B"acklund Transformations as exact integrable time-discretizations for the trigonometric Gaudin model

B¨ ac klund T ransformations as exact in teg r abl e time-discr e tizations for the trigono metric Gaudin mo de l Orlando Ragnisco, F ederico Zullo Dipartimen to di Fis ica, Univ ersit` a di Roma T re Istituto Nazionale di F isica Nucleare, sezione di Roma T re Via V asca Na v ale 84, 00146 Roma , Italy E-mail: ragnisco@fis.uniroma3.it, zullo@fis.uniroma3.it Abstract W e construct a t wo -parameter f amily of B¨ ac klund transformations for the trigono- metric classical Gaud in magnet. The approac h follo w s closely the one intro d uced b y E.Skly anin and V.Kuznetso v (1998,19 99) in a num b er of seminal pap ers, and tak es adv an tage of th e int imate relation b et ween the trigonometric and th e ra- tional case. As in the pap er b y A.Hone, V.Kuznetso v and one of th e authors (O.R.) (2001) the B¨ ac klund transformations are presente d as explicit s y m plectic maps, starting from their Lax represen tation. The (exp ected) connection with the xxz Heisen b erg chai n is established and the r ational ( xxx ) case is reco v ered in a suitable limit. It is sh o wn ho w to obtain a “ph ysical” transformation mapping real v ariables in to r eal v ariables. The in terp olating Hamiltonian flow is d eriv ed and some numerical iterations of the m ap are pr esen ted. KEYW ORDS: B¨ a cklund T ransformations, Integrable maps, Gaudin systems, L a x rep- resen tation, r -matrix. 1 1 In tro ducti on B¨ ac klund transformations are now ada ys a widespread useful to ol related to the theory of nonlinear differen tia l equations. The first historical evidence of their mathematical significance w as giv en by Bianchi [3] and B¨ ac klund [2] on their works o n surfaces of con- stan t curv ature. A simple approach to understand their imp ortance can b e to regard them a s a mec hanism allowing to endo w a giv en nonlinear differen tial equation with a nonlinear sup erp o sition principle yielding a set o f solutions throug h a merely algeb r a ic pr o c e dur e [1 9],[1],[12]. B¨ ac klund transformatio ns are indeed parametric families o f dif- ference equations enco ding the whole set of symmetries of a giv en in tegrable dynamical system. F or finite-dimensional in tegrable systems the techniq ue of B¨ ac klund transfor- mations leads to the construction o f in tegrable P oisson maps that discretize a f a mily of con tinu o us flo ws [27],[25],[24],[22],[10 ],[9]. Actually in the last tw o decades n umerous results ha ve app eared in the field of exact discretization of many - b o dy in tegrable sys- tems emplo ying the B¨ ac klund transformations t o ols [17 ],[24],[16],[9],[10],[15],[22 ]. F or the r ational G audin mo del suc h discretization has b een obtained ten y ears ago in [8]; afterw ards, these results ha ve b een used for constructing an in tegrable discretization of classical dynamical systems (as the Lag r a nge top) connected to G audin mo del through In¨ on u-Wigner contractions [13],[11],[14]. The aim of the presen t work is to construct B¨ ack lund transformations f o r the Gaudin mo del in the partially anisotropic ( xxz ) case, i.e. for t he trigonometric Gaudin mo del. W e p oin t out that partial results on this issue ha v e already b een giv en in [1 8]. The pap er is org anized as follows . In Section (2) we review the main features of the trig onometric G a udin mo del from the p oin t o f view of its integrabilit y structure. F or the sak e of completeness , in Section (3) w e briefly recall the preliminary r esults on B¨ acklund T ransformations (BTs) for trigonometric Gaudin g iv en in [18]. In Section (4) the e xplicit form of BTs is g iv en; it is sho wn that they a re indeed a trigonometric generalization of the r a tional ones (see [8]) whic h can be reco v ered in a suitable (“small a ngle” ) limit. The simple cticity