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¨

• te, 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)

site j

ω(G, j) , Z =

• subgraph G

Π(G)

◮ Local Boltzmann weights t u1 u2 v w2 w1

• 2. Yang-Baxter Equations (YBE)

◮ Plaquette diagram (“ˇ

R-matrix”)

t(λ) +

• + . . .

:= λ +u1(λ)

• +w2(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 2θ t(λ) = − cos 2λ + sin 5θ 2 − sin 3θ 2 − sin θ 2 u1(λ) = 2 sin θ cos 3θ − π 4 − λ

• u2(λ)

= 2 sin θ cos 3θ − π 4 + λ

• v(λ)

= −

• cos 2λ + sin 3θ

2

• w1(λ)

= −

• cos(θ − 2λ) + sin θ

2

• w2(λ)

= −

• cos(θ + 2λ) + sin θ

2

• .

• 4. Three physical regimes

[Nienhuis et al.]

◮ Regime I : 0 < θ < π

◮ Central charge : ceff = 1 − 6(1−g)2

g

, g = 2θ

π

◮ Conformal dimensions : h, ¯

h = 1

4

• e

√g ± m√g

2 , e, m ∈ Z

Simple Coulomb gas (= compactified GFF)

• 4. Three physical regimes

[Nienhuis et al.]

◮ Regime I : 0 < θ < π

◮ Central charge : ceff = 1 − 6(1−g)2

g

, g = 2θ

π

◮ Conformal dimensions : h, ¯

h = 1

4

• e

√g ± m√g

2 , e, m ∈ Z

Simple Coulomb gas (= compactified GFF)

◮ Regime II : −π < θ < − π 3

◮ Central charge : ceff = 3

2 − 6(1/2−2g)2 g

, g = π+θ

2π

◮ Conformal dimensions :

h ∈

• (e/√2g+m√2g)

2

8

, (e/√2g+m√2g)

2

8

+ 1

2

• ,

e ≡ m [2] h = (e/√2g+m√2g)

2

8

+ 1

16 ,

e ≡ m + 1 [2]

Coulomb gas + Ising

• 4. Three physical regimes

[Nienhuis et al.]

◮ Regime I : 0 < θ < π

◮ Central charge : ceff = 1 − 6(1−g)2

g

, g = 2θ

π

◮ Conformal dimensions : h, ¯

h = 1

4

• e

√g ± m√g

2 , e, m ∈ Z

Simple Coulomb gas (= compactified GFF)

◮ Regime II : −π < θ < − π 3

◮ Central charge : ceff = 3

2 − 6(1/2−2g)2 g

, g = π+θ

2π

◮ Conformal dimensions :

h ∈

• (e/√2g+m√2g)

2

8

, (e/√2g+m√2g)

2

8

+ 1

2

• ,

e ≡ m [2] h = (e/√2g+m√2g)

2

8

+ 1

16 ,

e ≡ m + 1 [2]

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) α L 1 ◮ Scaling limit

TL(α) ∼ constL exp(− sin α H) exp(i cos α P)

◮ Conformal invariance

H = 2π L (L0 + ¯ L0 − c 12) , P = 2π L (L0 − ¯ L0) , where Ln, ¯ Ln are Virasoro generators

◮ CFT prediction for eigenvalues of TL(α)

− log ΛL(α) ≃ Lf∞ − 2π L

• ieiα

h − c 24

• − ie−iα

¯ h − c 24

• 6. Relation between α and λ

◮ YBE −

→ Bethe Ansatz − → asymptotics of ΛL

◮ Result

− log Λ ≃ Lf∞ + 2π L

• eiρ(θ)λ

h − c 24

• + e−iρ(θ)λ

¯ h − c 24

• For example, in regime I, ρ(θ) = 2π/(3π − 3θ).

◮ Simple relation

α = π 2 − ρ(θ)λ , |λ| < π 2ρ

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 {zi} :

• i j∈∂P

(zi − zj) F zi + zj 2

• = 0 .

• 8. Definition of the discrete parafermion

[Smirnov, Cardy-Rajabpour-Riva-YI]

◮ Introduce a pair of defects at 0 and z, and let

ψs(z) = 1 Z

• G| [0 and z carry defects]

Π(G) e−isW (z)

◮ Example configuration (W (z) = −π)

z

• 9. Discrete holomorphicity equations

◮ Around a plaquette : ψ(z)δz = 0. ◮ Linear equations on the Boltzmann weights

A(θ, s, α)    t . . . w2    = 0

◮ Singularity condition : det A(θ, s, α) = 0 ⇔ s = 3θ−π 2π ◮ Solution for Boltzmann weights = solution of YBE !

One recovers the relation λ ↔ α

• 10. Relation to SLE

[Smirnov]

◮ Prove (or assume) convergence of ψs to an analytic function ◮ In the upper half plane H, ψsH solves a boundary value

problem : Arg ψsH = πs 2 sgn(z) for real z ⇒ ψsH = const zs

◮ If gt is the conformal map

gt : H\γt → H then (g′

t/gt)s is a martingale ◮ Consequence : the driving function Wt is Brownian

Wt = √κBt, with s = 6−κ

2κ

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 !