hep-th 2012-07-30 0

Generalized Landau-Lifshitz models on the interval

We study the classical generalized gl(n) Landau-Lifshitz (L-L) model with special boundary conditions that preserve integrability. We explicitly derive the first non-trivial local integral of motion, which corresponds to the boundary Hamiltonian for …

Authors: Anastasia Doikou, Nikos Karaiskos

Generalized Landau-Lifshitz mo dels on the in terv al Anastasia Doik ou and Nik os Karaisk os Department of Enginee r ing Sciences, Universit y o f P atras, Physics Division GR-2650 0 Patras, Greece E-mail : { adoikou, nk araiskos } @ upatras.gr Abstract W e study the classical generalized gl n Landau-Lifshitz (L-L) mo del with special b oundary conditions that preserv e in tegrabilit y . W e explicitly deriv e the first non-tr ivial lo cal in tegral o f motion, whic h correspo nd s to the b oundary Hamiltonian for the sl 2 L-L mo del. No v el expressions of the mo dified Lax pairs asso ciat ed to the in tegrals of motion are also extracted. The r elev ant equations of motion with the corresp onding b o un dary conditions are determined. Dynamical inte grable b oundary conditions ar e also examined within t h is spirit. Then the generalized isotropic and anisotropic gl n Landau-Lifshitz mo dels are considered, and no v el expressions o f the b oundary Hamiltonians a nd the relev an t equations of motion and b oundary conditions are deriv ed. Con ten ts 1 In tro duction 2 2 The isotropic sl 2 Landau- Lifshitz mo del 3 2.1 The Lax pair form ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 In tegrals of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Implemen ting in tegrable b oundaries 7 3.1 Algebraic setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 The b oundary Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.1 Dynamical b oundary conditions . . . . . . . . . . . . . . . . . . . 9 4 The mo d ified Lax pair 10 4.1 Reviewing the construction . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 Mo dified Lax pairs fo r the L-L mo del . . . . . . . . . . . . . . . . . . . . 11 4.2.1 Dynamical b oundary conditions . . . . . . . . . . . . . . . . . . . 12 5 In tegrable con tin uum limit 13 5.1 The op en XXX c hain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.1.1 Dynamical b oundary conditions . . . . . . . . . . . . . . . . . . . 15 6 Generalized b ound ary L-L mo dels 16 6.1 The b oundary anisotropic sl 2 L-L mo del . . . . . . . . . . . . . . . . . . 16 6.2 The b oundary isotropic gl n L-L mo del . . . . . . . . . . . . . . . . . . . 17 6.3 The b oundary anisotropic gl n L-L mo del . . . . . . . . . . . . . . . . . . 19 7 Boundary symmetries 20 8 Discussion 23 A Deriv ation of lo c al integrals of motion 24 B The op en spin ch ain Hamiltonian 26 C The modified V -op erator 27 1 1 In t ro duction Numerous in ves tigations ha v e b een devoted to the issue of incorp orating non-trivial con- ditions that preserv e integrabilit y b oth in discre te [1]–[10], and contin uum in tegrable systems [1], [11]–[18]. The central purp ose of the presen t ar t ic le is the study of classical in tegrable mo dels when g eneral b oundaries that preserv e integrabilit y are implemen ted. Among the v arious classes of in tegrable mo dels w e choose to consider here a proto type mo del, that is the generalized classical con tin uum Heisen b erg or Landau- Lifshitz (L-L) mo del. This mo del ma y b e though t of as the immediate classical analogue of the XXX (XYZ for the anisotropic case) quan tum spin chain [19, 20], whereas higher ra nk gen- eralizations ma y b e seen as contin uum limits of kno wn high ra nk quan tum spin c hains. Although m uc h atten tion has b een dev oted to the in v estigation of the quan tum mo dels with in tegra ble b oundary conditions not m uc h progress has b een made –from the al- gebraic p oin t of view– o n their classical con tin uum coun terparts. Here, w