Dimer models: monomers, arctic curve and CFT Nicolas Allegra (Groupe de physique statistique, IJL Nancy) July 2, 2015 Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 1 / 32
1 Critical phenomena on rectangle geometry Generalities about boundary conditions Critical free energy and CFT Correlation functions 2 Dimer models Free boson theory Free fermion theory Corner free energy and exponents 3 Arctic circle phenomena and curved Dirac field Arctic Circle Exact Calculations Asymptotic and field theory correspondence: toy model 4 Conclusions Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 2 / 32
Critical systems: Example of the Ising model Free Free Free Free Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 2 / 32
Critical systems: Example of the Ising model Free Free Free Free Some interesting questions Change of boundary conditions → change of the critical behavior Expression of the free energy at the critical point Magnetization profile at the critical point Spin/energy correlation exponents close to a surface or corner Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 2 / 32
Critical systems: Example of the Ising model Free Free Free Free Some interesting questions Change of boundary conditions → change of the critical behavior Expression of the free energy at the critical point Magnetization profile at the critical point Spin/energy correlation exponents close to a surface or corner Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 2 / 32
Ising model → c = 1 / 2 CFT Free Free ++++ - - - - Free Free Free Free Boundary condition changing operators (bcc) ( − − − ) to (+ + +) or (Free) to (+ + +) or ( − − − ) to (Free) Ψ bcc primary operators of the c = 1 / 2 CFT Kac table c = 1 / 2 → h bcc = { 0 , 1 / 2 , 1 / 16 } Ψ + free = σ and Ψ + − = ϵ with h + free = 1 / 2 and h + − = 1 / 16 Boundary conformal field theory (Cardy ’84) Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 3 / 32
Ising model → c = 1 / 2 CFT Free Free ++++ - - - - Free Free Free Free Boundary condition changing operators (bcc) ( − − − ) to (+ + +) or (Free) to (+ + +) or ( − − − ) to (Free) Ψ bcc primary operators of the c = 1 / 2 CFT Kac table c = 1 / 2 → h bcc = { 0 , 1 / 2 , 1 / 16 } Ψ + free = σ and Ψ + − = ϵ with h + free = 1 / 2 and h + − = 1 / 16 Boundary conformal field theory (Cardy ’84) Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 3 / 32
Ising model → c = 1 / 2 CFT Free Free ++++ - - - - Free Free Free Free Boundary condition changing operators (bcc) ( − − − ) to (+ + +) or (Free) to (+ + +) or ( − − − ) to (Free) Ψ bcc primary operators of the c = 1 / 2 CFT Kac table c = 1 / 2 → h bcc = { 0 , 1 / 2 , 1 / 16 } Ψ + free = σ and Ψ + − = ϵ with h + free = 1 / 2 and h + − = 1 / 16 Boundary conformal field theory (Cardy ’84) Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 3 / 32
Ising model → c = 1 / 2 CFT Free Free ++++ - - - - Free Free Free Free Boundary condition changing operators (bcc) ( − − − ) to (+ + +) or (Free) to (+ + +) or ( − − − ) to (Free) Ψ bcc primary operators of the c = 1 / 2 CFT Kac table c = 1 / 2 → h bcc = { 0 , 1 / 2 , 1 / 16 } Ψ + free = σ and Ψ + − = ϵ with h + free = 1 / 2 and h + − = 1 / 16 Boundary conformal field theory (Cardy ’84) Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 3 / 32
Ising model → c = 1 / 2 CFT Free Free ++++ - - - - Free Free Free Free Boundary condition changing operators (bcc) ( − − − ) to (+ + +) or (Free) to (+ + +) or ( − − − ) to (Free) Ψ bcc primary operators of the c = 1 / 2 CFT Kac table c = 1 / 2 → h bcc = { 0 , 1 / 2 , 1 / 16 } Ψ + free = σ and Ψ + − = ϵ with h + free = 1 / 2 and h + − = 1 / 16 Boundary conformal field theory (Cardy ’84) Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 3 / 32
Ising model → c = 1 / 2 CFT Free Free ++++ - - - - Free Free Free Free Boundary condition changing operators (bcc) ( − − − ) to (+ + +) or (Free) to (+ + +) or ( − − − ) to (Free) Ψ bcc primary operators of the c = 1 / 2 CFT Kac table c = 1 / 2 → h bcc = { 0 , 1 / 2 , 1 / 16 } Ψ + free = σ and Ψ + − = ϵ with h + free = 1 / 2 and h + − = 1 / 16 Boundary conformal field theory (Cardy ’84) Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 3 / 32
