Critical interfaces in random media: random bond Potts model and - PowerPoint PPT Presentation

Critical interfaces in random media: random bond Potts model and logarithmic CFTs Raoul Santachiara LPTMS (Orsay) GGI, Florence 2008 In collaboration: Jesper Jacobsen, Pierre Le Doussal, Kay Wiese: LPTENS,Paris Marco Picco: LPTHE,Paris October 28, 2008 R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 1 / 24

Outlines Outline Pure critical Ising and 3 − states Potts model: geometrical exponents 1 Random bond Potts Model: perturbed CFT approach 2 Geometric exponents in the random Potts model: perturbative CFT 3 computation and logarithmic correlation functions Numerical studies:Montecarlo and Transef Matrix methods 4 Conclusions 5 R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 2 / 24

Pure critical Ising and 3 − states Potts model: geometrical exponents ISING MODEL: H = − J � < ij > σ i σ j Critical point ⇒ Local Scale Invariance ⇒ CFT R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 3 / 24

Pure critical Ising and 3 − states Potts model: geometrical exponents ISING MODEL, Local observables: Minimal M 3 , Unitary grid 1/2 0 ε Id 1/16 1/16 ∆ n , m = − 1 + 16 m 2 − 24 mn + 9 n 2 σ σ 1/2 0 48 ε 1 ≤ n ≤ 3 1 ≤ m ≤ 2 Id Energy and Spin-Spin correlation functions: c = 1 / 2 { φ } = { I , σ, ε } { ∆ } = { 0 , 1 / 16 , 1 / 2 } < σ ( z ) σ (0) > = | z | − 1 / 4 < ε ( z ) ε (0) > = | z | − 2 σσ → I + ε , εε → I R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 4 / 24

Geometrical description of phase transitions Geometrical objects.. Stochastic (FK) clusters: Bond between equal spin with prob. � � � � � � � � � � � � p = 1 − e − K � � � � � � � � Geometric (G) clusters: p = 1 �� �� � � � � � � �� �� � � �� �� � � � � � � � � � � � � Taken from Wolfhard Janke, KITP2006 show fractal behaviour and critical scaling Distribution of FK, G → Ising, q = 1 tricritical Potts critical exponent In 3 D Ising: different percolation temperature.. ..also in 2D non-minimal spin models? (M.Picco, A. Sicilia,RS, in progress) R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 5 / 24

Geometrical description of phase transitions ...random interfaces and geometric exponents Prob. two points belong to the perimeter of the same FK,G cluster: H. Blote,Y. Knops,B. Nienhuis (1992) 1 ∝ < φ FK , G ( z 1 ) φ FK , G ( z 2 ) > = | z 1 − z 2 | 4∆ FK , G φ FK = φ 1 , 0 , φ G = φ 0 , 1 I. Rushkin, E. Bettelheim, I. A. Gruzberg, P. Wiegmann (2007) Extended Kac Table, logarithmic minimal model P. Pearce, J. Rasmussen, J.Zuber (2006),Y.Saint-Aubin,P. Pearce, J. Rasmussen (2008) fractal dimensions d FK , G = 2 − 2∆ FK , G f d FK = 5 / 3( SLE 16 / 3 ), d G f = 11 / 8 ( SLE 3 ) f A.Coniglio,A den Nijs, J. Cardy, B. Duplantier, B. Nienhuis , H. Saleur,C. Vanderzande R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 6 / 24

Geometrical description of phase transitions 3 − states Potts model , H = − J � < ij > δ σ i ,σ j : Y. Deng,H. Blote, B. Nienhuis √ Critical at β c : e β c J = 1 + 3 33/40 1/5 7/24 143/120 d FK = 8 / 5( SLE 24 / 5 ), d G f = 17 / 12 ( SLE 10 / 3 ) f R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 7 / 24

Random bond Potts Model: perturbed CFT approach General q -states Potts model: � e β P � ij � J ij δ σ i σ j ∼ � � Z = � 1 − p ij + p ij δ σ i σ j � { σ i } { σ i } � ij � For J ij = J , p = 1 − e − β J : p |G| (1 − p ) |G| q ||G|| , � Z ∼ G R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 8 / 24

Random bond Potts Model: perturbed CFT approach Bond disorder and perturbed CFT J ij = J + δ J ij : Gaussian random variables: β 2 δ J 2 ij = g 0 weak disorder: √ g 0 ≪ β J Near the β c : � d 2 x ε ( x ) δ J ( x ) H = H pure + β H pure → Minimal CFT with 3 √ q = 2 cos( π/ (2 ǫ − 4)) c = 1 − (2 ǫ + 3)( ǫ + 2) ǫ : RG regularitation parameter. ǫ = 0 , 1 → Ising and 3 − states Potts R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 9 / 24

Random bond Potts Model: perturbed CFT approach Bond disorder: replica approach n � � � − β H a exp a =1 n n � � � � � H a d 2 x ε a ( x ) ε b ( x ) = exp − β pure + g 0 a =1 a , b =1 4∆ ε = 2 ǫ + 6 2 ǫ + 3 ǫ = 0 (Ising) → 4∆ ε = 2, disorder is marginal ǫ = 1 (3 − state Potts) → 4∆ ε < 2, disorder is relevant R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 10 / 24

Random bond Potts Model: perturbed CFT approach β g*~ ε g Disordered fixed point Pure Model β ( g ) = (2 − 4∆ ǫ ) g + 4 π ( n − 2) g 2 + · · · Replica limit: n → 0, g ∗ = 1 − 2∆ ǫ , conformal symmetry restored 4 π Perturbative computation in g and ǫ − expansion around the Ising model analogous to the ǫ -expansion for φ 4 scalar field theory around the gaussian model R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 11 / 24

