Defects in the discrete non-linear Schrodinger model

Defects in the discrete non-linear Sc hr¨ odinger mo del Anastasia Doik ou Univ ersity of Patras, Department of Engineering Sciences, Physics Divisio n GR-2650 0 Patras, Gr e e ce E-mail : adoiko u @ upatr as.gr Abstract The discrete non-linear Sc hr¨ odinger (NLS) mo del in the prese nce of an in tegrable defect is examined. The pr o blem is view ed from a purely algebraic p oin t o f view, starting fro m the fundamen tal a lgebraic relations that rule the mo del. T he first c harges in inv olution are explicitly constructed, as w ell as the corresp onding Lax pairs. These lead to sets of difference equations, whic h include particular terms corresp onding to the impurit y p oin t. A first glimpse regar ding t he corresponding contin uum limit is also provided. Con t en ts 1 In t ro duction 1 2 The p erio dic DNLS model 2 3 Lo cal In tegrals of motion 3 3.1 The Lax pair formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 The DNLS mo del with integrable defect 7 4.1 Lo cal In tegra ls of motio n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 The asso ciated Lax pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 The con tinuum limit: a first glance 12 6 Discussion 15 1 In tro duction The presence of defects in 1 + 1 integrable field theories has b een the sub ject of in tense researc h during the recen t y ears ( see e.g. [1]–[11]). It is w ell established b y now that the requiremen t of integralit y leads to a set of sev ere algebraic constrain ts that should b e satisfied b y the asso ciated degrees of freedom as w ell as the relev an t ph ysical quan t ities, suc h as scattering matrices, at the quan tum level ( see e.g. [1, 2]). In in tegra ble field theories the defect is usually in tro duced as a discon t inuit y together with suitable sew ing conditions [3]–[9]. In this case no systematic alg ebraic description exist so far with the exception of some recen t a t tempts [9], but aga in the issue of in tegrabilit y is not fully resolv ed to our understanding (see also [10]). In the presen t study w e start our inv estigation using an inte grable mo del on the one dimensional lattice, that is the discrete non-linear Schr¨ odinger (NLS) mo del, and imp ose an ultra-lo cal integrable defect (see also relev an t considerations in [11]). W e by construction deal with an in tegrable system by a priori imp osing the necessary algebraic constrain ts that ensure the integrabilit y of the mo del. W e then explicitly construct the first in tegrals of 1 motion as w ell as the relev an t Lax pairs. The corresp onding equations of mo t ion are also deriv ed. The p ertinen t question in this frame is whether and how in tegrability is preserv ed in the contin uum limit. In ot her w ords is the underlying algebra that ensures in tegrability mo dified and how? W e mak e a first attempt to answ er these questions b y considering the con tinuum limits of the first couple of in tegrals of motion as w ell as the corresp onding Lax pairs; then certain con tinuit y or sewing conditions are naturally induced via the pro cess. These preliminarily results on the con tinuum limit, provide a first in sigh t on ho w a systematic con tinuum pro cess should b e formulated within this contex t. The outline of the presen