Free energy decomposition F = L 2 f bulk + Lf surface + f corner f bulk and f surface can be obtained by BA/TM but not f corner Corner free energy with a bcc operator φ b θ h ¯ φ b � � �� π c θ π − π CFT predicts f corner = θ h bcc + log L universal (Cardy Peschel) θ 24 Valid close to criticality L → ξ Nice way to compute c and ξ (Vernier Jacobsen ’12) Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 4 / 32
Free energy decomposition F = L 2 f bulk + Lf surface + f corner f bulk and f surface can be obtained by BA/TM but not f corner Corner free energy with a bcc operator φ b θ h ¯ φ b � � �� π c θ π − π CFT predicts f corner = θ h bcc + log L universal (Cardy Peschel) θ 24 Valid close to criticality L → ξ Nice way to compute c and ξ (Vernier Jacobsen ’12) Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 4 / 32
Free energy decomposition F = L 2 f bulk + Lf surface + f corner f bulk and f surface can be obtained by BA/TM but not f corner Corner free energy with a bcc operator φ b θ h ¯ φ b � � �� π c θ π − π CFT predicts f corner = θ h bcc + log L universal (Cardy Peschel) θ 24 Valid close to criticality L → ξ Nice way to compute c and ξ (Vernier Jacobsen ’12) Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 4 / 32
Free energy decomposition F = L 2 f bulk + Lf surface + f corner f bulk and f surface can be obtained by BA/TM but not f corner Corner free energy with a bcc operator φ b θ h ¯ φ b � � �� π c θ π − π CFT predicts f corner = θ h bcc + log L universal (Cardy Peschel) θ 24 Valid close to criticality L → ξ Nice way to compute c and ξ (Vernier Jacobsen ’12) Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 4 / 32
Bulk surface and corner correlations � ) r ( σ b ) 0 ( σ b � Correlations and scaling dimensions (spin σ and energy ϵ ) ⟨ σ b ( 0 ) σ b ( r ) ⟩ ∼ r − x σ b − x σ ⟨ ϵ b ( 0 ) ϵ b ( r ) ⟩ ∼ r − x ϵ b − x ϵ b b ⟨ σ b ( 0 ) σ s ( r ) ⟩ ∼ r − x σ b − x σ ⟨ ϵ b ( 0 ) ϵ s ( r ) ⟩ ∼ r − x ϵ b − x ϵ s s ⟨ σ b ( 0 ) σ c ( r ) ⟩ ∼ r − x σ b − x σ ⟨ ϵ b ( 0 ) ϵ c ( r ) ⟩ ∼ r − x ϵ b − x ϵ c c x b , x s , x c define bulk, surface and corner dimension of the operator x c and x s related by x c = ( π/θ ) x s Valid for all the primary operators of the CFT Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 5 / 32
Bulk surface and corner correlations � ) r ( σ b ) 0 ( σ b � Correlations and scaling dimensions (spin σ and energy ϵ ) ⟨ σ b ( 0 ) σ b ( r ) ⟩ ∼ r − x σ b − x σ ⟨ ϵ b ( 0 ) ϵ b ( r ) ⟩ ∼ r − x ϵ b − x ϵ b b ⟨ σ b ( 0 ) σ s ( r ) ⟩ ∼ r − x σ b − x σ ⟨ ϵ b ( 0 ) ϵ s ( r ) ⟩ ∼ r − x ϵ b − x ϵ s s ⟨ σ b ( 0 ) σ c ( r ) ⟩ ∼ r − x σ b − x σ ⟨ ϵ b ( 0 ) ϵ c ( r ) ⟩ ∼ r − x ϵ b − x ϵ c c x b , x s , x c define bulk, surface and corner dimension of the operator x c and x s related by x c = ( π/θ ) x s Valid for all the primary operators of the CFT Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 5 / 32
Bulk surface and corner correlations � ) r ( σ b ) 0 ( σ b � Correlations and scaling dimensions (spin σ and energy ϵ ) ⟨ σ b ( 0 ) σ b ( r ) ⟩ ∼ r − x σ b − x σ ⟨ ϵ b ( 0 ) ϵ b ( r ) ⟩ ∼ r − x ϵ b − x ϵ b b ⟨ σ b ( 0 ) σ s ( r ) ⟩ ∼ r − x σ b − x σ ⟨ ϵ b ( 0 ) ϵ s ( r ) ⟩ ∼ r − x ϵ b − x ϵ s s ⟨ σ b ( 0 ) σ c ( r ) ⟩ ∼ r − x σ b − x σ ⟨ ϵ b ( 0 ) ϵ c ( r ) ⟩ ∼ r − x ϵ b − x ϵ c c x b , x s , x c define bulk, surface and corner dimension of the operator x c and x s related by x c = ( π/θ ) x s Valid for all the primary operators of the CFT Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 5 / 32
Bulk surface and corner correlations � ) r ( σ b ) 0 ( σ b � Correlations and scaling dimensions (spin σ and energy ϵ ) ⟨ σ b ( 0 ) σ b ( r ) ⟩ ∼ r − x σ b − x σ ⟨ ϵ b ( 0 ) ϵ b ( r ) ⟩ ∼ r − x ϵ b − x ϵ b b ⟨ σ b ( 0 ) σ s ( r ) ⟩ ∼ r − x σ b − x σ ⟨ ϵ b ( 0 ) ϵ s ( r ) ⟩ ∼ r − x ϵ b − x ϵ s s ⟨ σ b ( 0 ) σ c ( r ) ⟩ ∼ r − x σ b − x σ ⟨ ϵ b ( 0 ) ϵ c ( r ) ⟩ ∼ r − x ϵ b − x ϵ c c x b , x s , x c define bulk, surface and corner dimension of the operator x c and x s related by x c = ( π/θ ) x s Valid for all the primary operators of the CFT Allegra Dimer models: monomers, arctic curve and CFT July 2, 2015 5 / 32
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