New bootstrap solutions in two-dimensional percolation models Raoul - PowerPoint PPT Presentation

New bootstrap solutions in two-dimensional percolation models Raoul Santachiara LPTMS CNRS, Universit´ e Paris-Saclay, 91405 Orsay, France Annecy, 2020

Q -Potts random cluster model Probability ( G ) = p #bonds (1 − p ) #edges without bond Q #clusters #bonds = 11 #edges without bond = 5 #clusters = 5 G : Prob( G ) = p 11 (1 − p ) 5 Q 5 Study object : the connectivity properties of clusters Ex: ∃ infinite cluster (connecting 0 ↔ ∞ )?

Percolation transition √ Q √ Q + 1 , Q ∈ [0 , 4] p c = Clusters − − − − → scaling conformal random fractals Fractal dimension of cluster, curves, pivotal bonds · · · ( Di Francesco, Saleur, Zuber ’87 , De Nijs, Duplantier, Nienhuis, Saleur...’89 ) Crossing probabilities (Cardy formula) ( Cardy ’92 ) New set-up in probability theory and complex analysis (ex: SLE, lattice parafermion etc..) ( Werner, Smirnov, Bernard, Bauer’01 )

Potts CFT ? A 30 y.o. open problem... What was known: The CFT torus partition function: central charge, and the set of Virasoro representations (spectre S Potts ) The representation properties of certain boundary fields What was NOT known: The structure constants, necessary to compute all the field correlation function The fine structure of the Virasoro representations Why is difficult and at the same time interesting: So far NO consistent CFT that is non-unitary, non-rational and logarithmic has been found

...towards a solution! 2D Bootstrap approach to four-point connectivities ( Picco, Ribault, Santachiara ’15,’16, ’19 ) ( Ninevisat, Ribault, Samuelsson, Liu, He, Jacobsen, Saleur ’18 ’19 ’20 ) key inputs Monte-Carlo simulations Transfer matrix simulations Representation theory of Affine Temperley-Algebra

Potts CFT central charge: non-unitarity ( Di Francesco, Saleur, Zuber ’87 , De Nijs, Duplantier, Nienhuis, Saleur...’89 ) 1 c ∈ [ − 2 , 1] central charge 0 Unitary CFT series: − 1 Ising Perco. 6 − 2 c = 1 − p = 2 , 3 · · · 0 1 2 3 4 p ( p + 1) Q Spanning tree 3,4-Potts Local and positives Boltzmann weights − − − − → scaling unitary CFT Potts model is not local (or local but with complex Boltzmann weights) − scaling non-unitary CFT − − − →

Potts CFT spectre S Potts : non-rational S Potts = { V D 1 , 1 , V D , V N 2 , V N , V N 2 , 0 , V N 3 , 0 , V N 2 , 1 , V N 2 , V N 1 , 2 , ... 2 , ... 2 , ... } 0 , 1 0 , 3 2 , 1 4 , 1 � �� � � �� � � �� � Termal sector Magnetic Sector Other sectors V D z ) → (∆ r , s , ∆ r , s ) , V N r , s ( z , ¯ r , s ( z , ¯ z ) → (∆ r , s , ∆ r , − s ) Correlation lenght ν = (2 − 2∆ 1 , 2 ) − 1 → energy field V N 1 , 2 connectivity field V N Order parameter β = ∆ 0 , 1 2 / (2 − 2∆ 1 , 2 ) → 0 , 1 2 − 4∆ 0 , 1 p 12 = | x | 2 1 2

Potts CFT representations: Logarithmic Log CFT: not semi-simple (indecomposable but not irreducible) Virasoro representation Rank 2 example � � V 1 | L 0 V 1 � � � ∆ � � V 1 | L 0 V 2 � 1 = , � V 2 | L 0 V 1 � � V 2 | L 0 V 2 � 0 ∆ � V 1 ( x ) V 1 (0) � = 0 � V 1 ( x ) V 2 (0) � ∼ | x | − 4∆ � V 2 ( x ) V 2 (0) � ∼ ln | x | 2 | x | − 4∆