of the transformations is also discussed in the same Section and the pro of allows us to elucidate the (expected) link b et w een the Darb oux-dressing matrix and the elemen tary Lax matrix for the xxz Heisen b erg magnet o n the la t tice. W e end the Section by men tioning an op en question, namely the construction of an explicit generating function for these B¨ ac klund transformat ions. In Section (5) w e will sho w how our map can lead, with an appropria te c hoice of B¨ ac klund parameters, to ph ysical transformations, i.e. transformations from real v ariables to real v ariables. In the last Section w e sho w ho w a suitable con tinuous limit yields the in terp olating Hamilto nian flow and finally presen t n umerical examples of iteration of the map. 2 Gaudin magnet in the trigono met ric case F or a full account of the in tegrabilit y structure of the classical and quan tum Gaudin mo del we refer the reader to the fundamen tal contributions by Semeno v-Tian-Shanski [26] a nd Bab elon-Bernard-T alon [4]. In this section w e briefly recall t he main features 2 of the trigonometric G audin magnet. The Lax matrix of the mo del is giv en b y the express ion: L ( λ ) =  A ( λ ) B ( λ ) C ( λ ) − A ( λ )  (1) A ( λ ) = N X j =1 cot( λ − λ j ) s 3 j , B ( λ ) = N X j =1 s − j sin( λ − λ j ) , C ( λ ) = N X j =1 s + j sin( λ − λ j ) . (2) In (1) and (2) λ ∈ C is the sp ectral pa r a meter, λ j are arbitrary real parameters o f the mo del, while  s + j , s − j , s 3 j  , j = 1 , . . . , N , are the dynamical v a riables of the system ob eying to ⊕ N sl ( 2) algebra, i.e.  s 3 j , s ± k  = ∓ iδ j k s ± k ,  s + j , s − k  = − 2 iδ j k s 3 k , (3) By fixing the N Casimirs  s 3 j  2 + s + j s − j . = s 2 j one obtains a symplectic manifold giv en b y the direct sum of the corresp onden t N t wo-spheres . Reform ulating the Poiss on structure in terms of the r -matrix formalism amoun ts to state t hat the Lax matrix satisfies the line ar r -ma t r ix Poiss o n alg ebra (see aga in [26], [4]) :  L ( λ ) ⊗ 1 , 1 ⊗ L ( µ )  =  r t ( λ − µ ) , L ( λ ) ⊗ 1 + 1 ⊗ L ( µ )  , (4) where r t ( λ ) stands for the tr igonometric r matrix [5]: r t ( λ ) = i sin( λ )     cos( λ ) 0 0 0 0 0 1 0 0 1 0 0 0 0 0 cos( λ )     , (5) Equation (4) en tails the following Poiss on brack ets for the functions (2): { A ( λ ) , A ( µ ) } = { B ( λ ) , B ( µ ) } = { C ( λ ) , C ( µ ) } = 0 , { A ( λ ) , B ( µ ) } = i cos( λ − µ ) B ( µ ) − B ( λ ) sin( λ − µ ) , { A ( λ ) , C ( µ ) } = i C ( λ ) − cos( λ − µ ) C ( µ ) sin( λ − µ ) , { B ( λ ) , C ( µ ) } = i 2( A ( µ ) − A ( λ )) sin( λ − µ ) . (6) The determinant of the Lax matrix is the generating function of the integrals of motio n: − det( L ) = A 2 ( λ ) + B ( λ ) C ( λ ) = N X i =1  s 2 i sin 2 ( λ − λ i ) + H i cot( λ − λ i )  − H 2 0 (7) 3 where the N Hamiltonians H i are of the for m: H i = N X k 6 = i 2 cos( λ i − λ k ) s 3 i s 3 k + s + i s − k + s − i s + k sin( λ i − λ k ) (8) Note that only N − 1 among these Hamiltonians are indep endent, b ecause of P i H i = 0. Another in tegra l is giv en by H 0 , the pro j ection of the total spin on t he z axis: H 0 = N X j =1 s 3 j . = J 3 (9) The Hamiltonians H i are in in volution for the P oisson brac ke t ( 3): { H i , H j } = 0 i, j = 0 , . . . , N − 1 (10) The corresp onding Ha milto nian flow s a r e then giv en b y: ds 3 j dt i = { H i , s 3 j } ds ± j dt i = { H i , s ± j } (11) In the xxx model a remark able Hamiltonian is f o und by taking a linear combination of the in tegrals corresp onding to (8) in the ra tional case [6]. It describ es a mean field spin-spin in tera ctio n: H r = 1 2 N X i 6 = j s i · s j Where the notation for the b old sym bo l s i is s i = ( s 1 i , s 2 i , s 3 i ) with s + i = s 1 i + is 2 i and s − i = s 1 i − is 2 i . The na t ur a l tr ig onometric g eneralization of this Hamiltonia n can b e found b y taking the linear combination of 