Nested Bethe ansatz for orthogonal and symplectic open spin chains - PowerPoint PPT Presentation

Nested Bethe ansatz for orthogonal and symplectic open spin chains Allan Gerrard in collaboration with Vidas Regelskis and Curtis Wendlandt University of York RAQIS 2018, 10th September 2018 1/19

Historical timeline gl 2 closed chain (ABA) - Faddeev–Sklyanin–Takhtadjan’79 gl N closed chain (NBA) - Kulish–Reshetikhin’81 sp 2 n closed chain (NBA) - Reshetikhin’85 so 2 n closed chain (NBA) - de Vega–Karowski’87 gl 2 open chain (ABA) - Sklyanin’88 gl N open chain (NBA) - Martin–Galleas’04; Belliard–Ragoucy’09 osp M | 2 n open chain (Analytical BA) - Doikou et. al.’03 so 2 n open chain (NBA) - Gombor–Palla’16 2/19

Historical timeline gl 2 closed chain (ABA) - Faddeev–Sklyanin–Takhtadjan’79 gl N closed chain (NBA) - Kulish–Reshetikhin’81 sp 2 n closed chain (NBA) - Reshetikhin’85 so 2 n closed chain (NBA) - de Vega–Karowski’87 gl 2 open chain (ABA) - Sklyanin’88 gl N open chain (NBA) - Martin–Galleas’04; Belliard–Ragoucy’09 osp M | 2 n open chain (Analytical BA) - Doikou et. al.’03 so 2 n open chain (NBA) - Gombor–Palla’16 2/19

Historical timeline gl 2 closed chain (ABA) - Faddeev–Sklyanin–Takhtadjan’79 gl N closed chain (NBA) - Kulish–Reshetikhin’81 sp 2 n closed chain (NBA) - Reshetikhin’85 so 2 n closed chain (NBA) - de Vega–Karowski’87 gl 2 open chain (ABA) - Sklyanin’88 gl N open chain (NBA) - Martin–Galleas’04; Belliard–Ragoucy’09 osp M | 2 n open chain (Analytical BA) - Doikou et. al.’03 so 2 n open chain (NBA) - Gombor–Palla’16 2/19

Notation and definitions Throughout, ± will distinguish the orthogonal and symplectic cases. � with upper sign . so 2 n g 2 n = sp 2 n with lower sign . The gl n invariant R -matrix (Yang’68), R ( u ) := I − P u ∈ End ( C n ⊗ C n ) . The g 2 n invariant R -matrix (Zamolodchikov’78), R ( u ) := I − P Q κ − u ∈ End ( C 2 n ⊗ C 2 n ) , u − where Q = P t , Q 2 = 2 nQ and κ = n ∓ 1. 3/19

The orthogonal/symplectic open spin chain The state space of the chain is given by, M = L 1 ( λ 1 ) ⊗ · · · ⊗ L ℓ ( λ ℓ ) ⊗ M ℓ +1 ( µ ) . Each L i ( λ i ) is a highest weight g 2 n module of weight  ( k i , 0 , . . . , 0 ) for so 2 n ,    � �� �  n − 1 λ i = (1 , . . . , 1 , 0 , . . . , 0 ) for sp 2 n .    � �� � � �� �  n − k i k i M ℓ +1 ( µ ) is a one-dimensional vector space corresponding to one of two distinct diagonal boundary types  diag(1 , . . . , 1 , − 1 , . . . , − 1 , 1 , . . . , 1 ) g 2 p ⊕ g 2 q    � �� � � �� � � �� �  p 2 q p K = diag(1 , . . . , 1 , − 1 , . . . , − 1 ) gl n    � �� � � �� �  n n 4/19

