The Elliptic Gaudin Model with Boundary Nenad Manojlovi c - PowerPoint PPT Presentation

Outline Introduction XYZ Heisenberg spin chain Elliptic Gaudin model The Elliptic Gaudin Model with Boundary Nenad Manojlovi´ c Departamento de Matem´ atica da Faculdade de Ciˆ encias e Tecnologia Universidade do Algarve MPHYS10 Belgrade, Serbia 13 September 2019 N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline Introduction XYZ Heisenberg spin chain Elliptic Gaudin model Introduction 1 XYZ Heisenberg spin chain 2 XYZ Lax Operator Reflection Equation Elliptic Gaudin model 3 Gaudin model as the quasi-classical limit Gaudin model with boundary terms N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline Introduction XYZ Heisenberg spin chain Elliptic Gaudin model Introduction 1 XYZ Heisenberg spin chain 2 XYZ Lax Operator Reflection Equation Elliptic Gaudin model 3 Gaudin model as the quasi-classical limit Gaudin model with boundary terms N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline Introduction XYZ Heisenberg spin chain Elliptic Gaudin model Introduction 1 XYZ Heisenberg spin chain 2 XYZ Lax Operator Reflection Equation Elliptic Gaudin model 3 Gaudin model as the quasi-classical limit Gaudin model with boundary terms N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline Introduction XYZ Heisenberg spin chain Elliptic Gaudin model Outline Introduction 1 XYZ Heisenberg spin chain 2 XYZ Lax Operator Reflection Equation Elliptic Gaudin model 3 Gaudin model as the quasi-classical limit Gaudin model with boundary terms N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline Introduction XYZ Heisenberg spin chain Elliptic Gaudin model Elliptic Gaudin Model The elliptic model has interesting algebraic, geometrical and functional structures. Both the rational and the trigonometric models can be obtained as appropriate limits of the elliptic one. Some of the results obtained may be relevant for some other systems. N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline Introduction XYZ Heisenberg spin chain Elliptic Gaudin model Elliptic Gaudin Model The elliptic model has interesting algebraic, geometrical and functional structures. Both the rational and the trigonometric models can be obtained as appropriate limits of the elliptic one. Some of the results obtained may be relevant for some other systems. N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline Introduction XYZ Heisenberg spin chain Elliptic Gaudin model Elliptic Gaudin Model The elliptic model has interesting algebraic, geometrical and functional structures. Both the rational and the trigonometric models can be obtained as appropriate limits of the elliptic one. Some of the results obtained may be relevant for some other systems. N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline XYZ Lax Operator Introduction RE XYZ Heisenberg spin chain Elliptic Gaudin model Outline Introduction 1 XYZ Heisenberg spin chain 2 XYZ Lax Operator Reflection Equation Elliptic Gaudin model 3 Gaudin model as the quasi-classical limit Gaudin model with boundary terms N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline XYZ Lax Operator Introduction RE XYZ Heisenberg spin chain Elliptic Gaudin model R-matrix of the XYZ chain The R-matrix of the XYZ chain is given by 3 � W α ( λ, η, κ ) σ α ⊗ σ α , R ( λ, η, κ ) = ✶ + α =1 were we use ✶ for the identity matrix, W 1 ( λ, η, κ ) = cn ( λ + η, κ ) sn ( η, κ ) sn ( λ + η, κ ) cn ( η, κ ) , W 2 ( λ, η, κ ) = dn ( λ + η, κ ) sn ( η, κ ) sn ( λ + η, κ ) dn ( η, κ ) , sn ( η, κ ) W 3 ( λ, η, κ ) = sn ( λ + η, κ ) , the functions sn ( λ, κ ), cn ( λ, κ ), and dn ( λ, κ ) are the usual Jacobi elliptic functions, λ is a spectral parameter, η is a quasi-classical parameter, κ is the modulus and N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline XYZ Lax Operator Introduction