Random bond Potts Model: perturbed CFT approach Energy and Spin disordered average correlation functions A. Ludwig 1987, Vl. Dotsenko, M. Picco and P. Pujol, 1995 < O (0) O ( R ) > = < O (0) O ( R ) > 0 + < S I O (0) O ( R ) > 0 + n +1 � 2 < S 2 d 2 x � ε a ( x ) ε b ( x ) I O (0) O ( R ) > 0 + · · · S I = g 0 a , b =1 1 1 < ε (0) ε ( x ) > = < σ (0) σ ( x ) > = | x | 4∆ ∗ | x | 4∆ ∗ ε σ 2∆ ∗ ε = 2∆ ε + 0( ǫ ) ∼ 2∆ ε +0 . 36 + 0( ǫ 3 ) 2∆ ∗ σ = 2∆ σ + 0( ǫ 3 ) ∼ 2∆ σ +0 . 00264 + 0( ǫ 4 ) R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 12 / 24

Geometric exponents in the random Potts model: perturbative CFT computation and logarithmic correlation functions Renormalization of the operator Φ 1 , 0 Second order diagrams:     10 ( z 1 ) g 2 � � 0 � �  → Φ a Φ a ε b ( z 2 ) ε c ( z 2 ) ǫ d ( z 3 ) ε e ( z 3 ) 10 ( z 1 )  2! z 2 z 3 b � = c d � = e We have to consider the following integral: � � Φ 10 ( z 1 ) ε ( z 2 ) ε ( z 3 )Φ 10 ( ∞ ) � � ε ( z 2 ) ε ( z 3 ) � z 2 , z 3 R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 13 / 24

Geometric exponents in the random Potts model: perturbative CFT computation and logarithmic correlation functions Appearence of logarithmics.. < Φ 10 ( z 1 ) ε ( z ) ε ( z 2 )Φ 10 ( z 3 ) > = · · · η c 1 ( η − 1) c 2 H ( η ) Hypergeometric differential equation: ′′ ( η )+( a (∆ 12 , c 1 ) − b ( c 1 , c 2 , ∆ 12 ) η ) H ′ ( η ) − c (∆ 12 , c 1 , c 2 ) H ( η ) = 0 η (1 − η ) H 3 2(2∆ 12 + 1)( c 1 ( c 1 − 1)) = ∆ 10 − c 1 3 2(2∆ 12 + 1)( c 2 ( c 2 − 1)) = ∆ 12 − c 2 R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 14 / 24

Geometric exponents in the random Potts model: perturbative CFT computation and logarithmic correlation functions Solutions of the hypergeometric diff eq: H ( η ) = a 1 H 1 ( η ) + a 2 (ln( η ) H 1 ( η ) + H 2 ( η )) Consistent with the OPE: Gurarie (1994) � ′ ( z ) + 1 � ′ ( z ) φ 1 , 0 ( η ) ε (0) = η − ∆ 1 , 2 − ∆ 1 , 0 +1 W ( z ) ln( z ) + W z ∂ z W Imposing simple monodromy: 2 p =2 = Γ( 1 3 ) 6 | u | � 3 � 2 � − 1 3 , 2 �� � G ( u ) � 2 F 1 3 ; 2; u � 27 π 2 | 1 − u | 2 � 2 Γ( 1 3 ) 8 1 3 , 4 | u | � � � 3 � G 2 , 0 − 1 3 , 2 � √ � � 3 + 2 F 1 3 ; 2; u u 2 , 2 � | 1 − u | 2 − 1 , 0 54 3 π 3 � � + c . c . , R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 15 / 24

Geometric exponents in the random Potts model: perturbative CFT computation and logarithmic correlation functions Coulomb gas � � d 2 zV + + µ − d 2 zV − S = S 0 + µ + α + α − = − 1 α + + α − = 2 α 0 V ± = : exp ( i α ± ϕ ( z )) : 1 − 12 α 2 = < ϕ ( z ) ϕ (0) > = − 4 log | x / L | c 0 3 √ q = 2 cos( π/ (2 ǫ − 4)) c = 1 − ( ǫ + 2)(2 ǫ + 3) Operators Φ n , m ( z ) written in terms of vertex operators Φ nm ( z ) → V nm ( z ) =: exp ( i α nm ϕ ( z )) : 1 − n α − + 1 − m α nm = α + 2 2 R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 16 / 24

Geometric exponents in the random Potts model: perturbative CFT computation and logarithmic correlation functions Results: � � Φ 10 ( z 1 ) ǫ ( z 2 ) ǫ ( z 3 )Φ 10 ( ∞ ) � � ǫ ( z 2 ) ǫ ( z 3 ) � z 2 , z 3 Coulomb gas (+procedure of ”regularitation” logarithmic cf) → � 10 ( ∞ ) � | z 2 − z 3 | − 4∆ 12 I = N � V 10 ( z 1 ) V 1 , 2 ( z 2 ) V 1 , 2 ( z 3 ) V + ( u ) V ¯ z 2 , z 3 , u how to compute that? see Dotsenko, Picco, Pujol, (1995)! From RG: ǫ 2 2∆ 10 + I 9˜ − 2 ǫ 2∆ ∗ = ˜ ǫ = 10 16 π 3(3 + 2 ǫ ) 2 = 8 p =3 d FK = 5 − 0 . 01433 → 5+0 . 01433 f R. Santachiara (LPTMS,Orsay) Critical interfaces in random media: October 28, 2008 17 / 24