e consider the classical con tin uum case, we iden tify the b oundary Hamiltonian, a nd the relev ant b ound- ary Lax pair, for the sl 2 L-L mo del, utilizing primarily the a lgebra that rules the mo del, that is the classical reflection alg e bra. Our study concerns not only typical c -n um b er reflection matrices, but a ls o dynamical reflection matrices, whic h giv e rise to dynamical t yp e b oundary conditions for the aforemen tioned mo del. The generalized gl n L-L mo dels are also examined within this spirit. It is w orth noting that the significance of the particular study stems primarily from the fact that it prov ides nov el results for a wide class of classical in tegrable mo dels asso ciated t o the gl n algebra. No te that suc h anisotropic (trigonometric) models ma y b e appropria t e ly mapp ed to A (1) n − 1 affine T o da field theories (see e.g. [21]). Moreo ve r, the prese n t in ves tigation provides a first systematic description on the issue of in tegrable con tin uum limits of discrete in tegrable mo dels , that con tain b oundary type terms or other distinct lo cal terms, such as the ones arising also in the case of integrable defects. The o u tline of the ar t icle is as follo ws: in the next section w e briefly review the mo del with p erio dic b oundary conditions, as w ell as the relev an t fundamental ingredien ts (see also [21]). The Lax pa ir form ulation, and the construction of the asso ciated in tegrals of motion through the asso ciated a lgebras are review ed. In section 3 w e review Skly a nin ’s generic algebraic f rame [1] describing classical mo dels with b oundaries that pr eserv e in tegrabilit y . Based on this framew ork w e explicitly deriv e t h e asso ciated Hamiltonian with su itable in tegrable boundary terms reco v ering also some of the expressions presen ted in [1]. Note that in [1] only diagonal b oundary terms w ere treated, whereas here the most general b oundary terms that prese rv e in tegrabilit y are deriv ed. Dynamical b oundaries are also examined within this con text. Note that our results are consisten t with the classical con tin uum limits o f the relev ant quan t um discrete Hamiltonians. In section 4 w e review the construction of mo difie d Lax pairs in the presence of in tegrable b oundaries 2 discusse d in [2]. Relying on this framew ork, a n d using generic solutions of the reflection equation – c -n umber and dynamical– w e are able to deriv e the Lax pair asso c iated to the extracted Hamiltonians. This w ay the consistency of the whole pro cedure is fully ensured. In section 5 t he con tin uum limit of t he XXZ op en spin c hain is considered leading to the b oundary an t it ropic sl 2 L-L mo del. This furt her ensures the v alidity of the con tin uum limit pro cess follo w ed. W e then examine the isotropic and anisotropic gl n L-L mo dels a s con tin uum limits of the gl n and U q ( gl n ) op en spin c hains resp ectiv ely , and obtain the classical con tinuum Hamiltonians, and the asso ciated equations of motio n and b oundary conditio ns. 2 The isot ropic s l 2 Landau-Lifshit z mo del Let us briefly review the con tin uous isotropic Landau-Lifshitz (L-L) mo del with peri- o dic b oundary conditions, associated t o the sl 2 classical algebra (see also [21]). After in tro ducing the basic ingredien ts of the mo del and setting up our notations, w e recall the Lax pair formulation for the classical in tegra ble Hamiltonian system, and discuss the systematic means of constructing the whole tow er of lo cal in tegrals in in v olution. The ph ysical quan tities of the mo del are describ ed b y vec tor-v alued functions ~ S ( x ) = ( S 1 ( x ) , S 2 ( x ) , S 3 ( x )) taking v alues on the unit 2-sphere ~ S 2 ( x ) = 3 X i =1 S 2 i ( x ) = 1 . (2.1) Note that throughout the text w e shall also use the following combinations of S i ( x ) S ± ( x ) = 1 2 ( S 1 ( x ) ± iS 2 ( x )) . (2.2) The equations of motio n associated to the isotropic Landau-Lifshitz mo del, whic h is our main in terest here, are of the form: ∂ ~ S ∂ t = i ~ S ∧ ∂ 2 ~ S ∂ x 2 . (2.3) The fields S i ( x ) ob e y b oundary conditions, whic h are tak en to b e suc h that either S i ( x ) b ecome p erio dic, i.e. S i ( x + 2 L ) = S i ( x ), or consider t h e fields and their deriv ative s to b e zero at the endp oin ts (Sc h w artz b oundary conditions). The Pois son structure of the phase space f or the ph ysical quantities S i ( x ) is g iv en b y the P oisson brac k ets { S a ( x ) , S b ( y ) } = 2 iε abc S c ( x ) δ ( x − y ) , (2.4) 3 where ε abc is the tot a lly antisy mmetric Levi-Civita tensor with v alue ε 123 = 1. The Hamiltonian of the mo del is g iv en by H = − 1 4 Z  ∂ S 1 ∂ x  2 +  ∂ S 2 ∂ x  2 +  ∂ S 3 ∂ x  2 ! dx. (2.5) The equations of motion in the Hamiltonian form are expressed as ∂ ~ S ∂ t = { H , ~ S } . (2.6) Other ph ysical in tegrals of motion include the momen tum, whic h is giv en by P = Z S 1 ∂ S 2 ∂ x − S 2 ∂ S 1 ∂ x 1 + S 3 dx, (2.7) and the total spin of the mo del in the case where p erio dic b oundaries are considered. 2.1 The Lax pair form ulation Within the Lax pair formulation of a classic al in tegrable Hamiltonia n system one first defines the auxiliary linear differen tial problem, whic h reads as ∂ ∂ x Ψ( x, t ) = U ( x, t, λ )Ψ ( x , t ) ∂ ∂ t Ψ( x, t ) = V ( x, t, λ )Ψ ( x , t ) . (2.8) In general, U and V a re n × n matrices. Their en tries con tain dynamical fields, t heir deriv ativ es and p ossibly , the sp e ctral parameter λ . The compatibility condition of these t w o equations leads to the so-called zero curv ature condition ∂ t U − ∂ x V + [ U , V ] = 0 , (2.9) whic h pro vides the equations of motion o f the system under consideration. One then constructs the mono drom y matrix T ( x, y , λ ) = P ex p  Z x y U ( z ) dz  , (2.10) b eing a solution of the equation ( 2.8). Assume tha t U ob ey s the classical linear Poiss on algebraic relation [21] { U a ( x, λ ) , U b ( y , µ ) } = [ r ab ( λ − µ ) , U a ( x, λ ) + U b ( y , µ )] δ ( x − y ) , (2.11) then it follow s that the mono drom y matrix satisfies the quadratic algebraic relation, { T a ( x, y , t, λ ) , T b ( x, y , t, µ ) } = [ r ab ( λ − µ ) , T a ( x, y , t, λ ) T b ( x, y , t, µ )] . (2.12) 4 r ab is the classical r -matrix corresp onding to the Hamiltonian system, and satisfies the classical Y ang-Baxter equation [22]. The conserv ed ch arges ma y b e obta ine d via the expansion of t ( λ ) = tr T ( λ ) in p o w ers of the sp ectral pa r ame ter, λ . It can also b e sho wn via (2.12) that these c harges are in in v olution, t ha t is they satisfy [21] { t ( λ ) , t ( µ ) } = 0 . (2.13) In the case of the L-L mo del, the classical r -matrix has the simple form [23] r ( λ ) = P λ , (2.14) where P is the p ermutation op erator: P ( ~ a ⊗ ~ b ) = ~ b ⊗ ~ a . W e shall restrict ourselv es fo r the mo ment in the case of t h e sl 2 L-L mo del. In this case, the p erm utation and Lax op erators a re resp e ctiv ely P =      1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1      , U ( x ) = 1 λ S 3 2 S − S + − S 3 2 ! ≡ 1 2 λ S . (2.15) W e also note here the explicit form of the V -op erator V ( x ) = 1 2 λ 2 S − 1 2 λ ∂ S ∂ x S . (2.16) Inserting the Lax pair op erators (2.1 5 ) and (2.16) in t o the zero curv ature condition (2.9) yields exactly the equations of motion (2.3). 