t article is as follow s: In the next section w e briefly review the discrete NLS mo del with p erio dic b oundary conditions. In section 3 the first three in tegra ls of motio n are deriv ed together with the corresp onding Lax pairs. W e also deriv e the equations of motion asso ciated to the third c harge, whic h is the t ypical Hamilto nian. In the next section w e consider the discrete NLS mo del in the presence of an integrable lo cal defect. W e in t r o duce a suitable Lax op erator asso ciated to the defect p oin t so that the system is b y construction inte grable. W e then deriv e the first inte grals of mot io n as w ell as the corr esp onding La x pairs. These are no v el express ions t ha t contain non- trivial con tributio ns due to t he presence of the inte grable defect. The related equations of motion for the Hamiltonian are also derived . P articular emphasis is giv en exactly on the defect p oin t where the equations of motion are of a completely different form compared t o the other p oin ts due to the structural dissimilarit y of the asso ciated Lax op erator. Finally , in section 5 w e prov ide a first glimpse on the contin uum limits of the deriv ed phy sical quan tities. Certain sewing conditions naturally arise as con tin uity requiremen t s, ensuring the P oisson comm utativity of the first t w o con tinuum c harges. 2 The p erio d ic DNLS mo del W e shall briefly revie w the discrete NLS mo del with perio dic b oundary conditio ns. W e shall repro duce the first three lo cal integrals of mot ion and the asso ciated Lax pairs. In the subseque n t sections we shall r epeat these deriv ations in the presence of an in tegra ble defect. Our ult imat e aim is to mak e a con ta ct with recen t results on defects arising in 1+ 1 in tegra ble field theories (see e.g. [8]). W e aim at taking the appropria t e con tin uum limit, that will pro vide t he classical con tinuum mo dels with defects that are still integrable. This 2 is conceptually and t echnic ally a very in triguing problem, and will b e pursued in full detail in forthcoming in v estigations. Here, ho w ev er w e pro vide some preliminary results regarding the con tin uum limit of the first tw o integrals of motion. 3 Lo cal In teg r als of motion Our main aim in this section is to extract the first inte grals of motion for the p erio dic discrete NLS mo del. The asso ciated L ax operat o r is giv en b y (see e.g. [12 ]) : L aj ( λ ) = λD j + A j = λ + N j x j − X j 1 ! (3.1) where N j = 1 − x j X j . W e shall fo cus here, mainly for simplicit y , on the classical case. How ev er, we hav e to note tha t o ur results are v alid in the quan tum case as w ell. The L matrix satisfies the fundamen tal algebraic relation [13] 1 { L a ( λ 1 ) , L b ( λ 2 ) } = h r ab ( λ 1 − λ 2 ) , L a ( λ 1 ) L b ( λ 2 ) i . (3.3) In this case the r -matrix is the fa miliar sl 2 Y angian matrix [14]: r ( λ ) = 1 λ P , (3.4) P is the permutation op erator: P  ~ a ⊗ ~ b  = ~ b ⊗ ~ a . The form ula (3.3) is then realized b y the follo wing relations: { x i , X j } = δ ij , (3.5) that is x, X are canonical v a riables. The discrete mo del with N sites, and p erio dic b oundary conditions is ass o ciated to the transfer matrix defined as [13, 15, 1 6]: t ( λ ) = T r a T a ( λ ) where T a ( λ ) = L aN ( λ ) L aN − 1 ( λ ) . . . L a 1 ( λ ) . (3.6) 1 The same L -o pera tor ho lds for the quantum cas e as w ell. It then satisfies: R 12 ( λ 1 − λ 2 ) L 1 ( λ 1 ) L 2 ( λ 2 ) = L 2 ( λ 2 ) L 1 ( λ 1 ) R 12 ( λ 1 − λ 2 ) , (3.2) where R ( λ ) = λ I + P . 