Potts CFT 3-pt connectivity: the Delfino-Viti conjecture 1 2 3 √ Constant (0 , 1 2 ) p 123 = , Constant = 2 C (0 , 1 2 ) , (0 , 1 ∆ 0 , 1 2 ) | x 12 x 13 x 23 | 2 ( Delfino-Viti ’10 )

c ≤ 1 Liouville structure constant Shift relations: C ( r 3 +2 , s 3 ) C ( r 3 , s 3 +2) ( r 1 , s 1 ) , ( r 2 , s 2 ) ( r 1 , s 1 ) , ( r 2 , s 2 ) = Product of Γ , = Product of Γ C ( r 3 , s 3 ) C ( r 3 , s 3 ) ( r 1 , s 1 ) , ( r 2 , s 2 ) ( r 1 , s 1 ) , ( r 2 , s 2 ) ( Teschner ’95 ) Admits an unique solution (product of double Γ 2 ): for c ≥ 25: C DOZZ : Liouville theory for 2D quantum gravity for c ≤ 1: C , used in Delfino-Viti conjecture ( Schomerus ’03, Kostov, Petkova, Zamolodchikov ’05 ) ( Delfino,Picco, S., Viti 2012, Dotsenko 2013 ) Liouville c ≤ 1 theory ( Ribault, S.’15 ) ( Gavrilenko, S.’18 ) Generalized to other three-point observables ( Estienne, Ihklef, Jacobsen Saleur, ’15 )

Potts CFT and 4 − pts functions: an ambitious project 1 2 1 2 1 2 1 2 3 4 4 3 4 3 4 3 p 1234 p 12;34 p 14;23 p 13;24 � 2 � � �� 2 � z = x 12 x 34 − 4∆ 0 , 1 � � � D ( r , s ) p σ = | x 13 x 24 | � F r , s 2 � � σ x 13 x 24 � ( r , s ) ∈S σ σ = 1234 , 12; 34 , 14; 23 , 13; 24 V 0 , 1 2 (0) V 0 , 1 2 ( ∞ ) [∆ r , s ] V 0 , 1 2 ( z ) V 0 , 1 2 (1)

4-pts functions: crossing relations z 1 − z 1 / z z / ( z − 1) 1 − 1 / z 1 / (1 − z ) id (13) (23) (12) (123) (132) (13)(24) (24) (14) (34) (243) (234) (12)(34) (1234) (1342) (1324) (134) (143) (23)(14) (1432) (1243) (1423) (142) (124) p 1234 (1 ↔ 3) = p 1234 , p 13;24 (1 ↔ 3) = p 13;24 , p 12;34 (1 ↔ 3) = p 14;23 � � 2 � |F r , s ( z ) | 2 − |F r , s (1 − z ) | 2 � � D ( r , s ) = 0 1234 ( r , s ) ∈S 1234 � � 2 � |F r , s ( z ) | 2 − |F r , s (1 − z ) | 2 � � D ( r , s ) = 0 13;24 ( r , s ) ∈S 13;24 · · ·

A new (not-log) bootstrap solution ( Picco, Ribault, R.S ’16,’19 ) � ( − 1) rs C ( r , s ) C ( r , − s ) F s r , s (1 − z ) F s r , − s (1 − ¯ R 1 = z ) ( r , s ) ∈ ( 2 Z , Z + 1 2 ) � C ( r , s ) C ( r , − s ) F s r , s ( z ) F s R 2 = r , − s (¯ z ) ( r , s ) ∈ ( 2 Z , Z + 1 2 ) � ( − 1) rs C ( r , s ) C ( r , − s ) F s r , s ( z ) F s R 3 = r , − s (¯ z ) ( r , s ) ∈ ( 2 Z , Z + 1 2 ) � � z − 4∆ (0 , 1 R 1 ( z ) = R 3 (1 − z ) = | 1 − z | 2 ) R 1 z − 1 � � z − 4∆ (0 , 1 2 ) R 3 R 2 ( z ) = R 2 (1 − z ) = | 1 − z | z − 1

An educated guess... 2 R 1 = p 1234 + Q − 2 p 12;34 2 R 2 = p 1234 + Q − 2 p 13;24 2 R 3 = p 1234 + Q − 2 p 14;23 Exact for Q = 0 , 3 , 4. ( Q = 4, Ashkin-Teller model, Zamolodchikov ’86 ) Very good agreement with Monte-Carlo simultations Q = 1 θ = 0 θ = π/ 6 θ = π/ 4 1 . 2 θ = π/ 3 1 . 1 1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 ρ