The monodromy matrix The double-row monodromy matrix S ( u ) ∈ End ( C 2 n ⊗ M ) is S a ( u ) ≡ L a 1 ( u ) · · · L a ℓ ( u ) K a ( u ) L t a ℓ ( κ − u ) · · · L t a 1 ( κ − u ) Lax operators L ai ( u ) ∈ End ( C 2 n ⊗ L i ( λ i )) are constructed via fusion and satisfy R ab ( u − v ) L ai ( u ) L bi ( v ) = L bi ( v ) L ai ( u ) R ab ( u − v ) . Boundary Lax operator K ( u ) ∈ End ( C 2 n ) is a diagonal matrix . The monodromy matrix S ( u ) satisfies the reflection equation R ab ( u − v ) S a ( u ) R ab ( u + v ) S b ( v ) = S b ( v ) R ab ( u + v ) S b ( u ) R ab ( u − v ) . Problem Diagonalise τ ( u ) := tr S ( u ) on the spin chain M . 5/19

The monodromy matrix The double-row monodromy matrix S ( u ) ∈ End ( C 2 n ⊗ M ) is S a ( u ) ≡ L a 1 ( u ) · · · L a ℓ ( u ) K a ( u ) L t a ℓ ( κ − u ) · · · L t a 1 ( κ − u ) Lax operators L ai ( u ) ∈ End ( C 2 n ⊗ L i ( λ i )) are constructed via fusion and satisfy R ab ( u − v ) L ai ( u ) L bi ( v ) = L bi ( v ) L ai ( u ) R ab ( u − v ) . Boundary Lax operator K ( u ) ∈ End ( C 2 n ) is a diagonal matrix . The monodromy matrix S ( u ) satisfies the reflection equation R ab ( u − v ) S a ( u ) R ab ( u + v ) S b ( v ) = S b ( v ) R ab ( u + v ) S b ( u ) R ab ( u − v ) . Problem Diagonalise τ ( u ) := tr S ( u ) on the spin chain M . 5/19

The symmetry relation The entries of S ( u ) are not algebraically independent, which is in part summarised by the symmetry relation (Guay–Regelskis’16), S t ( u ) = γ S ( κ − u ) ± S ( u ) − S ( κ − u ) + tr( K ( u )) S ( u ) − tr( S ( u )) . 2 u − κ 2 u − 2 κ where γ = +1 or γ = − 1, depending on the boundary type. Multiplication by a certain scalar factor S ( u ) = g ( u ) S ( u ) leads to a “boundary independent” symmetry relation � � 1 S ( κ − u ) ± S ( u ) 2 u − κ − tr S ( u ) S t ( u ) = − 1 ± 2 u − 2 κ. 2 u − κ 6/19

The symmetry relation The entries of S ( u ) are not algebraically independent, which is in part summarised by the symmetry relation (Guay–Regelskis’16), S t ( u ) = γ S ( κ − u ) ± S ( u ) − S ( κ − u ) + tr( K ( u )) S ( u ) − tr( S ( u )) . 2 u − κ 2 u − 2 κ where γ = +1 or γ = − 1, depending on the boundary type. Multiplication by a certain scalar factor S ( u ) = g ( u ) S ( u ) leads to a “boundary independent” symmetry relation � � 1 S ( κ − u ) ± S ( u ) 2 u − κ − tr S ( u ) S t ( u ) = − 1 ± 2 u − 2 κ. 2 u − κ 6/19

Nesting procedure - gl n open spin chain In the gl n case,   a ( u ) B ( u )   S ( gl n ) ( u ) =  .   C ( u ) D ( u )  As an ( n − 1) × ( n − 1) matrix of operators, D ( u ) satisfies R ′ ab ( u − v ) D a ( u ) R ′ ab ( u + v ) D b ( v ) = D b ( v ) R ′ ab ( u + v ) D a ( u ) R ′ ab ( u − v ) . Creation operators B a i ( u i ) give rise to the Bethe vector: Φ( u ) = B a 1 ( u 1 ) · · · B a m ( u m ) · Φ ′ a 1 ,..., a m . where u = ( u 1 , . . . , u m ) and a 1 , . . . , a m label auxiliary spaces, each being a copy of C n , and Φ ′ a 1 ,..., a m is a “nested” Bethe vector for the residual gl n − 1 open spin chain (Belliard–Ragoucy’09). 7/19