2.2 In tegrals of motion As already men tio ned, t h e lo c al in tegrals of motion ma y b e extracted throug h the ex- pansion of the trace o f the mono drom y matrix in p o w ers of the sp ec tral parameter λ . A k ey prop e rty of the mono dromy matrix that is crucial in what follo ws is that is satisfies: ∂ ∂ x T ( x, y , λ ) = U ( x, λ ) T ( x, y , λ ) . (2.17) Let us consider the follow ing ansatz for the mono drom y matrix: T ( x, y , λ ) = (1 + W ( x, λ )) e Z ( x,y ,λ ) (1 + W ( y , λ )) − 1 , (2.18) W and Z are purely off-diagona l and diagonal matrices resp ectiv ely . W e also assume that W , Z are expresse d as: W ( x, λ ) = ∞ X n =0 λ n W n ( x ) , Z ( x, y , λ ) = ∞ X n = − 1 λ n Z n ( x, y ) . (2.19) 5 Our main aim hence forth is t o iden tify the elemen ts W n , Z n , and hence the in tegrals of motion. It is tec hnically con v enien t to split the Lax op erator in to a diagonal a n d an off-diagonal part as U = U d + U a ≡ 1 2 λ S z 0 0 − S z ! + 1 λ 0 S − S + 0 ! . (2.20) Substituting the ansatz (2.18) in to the relation (2.17), and splitting the resulting equation in to a diagonal and an off-diagonal part one obtains dW dx + W U d − U d W + W U a W − U a = 0 , ∂ Z ∂ x = U d + U a W . (2.21) Plugging in the explicit expressions for U a , U d , w e end up with the f ollo wing relation f o r W dW dx + 1 λ W S 3 σ 3 − 1 λ ( S − σ + + S + σ − ) + 1 λ W ( S − σ + + S + σ − ) W = 0 , (2.22) with σ 3 , σ ± b eing the fa miliar 2 × 2 Pauli matrices: σ 3 = 1 0 0 − 1 ! , σ + = 0 1 0 0 ! , σ − = 0 0 1 0 ! . (2.23) Inserting the expansion (2 .19 ) in the equation a bov e, one determines all W n ’s. W e are only in terested in the first three terms of the expansion, whic h a re giv en b y: O (1 /λ ) : W 0 = 0 − ¯ a a 0 ! , a = 1 − S 3 2 S − = 2 S + 1 + S 3 , O ( λ 0 ) : W 1 = 0 − ¯ a ′ − a ′ 0 ! , O ( λ ) : W 2 = 0 − ¯ a ′′ + ( ¯ a ′ ) 2 S + a ′′ − ( a ′ ) 2 S − 0 ! ≡ 0 − ¯ b b 0 ! . (2.24) The first three terms suffice in order to compute t h e first t w o in tegrals of motion, namely the momen tum and the Hamilto nia n. T o complete the computatio n w e should also determine Z , pro vided b y the follo wing equation ∂ Z ∂ x = U d + U a W . (2.25) Substituting t he relev an t expansion of W and the explicit forms of U d , U a in to the equa- tion ab o v e w e conclude Z ( x, y , λ ) = 1 2 λ ( x − y ) σ 3 + ∞ X n =1 λ n − 1 Z x y ( S + σ − + S − σ + ) W n ( z ) dz . (2.26) 6 By using the exact expressions for W n , one ma y finally determine Z , order by order. In particular, as noted ab o v e, at order O (1 /λ ) we found Z − 1 = 1 2 ( x − y ) σ 3 . (2.27) Here w e set x = L and y = − L . The explicit fo rm of Z − 1 is imp ortan t since it indicates the leading contribution of e Z as λ → 0, a result whic h will b e gr eat ly used b elow and in the forthcoming sections. Mov ing on to the next orders, one naturally a rriv es at the first t w o in tegra ls of motio n, as exp ected. More sp ecifically , b y w orking with the con v en tions defined ab o v e one concludes that ( Z 0 ) 11 ∝ P, a nd ( Z 1 ) 11 ∝ H . (2.28) Extra care is needed while computing the in tegr a ls of motion ab o v e, taking in to accoun t the Sc hw artz b oundary conditions at the endp o in ts for the fields and their deriv ativ es, namely ( S i ( ± L ) , S ′ i ( ± L )) → 0. Note that in the case of non-trivial b oundaries extra terms, emerging from the bulk part, should b e tak en into accoun t, giv en that the fields do not v anish in that case at the endp oin ts. This will b e transparent in the subsequen t sections. 3 Implemen ting in tegrable b oundarie s W e shall no w briefly describ e the relev ant mac hinery needed to implemen t sp ecial b ound- aries into a classical mo del in a w a y that in tegr a bilit y is ensured. After reviewing the general setting w e fo c us on t he particular example of interest, that is the Landau-Lifshitz mo del with integrable b oundaries. 