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

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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 L0
• Two dimensions collide J. Cardy 2013

φ(0)φ(r) =

• i

Ci r2∆i , C1 ∼ −C2 → ∞, C1(∆1 − ∆2) stays finite contribution r−2∆1 log r

• Insight from discrete symmetries

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Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions

Potts model

• Q-state Potts model : discrete SQ symmetry

Z =

• {σ}
• (i,j)∈E

eKδσi,σj , σ = 1, . . . , Q

• Fortuin-Kasteleyn clusters (Q ∈ R) :

Z =

• A⊆E

(eK − 1)|A|Qk(A)

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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

• General form : O(σ) =

Q

• a=1

Oaδσ,a

• Action of the symmetric group SQ :

(pO)(σ) =

Q

• a=1

Oaδσ,p(a) =

Q

• a=1

Oaδp−1(σ),a

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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

• General form : O(σ) =

Q

• a=1

Oaδσ,a

• Action of the symmetric group SQ :

(pO)(σ) =

Q

• a=1

Oaδσ,p(a) =

Q

• a=1

Oaδp−1(σ),a

• Two irreps :

t(σ) =

Q

• a=1

δσ,a, ta(σ) = δσ,a − 1 Q

• Identity and magnetization operator
• decomposition with Young diagrams L(1)

Q = [Q] ⊕ [Q − 1, 1]

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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 × . . . = QN tensors : L(N)

Q

= Span

• Oa1,...,aN (σ1, . . . , σN) =

N

• i=1

δσi,ai

• Action of p ∈ SQ : O{ai} = O{p(ai)}

SQ symmetry :

• Choose Young tableau λQ with at least Q − N boxes in the first row
• Compute Young symmetrizer eλQ
• May have to specify SN representation
• Generate invariant subspace of operators

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Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions

Result for L(2)

Q , operators acting on σ1 and σ2 :

• Subspace with σ1 = σ2 isomorphic to L(1)

Q

• Symmetric operators under σ1 ↔ σ2

t[Q](σ1, σ2) = δσ1=σ2

...

t[Q−1,1],[2]

a

(σ1, σ2) = δσ1=σ2

• δσ1,a + δσ2,a − 2

Q

• ...

t[Q−2,2]

a,b

(σ1, σ2) = δσ1=σ2

• δσ1,aδσ2,b + δσ1,bδσ2,a

...

− 1 Q − 2

• t[Q−1,1],[2]

a

+ t[Q−1,1],[2]

b

• −

2 Q(Q − 1)t[Q]

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Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions

Result for L(2)

Q , operators acting on σ1 and σ2 :

• Subspace with σ1 = σ2 isomorphic to L(1)

Q

• Symmetric operators under σ1 ↔ σ2

t[Q](σ1, σ2) = δσ1=σ2

...

t[Q−1,1],[2]

a

(σ1, σ2) = δσ1=σ2

• δσ1,a + δσ2,a − 2

Q

• ...

t[Q−2,2]

a,b

(σ1, σ2) = δσ1=σ2

• δσ1,aδσ2,b + δσ1,bδσ2,a

...

− 1 Q − 2

• t[Q−1,1],[2]

a

+ t[Q−1,1],[2]

b

• −

2 Q(Q − 1)t[Q]

• Anti-symmetric operators under σ1 ↔ σ2

t[Q−1,1],[1,1]

a

(σ1, σ2) = δσ1=σ2 (δσ1,a − δσ2,a)

...

t[Q−2,1,1]

a,b

(σ1, σ2) = δσ1=σ2

• δσ1,aδσ2,b − δσ1,bδσ2,a

...

− 1 Q

• t[Q−1,1],[1,1]

a

− t[Q−1,1],[1,1]

b

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Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions

Result for L(2)

Q , operators acting on σ1 and σ2 :

• L(2)

Q

decomposition : L(2)

Q = L(1) Q ⊕ ...

⊕ 2

...

⊕

...

⊕

...

• Ok with hook formula

Q2 = Q + 1 + 2(Q − 1) + Q(Q − 3) 2 + (Q − 1)(Q − 2) 2

• Subtracted operators correspond to Young diagrams with boxes

removed

• Poles are related to logarithmic features

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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
• t[Q−1,1]

a

t[Q−1,1]

b

• = 1

Q

• δa,b − 1

Q

• P
• For N ≥ 2 each operators act on N spins in the same neighbourhood
• Symmetric correlation functions N = 2 :
• t[Q−2,2]

a,b

(0)t[Q−2,2]

c,d

(r)

• ∝
• (2δδ) −

1 Q − 2 (4δ) + 2 (Q − 2)(Q − 1)

• 

 P       + P         

• Anti-symmetric correlation functions N = 2 :
• t[Q−2,1,1]

a,b

(0)t[Q−2,1,1]

c,d

(r)

• ∝
• (2δδ) − 1

Q (4δ)

• 

 P       − P         

• Correlation functions defined for Q ∈ R for finite size
• t[Q−2,1,1]

a,b

(0)t[Q−2,1,1]

c,d

(r)

• ∝
• (2δδ) − 1

Q(4δ) ˜ Z(Q) Z(Q)

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Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions

Scale invariance

• Power-law correlation functions
• t[Q−2,1,1]

a,b

(0)t[Q−2,1,1]

c,d

(r)

• ∝
• (2δδ) − 1

Q(4δ)

• 1

r2∆λ2 (Q)

• on the cylinder from Jones-Temperley-Lieb representation theory

∆ = h k

N ,N + h− k N ,N

• 4-leg watermelon operator

...

→ P

• + P
• , ∆ = 2h0,2
• Other observables

...

→ P

• − P
• , ∆ = h−1/2,2 + h1/2,2
• Non-scalar, s = h−1/2,2 − h1/2,2 = 1
• 2d simplified, some configurations are forbidden : P
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Operator in the Potts model Correlation functions and non-scalar operators Log correlation functions

Numerics

• Transfer matrix methods
• Monte-Carlo simulation

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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

t[Q−2,2]

a,b

= regular part − 2 Q(Q − 1)ǫ, ǫ = t[Q],[2] −

• t[Q],[2]

φa,b = t[Q−2,2]

a,b

+ 2 Q(Q − 1)ǫ

• Correlation functions should be finite ∆ǫ = ∆2
• Indeed in 2d, ∆ǫ = ∆2 = 5/4

Contribution as Q → 1 : 1 Q − 1

• 1

r2∆ǫ(Q) − 1 r2∆2(Q)

• ∼ ∆ǫ(Q) − ∆2(Q)

Q − 1 log r r2∆2(1)

• energy and 4-leg watermelon operator mixed in Jordan cell

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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

P

• + P
• − P (

)2 P

• + P
• ∼ δr−2∆2 log r
• 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

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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

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