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:

C = − → W (C) = K #edges(C) n#loops(C)

◮ Scaling limit (K > Kc):

• CFT with c = 1 − 6(1−g)2

g

n = −2 cos πg , 0 < g < 1

◮ Three-point connectivity

• r2
• r3
• r1

P( r1, r2, r3) = C (| r1 − r2|| r2 − r3|| r1 − r3|)2h

The compact boson CFT

[Nienhuis, Di Francesco-Saleur-Zuber, Alcaraz ’80s]

◮ Free-field action:

A[Φ] = g 4π

• d2r (∇Φ)2 ,

Φ ≡ Φ + 2π , Φ = ϕ(z) + ϕ(¯ z)

◮ Vertex operators:

Vα,¯

α(z, ¯

z) = exp[iαϕ(z) + i ¯ αϕ(¯ z)]

◮ Lattice of charges:

αrs = (1−r)√g

2

− (1−s)

2√g ,

(r, s) ∈ Z2

◮ Spectrum of the O(n) model:

• 1. Electric operators

V1k = Vα1k,α1k , k ∈ Z (V = 1, ǫ . . . )

• 2. Watermelon operators

Wme = Vαme,α−m,e , m = 1, 2, 3 . . . e ∈ Z/m Wme = “(2m-leg defect) × exp(ieΦ)”

What is the operator algebra of the O(n) model?

◮ Operator Product Expansion (OPE):

Oa(z, ¯ z)Ob(0) ∼

z→0

• c

C c

ab z−ha−hb+hc ¯

z−¯

ha−¯ hb+¯ hc Oc(0)+. . . ◮ Defines an operator algebra

Oa × Ob =

• c

C c

ab Oc ◮ Questions

◮ Fusion rules and C c

ab for the V1k’s and Wme’s ?

◮ ⇒ consistent CFT with generic c < 1? ◮ Role of non-unitarity / indecomposability?

Other CFTs with infinite operator algebras

◮ Liouville CFT

◮ Spectrum = scalar vertex ops. Vα,α ◮ OPE constants by conformal bootstrap

[Dorn-Otto,Zamolodchikov-Zamolodchikov,Teschner, 90’s]

◮ Analytic continuation to c < 1: “time-like Liouville” ◮ OPE coefficient Cσσσ 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. V1k(z) ◮ Fusion rules V1m × V1n → V1,|m−n|+1 + · · · + V1,m+n−1 ◮ At c = 0 (polymers, perco.), T is a null-vector (TT = 0)

⇒ Logarithmic CFT [Gurarie, Rozansky-Saleur, Ludwig . . . ’90-’00s]

◮ Exact Jordan cells

λ 1 λ

• in the lattice transfer matrix

[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) = O1(0)O2(z, ¯ z)O3(1)O4(∞) =

• p,¯

p

Xp¯

p Fp(z|h1, . . . , h4)F¯ p(z|¯

h1, . . . , ¯ h4)

◮ Bases of conformal blocks

V1(0) V4(∞) V3(1) V2(z) Vp

← →

V1(0) V4(∞) V2(z) V3(1) Vq

Fp(z|h1, . . . , h4)

• Fq(z|h1, . . . , h4)

Change of basis: Fp(z) =

• q

βpq Fq(z)

OPE constants from four-point functions (2/2)

◮ Monodromy invar. ⇒

∀q = ¯ q

• p,¯

p

βpq ¯ β¯

p¯ qXp¯ p = 0

(M)

◮ Computing steps:

• 1. Matrix elements βpq are known, solve (M) for Xp¯

p

• 2. Interpret Xp¯

p as C(O1, O2, Op¯ p) × C(Op¯ p, O3, O4)

• 3. Extract ratios of the form C(O1, O2, Op¯

p)/C(O1, O2, Op′¯ p′)

◮ Variants

◮ Minimal models:

[Dotsenko-Fateev ’85]

◮ Fp = Qp j=1

H dwj . . .

◮ βpq obtained by contour deformation ◮ Liouville:

[Teschner ’95]

◮ Set O4 = V12 or O4 = V21, then Fp = hypergeometric ◮ Obtain recursion relations

C(α1, α2, α + b/2) C(α1, α2, α − b/2) = f , C(α1, α2, α + b−1/2) C(α1, α2, α − b−1/2) = g

◮ Unique solution CL(α1, α2, α3) as an explicit special function

Some OPE constants in the O(n) model

[B. Estienne, YI]

◮ Method:

◮ Start with correlation function

C(z, ¯ z) = O1(0)O2(z, ¯ z)O3(1)O4(∞) (Oj ∈ {V1k}∪{Wme})

◮ The Wme’s have integer spin ⇒ monodromy unchanged ◮ Apply Teschner’s bootstrap with V12

• V21 /

∈ spectrum!

◮ Results:

◮

C(Wme, Wme, V1k) =

• CL(αme, αme, α1k)CL(α−m,e, α−m,e, α1k)

◮

C(O1, O2, Wm,e+1) C(O1, O2, Wm,e−1) =

• CL(α1, α2, αm,e+1)

CL(α1, α2, αm,e−1) × CL(¯ α1, ¯ α2, α−m,e+1) CL(¯ α1, ¯ α2, α−m,e−1)

Numerical transfer-matrix study

[B. Estienne, YI] C(W12, W10, W10)/C(W10, W10, W10)

n

• 3. Signature of Logarithmic

CFT

A surprising result

◮ Both analytical/numerical comp. ⇒ W12W10W10 = 0 . . . ◮ . . . in contradiction with general CFT argument:

• 1. Wme has conformal weights (hme, hme + me)
• 2. ⇒ (L−2 − gL2

−1)W12 is a “null vector”

• 3. [Null-vector cond.] ⇒ [PDE on W12 . . .] ⇒ [fusion rules]
• 4. Resulting fusion rule: W12 × (φr1s1 ⊗ ¯

φ¯

r1¯ s1) → (φr1s1±1 ⊗ ¯

φ¯

r2¯ s2)

◮ 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:

◮ |V12 has a null vector at level 2:

|χ12 = (L−2 − gL2

−1)|V12 ,

χ12|χ12 = 0

◮ Two-fold degeneracy: |χ12 and |W1,−2

(h12 + 2, h12)

◮ On the lattice:

◮ Transfer matrix of loop model, acting on connectivity patterns ◮ Periodic Temperley-Lieb algebra, generic q

|α = ej =

◮ Dj = repr. with 2j strings

• Dj = repr. with ≤ 2j strings

◮ H = −

j ej has Jordan cells in

D1: Ek 1 Ek

• ◮ Energies: Ek ≃ Egs + 2πvF

L (2h1k + 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−1V11 = 0?) ◮ Compute indecomposability constants bk ◮ Understand spatial dependence of 1,2,3-point functions ◮ Obtain full set of fusion rules/OPE constants

Thank you for your attention!