3 T is the mono drom y matrix also satisfying the quadrat ic algebraic relation ( 3 .3). In the notation L 0 i , the index a denotes the auxiliary space , whereas the index i denotes the i th site on the one dimensional lattice deriv ed by (3.6). As will b e transparen t later in the text in the con t inuum limit the discrete index i will b e replaced by the con tinuum co ordinate x . The transfer matrix t ( λ ) as is w ell kno wn prov ides all the c ha rges in in v olution. Indeed, via (3.3) one readily sho ws that n t ( λ ) , t ( µ ) o = 0 , (3.7) hence the system is b y construction integrable. T o deriv e the lo c al integrals of motion one should expand the ln t ( λ ) in pow ers o f λ or 1 λ . In this case w e expand in p ow ers o f 1 λ , b ecause the L -matrix (3.1) reduces to the degenerate matrix D at λ → ∞ . Let us now expand the mono drom y matrix: T ( λ → ∞ ) ∝ D N . . . D 1 + 1 λ N X i =1 D N . . . D i +1 A i D i − 1 . . . D 1 + 1 λ 2 X i>j D N . . . D i +1 A i D i − 1 . . . D j +1 A j . . . D 1 + 1 λ 3 X i>j >k D N . . . D i +1 A i D i − 1 . . . D j +1 A j . . . D k +1 A k . . . D 1 + . . . (3.8) T aking in to accoun t the la tter expansion and the definition of the tra nsfer matrix w e con- clude: ln t ( λ → ∞ ) ∝ 1 λ H 1 + 1 λ 2 H 2 + 1 λ 3 H 3 + . . . , (3.9) where t he extracted inte grals of motion ha ve the following familiar f orm (see also e.g. [12, 17 ]) H 1 = N X i =1 N i , H 2 = − N X i =1 x i +1 X i − 1 2 N X i =1 N 2 i H 3 = − N X i =1 x i +2 X i + N X i =1 ( N i + N i +1 ) x i +1 X i + 1 3 N X i =1 N 3 i . (3.10) 4 The latter pro vide the first in tegrals of motion (n um b er of particles, momen tum and Hamil- tonian resp ectiv ely) of the whole hierarc hy for the NLS mo del. It is clear tha t the con tinuum limits of the ab ov e quan tities pro vide t he corresp onding integrals of motion of the con tin uum NLS model [12, 17]. The latter expressions are v alid in the quantum case as w ell (see e.g. [12, 17]). 3.1 The Lax pair form ulation Let us no w br iefly review ho w t he Lax pair asso ciated to eac h lo cal in tegral of motion is deriv ed via the r -matrix form ula tion (see also [13]). In tro duce first the Lax pair ( L, A ) for discrete in tegra ble mo dels, and the asso ciated discrete auxiliary linear problem (see e.g. [13]) ψ j +1 = L j ψ j ˙ ψ j = A j ψ j . (3.11) F rom t he latt er equ ations one ma y immediately obtain the discrete zero curv ature condition as a compatibility condition: ˙ L j = A j +1 L j − L j A j . (3.12) Recall tha t the index j denotes the site on an one dimensional lattice, and it will b e replaced in the contin uum limit b y the con tin uum co ordinate x . In the contin uum limit as will b e clear the equations (3.11), ( 3 .12) reduce to the contin uum linear auxiliary problem and the con tinuum zero curv a ture condition resp ectiv ely (see section 5). Let us introduce at this p oint some useful notation. W e define for i > j : T a ( i, j ; λ ) = L ai ( λ ) L ai − 1 ( λ ) . . . L aj ( λ ) . (3.13) T o b e able t o construct the Lax pair w e should