8: N X i =1 sin(2 λ i ) 2 H i giving H t = 1 2 N X i 6 = j cos( λ i + λ j )  s 1 i s 1 j + s 2 i s 2 j + cos( λ i − λ j ) s 3 i s 3 j  (12) 3 A first approac h to Darb oux-dres s ing matrix In this Section, for the sake of completeness , w e recall t he results already app eared in [18]. The leading observ ation is that by p erforming the “unifo rmization” mapping: λ → z . = e iλ the N -sites trigono metric Lax ma t r ix tak es a rational form in z that corresponds to the 2 N -sites rational L a x mat r ix plus a n additional reflection symmetry (see also [7]); in fact, b y p erforming the substitution (3), the Lax matr ix (1) b ecomes: 4 L ( z ) = iJ 3 + N X j =1 L j 1 z − z j − σ 3 L j 1 z + z j σ 3 ! , (13) where σ 3 is the P auli matrix diag (1 , − 1) and the matrices L j 1 , j = 1 , . . . , N , are give n b y: L j 1 = iz j s 3 j s − j s + j − s 3 j ! So, equation (13) entails the follo wing inv olution on L ( z ): L ( z ) = σ 3 L ( − z ) σ 3 (14) Constructing a B¨ ac klund transformation for the T rigonometric Gaudin System (TGS) amoun ts to build up a P oisson map for t he field v a riables of the mo del (2) suc h t ha t the in tegrals of motion (8) are preserv ed. A t the lev el of Lax matrices, this transformat ion is usually seek ed as a similarit y transformation b etw een an old , or “undressed”, Lax matrix L , and a ne w , or “dressed” one, say ˜ L : L ( z ) → D ( z ) L ( z ) D − 1 ( z ) ≡ ˜ L ( z ) (15) But L and ˜ L hav e to enjoy the same reflection symmetry (14 ) t o o: to preserv e this in v olutio n the D arb oux dressing mat r ix D has to share with L the prop ert y (14); the elemen tary dressing matrix D is then obtained by requiring the exis tence of only one pair of opp osite p oles for D in the complex pla ne of the sp ectral parameter. W e will sho w in the next Section that, thanks t o this constrain t, one recov ers the form of the Lax matrix for the elemen tary xxz Heisen b erg spin c hain: on the other hand, this is quite natura l if one recalls that for the ra t io nal Ga udin mo del the elemen tary Darb oux-dressing matrix is giv en b y the Lax matrix for the elemen tary xxx Heisen b erg spin c hain [8],[10]. The previous observ ations lead to the follo wing D arb oux matrix: D ( z ) = D ∞ + D 1 z − ξ − σ 3 D 1 z + ξ σ 3 (16) By taking the limit z → ∞ in (16) it is readily seen t ha t D ∞ has to b e a diagonal matrix. In order to ensure that L and ˜ L ha v e t he same rational structure in z , we rewrite equation (15) in the form: ˜ L ( z ) D ( z ) = D ( z ) L ( z ) (17) No w it is clear that b oth sides ha v e the same residues at the po les z = z j , z = ξ j (it is unneces sary to lo o k at the p oles in z = − z j and z = − ξ j b ecause of the symmetry (14), so that the fo llo wing set o f equations ha ve to b e satisfied: ˜ L ( j ) 1 D ( z j ) = D ( z j ) L ( j ) 1 , (18) ˜ L ( ξ ) D 1 = D 1 L ( ξ ) . (19) 5 In principle, equations (18), (19) yield a Darb oux matrix depending b oth on the old (un tilded) v ariables and t he new (tilded) ones, implying in turn an implicit relationship b et we en the same v ariables. T o g et an explicit relationship one has to resort to the so- called sp ectralit y prop ert y [10] [9]. T o this aim w e need to force the determinan t of the Darb oux matrix D ( z ) to hav e, b esides the pa ir o f p o les at z = ± ξ , a pair of opp osite nondynamic al zero es, say a t z = ± η , and to allow the matrix D 1 to b e prop ortio nal t o a pro jector [18]. Again b y symmetry it suffices to consider just o ne of these zero es. If η is a zero of det D ( z ), then D ( η ) is a rank one matrix, p o ssessin g a one dimensional k ernel | K ( η ) i ; the equation (17) : ˜ L ( η ) D ( η ) = D ( η ) L ( η ) (20) en tails D ( η ) L ( η ) | K ( η ) i = 0 . (21) This equation in turn a llo ws to infer that | K ( η ) i is an eigen vec to r for the Lax matrix L ( η ): L ( η ) | K ( η ) i = µ ( η ) | K ( η ) i , (22) This relat io ns giv es a direct link b et w een the pa r a meters a pp earing in the dressing matrix D and the old dynamical v aria bles in L . Because of (19) we ha v e another one dimensional k ernel | K ( ξ ) i of D 1 , suc h that: L ( ξ ) | K ( ξ ) i = µ ( ξ ) | K ( ξ ) i . (23) In [1 8] w e hav e sho wn how the t w o sp ectrality conditions (22), (23) enable to write D in terms of the o ld dynamical v ariables and of the tw o B¨ a c klund parameters ξ and η . The explicit expression of the Darb oux dressing matrix is giv en by: D ( z ) = β z z 2 − ξ 2   z ( p ( η ) η − p ( ξ ) ξ ) b + ( p ( ξ ) η − p ( η ) ξ ) η ξ bz ξ 2 − η 2 b bp ( ξ ) p ( η ) ( ξ 2 − η 2 ) ηξ b ( p ( η ) η − p ( ξ ) ξ ) z + bz ( p ( ξ ) η − p ( η ) ξ ) ηξ   . (24) In this expression β is a global multiplicativ e fa ctor, inessen tial with resp ect to the form of the BT, b is an undeterminate parameter that in Section (4) w e will fix in order to reco ve r the form of the Lax matrix for the discrete xxz Heisen b erg spin c hain. The functions p ( η ) and p ( ξ ) c haracterize completely the k ernels of D ( η ) and D ( ξ ): in fact w e hav e the fo llowing formulas [18]: | K ( ξ ) i =  1 p ( ξ )  | K ( η ) i =  1 p ( η )  (25) As | K ( ξ ) i and | K ( η ) i are resp ectiv ely eigen vec to rs of L ( ξ ) and L ( η ), p ( ξ ) and p ( η ) m ust satisfy: p ( ξ ) = µ ( ξ ) − A ( ξ ) B ( ξ ) , p ( η ) = µ ( η ) − A ( η ) B ( η ) (26) with A ( z ), B ( z ), C ( z ) g iv en by (2 ) and µ 2 ( z ) = A 2 ( z ) + B ( z ) C ( z ). 6 4 Explicit map and an e qu iv alen t appro ac h to Darb oux- dressin g m atr i x The matrix (24) con tains just one set of dynamical v ariables so that the relation (15) giv es now an explicit map b etw een the v a riables  ˜ s + j , ˜ s − j , ˜ s 3 j  and  s + j , s − j , s 3 j  . The map is easily found b y (18); it reads: ˜ s 3 k = p ( ξ ) p ( η ) ( ξ 2 − η 2 ) (( z k 2 − η 2 ) p ( ξ ) ξ − ( z k 2 − ξ 2 ) p ( η ) η ) s − k z k ∆ k + ( ξ 2 − η 2 ) (( z k 2 − ξ 2 ) p ( ξ ) η − p ( η ) ξ ( z k 2 − η 2 )) s + k z k ∆ k + s 3 k h p ( ξ ) p ( η ) (( ξ 2 + z k 2 ) ( η 2 + z k 2 ) − ( η 2 + ξ 2 ) − 8 η 2 ξ 2 z k 2 ) ∆ k + −  η ξ ( ξ 2 − z k 2 ) ( η 2 − z k 2 )  p ( ξ ) 2 + p ( η ) 2  i ∆ k (27a) ˜ s + k = − b 2 p ( ξ ) 2 p ( η ) 2 ( η 2 − ξ 2 ) 2 s − k z 2 k ξ η ∆ k + b 2 (( z k 2 − ξ 2 ) p ( ξ ) η − p ( η ) ξ ( z k 2 − η 2 )) 2 s + k η ξ ∆ k + 2 b 2 p ( ξ ) p ( η ) ( ξ 2 − η 2 ) (( z k 2 − ξ 2 ) p ( ξ ) η − p ( η ) ξ ( z k 2 − η 2 )) s 3 k z k η ξ ∆ k (27b) ˜ s − k = − ( η 2 − ξ 2 ) 2 s + k z 2 k ξ η b 2 ∆ k + (( z k 2 − η 2 ) p ( ξ ) ξ − ( z k 2 − ξ 2 ) p ( η ) η ) 2 s − k ξ η b 2 ∆ k + 2 ( ξ 2 − η 2 ) (( z k 2 − η 2 ) p ( ξ ) ξ − ( z k 2 − ξ 2 ) p ( η ) η ) s 3 k z k ξ η b 2 ∆ k (27c) where ∆ k is prop ortio na l to the determinan t of D ( z k ), i.e. ∆ k = ( z 2 k − ξ 2 )( z 2 k − η 2 )( p ( ξ ) η − p ( η ) ξ )( p ( η ) η − p ( ξ ) ξ ) (28) F orm ulas ( 2 7a), (27b), (27 c) define a tw o-parameter B¨ ac klund transformation, the parameters b eing ξ and η : as we will sho w in the next section, it is a crucial p oin t to ha v e a two -parameter family of tr ansformations when lo oking f or a phy sical ma p from real v ariables to real v ariables. As men tioned in the previous Section, we no w sho w that indeed, b y p osing: b = i p η ξ (29) the expression (24) of the dressing matrix go es in to the express io n of the elemen tary Lax matrix for the classical, partially anisotropic, Heisen b erg spin c hain on the lat t ice [5]. 