Logarithmic correlations in percolation and other geometrical critical phenomena Jesper L. Jacobsen 1 , 2 1 Laboratoire de Physique Théorique, École Normale Supérieure, Paris 2 Université Pierre et Marie Curie, Paris Statistical Mechanics, Integrability and Combinatorics, Galileo Galilei Institute, 26 June 2015 Collaborators: Romain Couvreur (ENS), Hubert Saleur (Saclay), Romain Vasseur (Berkeley) Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 1 / 26
Introduction Logarithms in critical phenomeana Scale invariance ⇒ correlations are power-law or logarithmic Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 2 / 26
Introduction Logarithms in critical phenomeana Scale invariance ⇒ correlations are power-law or logarithmic Two possibilities for logarithms: Marginally irrelevant operator: 1 Gives logs upon approach to fixed point theory. Dilatation operator not diagonalisable: 2 Logs directly in the fixed point theory. Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 2 / 26
(2) Non-diagonalisable dilatation operator Happens when dimensions of two operators collide Resonance phenomenon produces a log from two power laws Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 3 / 26
(2) Non-diagonalisable dilatation operator Happens when dimensions of two operators collide Resonance phenomenon produces a log from two power laws Cf. Frobenius method for solving second-order differential equations. When the two roots of the indicial equation collide, a log is produced in one solution. Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 3 / 26
(2) Non-diagonalisable dilatation operator Happens when dimensions of two operators collide Resonance phenomenon produces a log from two power laws Cf. Frobenius method for solving second-order differential equations. When the two roots of the indicial equation collide, a log is produced in one solution. Where do such logarithms appear? CFT with c = 0 [Gurarie, Gurarie-Ludwig, Cardy, . . . ] Percolation, self-avoiding polymers ( c → 0 catastrophe) Quenched random systems (replica limit catastrophe) Logarithmic minimal models [Pearce-Rasmussen-Zuber, Read-Saleur] For any d ≤ d uc , the upper critical dimension Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 3 / 26
Logarithms and non-unitarity [Cardy 1999] Standard unitary CFT Expand local density Φ( r ) on sum of scaling operators ϕ ( r ) A ij � � Φ( r )Φ( 0 ) � ∼ r ∆ i +∆ j ij A ij ∝ δ ij by conformal symmetry [Polyakov 1970] A ii ≥ 0 by reflection positivity Hence only power laws appear Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 4 / 26
Logarithms and non-unitarity [Cardy 1999] Standard unitary CFT Expand local density Φ( r ) on sum of scaling operators ϕ ( r ) A ij � � Φ( r )Φ( 0 ) � ∼ r ∆ i +∆ j ij A ij ∝ δ ij by conformal symmetry [Polyakov 1970] A ii ≥ 0 by reflection positivity Hence only power laws appear The non-unitary case Cancellations may occur Suppose A ii ∼ − A jj → ∞ with A ii (∆ i − ∆ j ) finite Then leading term is r − 2 ∆ i log r Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 4 / 26
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Geometrical models Q -state Potts model Definition in terms of spins σ i = 1 , 2 , . . . , Q � � e K δ σ i ,σ j Z = { σ } ( ij ) ∈ E Reformulation in terms of Fortuin-Kasteleyn clusters � Q k ( A ) ( e K − 1 ) | A | Z = A ⊆ E Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 5 / 26
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Geometrical models Q -state Potts model Definition in terms of spins σ i = 1 , 2 , . . . , Q ( colours ) � � e K δ σ i ,σ j Z = { σ } ( ij ) ∈ E Reformulation in terms of Fortuin-Kasteleyn clusters ( black ) � Q k ( A ) ( e K − 1 ) | A | Z = A ⊆ E Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 5 / 26
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Geometrical models Q -state Potts model Definition in terms of spins σ i = 1 , 2 , . . . , Q ( colours ) � � e K δ σ i ,σ j Z = { σ } ( ij ) ∈ E Reformulation in terms of Fortuin-Kasteleyn clusters ( black ) � Q k ( A ) ( e K − 1 ) | A | Z = A ⊆ E Here shown for Q = 3 The limit Q → 1 is percolation Surrounding loops ( grey ) satisfy the Temperley-Lieb algebra Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 5 / 26
Logarithmic correlation functions for 2 ≤ d ≤ d uc Reminders 2 and 3-point functions in any d from global conformal invariance This is supposing only conformal invariance! Extra discrete symmetries must be taken into account as well Physical operators are irreducible under such symmetries [Cardy 1999] O( n ) symmetry for polymers ( n → 0) S n replica symmetry for systems with quenched disorder ( n → 0) Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 6 / 26
Logarithmic correlation functions for 2 ≤ d ≤ d uc Reminders 2 and 3-point functions in any d from global conformal invariance This is supposing only conformal invariance! Extra discrete symmetries must be taken into account as well Physical operators are irreducible under such symmetries [Cardy 1999] O( n ) symmetry for polymers ( n → 0) S n replica symmetry for systems with quenched disorder ( n → 0) Correlators in bulk percolation in any dimension 2 and 3-point functions in bulk percolation Limit Q → 1 of Potts model with S Q symmetry Structure for any d ; but universal prefactors only for d = 2 Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 6 / 26
Symmetry classification of operators N -spin operators irreducible under S Q and S N symmetries Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 7 / 26
Symmetry classification of operators N -spin operators irreducible under S Q and S N symmetries Operators acting on one spin Most general one-spin operator: O ( r i ) ≡ O ( σ i ) = � Q a = 1 O a δ a ,σ i � � 1 δ a ,σ i − 1 δ a ,σ i = + Q Q ���� ���� � �� � reducible invariant ϕ a ( σ i ) Dimensions of representations: ( Q ) = ( 1 ) ⊕ ( Q − 1 ) Identity operator 1 = � a δ a ,σ i Order parameter ϕ a ( σ i ) satisfies the constraint � a ϕ a ( σ i ) = 0 Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 7 / 26
Operators acting symmetrically on two spins Q × Q matrices O ( r i ) ≡ O ( σ i , σ j ) = � Q � Q b = 1 O ab δ a ,σ i δ b ,σ j a = 1 The Q operators with σ i = σ j decompose as before: ( 1 ) ⊕ ( Q − 1 ) � � Other Q ( Q − 1 ) Q ( Q − 3 ) operators with σ i � = σ j : ( 1 ) + ( Q − 1 ) + 2 2 Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 8 / 26
Operators acting symmetrically on two spins Q × Q matrices O ( r i ) ≡ O ( σ i , σ j ) = � Q � Q b = 1 O ab δ a ,σ i δ b ,σ j a = 1 The Q operators with σ i = σ j decompose as before: ( 1 ) ⊕ ( Q − 1 ) � � Other Q ( Q − 1 ) Q ( Q − 3 ) operators with σ i � = σ j : ( 1 ) + ( Q − 1 ) + 2 2 Easy representation theory exercise E = δ σ i � = σ j = 1 − δ σ i ,σ j � � φ a = δ σ i � = σ j ϕ a ( σ i ) + ϕ a ( σ j ) 1 2 ˆ ψ ab = δ σ i , a δ σ j , b + δ σ i , b δ σ j , a − Q − 2 ( φ a + φ b ) − Q ( Q − 1 ) E Jesper L. Jacobsen (LPTENS) Logarithmic correlations GGI, 26 June 2015 8 / 26
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