first form ulate the following Pois son structure [13]: n T a ( λ ) , L bj ( µ ) o = T a ( N , j + 1; λ ) r ab ( λ − µ ) T a ( j, 1; λ ) L bj ( µ ) − L bj ( µ ) T a ( N , j ; λ ) r ab ( λ − µ ) T a ( j − 1 , 1; λ ) . (3.14) 5 It then immediately f ollo ws for the generating function of the l o c al integrals of motion: n ln t ( λ ) , L bj ( µ ) o = t − 1 ( λ ) T r a  T a ( N , j + 1; λ ) r ab ( λ − µ ) T a ( j, 1; λ )  L bj ( µ ) − L bj ( µ ) t − 1 ( λ ) T r a  T a ( N , j ; λ ) r ab ( λ − µ ) T a ( j − 1 , 1; λ )  . (3.15) Recalling the classical equation of motion ˙ L j ( µ ) = n ln t ( λ ) , L j ( µ ) o , (3.16) and comparing with expression (3.1 5) we obtain A j ( λ, µ ) = t − 1 ( λ ) tr a h T a ( N , j ; λ ) r ab ( λ − µ ) T a ( j − 1 , 1; λ ) i , (3.17) where the relev an t classical r -matr ix is giv en in (3.4). Substituting the r -matrix in t o the latt er expression we conclude that A j ( λ, µ ) = t − 1 ( λ ) λ − µ T ( j − 1 , 1; λ ) T ( N , j ; λ ) . (3.18) Expansion o f the latter expression in p o w ers of 1 λ pro vides the Lax pairs asso ciated to eac h one of the lo cal integrals of motion (see a lso [18]), i.e.: A (1) j ( µ ) = 1 0 0 0 ! , A (2) j ( µ ) = µ x j − X j − 1 0 ! , A (3) j = µ 2 + x j X j − 1 µx j − x j N j + x j +1 − µX j − 1 + X j − 1 N j − 1 − X j − 2 − x j X j − 1 ! . (3.19) Both the Lax pair via the zero curv ature condition and the Hamiltonian description giv e rise to the same equations of motion. Consider fo r instance the equations of motion asso ciated to H 3 (and the La x pair L, A (3) ). Indeed fro m ˙ x j = { H 3 , x j } , ˙ X j = { H 3 , X j } , (3.20) and via the zero curv ature condition for the pa ir L, A (3) w e obta in the follow ing set of difference equations: ˙ x j = x j +2 − 2 x j +1 N j − x j +1 N j +1 + x j N 2 j + x 2 j X j − 1 + x j +1 ˙ X j = − X j − 2 + 2 X j − 1 N j + X j − 1 N j − 1 − X j N 2 j − X 2 j x j +1 − X j − 1 . (3.21) With this w e conclude our brief review on the p erio dic discrete NLS mo del. 6 4 The DNL S mo de l with in te grable defect W e shall henceforth fo cus on the discrete NLS mo del in the presence of an integrable defect. W e shall basically extract the lo cal integrals of motion and the corresp onding L a x pairs for the aforemen tioned mo del, and shall derive the mo dified equations o f motio n due to the presence of the defect. Let us first describ e the algebraic setting fo r the defect p er se. Introduce the Lax op erator asso ciated to the defect, which is lo cated at a particular site say n : ˜ L an = λ + ˜ A an = λ + α n β n γ n δ n ! , (4.1) the index n simply denotes the p osition of t he defect on the o ne dimensional spin c hain. Note t ha t the ˜ L matrix is required t o ob ey the same P oisson bra ck et structure with the bulk matrices L (3.1 ) so that integrabilit y is ensured. The en tr ies of the ab ov e ˜ L matrix ma y b e parameterized as (see e.g. [19], and references therein) α n = − δ n = 1 2 cos(2 θ n ) , β n = 1 2 sin(2 θ n ) e 2 iφ n , γ n = 1 2 sin(2 θ n ) e − 2 iφ n , (4.2) the fields θ n , φ n ma y b e rewritten in terms of the canonical v aria bles p n and q n as cos(2 θ n ) = p n , φ n = q n { q n , p n } = i. (4.3) It is then immediately sho wn via the algebraic relation (3 .3) that the elemen ts α n , β n , γ n , δ n satisfy the fo llo wing exc