7 Ob viously tw o matrices differing only for a glo bal multiplicativ e factor giv e rise to t he same similarit y transformation. So we omit the term β z z 2 − ξ 2 in (24), and, taking into accoun t (29 ), w e write for the diagonal part D d of (24): D d = i 2  ( p ( ξ ) − p ( η ))( v − w )1 + ( p ( ξ ) + p ( η ))( v + w ) σ 3  (30) where v ( ξ , η ) and w ( ξ , η ) are give n b y: v ( ξ , η ) = z ξ √ η ξ − η √ η ξ z w ( ξ , η ) = ξ √ η ξ z − z η √ η ξ = − v ( η , ξ ) (31) W e substitute: ξ → e iζ 1 η → e iζ 2 z → e iλ (32) and tak e a suitable redefinition of the B¨ ack lund para meters to clarify the structure of the D matrix: λ 0 . = ζ 1 + ζ 2 2 µ . = ζ 1 − ζ 2 2 (33) With these p ositions it is simple to find that v − w = 4 ie iλ 0 sin( λ − λ 0 ) cos( µ ) and v + w = 4 ie iλ 0 cos( λ − λ 0 ) sin( µ ). So, considering equation (30) jointly with the o ff- diagonal part of (24), the dr essing matrix can b e written as: D ( λ ) = α h sin( λ − λ 0 )1 + p ( ζ 1 ) + p ( ζ 2 ) p ( ζ 1 ) − p ( ζ 2 ) tan( µ ) cos( λ − λ 0 ) σ 3 + + 2 sin( µ ) p ( ζ 2 ) − p ( ζ 1 )  0 1 − p ( ζ 1 ) p ( ζ 2 ) 0  i (34) where α is the globa l factor 2 e iλ 0 ( p ( ζ 2 ) − p ( ζ 1 )). Observ e that in formula (34), with some abuse of notatio n, p ( ζ 1 ) ( p ( ζ 2 )) stands of course for p ( ξ ) | ξ = e iζ 1  p ( η ) | η = e iζ 2  . A last c hange of v ariables allo ws t o iden tify the dressing ma t rix with the elemen ta ry Lax matr ix of the classical xxz Heisen b erg spin c hain o n the lattice, and furthermore to recov er the form of the Da r b oux matrix for the r ational Gaudin mo del [8][20] in the limit of smal l angles . Namely , w e in tro duce t wo new functions, P and Q , by letting p ( ζ 1 ) = − Q p ( ζ 2 ) = 2 sin( µ ) P − Q. (35) Then equation (34) b ecomes: D ( λ ) = α  sin( λ − λ 0 − µ ) + P Q cos( λ − λ 0 ) P cos( µ ) Q sin( 2 µ ) − P Q 2 cos( µ ) sin( λ − λ 0 + µ ) − P Q cos( λ − λ 0 )  (36) Ob viously no w w e can rep eat the arg umen t made b efo r e ab o ut sp ectrality ; indeed no w D   λ = λ 0 + µ and D   λ = λ 0 − µ are rank one mat r ices. So if Ω + and Ω − are resp ectiv ely the 8 k ernels of D ( λ 0 + µ ) and D ( λ 0 − µ ) one has a gain tha t Ω + and Ω − are eigen vec t o rs of L ( λ 0 + µ ) and L ( λ 0 − µ ) with eigen v alues γ + and γ − where γ ± = γ ( λ )    λ = λ 0 ± µ and w e hav e set (7) γ 2 ( λ ) ≡ A 2 ( λ ) + B ( λ ) C ( λ ) = − det( L ( λ )) (37) The t w o k ernels are giv en by: Ω + =  1 − Q  Ω − =  P 2 sin( µ ) − P Q  (38) and the eigen v ectors relations yields the following expression of P a nd Q in terms of the old v a riables only: Q = Q ( λ 0 + µ ) = A ( λ ) − γ ( λ ) B ( λ )    λ = λ 0 + µ 1 P = Q ( λ 0 + µ ) − Q ( λ 0 − µ ) 2 sin( µ ) (39) The explicit map can b e found b y equating the residues at the poles λ = λ k in ( 1 7), that is b y the relation: ˜ L k D k = D k L k (40) where L k =  s 3 k s − k s + k − s 3 k  , D k = D ( λ = λ k ) (41) or by p erforming the needed c hanges of v a r iables in (27a), (27b), (27c). An yw ay no w the map reads: ˜ s 3 k = 2 cos 2 ( µ ) − (cos 2 ( µ ) + cos 2 ( δ k 0 ))(1 − 2 P Q sin( µ ) + P 2 Q 2 ) ∆ k s 3 k + + P cos( µ )(sin( δ k + ) − P Q cos( δ k 0 )) ∆ k s + k + − Q cos( µ )(2 sin( µ ) − P Q )(sin( δ k − ) + P Q cos( δ k 0 )) ∆ k s − k (42a) ˜ s + k = (sin( δ k + ) − P Q cos( δ k 0 )) 2 ∆ k s + k − ( Q 2 cos 2 ( µ )(2 sin ( µ ) − P Q )) 2 ∆ k s − k + + 2 Q cos( µ )(2 sin( µ ) − P Q )(sin( δ k + ) − P Q cos( δ k 0 )) ∆ k s 3 k (42b) ˜ s − k = (sin( δ k − ) + P Q cos( δ k 0 )) 2 ∆ k s − k − P 2 cos 2 ( µ ) ∆ k s + k + − 2 P cos( µ )(sin( δ k − ) + P Q cos( δ k 0 )) ∆ k s 3 k (42c) 9 where for ty p esetting brevit y w e ha v e put:  δ k 0 = λ k − λ 0 δ k ± = λ k − λ 0 ± µ (43) and w e hav e denoted by ∆ k the determinan t of D ( λ k ), that is: ∆ k := sin( λ k − λ 0 − µ ) sin( λ k − λ 0 + µ )(1 − 2 P Q sin( µ ) + P 2 Q 2 ) A t this p oin t w e can sho w that for “small” λ 0 and µ one obtains, at first order, the B¨ ac klund for the rational G audin mo del, indep endently found b y Skly anin [20] o n one hand and Hone, Kuznetsov and R agnisco [8] on the other, as the comp osition of t wo one-parameter B¨ acklun ds. So let us tak e λ 0 → hλ 0 , µ → hµ and λ → hλ where h is the expansion parameter. O ne has: cot( λ − λ k ) = 1 h ( λ − λ k ) + O ( h ) 1 sin( λ − λ k ) = 1 h ( λ − λ k ) + O ( h ) , so that Q = q r + O ( h 2 ), where the sup erscript r stands for “rat io nal” . Thus , q r coincides with the v aria ble q that one finds in the rational case [8]. F or the v ariable P one has: P = h ( p r + O ( h 2 )) where p r = 2 µ q r ( λ 0 + µ ) − q r ( λ 0 − µ ) T aking into a ccoun t t hese express ions, it is straigh tforward to see that the matrix (36) has the expansion: D ( λ ) = hD r ( λ ) + O ( h 3 ) (44) where D r ( λ ) =  λ − λ 0 − µ + p r q r p r q r (2 µ − p r q r ) λ − λ 0 + µ − p r q r  . (45) The limit o f “small angles” in (27 a), (27 b), (27c) ob viously leads to the rational map of [8]. 4.1 Symplecticit y In this subsection we face the question of t he simplecticit y of our map; the corre- sp ondence with the rational B¨ ac klund in the limit of “small angles” sho ws that the transformations are surely canonical in this limit. Indeed, as our map is explicit, we could chec k by brute-fo ce calculations whether the P oisson structure ( 3) is preserv ed b y tilded v ariables. How ev er w e will f ollo w a finer argumen t due to Skly anin [21]. Supp ose that D ( λ ) ob eys the quadr atic P oisson brac k et, that is { D 1 ( λ ) , D 2 ( τ ) } = [ r t ( λ − τ ) , D 1 ( λ ) ⊗ D 2 ( τ )] (46) where as usually D 1 = D ⊗ 1 , D 2 = 1 ⊗ D . Consider the relatio n ˜ L ( λ ) ˜ D ( λ − λ 0 ) = D ( λ − λ 0 ) L ( λ ) (47) 10 in an extended phase space, where the entries of D P oisson comm utes with t ho se o f L . Note that in (4 7) w e hav e used tilded v ariables also for D ( λ ) ( in its l.h.s.) b ecause (47) is indeed the B¨ ac klund transformatio n in this extended phase space, whose co ordinates are ( s 3 j , s ± j , P , Q ), so that w e ha v e also a ˜ P and a ˜ Q . The k ey observ ation is that if b oth L a nd D hav e t he same Poiss o n structure, given b y equation ( 46), then this prop erty holds true for LD and D L as w ell, b ecause in this extended space the en tries of D Poiss on commute with the entries of L . This means that the transformation (47) defines a “cano nical” tra nsfor mat ion. Skly anin sho w ed [21] tha t if one now restricts the v ar ia bles on the constraint manifold ˜ P = P and ˜ Q = Q the symplecticit y is preserv ed; ho w eve r this constrain t leads to a dep endence of P and Q o n the entries of L , that f o r consistency m ust b e the same as the one giv en by the equation (47) on this constrained manifold. But there (47) is j ust giv en by the usual BT: ˜ L ( λ ) D ( λ − λ 0 ) = D ( λ − λ 0 ) L ( λ ) so that the map preserv es the sp ectrum of L ( λ ) and is canonical. What remains to sho w is that indeed (4 6) is fullfilled b y our D ( λ ). Ob viously D ( λ ) cannot hav e t his P oisson structure for an y Poisson brac ke t b et w een P and Q . In the rational case the Darb oux matrix has the P oisson structure imp osed b y the rat io nal r - matrix provided P and Q are canonically conjugated in the extended space [21] (and this is wh y they w ere called P and Q ); in the trigo no metric case P and Q are no longer canonically conjugated but obv iously o ne recov ers this prop ert y at order h in the “small angle” limit. First note that D ( λ ) can b e con ve niently written a s: D ( λ ) = α cos( µ ) h sin( λ )1 + a cos( λ ) σ 3 +  0 b c 0  i (48) where the co efficien ts a, b, c are giv en by: a = P Q − sin( µ ) cos( µ ) , b = P , c = 2 Q sin( µ ) − P Q 2 (49) Inserting (48) in (46) we ha v e the following constrain ts: { α, αa } = 0 = ⇒ α = α ( P Q ) (50) { α, αb } = − α 2 ab = ⇒ { α, P } = αP sin( µ ) − P Q cos( µ ) (51) { α, αc } = α 2 ac = ⇒ { α, Q } = − αQ sin( µ ) − P Q cos( µ ) (52) All remaining relations, na mely { αb, α c } = 2 α 2 a { αa, αb } = α 2 b { αa, αc } = − α 2 c (53) giv e the same constrain t, i.e.: { Q, P } = 1 + P 2 Q 2 − 2 P Q sin( µ ) cos( µ ) (54) 11 This express io n can b e used to find, after a simple integration, α ( P Q ) = k p (1 + P 2 Q 2 − 2 P Q sin( µ )) so