as an explicit special function

Some OPE constants in the O( n ) model [B. Estienne, YI] ◮ Method: ◮ Start with correlation function C ( z , ¯ z ) = �O 1 (0) O 2 ( z , ¯ z ) O 3 (1) O 4 ( ∞ ) � ( O j ∈ {V 1 k }∪{W me } ) ◮ The W me ’s have integer spin ⇒ monodromy unchanged ◮ Apply Teschner’s bootstrap with V 12 � V 21 / ∈ spectrum! ◮ Results: � C ( W me , W me , V 1 k ) = C L ( α me , α me , α 1 k ) C L ( α − m , e , α − m , e , α 1 k ) ◮ � C ( O 1 , O 2 , W m , e +1 ) C L ( α 1 , α 2 , α m , e +1 ) C L ( α 1 , α 2 , α m , e − 1 ) × C L (¯ α 1 , ¯ α 2 , α − m , e +1 ) C ( O 1 , O 2 , W m , e − 1 ) = ◮ C L (¯ α 1 , ¯ α 2 , α − m , e − 1 )

Numerical transfer-matrix study [B. Estienne, YI] C ( W 12 , W 10 , W 10 ) / C ( W 10 , W 10 , W 10 ) n

3. Signature of Logarithmic CFT

A surprising result ◮ Both analytical/numerical comp. ⇒ �W 12 W 10 W 10 � � = 0 . . . ◮ . . . in contradiction with general CFT argument: 1. W me has conformal weights ( h me , h me + me ) 2. ⇒ ( L − 2 − gL 2 − 1 ) W 12 is a “null vector” 3. [Null-vector cond.] ⇒ [PDE on �W 12 . . . � ] ⇒ [fusion rules] 4. Resulting fusion rule: W 12 × ( φ r 1 s 1 ⊗ ¯ s 1 ) → ( φ r 1 s 1 ± 1 ⊗ ¯ φ ¯ φ ¯ s 2 ) r 1 ¯ r 2 ¯ ◮ What could explain the violation of fusion rule: ◮ Null-vector does not decouple? ◮ Three-point function has non-standard form? ◮ ∃ non-normalisable states in the theory? ◮ Signatures of log CFTs ≡ models with indecomposable reps of Virasoro algebra!

Insight from the lattice [B. Estienne, YI] ◮ In the continuum: ◮ |V 12 � has a null vector at level 2: | χ 12 � = ( L − 2 − gL 2 − 1 ) |V 12 � , � χ 12 | χ 12 � = 0 ◮ Two-fold degeneracy: | χ 12 � and |W 1 , − 2 � ( h 12 + 2 , h 12 ) ◮ On the lattice: ◮ Transfer matrix of loop model, acting on connectivity patterns ◮ Periodic Temperley-Lieb algebra, generic q | α � = e j = � ◮ D j = repr. with 2 j strings D j = repr. with ≤ 2 j strings � E k � ◮ H = − � 1 j e j has Jordan cells in � D 1 : 0 E k ◮ Energies: E k ≃ E gs + 2 π v F L (2 h 1 k + k )

Summary and Perspectives ◮ Summary: ◮ Computed some OPE constants involving watermelon ops ◮ Method = extension of standard Dotsenko-Fateev approach ◮ Usual fusion rules are broken! ◮ O (n) model = bulk log CFT with generic c < 1 ◮ Infinite number of 2 × 2 Jordan cells ◮ Perspectives: ◮ Determine which null vectors fully decouple (e.g. L − 1 V 11 = 0?) ◮ Compute indecomposability constants b k ◮ Understand spatial dependence of 1,2,3-point functions ◮ Obtain full set of fusion rules/OPE constants

Thank you for your attention!