hange relations: { α n , β n } = β n { α n , γ n } = − γ n { β n , γ n } = 2 α n (4.4) whic h are the typical sl 2 exc hange relations. F or simplicit y , and in order to av oid un w an ted b oundary effects we shall consider the defect aw ay fro m the ends of the o ne dimensional lat t ice mo del. Inserting the defect at the n site of the o ne dimensional lattice the corresp onding mono dro m y matrix is expressed as: T a ( λ ) = L aN ( λ ) L aN − 1 ( λ ) . . . ˜ L an ( λ ) . . . L a 1 ( λ ) . (4.5) 7 Note that due to the fact that the ˜ L -op erator is r equired to satisfy the same f undamen tal algebraic relat io n as the mono dro m y matrix, the trace of it –the transfer matrix– pro vides a family of P oisson comm uting op erators. Ha ving the latter expression at our dispo sal w e ma y no w construct the desired ph ysical quantities . 4.1 Lo cal In tegrals of motion First we wish to extract the asso ciated lo cal integrals of motion. T hey ar e obtained, as in the previous section, f rom the expansion of ln t ( λ ). Let us first prese n t the expansion of the relev ant mono drom y matrix: T ( λ → ∞ ) ∝ D N . . . D 1 + 1 λ N X n 6 = i =1 D N . . . D i +1 A i D i − 1 . . . D 1 + D N . . . D n +1 ˜ A n D n − 1 . . . D 1 ! + 1 λ 2 X i>j D N . . . D i +1 A i D i − 1 . . . D j +1 A j . . . D 1 + 1 λ 2 X n>j D N . . . D n +1 ˜ A n D n − 1 . . . D j +1 A j . . . D 1 + 1 λ 2 X j >n D N . . . D j +1 A j D j − 1 . . . D n +1 ˜ A n . . . D 1 + . . . (4.6) The tec hnical details are omitted for brevity , and we directly pro vide the final expressions : log t ( λ ) = 1 λ H 1 + 1 λ 2 H 2 + 1 λ 3 H 3 + . . . (4.7) 8 W e shall write do wn here the first three terms of the expansion, whic h after some tedious computations ar e given b y ( the express ions b elo w hold at the quan tum lev el as well): H 1 = X j 6 = n N j + α n H 2 = − X j 6 = n, n − 1 x j +1 X j − 1 2 X j 6 = n N 2 j − x n +1 X n − 1 − β n X n − 1 + γ n x n +1 − α 2 n 2 H 3 = − X j 6 = n, n ± 1 x j +1 X j − 1 + X j 6 = n, n − 1 ( N j + N j +1 ) x j +1 X j + 1 3 X j 6 = n N 3 j + ˜ x n,n +1 N n − 1 X n − 1 + ˜ X n,n − 1 x n +1 N n +1 + α n ˜ x n,n +1 X n − 1 + α n ˜ X n,n − 1 x n +1 − ˜ x n,n +1 X n − 2 − x n +2 ˜ X n,n − 1 + α n 3 3 (4.8) where w e define ˜ x n,n +1 = x n +1 + β n ˜ X n,n − 1 = X n − 1 − γ n . (4.9) It is clear that as w e consider higher orders in the expansion, the terms asso ciated to the defect become less and less lo cal. And although the defect is attached to a particular site n , its effect to higher in tegrals of motion b ecomes highly non-lo cal. A similar behav ior is naturally exp ected when deriving the relev an t Lax pairs a s will b e t r ansparen t in the subseque n t section. 4.2 The asso ciated Lax pair In this case one has to distinguish v ario us cases due to the pres ence of the impurity . More precisely as w e consider higher order express ions w e need to tak e into accoun t more and more p oints around the defect in order to include all the p ossible in teractions. F or instance, to deriv e A (2) w e consider the “bulk” p o ints and separately the p oint n, n + 1. F or A (3) w e separately ev aluate t he op erator for the p oin ts n, n ± 1 , n + 2 an so on. Indeed, the main observ ation is that the presence of the defect describ ed b y t he La x op erato r ˜ L n induces