The correct statistical interpretation ( Samuelsson, Liu, He, Jacobsen, Saleur ’18,’19 ) p 12;34 ˜ 1 2 1 2 2(3 Q − 10) 2 − + · · · ( Q − 2)( Q 2 − 4 Q + 2) Q − 2 4 3 4 3 R 1 = p 1234 + ˜ p 12;34 Bootstrap solution: p , q →∞ D(p,q) RSOS models lim

The Potts bootstrap log solutions ( Ninesvivat, Ribault ’20, Samuelsson, Liu, He, Jacobsen, Saleur ’20 ) � 2 ) � 2 � � 2 � C (0 , n � � p 1234 + p 12;34 = � F 0 , n + � 2 n ∈ 2 N +1 2 � 2 ) � 2 � C (2 , n F 2 , n 2 F 2 , − n + 2 + Q − 2 n ∈ 2 N +1 4 � 2 ) � 2 � C (2 , n − F 4 , n 2 F 4 , − n 2 + ... ( Q − 1)( Q − 2)( Q 2 − 4 Q + 2) n ∈ 2 N +1 � 4 ) � 2 � 3 ) � 2 � � D (4 , 1 D (6 , 1 + F 4 , n 4 F 4 , − n 4 + F 6 , n 3 F 6 , − n 3 + · · · n ∈ 4 N +1 n ∈ 3 N +1 � D (2 , 0) � 2 � D (2 , n ) � 2 � 2 , n | 2 + · · · |G reg + F 2 , 0 F 2 , 0 + n ∈ N ∗

Origin of log structures (I) Null-vectors: If ( r , s ) ∈ ( N ∗ , N ∗ ) ∃ η rs ∈ V r , s of zero norm � η rs | η rs � = 0 η r , s = 0, always true for positive definite inner-product � ... | ... � (unitary CFT) V r , s → V r , s [ η rs ] implies fusion rules Example: � V 1 , 2 V r , s V r ′ , s ′ � = 0 if r � = r ′ & s ′ � = s ± 1 η r , s � = 0, possible if inner-product � ... | ... � is non definite (non-unitary CFT) Example: � V 1 , 1 V r , s V r ′ , s ′ � � = 0 for r � = r ′ & s ′ � = s

Origin of log structures (II) ( R.S., Viti ’13 ) A fixed c , log-conformal block con be obtained by a limit of no-log ones � C ( r , s + ǫ ) � 2 � � C ( r , − s − ǫ ) � 2 � � 2 + � � � 2 � F s � F s r , s + ǫ r , − s − ǫ for r , s ∈ N ∗ → 0 ǫ 2 + 0 ǫ + |G reg r , s ( x ) | 2 V r , − s V r , − s , and η r , s ¯ η r , s form the rank 2 log-partner

Potts CFT on a torus 0 . 9 numerical 0 . 85 24 p 12 ( r ) 5 0 . 8 r 0 . 75 10 − 4 10 − 3 10 − 2 10 − 1 10 0 r / N ( Javerzat, Picco, R.S. ’18’ 19 )   � � � 2 � r √ Γ( 7 � 5 4 ) � � p 12 = 1 4 5 e − 5 π e − 53 24 π M 4  + 4 π 24 + O 1 +  (2 π ) 3 N 5 Γ( 1 N 9 4 ) r 24 � 2 � � �� r + O . N

Conclusions The two boostrap solutions are probably the first of a hierarchy of solutions describing the scaling limit of Temperlie-Lieb algebra models Some of the structure constants have been obtained numerically using bootstrap. Analytic expressions will be probably soon derived Another class of 2D systems hints to new boostrap solutions: the disordered fixed points (i.e. Potts bond quenched disorder) We are exploring new 2D critical points in non-integrable deformation of percolation model. Results are quite promising..

Transfer matrix and Temperley-Lieb algebra � Transfer Matrix M � � Probability( G ) = Tr Z = G � Row state � � � �� � � H 1 H 2 · · · V 1 V 2 · · · | � = = | � � � � �� � � Transfer Matrix e 2 i e 2 i − 1 Id 2 i Id 2 i − 1 H i = Q Id 2 i − 1 + e 2 i − 1 , V i = Id 2 i + Q e 2 i e 2 i = Q e i , e i − 1 e i e i +1 = e i , e i e j = e j e i for | i − j | > 2