Lattice models Diagrammatic calculus and functional relations Conclusion Functional relations for the A ( 1 ) models 2 Alexi Morin-Duchesne Recent Advances in Quantum Integrable Systems Annecy-le-Vieux, 13/09/2018 Joint work with Paul A. Pearce and Jørgen Rasmussen
Lattice models Diagrammatic calculus and functional relations Conclusion Outline • The A ( 1 ) lattice models 2 � A family of models: loop, vertex and dimer models � Diagrammatic algebras • Diagrammatic calculus and functional relations � Planar identities � Wenzl-Jones projectors � Fused transfer matrices � Fusion hierarchy relations of s ℓ 3 type � Y -systems
Lattice models Diagrammatic calculus and functional relations Conclusion The A ( 1 ) loop model 2 • The elementary face operator: (Nienhuis, Warnaar ’93) = sin ( λ − u ) � � + sin u � � + + + + + u sin λ sin λ • u is the spectral parameter • Loop fugacities: β = 2 cos λ = q + q − 1 � contractible: � non contractible: α • Vacancies are preserved • Weights and partition functions: W σ = α n α β n β � � w f Z = W σ A loop configuration σ f σ on the 12 × 12 torus
Lattice models Diagrammatic calculus and functional relations Conclusion Crossing symmetry and transfer tangles • No crossing symmetry: � = u λ − u � � � � = sin ( λ − u ) + sin u + + + + + u sin λ sin λ λ − u = sin u � � + sin ( λ − u ) � � + + + + + sin λ sin λ
Lattice models Diagrammatic calculus and functional relations Conclusion Crossing symmetry and transfer tangles • No crossing symmetry: � = u λ − u � � � � = sin ( λ − u ) + sin u + + + + + u sin λ sin λ λ − u = sin u � � + sin ( λ − u ) � � + + + + + sin λ sin λ • Two elementary transfer tangles: T 1 , 0 ( u ) = T 0 , 1 ( u ) = . . . . . . u u u λ − u λ − u λ − u • The identity strand: = + • They are assigned labels corresponding to the two fundamental representations of s ℓ ( 3 ) .
b b b b b b b b b b b b b b b Lattice models Diagrammatic calculus and functional relations Conclusion s ℓ 3 representations • Irreducible representations ... and Young diagrams: ... ( m , n ) ← → ���� � �� � m n ... • Tensor products of irreducible representations: ( 1 , 0 ) ... ( m , n ) ⊗ ( 1 , 0 ) = ( m + 1 , n ) ... ⊕ ( m − 1 , n + 1 ) ∅ ⊕ ( m , n − 1 ) ( 0 , 1 ) The s ℓ 3 weight lattice • Equivalently: ⊗ ⊕ ⊕ = � �� � ���� ���� ���� � �� � � �� � � �� � ���� m m + 1 m − 1 m n n n + 1 n − 1
b b b b b b b b b b b b b b b Lattice models Diagrammatic calculus and functional relations Conclusion s ℓ 3 representations • Irreducible representations ... and Young diagrams: ... ( m , n ) ← → ���� � �� � m n ... • Tensor products of irreducible representations: ( 1 , 0 ) ... ( m , n ) ⊗ ( 1 , 0 ) = ( m + 1 , n ) ... ⊕ ( m − 1 , n + 1 ) ∅ ⊕ ( m , n − 1 ) ( 0 , 1 ) The s ℓ 3 weight lattice • Equivalently: ⊗ ⊕ ⊕ = � �� � ���� ���� ���� � �� � � �� � � �� � ���� m m + 1 m − 1 m n n n + 1 n − 1
Lattice models Diagrammatic calculus and functional relations Conclusion Dilute Temperley-Lieb algebra � The periodic dilute Temperley-Lieb algebra, pdTL N ( α, β ) , is the linear span of connectivity diagrams: a 1 = a 2 = � Product of two connectivity diagrams: a 1 a 2 = = β = β a 3 � More examples of products for N = 6: = α = 0
Lattice models Diagrammatic calculus and functional relations Conclusion A smaller algebra • pdTL N , v ( α, β ) : subalgebra of pdTL N ( α, β ) with connectivities that have v preserved vacancies • The algebra A N ( α, β ) is the direct sum of these subalgebras: N � A N ( α, β ) = pdTL N , v ( α, β ) v = 0 • Examples: ∈ pdTL N , 1 ( α, β ) ⊂ A N ( α, β ) ∈ A N ( α, β ) / • The transfer tangles are elements of A N ( α, β ) : T ( 1 , 0 ) ( u ) , T ( 0 , 1 ) ( u ) ∈ A N ( α, β )
Lattice models Diagrammatic calculus and functional relations Conclusion The A ( 1 ) vertex model 2 • A configuration of the 15-vertex model: (Jimbo ’86) � The 15 admissible vertices and their Boltzmann weights: s 1 (− u ) s 0 ( u ) s 0 ( u ) 1 s k ( u ) = sin ( k λ + u ) e i u e − i u s 1 (− u ) sin λ