non- trivial “ b oundar y” t yp e effects on to the neigh b oring operat o rs A j around the defec t p oint. 9 More precisely , the generic expression for A j the sites n, n + 1 o r instance are given as: A n ( λ, µ ) = t − 1 ( λ ) λ − µ L n − 1 ( λ ) . . . L 1 ( λ ) L N ( λ ) . . . ˜ L n ( λ ) A n +1 ( λ, µ ) = t − 1 ( λ ) λ − µ ˜ L n ( λ ) . . . L 1 ( λ ) L N ( λ ) . . . L n +1 ( λ ) (4.10) and so on for p oin ts around the defect. The non-trivial “b oundary” effects are due to t he fact that the ˜ L op erator is lo cated near or on the edges of the sequence of the Lax op erators in the latter expressions. After some quite t edious computations w e conclude that: the Lax pair A (1) j remains the same as in (3.1 9) for all sites, A (2) j for j 6 = n, n + 1 is giv en b y expression (3.19), whereas A (2) n = µ β n + x n +1 − X n − 1 0 ! , A (2) n +1 = µ x n +1 γ n − X n − 1 0 ! (4.11) Also A (3) j for j 6 = n, n ± 1 , n + 2 is given b y (3.19) and: A (3) n − 1 = µ 2 + x n − 1 X n − 2 µx n − 1 + ˜ x n,n +1 − N n − 1 x n − 1 − µX n − 2 − X n − 3 + N n − 2 X n − 2 − X n − 2 x n − 1 ! A (3) n = µ 2 + ˜ x n,n +1 X n − 1 µ ˜ x n,n +1 + x n +1 − N n +1 x n +1 + f − µX n − 1 − X n − 2 + N n − 1 X n − 1 − ˜ x n,n +1 X n − 1 ! A (3) n +1 = µ 2 + x n +1 ˜ X n,n − 1 µx n +1 + x n +2 − N n +1 x n +1 − µ ˜ X n,n − 1 − X n − 1 + N n − 1 X n − 1 + g − ˜ X n,n − 1 x n +1 ! A (3) n +2 = µ 2 + x n +2 X n +1 µx n +2 + x n +3 − N n +2 x n +2 − µX n +1 − ˜ X n,n − 1 + N n +1 X n +1 − X n +1 x n +2 ! (4.12) where w e define f = x n +2 − x n +1 − α n ( β n + 2 x n +1 ) g = X n − 1 − X n − 2 − α n ( γ n − 2 X n − 1 ) . (4.13) Ha ving b een able to explicitly deriv e the first in tegrals of motion a s w ell as the asso ciated Lax pairs w e ma y no w iden tify the corresp onding difference equations of motion, and c hec k the consistency of the approache s follo w ed. Indeed, b oth descriptions, i.e. the Hamiltonian as w ell as the zero curv ature condition pro vide as exp ected the same equations of motion. 10 Let us now fo cus on the third c harge, and extract the relev an t equations of motion. These are giv en for j 6 = n, n ± 1 , n ± 2 b y equations (3.21 ), whereas fo r the p oin ts around the impurit y w e obtain: ˙ x n − 2 = ˜ x n,n +1 − 2 x n − 1 N n − 2 − x n − 1 N n − 1 + x n − 2 N 2 n − 2 + X n − 3 x 2 n − 2 + x n − 1 ˙ X n − 2 = − X n − 4 + 2 X n − 3 N n − 2 + X n − 3 N n − 3 − X n − 2 N 2 n − 2 − x n − 1 X 2 n − 2 − X n − 3 ˙ x n − 1 = x n +1 − 2 ˜ x n,n +1 N n − 1 − N n +1 x n +1 + x n − 1 N 2 n − 1 + x 2 n − 1 X n − 2 + ˜ x n,n +1 + f ˙ X n − 1 = − X n − 3 + 2 X n − 2 N n − 1 + X n − 2 N n − 2 − X n − 1 N 2 n − 1 − ˜ x n,n +1 X 2 n − 1 − X n − 2 ˙ x n +1 = x n +3 − 2 x n +2 N n +1 − x n +2 N n +2 + x n +1 N 2 n +1 + x 2 n +1 ˜ X n,n − 1 + x n +2 ˙ X n +1 = − X n − 1 + 2 ˜ X n,n − 1 N n +1 − X n − 1 N n − 1 − X n +1 N 2 n +1 − x n +2 X 2 n +1 − ˜ X n,n − 1 + g ˙ x n +2 = x n +4 − 2 x n +3 N n +2 − x n +3 N n +3 + x n +2 N 2 n +2 + x 2 n +2 X N +1 + x n +3 ˙ X n +2 = − ˜ X n,n − 1 + 2 X n +1 N n +2 + X n +1 N n +1 − X n +2 N 2 n +2 − x n +3 X 2 n +2 − X n +1 . (4.14) P articular attention is giv en to the defect p oint. In this case one has to tak e in to accoun t the defect degrees of freedom and the exc hange relations among the elemen ts α, β , γ , δ when considering