Non-scalar operators and logarithmic correlation functions for the - PowerPoint PPT Presentation

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Non-scalar operators and logarithmic correlation functions for the Potts model in arbitrary dimension RGP 2016 - IHP Romain Couvreur Laboratoire de Physique Théorique - École Normale Supérieure Institut de Physique Théorique - CEA Saclay October 2016 Collaborators : Jesper L. Jacobsen, Romain Vasseur 1 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Introduction • Scale Invariance : power-law and logarithmic correlation • c = 0 CFTs : percolation, disordered systems (IQHE,. . . ) • Logarithmic minimal models • Jordan cell in the dilatation operator L 0 • Two dimensions collide J. Cardy 2013 C i � � φ (0) φ ( r ) � = r 2∆ i , C 1 ∼ − C 2 → ∞ , C 1 (∆ 1 − ∆ 2 ) stays finite i contribution r − 2∆ 1 log r • Insight from discrete symmetries 2 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Potts model • Q -state Potts model : discrete S Q symmetry � � e Kδ σi,σj , Z = σ = 1 , . . . , Q { σ } ( i,j ) ∈ E • Fortuin-Kasteleyn clusters ( Q ∈ R ) : ( e K − 1) | A | Q k ( A ) � Z = A ⊆ E 3 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Operators acting on 1 spin σ : R. Vasseur, J. L. Jacobsen 2014 Q • General form : O ( σ ) = � O a δ σ,a a =1 • Action of the symmetric group S Q : Q Q � � ( p O )( σ ) = O a δ σ,p ( a ) = O a δ p − 1 ( σ ) ,a a =1 a =1 4 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Operators acting on 1 spin σ : R. Vasseur, J. L. Jacobsen 2014 Q • General form : O ( σ ) = � O a δ σ,a a =1 • Action of the symmetric group S Q : Q Q � � ( p O )( σ ) = O a δ σ,p ( a ) = O a δ p − 1 ( σ ) ,a a =1 a =1 • Two irreps : Q t a ( σ ) = δ σ,a − 1 � t ( σ ) = δ σ,a , Q a =1 • Identity and magnetization operator • decomposition with Young diagrams L (1) Q = [ Q ] ⊕ [ Q − 1 , 1] 4 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions General setup (short version) : Operator acting on N spins : Operators are Q × Q × . . . = Q N tensors : N � � L ( N ) � = Span O a 1 ,...,a N ( σ 1 , . . . , σ N ) = δ σ i ,a i Q i =1 Action of p ∈ S Q : O { a i } = O { p ( a i ) } S Q symmetry : • Choose Young tableau λ Q with at least Q − N boxes in the first row • Compute Young symmetrizer e λ Q • May have to specify S N representation • Generate invariant subspace of operators 5 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Result for L (2) Q , operators acting on σ 1 and σ 2 : • Subspace with σ 1 = σ 2 isomorphic to L (1) Q • Symmetric operators under σ 1 ↔ σ 2 t [ Q ] ( σ 1 , σ 2 ) = δ σ 1 � = σ 2 ... � δ σ 1 ,a + δ σ 2 ,a − 2 � ... t [ Q − 1 , 1] , [2] ( σ 1 , σ 2 ) = δ σ 1 � = σ 2 a Q � t [ Q − 2 , 2] ... ( σ 1 , σ 2 ) = δ σ 1 � = σ 2 δ σ 1 ,a δ σ 2 ,b + δ σ 1 ,b δ σ 2 ,a a,b 1 2 � � Q ( Q − 1) t [ Q ] � t [ Q − 1 , 1] , [2] + t [ Q − 1 , 1] , [2] − − a b Q − 2 6 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Result for L (2) Q , operators acting on σ 1 and σ 2 : • Subspace with σ 1 = σ 2 isomorphic to L (1) Q • Symmetric operators under σ 1 ↔ σ 2 t [ Q ] ( σ 1 , σ 2 ) = δ σ 1 � = σ 2 ... � δ σ 1 ,a + δ σ 2 ,a − 2 � ... t [ Q − 1 , 1] , [2] ( σ 1 , σ 2 ) = δ σ 1 � = σ 2 a Q � t [ Q − 2 , 2] ... ( σ 1 , σ 2 ) = δ σ 1 � = σ 2 δ σ 1 ,a δ σ 2 ,b + δ σ 1 ,b δ σ 2 ,a a,b 1 2 � � Q ( Q − 1) t [ Q ] � t [ Q − 1 , 1] , [2] + t [ Q − 1 , 1] , [2] − − a b Q − 2 • Anti-symmetric operators under σ 1 ↔ σ 2 ... t [ Q − 1 , 1] , [1 , 1] ( σ 1 , σ 2 ) = δ σ 1 � = σ 2 ( δ σ 1 ,a − δ σ 2 ,a ) a ... � t [ Q − 2 , 1 , 1] ( σ 1 , σ 2 ) = δ σ 1 � = σ 2 δ σ 1 ,a δ σ 2 ,b − δ σ 1 ,b δ σ 2 ,a a,b − 1 � � � t [ Q − 1 , 1] , [1 , 1] − t [ Q − 1 , 1] , [1 , 1] a b Q 6 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Result for L (2) Q , operators acting on σ 1 and σ 2 : • L (2) decomposition : Q ... ... ... L (2) Q = L (1) Q ⊕ ⊕ 2 ⊕ ⊕ ... • Ok with hook formula Q 2 = Q + 1 + 2( Q − 1) + Q ( Q − 3) + ( Q − 1)( Q − 2) 2 2 • Subtracted operators correspond to Young diagrams with boxes removed • Poles are related to logarithmic features 7 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Correlation functions for N = 1 , 2 , some examples • We can compute correlation functions with FK-clusters � � � � = 1 δ a,b − 1 � � t [ Q − 1 , 1] t [ Q − 1 , 1] P a b Q Q • For N ≥ 2 each operators act on N spins in the same neighbourhood • Symmetric correlation functions N = 2 :       � 1 2 � � � t [ Q − 2 , 2] (0) t [ Q − 2 , 2] ( r ) (2 δδ ) − Q − 2 (4 δ ) +  P  + P       ∝ a,b c,d ( Q − 2)( Q − 1)     • Anti-symmetric correlation functions N = 2 :       � (2 δδ ) − 1 � � � t [ Q − 2 , 1 , 1] (0) t [ Q − 2 , 1 , 1] ( r ) Q (4 δ )  P  − P       ∝ a,b c,d     • Correlation functions defined for Q ∈ R for finite size � ˜ � (2 δδ ) − 1 Z ( Q ) � � t [ Q − 2 , 1 , 1] (0) t [ Q − 2 , 1 , 1] ( r ) ∝ Q (4 δ ) a,b c,d Z ( Q ) 8 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Scale invariance • Power-law correlation functions � (2 δδ ) − 1 � 1 � � t [ Q − 2 , 1 , 1] (0) t [ Q − 2 , 1 , 1] ( r ) ∝ Q (4 δ ) a,b c,d r 2∆ λ 2 ( Q ) • on the cylinder from Jones-Temperley-Lieb representation theory ∆ = h k N ,N + h − k N ,N � � � � ... • 4-leg watermelon operator → P + P , ∆ = 2 h 0 , 2 � � � � ... • Other observables → P − P , ∆ = h − 1 / 2 , 2 + h 1 / 2 , 2 • Non-scalar , s = h − 1 / 2 , 2 − h 1 / 2 , 2 = 1 � � • 2d simplified, some configurations are forbidden : P 9 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Numerics • Transfer matrix methods • Monte-Carlo simulation 10 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Logarithmic correlation functions • How to deal with poles ? By mixing two operators • Percolation : 4 -leg watermelon operator diverges at Q → 1 2 ǫ = t [ Q ] , [2] − � t [ Q ] , [2] � t [ Q − 2 , 2] = regular part − Q ( Q − 1) ǫ, a,b 2 φ a,b = t [ Q − 2 , 2] + Q ( Q − 1) ǫ a,b • Correlation functions should be finite ∆ ǫ = ∆ 2 • Indeed in 2 d, ∆ ǫ = ∆ 2 = 5 / 4 Contribution as Q → 1 : � � 1 1 1 ∼ ∆ ǫ ( Q ) − ∆ 2 ( Q ) log r r 2∆ ǫ ( Q ) − r 2∆ 2 ( Q ) r 2∆ 2 (1) Q − 1 Q − 1 • energy and 4 -leg watermelon operator mixed in Jordan cell 11 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Logarithmic correlation functions • Predict scaling laws on the lattice with universal constant δ � � � � ) 2 + P − P ( P ∼ δr − 2∆ 2 log r � � � � + P P • Can be verified with Monte-Carlo • Whole new set of Jordan cell with non-scalar operators • Conjecture for the position of the poles gives the "full" log structure 12 / 13

Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions Conclusion • Classification of operators/LogCFTs in any d with discrete symmetries • Partial results, other symmetries can be considered • Jordan-cell of higher rank ? • Bestiary of interesting geometrical observables 13 / 13