Logarithmic correlations in geometrical critical phenomena Jesper L. - PowerPoint PPT Presentation

Logarithmic correlations in geometrical critical phenomena Jesper L. Jacobsen 1 , 2 1 Laboratoire de Physique Théorique, École Normale Supérieure, Paris 2 Université Pierre et Marie Curie, Paris Mathematical Statistical Physics Yukawa Institute, Kyoto 3 August 2013 Collaborators: R. Vasseur, H. Saleur, A. Gaynutdinov Jesper L. Jacobsen (LPTENS) Logarithmic correlations MSP , Kyoto, 03/08/2013 1 / 21

Introduction Logarithms in critical phenomeana Scale invariance ⇒ correlations are power-law or logarithmic Jesper L. Jacobsen (LPTENS) Logarithmic correlations MSP , Kyoto, 03/08/2013 2 / 21

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 MSP , Kyoto, 03/08/2013 2 / 21

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 MSP , Kyoto, 03/08/2013 3 / 21

Non-diagonalisable dilatation operator Happens when dimensions of two operators collide Resonance phenomenon produces a log from two power laws 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) Jesper L. Jacobsen (LPTENS) Logarithmic correlations MSP , Kyoto, 03/08/2013 3 / 21

Non-diagonalisable dilatation operator Happens when dimensions of two operators collide Resonance phenomenon produces a log from two power laws 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 ≤ upper critical dimension Jesper L. Jacobsen (LPTENS) Logarithmic correlations MSP , Kyoto, 03/08/2013 3 / 21

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 MSP , Kyoto, 03/08/2013 4 / 21

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 MSP , Kyoto, 03/08/2013 4 / 21

Application to geometrical models Q -state Potts model Hamiltonian H = J � � ij � δ ( σ i , σ j ) with σ i = 1 , 2 , . . . , Q Reformulation in terms of Fortuin-Kasteleyn clusters � Q k ( A ) ( e K − 1 ) | A | Z = A ⊆� ij � Jesper L. Jacobsen (LPTENS) Logarithmic correlations MSP , Kyoto, 03/08/2013 5 / 21

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Application to geometrical models Q -state Potts model Hamiltonian H = J � � ij � δ ( σ i , σ j ) with σ i = 1 , 2 , . . . , Q Reformulation in terms of Fortuin-Kasteleyn clusters ( black ) � Q k ( A ) ( e K − 1 ) | A | Z = A ⊆� ij � Jesper L. Jacobsen (LPTENS) Logarithmic correlations MSP , Kyoto, 03/08/2013 5 / 21

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Application to geometrical models Q -state Potts model Hamiltonian H = J � � ij � δ ( σ i , σ j ) with σ i = 1 , 2 , . . . , Q Reformulation in terms of Fortuin-Kasteleyn clusters ( black ) � Q k ( A ) ( e K − 1 ) | A | Z = A ⊆� ij � 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 MSP , Kyoto, 03/08/2013 5 / 21

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Application to geometrical models Q -state Potts model Hamiltonian H = J � � ij � δ ( σ i , σ j ) with σ i = 1 , 2 , . . . , Q Reformulation in terms of Fortuin-Kasteleyn clusters ( black ) � Q k ( A ) ( e K − 1 ) | A | Z = A ⊆� ij � Here shown for Q = 3 The limit Q → 1 is percolation Surrounding loops ( grey ) satisfy the Temperley-Lieb algebra Continuum limit described by (L)CFT or SLE κ Critical exponent in two dimensions exactly computable Jesper L. Jacobsen (LPTENS) Logarithmic correlations MSP , Kyoto, 03/08/2013 5 / 21

Logarithmic correlations in percolation Reminders 2 and 3-point functions fixed in any d by global conformal invariance alone [Polyakov 1970] 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 MSP , Kyoto, 03/08/2013 6 / 21

Logarithmic correlations in percolation Reminders 2 and 3-point functions fixed in any d by global conformal invariance alone [Polyakov 1970] 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 Two and three-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 MSP , Kyoto, 03/08/2013 6 / 21

Potts model Hamiltonian H = J � � ij � δ ( σ i , σ j ) with σ i = 1 , 2 , . . . , Q Operators must be irreducible under S Q symmetry [Cardy 1999] Jesper L. Jacobsen (LPTENS) Logarithmic correlations MSP , Kyoto, 03/08/2013 7 / 21

Potts model Hamiltonian H = J � � ij � δ ( σ i , σ j ) with σ i = 1 , 2 , . . . , Q Operators must be irreducible under S Q symmetry [Cardy 1999] 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 MSP , Kyoto, 03/08/2013 7 / 21

Operators acting 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 MSP , Kyoto, 03/08/2013 8 / 21

Operators acting 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 Q ( Q − 1 ) E ψ ab = δ σ i , a δ σ j , b + δ σ i , b δ σ j , a − Q − 2 ( φ a + φ b ) − Jesper L. Jacobsen (LPTENS) Logarithmic correlations MSP , Kyoto, 03/08/2013 8 / 21