RE XYZ Heisenberg spin chain Elliptic Gaudin model R-matrix of the XYZ chain σ α , α = 1 , 2 , 3, are the Pauli matrices � � δ α 3 δ α 1 − ıδ α 2 σ α = . δ α 1 + ıδ α 2 − δ α 3 This R-matrix satisfies the Yang-Baxter equation R 12 ( λ − µ ) R 13 ( λ ) R 23 ( µ ) = R 23 ( µ ) R 13 ( λ ) R 12 ( λ − µ ) . In the present case Yang-Baxter equation reduces to the following matrix equation 3 � ǫ αβγ ( W β ( λ − µ ) W γ ( λ ) − W α ( λ − µ ) W γ ( µ ) + W α ( λ ) W β ( µ ) α,β,γ =1 − W γ ( λ − µ ) W β ( λ ) W α ( µ )) σ α ⊗ σ β ⊗ σ γ = 0 . N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline XYZ Lax Operator Introduction RE XYZ Heisenberg spin chain Elliptic Gaudin model R-matrix of the XYZ chain σ α , α = 1 , 2 , 3, are the Pauli matrices � � δ α 3 δ α 1 − ıδ α 2 σ α = . δ α 1 + ıδ α 2 − δ α 3 This R-matrix satisfies the Yang-Baxter equation R 12 ( λ − µ ) R 13 ( λ ) R 23 ( µ ) = R 23 ( µ ) R 13 ( λ ) R 12 ( λ − µ ) . In the present case Yang-Baxter equation reduces to the following matrix equation 3 � ǫ αβγ ( W β ( λ − µ ) W γ ( λ ) − W α ( λ − µ ) W γ ( µ ) + W α ( λ ) W β ( µ ) α,β,γ =1 − W γ ( λ − µ ) W β ( λ ) W α ( µ )) σ α ⊗ σ β ⊗ σ γ = 0 . N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline XYZ Lax Operator Introduction RE XYZ Heisenberg spin chain Elliptic Gaudin model R-matrix of the XYZ chain σ α , α = 1 , 2 , 3, are the Pauli matrices � � δ α 3 δ α 1 − ıδ α 2 σ α = . δ α 1 + ıδ α 2 − δ α 3 This R-matrix satisfies the Yang-Baxter equation R 12 ( λ − µ ) R 13 ( λ ) R 23 ( µ ) = R 23 ( µ ) R 13 ( λ ) R 12 ( λ − µ ) . In the present case Yang-Baxter equation reduces to the following matrix equation 3 � ǫ αβγ ( W β ( λ − µ ) W γ ( λ ) − W α ( λ − µ ) W γ ( µ ) + W α ( λ ) W β ( µ ) α,β,γ =1 − W γ ( λ − µ ) W β ( λ ) W α ( µ )) σ α ⊗ σ β ⊗ σ γ = 0 . N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline XYZ Lax Operator Introduction RE XYZ Heisenberg spin chain Elliptic Gaudin model Some Properties of the R-matrix unitarity R ( λ ) R ( − λ ) = ρ ( λ, η, κ ) ✶ , where the function ρ ( λ, η, κ ) is given by � � 3 � ρ ( λ, η, κ ) = 1 + W α ( λ ) W α ( − λ ) α =1 = 4 sn 2 ( η, κ ) sn 2 ( λ, κ ) − sn 2 (2 η, κ ) sn 2 ( λ, κ ) − sn 2 ( η, κ ) . sn 2 (2 η, κ ) parity invariance R 21 ( λ ) = R 12 ( λ ); R t temporal invariance 12 ( λ ) = R 12 ( λ ); R ( λ ) = J 1 R t 1 ( − λ − 2 η ) J 1 , crossing symmetry where t 1 denotes the transpose in the second space and the two-by-two matrix J = σ 2 . N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline XYZ Lax Operator Introduction RE XYZ Heisenberg spin chain Elliptic Gaudin model Some Properties of the R-matrix unitarity R ( λ ) R ( − λ ) = ρ ( λ, η, κ ) ✶ , where the function ρ ( λ, η, κ ) is given by � � 3 � ρ ( λ, η, κ ) = 1 + W α ( λ ) W α ( − λ ) α =1 = 4 sn 2 ( η, κ ) sn 2 ( λ, κ ) − sn 2 (2 η, κ ) sn 2 ( λ, κ ) − sn 2 ( η, κ ) . sn 2 (2 η, κ ) parity invariance R 21 ( λ ) = R 12 ( λ ); R t temporal invariance 12 ( λ ) = R 12 ( λ ); R ( λ ) = J 1 R t 1 ( − λ − 2 η ) J 1 , crossing symmetry where t 1 denotes the transpose in the second space and the two-by-two matrix J = σ 2 . N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary

Outline XYZ Lax Operator Introduction RE XYZ Heisenberg spin chain Elliptic Gaudin model Some Properties of the R-matrix unitarity R ( λ ) R ( − λ ) = ρ ( λ, η, κ ) ✶ , where the function ρ ( λ, η, κ ) is given by � � 3 � ρ ( λ, η, κ ) = 1 + W α ( λ ) W α ( − λ ) α =1 = 4 sn 2 ( η, κ ) sn 2 ( λ, κ ) − sn 2 (2 η, κ ) sn 2 ( λ, κ ) − sn 2 ( η, κ ) . sn 2 (2 η, κ ) parity invariance R 21 ( λ ) = R 12 ( λ ); R t temporal invariance 12 ( λ ) = R 12 ( λ ); R ( λ ) = J 1 R t 1 ( − λ − 2 η ) J 1 , crossing symmetry where t 1 denotes the transpose in the second space and the two-by-two matrix J = σ 2 . N. Cirilo Ant´ onio, E. Ragoucy, I. Salom and N. Manojlovi´ c The Elliptic Gaudin Model with Boundary