that the D arb oux mat rix (36 ) is fixed (up to the constan t multiplic at iv e fa ctor k ) . As previously p oin ted out, it turns out that the Darb oux-dressing matrix (36) is formally equiv alen t to the elemen tary Lax matrix for the classical xxz Heisen b erg spin c hain on the lattice [5]. Moreo v er it has also the same (quadratic) P oisson brac ke t . This suggests that indeed D ( λ ) can b e recast in the form (see [5]): D ( λ ) = S 0 1 + i sin( λ )  S 1 σ 1 + S 2 σ 2 + cos( λ ) S 3 σ 3  (55) where the σ i are the P auli matrices and t he v ariables S i satisfies the following Poiss o n brac k et ([5]): { S i , S 0 } = J j k S j S k { S i , S j } = − S 0 S k (56) where ( i, j, k ) is a cyc lic p erm utation of (1 , 2 , 3) and J j k is an tisymmetric with J 12 = 0 , J 13 = J 23 = 1. Indeed it is straigh tfo rw ard to sho w that the link b et w een the tw o represen tations (48) and ( 5 5), up to the f a ctor cos ( µ ) sin( λ ) t ha t do es not affect neither (17) nor the P oisson brac ke t (46), is g iven b y : α = S 0 − iα 2 ( b + c ) = S 1 α 2 ( b − c ) = S 2 − iaα = S 3 (57) and the P oisson brac k ets (50), (51), (5 2), (53) corresp ond to those giv en in (56). An op en question regards the generating f unction of our BT. So far we ha v e not b een able t o write it in an y closed form; in our opinion t he question is ha r der than in the rational case (where the generating function is kno wn fr om [8]): in fact the rational map corresp onding to (27a), ( 27b), (27c) can b e written as the comp osition of t wo simpler one - parameter B¨ ack lund transfor ma t io ns, and this en tails the same prop erty to hold for the generating function; in the trigonometric case a fa ctorization of the B¨ ac klund tra nsformations cannot preserv e the symmetry (14): so probably one should lo ok for symmetry-violating generating functions such that their comp o sition enables symmetry to b e restored. 5 Ph ysical B¨ ac klund transformatio ns The transformatio ns w e ha v e found do not map, in general, real v ariables into real v ar ia bles. A sufficien t conditio n to ensure this pro p ert y is giv en by: ζ 1 = ¯ ζ 2 (58) whic h a mo unts te r equire that λ 0 and µ in (42a), (42b), (42c) b e, resp ectiv ely , real and imaginary n umbers. 12 Indeed w e claim that, if (58) holds, starting from a physic al solution of the dynamical equations, w e can find a new ph ysical solution with t wo real parameters. Let us pro v e the assertion. B¨ ac klund transformation are obtained by (40 ); starting from a real solution means start ing from an Hermitian L k . Th us, if the transformed ma t r ix ˜ L k has to b e Hermitian to o, the Darb oux matrix has to b e prop ortional to a n unitary matrix. W e will sho w tha t t his is t he case by choosing ζ 1 = ¯ ζ 2 and γ ( ζ 1 ) = − ¯ γ ( ζ 2 ) ( γ is the function defined in (3 7)). Note t ha t the conditio n on the γ ’s specifies their relat ive sign (the sheet on the Riemann surface), inessen tial for the sp ectrality prop erty . Hereafter w e assume the par ameter µ , defined in (32), to b e purely imag ina ry = iǫ , so t hat: ζ 1 = λ 0 + iǫ ( λ 0 , ǫ ) ∈ R 2 (59) The Darb oux matrix at λ = λ k can b e rewritten as: D k =  sin( v k − iǫ ) + P Q cos( v k ) P cosh( ǫ ) Q cosh( ǫ ) (2 i sinh( ǫ ) − P Q ) sin( v k + iǫ ) − P Q cos( v k )  (60) where v k ≡ λ k − λ 0 (w e are assuming that the parameters λ k of the mo del are real). W e recall that in (60): Q = Q ( ζ 1 ) = A ( ζ 1 ) − γ ( ζ 1 ) B ( ζ 1 ) = − C ( ζ 1 ) A ( ζ 1 ) + γ ( ζ 1 ) ; P = 2 i sinh( ǫ ) Q ( ζ 1 ) − Q ( ¯ ζ 1 ) . (61) F urthermore it is a simple matt er to sho w that A ( ζ 1 ) = ¯ A ( ¯ ζ 1 ); B ( ζ 1 ) = ¯ C ( ¯ ζ 1 ); C ( ζ 1 ) = ¯ B ( ¯ ζ 1 ) . (62) If the o ff - diagonal terms of D k D † k has to b e zero, then the following equation has to b e fullfilled: P (sin( v k − iǫ ) − ¯ P ¯ Q cos( v k )) = ¯ Q (2 i sinh( ǫ ) + ¯ P ¯ Q )(sin( v k − iǫ ) + P Q cos( v k )) (63) Using relations (61) and rearranging the terms, the previous equation b ecomes: ( 1 ¯ Q ( ζ 1 ) − 1 ¯ Q ( ¯ ζ 1 ) ) cosh( ǫ ) sin( v k ) + i ( 1 ¯ Q ( ζ 1 ) + 1 ¯ Q ( ¯ ζ 1 ) ) cos( v k ) sinh( ǫ ) = = ( Q ( ζ 1 ) − Q ( ¯ ζ 1 )) cosh( ǫ ) sin( v k ) + i cos( v k ) sinh( ǫ )( Q ( ζ 1 ) − Q ( ¯ ζ 1 )) (64) Note that the relations (62) g iv es γ 2 ( ζ 1 ) = γ 2 ( ¯ ζ 1 ), implying tha t γ 2 ( λ ) is a real function of its complex argument, consisten tly with the expansion (7). The c hoice: γ ( ζ 1 ) = − ¯ γ ( ¯ ζ 1 ) (65) en tails: ¯ Q ( ζ 1 ) = − 1 Q ( ¯ ζ 1 ) (66) With this constrain t the equation (64) holds to o. Moreov er (65) mak es the diagonal terms in D k D † k equal. This sho ws that, under the giv en assumptions, D k is a n unitary matrix. 13 6 In terp olatin g Ha milt onian fl ow The B¨ ac klund transformat ion can b e seen as a time discretization of a one-parameter ( λ 0 ) family of hamiltonian flo ws with the difference i ( ¯ ζ 1 − ζ 1 ) = 2 ǫ playing the role of the time-step. T o clarify this p o in t, let us tak e the limit ǫ → 0. W e ha v e: Q = A ( λ 0 ) − γ ( λ 0 ) B ( λ 0 ) + O ( ǫ ) ≡ Q 0 + O ( ǫ ) (67) P = − iǫ B ( λ 0 ) γ ( λ 0 ) + O ( ǫ 2 ) ≡ iǫP 0 + O ( ǫ 2 ) (68) and for the dressing matrix we can write: D ( λ ) = k sin( λ − λ 0 )1 + + iǫk  cos( λ − λ 0 )( P 0 Q 0 − 1) P 0 Q 0 (2 − P 0 Q 0 ) cos( λ − λ 0 )(1 − P 0 Q 0 )  + O ( ǫ 2 ) (69) Reorganizing the terms with the help of P 0 and Q 0 giv en in the equations (67) and (68) w e arr ive at the expression: D ( λ ) = k sin( λ − λ 0 )1 + − iǫk γ ( λ 0 )  A ( λ 0 ) cos( λ − λ 0 ) B ( λ 0 ) C ( λ 0 ) − A ( λ 0 ) cos( λ − λ 0 )  + O ( ǫ 2 ) (70) It is no w straig h tforward to show that in the limit ǫ → 0 the equation o f the map ˜ LD = D L t urns in to the Lax equation for a contin uous flow : ˙ L ( λ ) = [ L ( λ ) , M ( λ, λ 0 )] (71) where the time deriv ativ e is defined as: ˙ L = lim ǫ → 0 ˜ L − L ǫ (72) and the matrix M ( λ, λ 0 ) has the form i γ ( λ 0 ) A ( λ 0 ) cot( λ − λ 0 ) B ( λ 0 ) sin( λ − λ 0 ) C ( λ 0 ) sin( λ − λ 0 ) − A ( λ 0 ) cot( λ − λ 0 ) ! (73) The system (71) can b e cast in Hamiltonian f o rm: ˙ L ( λ ) = {H ( λ 0 ) , L ( λ ) } (74) with the Hamilton’s function g iv en by: H ( λ 0 ) = γ ( λ 0 ) = p A 2 ( λ 0 ) + B ( λ 0 ) C ( λ 0 ) (75) Quite remark ably , but not surprisingly , the Hamiltonian (75) characterizin g the inter- p olating flo w is (the square ro ot of ) the generating function (7) o f the whole set of 14 conserv ed quan tities. By choosing the parameter λ 0 to b e equal to an y of the p oles ( λ i ) of t he Lax matrix, t he map leads to N differen t maps { B T ( i ) } i =1 ..N , where B T ( i ) discretizes the flow corresp onding to the Hamiltonian H i , given b y equation (8). An y other integrable map for the trig onometric Gaudin mo del can b e, in principle, written in terms of the N maps { B T ( i ) } i =1 ..N . More explicitely , by p osing λ 0 = δ + λ i and taking the limit δ → 0, the Hamilton’s function (75) give s: γ ( λ 0 ) = s i δ + H i 2 s i + O ( δ ) (76) and the equations of motion tak e the f o rm: ˙ L ( λ ) = 1 2 s i { H i , L ( λ ) } (77) Accordingly , the in terp ola ting flo w encompasses all the comm uting flo ws of the system, so tha t the B¨ ac klund t r a nsformations turn out to b e a n e xact time-discr etizations o f suc h interpola t ing flo w. 6.1 Numerics -3 -2 -1 -3 0 -3 -2 -2 -1 1 -1 0 0 1 2 1 2 3 2 3 3 Figure 1: input par a meters: s + 1 = 2 + i , s − 1 = 2 − i , s 3 1 = − 2, s + 2 = 50 + 40 i , s − 2 = 50 − 40 i , s 3 2 = 70, λ 1 = π / 110, λ 2 = 7 π / 3 , λ 0 = 0 . 1, µ = − 0 . 002 i The figur es rep ort an example of itera t ion o f the map (42a), (42b), (42c). F or simplicit y w e t ak e N = 2 . The computations sho ws the first 1500 iterations: the plotted v aria bles are the ph ysical ones ( s x 1 , s y 1 , s z 1 ). 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