Lattice models Diagrammatic calculus and functional relations Conclusion The A ( 1 ) vertex model 2 • The vector space is ( C 3 ) ⊗ N , with the canonical basis: � 1 � 0 � 0 � � � | ↑ � = | 0 � = | ↓ � = 0 1 0 0 0 1 • The ˇ R ( u ) matrix: s 1 (− u ) 0 0 0 0 0 0 0 0 s 0 ( u ) 0 1 0 0 0 0 0 0 e i u 0 0 0 0 0 s 0 ( u ) 0 0 0 s 0 ( u ) 0 1 0 0 0 0 0 ˇ R ( u ) = = 0 0 0 0 s 1 (− u ) 0 0 0 0 u 0 0 0 0 0 1 0 s 0 ( u ) 0 e − i u 0 0 s 0 ( u ) 0 0 0 0 0 0 0 0 0 0 s 0 ( u ) 0 1 0 0 0 0 0 0 0 0 0 s 1 (− u ) • The local maps between the loop and vertex models: j j i → q 1 / 2 � ↑ i ↓ j | + q − 1 / 2 � ↓ i ↑ j | − − → | ↑ i �� ↑ j | + | ↓ i �� ↓ j | i i → q 1 / 2 | ↑ i ↓ j � + q − 1 / 2 | ↓ i ↑ j � − − → | 0 i � − → � 0 i | j i i
Lattice models Diagrammatic calculus and functional relations Conclusion Dimers on the hexagonal lattice • Bijection between dimer matchings and configurations of the fully packed loop model: (Kondev, de Gier, Nienhuis ’96)
Lattice models Diagrammatic calculus and functional relations Conclusion Dimers on the hexagonal lattice • Bijection between dimer matchings and configurations of the fully packed loop model: (Kondev, de Gier, Nienhuis ’96)
Lattice models Diagrammatic calculus and functional relations Conclusion Dimers on the hexagonal lattice • Bijection between dimer matchings and configurations of the fully packed loop model: (Kondev, de Gier, Nienhuis ’96) • Local maps: → → →
Lattice models Diagrammatic calculus and functional relations Conclusion Dimers on the hexagonal lattice • Bijection between dimer matchings and configurations of the fully packed loop model: (Kondev, de Gier, Nienhuis ’96) • Local maps: → → →
Lattice models Diagrammatic calculus and functional relations Conclusion Dimers on the hexagonal lattice • Bijection between dimer matchings and configurations of the fully packed loop model: (Kondev, de Gier, Nienhuis ’96) • Local maps: → → →
Lattice models Diagrammatic calculus and functional relations Conclusion Dimers on the hexagonal lattice • Bijection between dimer matchings and configurations of the fully packed loop model: (Kondev, de Gier, Nienhuis ’96) ← → • Local maps: → → →
Lattice models Diagrammatic calculus and functional relations Conclusion Dimers on the hexagonal lattice • Bijection between dimer matchings and configurations of the fully packed loop model: (Kondev, de Gier, Nienhuis ’96) ← → ← → • Local maps: → → → • Equivalent to the A ( 1 ) loop model at α = β = 1 with u = λ = π/ 3: 2 = + + + + λ
Lattice models Diagrammatic calculus and functional relations Conclusion Lattice models and representations • Family of A ( 1 ) lattice models: 2 � Loop model � Dimer model � Vertex model � RSOS model One specific A ( 1 ) A set of representations 2 ← → of the algebra A N ( α, β ) lattice model • To obtain the partition function, one must compute the eigenvalues of the transfer matrices T ( 1 , 0 ) ( u ) and T ( 0 , 1 ) ( u ) • Objective: find relations satisfied by T ( 1 , 0 ) ( u ) and T ( 0 , 1 ) ( u ) • By doing the calculations in A N ( α, β ) , we are solving all the A ( 1 ) models at once. 2
Lattice models Diagrammatic calculus and functional relations Conclusion Inversion identities • There are two local inversion identities: = s 1 ( u ) s 1 (− u ) = s 0 ( u ) s 3 (− u ) u − u u 3 λ − u • This is computed as follows: = u − u
Lattice models Diagrammatic calculus and functional relations Conclusion Inversion identities • There are two local inversion identities: = s 1 ( u ) s 1 (− u ) = s 0 ( u ) s 3 (− u ) u − u u 3 λ − u • This is computed as follows: = + + + + u − u + + + + + + + +
Lattice models Diagrammatic calculus and functional relations Conclusion Inversion identities • There are two local inversion identities: = s 1 ( u ) s 1 (− u ) = s 0 ( u ) s 3 (− u ) u − u u 3 λ − u • This is computed as follows: = + + + + u − u s 1 (− u ) s 1 ( u ) 1 × 1 1 × s 0 (− u ) 1 × 1 1 × s 0 (− u ) + + + + + s 0 ( u ) × 1 s 0 ( u ) s 0 (− u ) s 0 ( u ) × 1 s 0 ( u ) s 0 (− u ) s 1 (− u ) s 1 ( u ) + + + s 1 (− u ) s 0 (− u ) s 0 ( u ) s 1 ( u ) β s 0 ( u ) s 0 (− u )
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