3.1 Algebraic setting W e review Skly anin’s form ulation [1] in order to exhibit the in tegrability for class ical mo dels with sp ecial b oundaries . Supp ose tha t we hav e the mono drom y matrix o f the mo del at hand, endo wed with the Pois son structure (2.12). Let also K ± ( λ ) b e c -num b er (non-dynamical) represen ta tions of the classical reflection algebra [1, 11], 0 = [ r 12 ( λ 1 − λ 2 ) , K ( λ 1 ) K 2 ( λ 2 )] + K 1 ( λ 1 ) r 12 ( λ 1 + λ 2 ) K 2 ( λ 2 ) − K 2 ( λ 2 ) r 12 ( λ 1 + λ 2 ) K 1 ( λ 1 ) . (3.1) Our no tation is suc h that K + ( λ ) = K ( − λ, ξ + , k − ) a nd K − ( λ ) = K ( λ, ξ − , k + ); ξ ± , k ± are generic free b oundary parameters (see also b elo w in the text). One may then define a mo difie d transition matrix as [1] T ( x, y , λ ) = T ( x, y , λ ) K − ( λ ) ˆ T ( x, y , λ ) , (3.2) 7 where ˆ T ( λ ) = T − 1 ( − λ ). The mo dified mono drom y matrix satisfies the classical v ersion of the reflection equation [1, 11]: {T 1 ( λ 1 ) , T 2 ( λ 2 ) } = [ r 12 ( λ 1 − λ 2 ) , T ( λ 1 ) T 2 ( λ 2 )] + T 1 ( λ 1 ) r 12 ( λ 1 + λ 2 ) T 2 ( λ 2 ) − T 2 ( λ 2 ) r 12 ( λ 1 + λ 2 ) T 1 ( λ 1 ) . (3.3) The generalized transfer matrix reads as t ( x, y , λ ) = tr { K + ( λ ) T ( x , y , λ ) } , (3.4) and it is sho wn to satisfy [1] { t ( x, y , λ 1 ) , t ( x, y , λ 2 ) } = 0 , λ 1 , λ 2 ∈ C , (3.5) hence it may b e in terpreted as the g e nerating functional of the cons erv ed in tegrals of motion. By adopting t h e ansatz (2.18 ) and setting x = 0, y = − L for t he b oundary p oin ts, the generating functional of the lo c al in tegr a ls of motion takes the fo rm ln tr n K + ( λ ) T (0 , − L, λ ) K − ( λ ) ˆ T (0 , − L, λ ) o = ln tr { (1 + ˆ W (0)) − 1 K + ( λ )(1 + W (0)) e Z (0 , − L ) (1 + W ( − L )) − 1 K − ( λ )(1 + ˆ W ( − L )) e − ˆ Z (0 , − L ) } , (3.6) where ˆ W a r e the same as b e fore, but with λ → − λ . It is this expression that one expands in p o wers o f λ , in order to deriv e the lo cal in tegrals of motion. 3.2 The b oundary Hamiltonia n W e now pro ceed in deriving the in tegrals of motio n for the Landau-Lifshitz mo del with in tegrable b oundaries. W e shall b e using the follo wing c -n um b er represe n tation of the classical reflection algebra [3] K ( λ, ξ , k) = − λ + iξ 2k λ 2k λ λ + iξ ! . (3.7) As in the p erio dic case to deriv e the in tegrals of motion, one expands the generic ob ject (3.6) in p o w ers of λ . W e shall only presen t here the final results, whereas the tec hnical details o f the deriv ation can b e found in the App endix A. One should k eep in mind that extra terms, emerging directly from the bulk, cancel out suitably some purely b oundary terms pro viding ev en tually a quite simple b oundary contribution to the Hamiltonian. In t he case of op en b oundary conditions, the first in tegral of motion b ecomes trivial. This is actually exp ecte d, giv en that this is essen tially the momen tum, w hic h is not a 8 conserv ed quan tit y an ymore. The second in tegral of motion, i.e. the Hamiltonian is computed to b e (see also [1], where only diagonal b oundary t e rms are considered) I 1 = − 1 4 Z 0 − L ∂ ~ S ∂ x ! 2 dx − i 2 ξ −  2k − S 1 ( − L ) − S 3 ( − L )  + i 2 ξ +  2k + S 1 (0) − S 3 (0)  . (3.8) The first term in the e xpression ab o v e is just the bulk term, while the rest are total b oundary contributions. 3.2.1 Dynamical b oundary c on ditions W e shall now discuss the case of dynamical degrees of freedom, atta ched at the ends of the syste m. T o ac hiev e this w e shall consider the dynamical solution of the reflection equation [1]: K ( λ ) = L ( λ ) K ( λ ) L − 1 ( − λ ) , (3.9) where w e define L = λ I + S 3 2 S − S + − S 3 2 ! . (3.10) The L matrix satisfies the quadratic algebra (2.12) with the Y angian r -matrix. The elemen ts S z , S ± apparen tly satisfy the classical sl 2 algebra. Note that special limits of the generic L matrix lead to the D iscrete-Self-T rapping (DST) model, or the T o da mo del (see e.g. [17 ] and references therein), so this wa y one treats a generic class of dynamical b oundaries. W e shall consider here fo r simplicit y , but without loss of generality , the dynamical b oundary attache d to the left end of the system. The right end of the system will b e described by the trivial reflection matrix K + ∝ I . He nce, the generalized transfer matrix will b e of the form: t ( λ ) = tr a h T a ( λ ) K − a ( λ ) T − 1 a ( − λ ) i . (3.11) It will b e conv enien t for the follo wing computations to express the K -matrix a s : K − ( λ ) ∝ I + λ B + O ( λ 2 ) , B = − X Z Y X ! , (3.12) where w e define X = − 4 S 3 + 1 iξ −  2 S 2 3 − 1 − 4 k − S 3 ( S + + S − )  Y = 8 S + + 1 iξ −  8k − ( S + ) 2 − 2k − S 2 3 − 4 S 3 S +  Z = 8 S − + 1 iξ −  8k − ( S − ) 2 − 2k − S 2 3 − 4 S 3 S −  . (3.13) 9 It is clear that the elemen ts X , Y , Z contain the dynamical degrees of freedom attac hed to the b oundary . Expanding appropriately t he generalized transfer mat r ix as in the previous section, w e end up to the f o llo wing Hamiltonian with distinct dynamical terms attac hed to one b oundary: I 1 = − 1 4 Z 0 − L ∂ ~ S ∂ x ! 2 dx + 1 2 S + ( − L ) Z + 1 2 S − ( − L ) Y − 1 2 S 3 ( − L ) X . (3.14) Note that suc h b oundary terms would hav e emerged f o r the other end o f the system as w ell, after implemen ting a similar righ t dynamical reflection matrix. More precisely , a dynamical K + matrix for the right b oundary w ould lead t o extra b oundary terms in expressions (3.14) at x = 0 of exactly the same from of as the ones at x = − L , but with ξ − → − ξ + , k − → k + in (3.13). 4 The mo dified Lax pair When integrable b oundary conditions are implemen t ed, the Lax pairs asso c iated to the in tegrals of motion are accordingly mo dified. The systematic construction of the mo dified Lax pairs w as presen ted in [2]. In what follow s, w e briefly review the results of [2], and then apply the forma lism in the case of the sl 2 Landau-Lifshitz mo del with integrable b oundaries. 4.1 Reviewing the construction Recall first the construction of the V -op erator asso ciated to a giv en in tegral of mot io n for a classic al integrable mo del with p erio dic boundary conditions. Using (2.12) one form ulates the follow ing P oisson structure: n T a ( L, − L, λ ) , U b ( x, µ ) o = ∂ M ( x, λ, µ ) ∂ x + h M ( x, L, − L, λ, µ ) , U b ( x, µ ) i , (4.1) where w e define M ( x, λ, µ ) = T a ( L, x, λ ) r ab ( λ − µ ) T a ( x, − L, λ ) . (4.2) More details on the deriv ation of the latter formu la can b e found in [2 1 ]. Recalling now that t ( λ ) = tr T ( λ ) it naturally follows fro m (4.1) and (2.9) that n ln t ( λ ) , U ( x, λ ) o = ∂ V ( x, λ, µ ) ∂ x + h V ( x, λ, µ ) , U ( x, λ ) i , (4.3) with V ( x, λ, µ ) = t − 1 ( λ ) tr a  T a ( L, x, λ ) r ab ( λ, µ ) T a ( x, − L, λ )  . (4.4) 10 In the case of op en b oundary conditions one may pro v e that a generalized P o isson structure holds [2], i.e. n T a (0 , − L, λ ) , U b ( x, µ ) o = M ′ a ( x, λ, µ ) + h M a ( x, λ, µ ) , U b ( x, µ ) i , (4 .5 ) where w e no w define M ( x, λ, µ ) = T (0 , x, λ ) r ab ( λ − µ ) T ( x, − L, λ ) K − ( λ ) ˆ T (0 , − L, λ ) + T (0 , − L, λ ) K − ( λ ) ˆ T ( x, − L, λ ) r ab ( λ + µ ) ˆ T (0 , x, λ ) . (4.6) Finally , b earing in mind the definition of t ( λ ), and (4.5) w e conclude n ln t ( λ ) , U ( x, µ ) o = ∂ V ( x, λ, µ ) ∂ x + h V ( x, λ, µ ) , U ( x, µ ) i , (4.7) where V ( x, λ, µ ) = t − 1 ( λ ) tr a  K + a ( λ ) M a ( x, λ, µ )  . (4.8) This is the explicit form for the V -op erator in the case o f generic in tegrable b oundary conditions. One expands V in p o wers of λ in order to obtain the mo difie d op erator asso ciated to each integral of motion of the mo del under consideration. 4.2 Mo dified Lax pairs for the L -L mod e l W e are no w in the p osition to determine t he b oundary Lax pair for boundary L-L mo del. The classical r -matrix a sso ciated to the L-L mo del is prop ortional to the p erm utation op erator, a nd tr a P ab = I , then V can b e expressed in a simple form as V ( x, λ, µ ) = t − 1 ( λ ) λ − µ T ( x, − L, λ ) K − ( λ ) T − 1 (0 , − L, − λ ) K + ( λ ) T (0 , x, λ ) + t − 1 ( λ ) λ + µ T − 1 (0 , x, − λ ) K + ( λ ) T (0 , − L, λ ) K − ( λ ) T − 1 ( x, − L, − λ ) . (4.9) The latter expression is v alid for all classical mo dels asso ciated to t he Y angian