Nesting procedure - g 2 n open spin chain For g 2 n , we split the matrix S ( u ) into four n × n submatrices:   A ( u ) B ( u )       S ( u ) =       C ( u ) D ( u )   As an n × n matrix of operators, A ( u ) satisfies R ab ( u − v ) A a ( u ) R ab ( u + v ) A b ( v ) = A b ( v ) R ab ( u + v ) A a ( u ) R ab ( u − v ) + R ab ( u − v ) B a ( u ) U ab ( u + v ) C b ( v ) + B b ( v ) U ab ( u + v ) C a ( u ) R ab ( u − v ) , with the gl n -invariant R -matrix R ( u ), and U ( u ) := − P / u − Q / ( κ − u ). 8/19

Top level of nesting - creation operator B ( u ) matrix contains creation operators for the top-level excitations, that correspond to n th root vectors of g 2 n . We reinterpret B ( u ) as a row vector in two auxiliary spaces, n � j ∈ B ( u ) ⊗ ( C n ) ∗ ⊗ ( C n ) ∗ b n − i +1 , j ( u ) ⊗ e ∗ i ⊗ e ∗ β ˜ aa ( u ) := i , j =1 Bethe vector with m top-level excitations, � m 1 � � � Ψ( u ) = β ˜ a i a i ( u i ) R a j ˜ a i ( − u i − u j ) · Φ ˜ a m a m . a 1 a 1 ,... ˜ i =1 j = i − 1 a m a m ∈ ( C n ) ⊗ 2 m ⊗ M . where u = ( u 1 , . . . , u m ) and Φ ˜ a 1 a 1 ,... ˜ 9/19

Symmetry relation in block form In block form, the symmetry relation gives a linear relation between the A and D blocks of S ( u ), � � � tr A ( u ) � u 1 A ( κ − u ) ± A ( u ) D t ( u ) = − 1 ± , 2 u − κ − 2 u − κ 2 u − κ where brace brackets denote symmetrisation � u := f ( u ) + f ( κ − u ) . � f ( u ) In particular, the rescaled transfer matrix may be written in terms of the block A of S ( u ) only: � � u τ ( u ) := tr S ( u ) = 2 u − 2 κ p ( u ) tr A ( u ) . g ( u ) where p ( u ) = 1 / (2 u − κ ). 10/19

Exchange relation Using the symmetry relation, the AB exchange relation may be written � � v β ˜ � � v p ( v ) S ′ p ( v ) A a ( v ) a 1 a 1 ( u ) = β ˜ a 1 a 1 ( u ) a 1 a 1 ( v ; u ) a ;˜ � � v 1 p ( v ) β ˜ a 1 a 1 ( v ) � � w , p ( w ) S ′ + a 1 a 1 ( w ; u ) Res a ;˜ p ( u ) u − v w → u where S ′ a 1 a 1 ( v ; u ) = R t a 1 a ( u − v ) R t a 1 a ( κ − u − v ) A a ( v ) a ;˜ ˜ × R t a 1 a ( u − v ± 1) R t a 1 a ( κ − u − v ± 1) . ˜ 11/19

Exchange relation for multiple excitations Acting with the transfer matrix on the Bethe vector we find � m 1 � � � τ ( v ) · Ψ( u ) = β ˜ a i a i ( u i ) R a j ˜ a i ( − u i − u j ) i =1 j = i − 1 � v · Φ ˜ � p ( v ) tr S ′ a ( v ; u ) a m a m + UWT , × a 1 a 1 ,... ˜ where S ′ a ( v ; u ) is the nested monodromy matrix , m m � � S ′ R t R t a ( v ; u ) := a i a ( u i − v ) a i a ( κ − u i − v ) ˜ i =1 i =1 1 1 � � R t R t × A a ( v ) a i a ( u i − v ± 1) a i a ( κ − u i − v ± 1) ˜ i = m i = m and UWT stands for the unwanted terms. 12/19