classical r -matrix, prop ortional to the p erm utatio n op erator. Explicit computation shows tha t the V -op erator for an y p oin t x 6 = 0 , − L reduces to the familiar bulk op erator (2.16). In an y case, w e a re mostly in terested in computing the V -op erator exactly at the b oundary p oin ts, tha t is x b = (0 , − L ). W e shall only presen t the final results here, and p ostpone the hea vy tec hnical details of the computation un til the App endix C. A t t h e end p oin ts the V -op erator has the follow ing form (we ha v e m ultiplied the result of the expansion with 1 2 ) V b ( x b ) = V ( x b ) + δ V ( x b ) , (4.10) 11 where V is the bulk op erator (2 .16 ) and δ V ( − L ) = 1 2 µ " ∂ S ∂ x S + 2 iξ − k − ( S + ( − L ) − S − ( − L )) − k − S 3 ( − L ) − S − ( − L ) k − S 3 ( − L ) + S + ( − L ) − k − ( S + ( − L ) − S − ( − L )) !# δ V (0) = 1 2 µ " ∂ S ∂ x S − 2 iξ + k + ( S + (0) − S − (0)) − k + S 3 (0) − S − (0) k + S 3 (0) + S + (0) − k + ( S + (0) − S − (0)) . !# . (4.11) F rom the zero curv ature condition, and by requiring δ V = 0 (see mor e details on this argumen t in [2]), we obtain the equations of motion describ ed in (2.3), and the non- trivial b oundary conditio ns (see also [1] for only diagonal b oundary conditions):  S 2 ∂ S 3 ∂ x − S 3 ∂ S 2 ∂ x     x = − L = 1 iξ − S 2 ( − L )  S 3 ∂ S 1 ∂ x − S 1 ∂ S 3 ∂ x     x = − L = − 1 iξ − S 1 ( − L ) − 2k − iξ − S 3 ( − L )  S 1 ∂ S 2 ∂ x − S 2 ∂ S 1 ∂ x     x = − L = 2k − iξ − S 2 ( − L ) . (4.12) Of course one may easily c hec k that the same equations of motion, and b oundary con- ditions are extracted from the Hamilto nian (3.8) via (2.6 ). No te that in obtaining the b oundary conditions w e to ok into a c coun t that the deriv ativ e of the Casimir with resp ec t to x is zero. Note that similar equations of motion ar e obtained for the other end of t h e system ( x = 0), but are o mitt ed here for brevity . The en tailed b oundary conditions are as exp e cted mixed ones. 4.2.1 Dynamical b oundary c on ditions The explicit computation of the mo dified V -o perator in this case follow s exactly the previous section’s computations via t h e expression (4.9), so the result is quite straight- forw ard, as long as we ke ep in mind that the classical dynamical reflection matrix is no w expressed as in (3.12). Recall that w e restrict our attention here to one b oundary x b = − L , then the final expres sion for the b oundary op e rator is giv en as V b ( − L ) = V ( − L ) + δ V ( − L ) , (4.13) where w e define δ V ( − L ) = 1 2 µ " ∂ S ∂ x S + Z S + ( − L ) − Y S − ( − L ) − Z S 3 ( − L ) − 2 X S − ( − L ) Y S 3 ( − L ) + 2 X S + ( − L ) − Z S + ( − L ) + Y S − ( − L ) !# (4.14) where X , Y , Z are defined in (3.13). In this case the rele v ant b oundary conditions, en tailed from the conditions δ V = 0, read as  S 2 ∂ S 3 ∂ x − S 3 ∂ S 2 ∂ x     x = − L = i ( Z − Y ) 2 S 3 ( − L ) + X S 2 ( − L ) 12  S 3 ∂ S 1 ∂ x − S 1 ∂ S 3 ∂ x     x = − L = − X S 1 ( − L ) − Y + Z 2 S 3 ( − L )  S 1 ∂ S 2 ∂ x − S 2 ∂ S 1 ∂ x     x = − L = Y + Z 2 S 2 ( − L ) − i ( Y − Z ) 2 S 1 ( − L ) . (4.15) Needless to men tion that these b oundary conditions emerge also from the dynamical Hamiltonian (3.14) through (2.6). 5 In t egrable c o n tin uum limit W e shall describ e here a system atic means of obtaining classical contin uum limits o f quan tum discrete theories f or generic b oundary conditions along the lines discussed in [20]. Assume a collection of op erators assem bled in matrices L 1 i , a cting on “quantum ” Hilb ert spaces lab eled b y i a nd encapsulated in a matrix “acting” on the auxiliary space V 1 . F or any quan tum space q they ob ey the quadratic exc hange algebra [24, 25] R 12 L 1 q L 2 q = L 2 q L 1 q R 12 , (5.1) where op e rators acting on differen t quantum spaces comm ute, and R satisfies the Y ang - Baxter equation. The form of the mono drom y matrix T is then deduced from the co- mo dule structure of the YB algebra T a ≡ L aN L a 2 . . . L a 1 , (5.2) and th us naturally ob eys the same quadratic exc hange algebra (5.1). First consider that the R matrix has a classical limit as R = 1 + ~ r + O ( ~ 2 ) , (5.3) with r satisfying the classical Y ang-Baxter equation (see also e.g. [22, 20 ]) . W e ma y no w establish that T has a classical limit b y considering in addition the classical counterpart of L , whic h then satisfies the quadratic P oisson algebra emerging directly as a semi-classical limit of (5.1), after setting 1 ~ [ A, B ] → { A, B } . It reads { L a ( λ 1 ) , L b ( λ 2 ) } = [ r ab ( λ 1 − λ 2 ) , L a ( λ 1 ) L b ( λ 2 )] . (5.4) The classical disc rete mono drom y matrix is apparently of the same fo rm as in (5.2 ). The exc hange algebra for T ta kes the f orm { T a , T b } = [ r ab , T a T b ] . (5.5) This quadratic P oisson structure implies that the traces of p o wers of the mono drom y matrix tr ( T c ) generate Poisson-comm uting quan tities iden tified as classically integrable Hamiltonians. 13 No w that we hav e discus sed the classical limit we may pro ceed to t he con tinuum limit of discrete theories with op en b oundary conditions. In this case the modified mono drom y matrix has the form T ( λ ) = T ( λ ) K − ( λ ) T − 1 ( − λ ) , (5.6) where T is giv en b y (5.2), T satisfies the classical reflection equation (3.3) and K − is a c - n um b er solution of the reflection equation (3.1). In tro duce a suitable spacing parameter δ : O ( δ ) ∼ O ( 1 N ). Let us also express the L matrix as L an ( λ ) = 1 + δ U an ( λ ) + δ 2 U (2) an ( λ ) + . . . L − 1 an ( − λ ) = 1 − δ U an ( − λ ) + δ 2 ˜ U (2) an ( − λ ) + . . . (5.7) It then naturally follows for the mono drom y matrix and its in v erse T ( λ ) = 1 + δ X n U an ( λ ) + δ 2 X n>m U an ( λ ) U am ( λ ) + δ 2 X n U (2) an ( λ ) + . . . T − 1 ( − λ ) = 1 − δ X n U an ( − λ ) + δ 2 X nl l ′ j j ( x ) l lk ( x ) + 2 α X j >k l ′ j j ( x ) l k l ( x ) , (6.24) with corresp onding b oundary conditions n X j =1  l ′ j l ( − L ) l j k ( − L ) − l ′ k j ( − L ) l j l ( − L )  = − α X j l l j j ( − L ) l lk ( − L ) − α X j >k l j j ( − L ) l k l ( − L ) − ˜ C − h − Q − 1  δ 1 l l 1 k ( − L ) − δ 1 k l 1 l ( − L )  − Q  δ nl l nk ( − L ) − δ nk l nl ( − L ) i − ˜ C − h δ 1 l l nk ( − L ) − δ nk l 1 l ( − L ) + δ nl l 1 k ( − L ) − δ 1 k l nl ( − L ) i , (6.25) and similarly for the other end of the theory at x = 0. With this w e conclude our analysis on the generalized gl n b oundary L-L mo dels. No te that similar generalizations can b e applied in a straigh tforw ard manner in the elliptic case, but are omitted here for brevit y . 7 Boundary s ym metries W e fo cus here mainly on the isotropic gl n case, and briefly discuss the con tin uum ana- logues of earlier woks on bo un dary symm etries (see e.g [8]), although w e ha v e to men tion that this is a whole separate sub ject of intere st. W e shall extract b elo w the so-called b oundary non-lo cal c harges which a re realizations of the underlying classical r e flections algebra. W e shall first consider K + ∝ I , and K − pro vided b y the generic expression: K − ( λ ) = 1 λ + B . (7.1) 20 The L and L − 1 op erators are expressed as L 0 i ( λ ) = 1 + δ λ P 0 i L − 1 0 i ( − λ ) = 1 + δ λ P 0 i + δ 2 λ 2 P 2 0 i + . . . (7.2) W e now consider the mo dified mono drom y matrix T , and by expanding in p o w ers of 1 λ w e extract the b oundary non-lo cal c harges (see also e.g. [8]). Let us start with the discrete T T 0 ( λ ) = L 0 N ( λ ) . . . L 01 ( λ ) K − ( λ ) L − 1 01 ( − λ ) . . . L − 1 0 N ( − λ ) , (7.3) the expansion of the latter leads to T 0 ( λ ) = T (0) 0 + 1 λ T (1) 0 + 1 λ 2 T (2) 0 + . . . = B 0 + δ λ  X i P 0 i B 0 + B 0 X i P 0 i + 1  + δ 2 λ 2  X i>j P 0 i P 0 j B 0 + B 0 X iy dx dy P k m ( x ) P mp ( y ) B pl + n X m,p =1 B k m Z x

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