the equations of motion from the Hamiltonian. F rom the zero curv atur e condition on the other hand one has to bear in mind that exactly on the defect p oin t the L -op erator is mo dified to ˜ L , th us the condition ma y b e rewritten as: ˙ ˜ L n ( λ ) = A n +1 ( λ ) ˜ L n ( λ ) − ˜ L n ( λ ) A n ( λ ) (4.15) and the entailed equations of motion fo r the defect p oin t are giv en as: ˙ α n = − β n N n − 1 X n − 1 − γ n x n +1 N n +1 + β n X n − 2 + γ n x n +2 − α n β n X n − 1 − α n γ n x n +1 ˙ β n = 2 α n x n +1 N n +1 − 2 α n x n +2 + 2 β n x n +1 X n − 1 + β 2 n X n − 1 − β n γ n x n +1 + 2 α 2 n x n +1 + α 2 n β n ˙ γ n = 2 α n X n − 1 N n − 1 − 2 α n X n − 2 − 2 γ n x n +1 X n − 1 − γ n β n X n − 1 + 2 α 2 n X n − 1 + γ 2 n x n +1 − α 2 n γ n . T aking the contin uum limit of the discrete mo del under study is a significan t asp ect of the whole pro cess. It is an essen tial step tow ards understanding ho w in tegrabilit y can b e preserv ed in the con tin uum case. There is a discuss ion on the con tin uum NLS mo dels in [8], but there is no convin cing argumen t as far as we can understand on the issue of integrabilit y . 11 Both descriptions i.e. the Hamiltonian v ersus t he Lax pair formulation are needed in order to obtain a complete view of the problem at hand. It is tec hnically more conv enien t in man y cases to use the information from the Lax pair form ulation or vise v ersa, how ev er in most cases com binat io n of b oth desc riptions helps to completely describe the problem es p ecially when dealing with the con tin uum v ersion of a lattice in tegrable mo del. 5 The co n tin u um limit: a firs t glance In or der to pro ceed with the contin uum limit of the discrete NLS mo del let us first in tr o duce the spacing parameter ∆ in the L -matrix of the discrete NLS mo del as w ell as in the ˜ L matrix of the defect (index free notatio n): L ( λ ) = 1 + ∆ λ − ∆ 2 xX ∆ x − ∆ X 1 ! (5.1) ˜ L ( λ ) = ∆ λ + α β γ δ ! (5.2) where w e no w define: α = − δ = 1 2 cos(2∆ θ ) , β = 1 2 sin(2∆ θ ) e 2 iφ , γ = 1 2 sin(2∆ θ ) e − 2 iφ , (5.3) w e also define: θ e 2 iφ = y , θe − 2 iφ = Y , (5.4) the latter identific ations will b e used in the f ollo wing a nalysis. Notice t hat the sp ectral parameter λ is a lso suitably renormalized to ∆ λ in b oth L a nd ˜ L matrices in o rder to form ulate a sensib le con tinuum limit pro cess compatible also with the con tin uum linear algebra (se e also [19]). Moreov er, suc h a renormalization is necessary if w e wish the whole pro cess to b e compatible with the so called “p o w er counting” argumen t in tro duced in [19 ]. Ha ving in tro duced the a ppropriate spacing parameter in the L -op erato r s ab o v e w e ma y no w consider the con tin uum limit of the integrals of motio n of discrete NLS mo dels and the asso ciated Lax pairs. Before obtaining the con tinuum limit let us first in tro duce the follo wing notation. In par t icular, w e set: x j → x − ( x ) , X j → X − ( x ) , 1 ≤ j ≤ n − 1 , x ∈ ( −∞ , x 0 ) 12 x j → x + ( x ) , X j → X + ( x ) , n + 1 ≤ j ≤ N , x ∈ ( x 0 , ∞ ) . (5.5) where x 0 is the defect p osition in the con tin uum theory . Note also that in order to p erform the con tin uum limit we b ear in mind that: ∆ n − 1 X j =1 f j → Z x − 0 −∞ dx f − ( x ) ∆ N X j = n +1 f j → Z ∞ x + 0 dx f + ( x ) . (5.6) The con tin uum limit o f the first inte gral of mot io n is then giv en as: H (1) = − Z x − 0 −∞ dx x − ( x ) X − ( x ) − Z ∞ x + 0 dx x + ( x ) X + ( x ) . (5.7) Notice that in the first in tegral w e considered terms prop ort ional to ∆, whereas in the second in tegral the first non trivial contribution to the con tinuum limit is of o rder ∆ 2 . The resp ectiv e con tinuum quan tity reads t hen as H (2) = − Z x − 0 −∞ dx x − ′ ( x ) X − ( x ) − Z ∞ x + 0 dx x + ′ ( x ) X + ( x ) + x − ( x 0 ) X − ( x 0 ) − x + ( x 0 ) X − ( x 0 ) + x + ( x 0 ) Y ( x 0 ) − y ( x 0 ) X − ( x 0 ) + 1 2 y ( x 0 ) (5.8) the prime denotes deriv ative with resp ect to x . Note that in t he con tinuum limit 2 : L ( λ ) ∼ I + ∆ U ( λ ) (5.10) also the contin uum zero curv ature condition with Lax pair U , V takes the from: ˙ U − V ′ + h U , V i = 0 . (5.11) 2 Notice that the ˜ L matr ix in the contin uum limit may be expr e ssed as: ˜ L ∼ σ 3 + ∆ ˜ U (5.9) σ 3 the familiar P a uli matrix. W e co uld have chosen instead ¯ L = σ 3 ˜ L , whic h a lso s atisfies the quadratic relation (3.3), and has the exp ected contin uum b ehavior (5.10). Such a choice would slightly mo dify the defect terms in the lo cal integrals of motion. Note that s uc h modifications can be suitably implemen ted in the co n tinuum mono dromy matrix, but we shall disc us s this matter in detail elsewhere. 13 The Lax pair asso ciated to the first in tegral in quite t r ivial a nd coincides with the o ne in (3.19). The Lax pair asso ciated to the second in tegral of motion is g iven b y the follo wing expressions : V (2) ( µ, x ) = µ x − ( x ) − X − ( x ) 0 ! x ∈ ( −∞ , x − 0 ] , V (2) ( µ, x ) = µ x + ( x ) − X + ( x ) 0 ! x ∈ ( x + 0 , ∞ ) V (2) ( µ, x 0 ) = µ x + ( x 0 ) + y ( x 0 ) − X − ( x 0 ) 0 ! , V (2) ( µ, x + 0 ) = µ x + ( x 0 ) Y ( x 0 ) − X − ( x 0 ) 0 ! . (5.12) Due to contin uity r equiremen ts at the p oin ts x + 0 , x − 0 (see also a similar argumen t in [18], w e end up with the f ollo wing sewing conditions asso ciated to the defect p oint: y ( x 0 ) = x − ( x 0 ) − x + ( x 0 ) , Y ( x 0 ) = X − ( x 0 ) − X + ( x 0 ) . (5.13) Notice that the con tin uity argumen t may b e successfully applied to the points around the defect, how ev er as exp ected a discon tinuit y (jump) is observ ed exactly o n the defect p oint. It should b e emphasized that the L op erator is alt ered at x 0 (i.e. L → ˜ L ), leading to mo dification or discon tinuit y in the zero curv ature condition at x 0 , whic h accordingly lead to adjustmen ts in the induced equations o f motion. It is also quite straigh tforw ar d to show that if the sewing conditions (5.13) are v alid then {H 1 , H 2 } = 0 , (5.14) whic h is a first go o d indication of t he preserv ation o f the in tegrability in the contin uum case as w ell. Ho w ev er, this is someho w “on shell” inf o rmation, giv en that one requires (5.13) in order to pro v e the P oisson comm utativit y (5.14). Moreov er, the constrain ts (5 .1 3) pro vide a first hin t on the existence of a n underlying non-ultra lo cal algebra (see e.g. [20 ]) asso ciated to the defect p oin t. And although in the discrete case one deals with an ultra lo cal algebra there is a n indication that in the con tin uum limit one has to consider a generalized non-ultra lo cal algebra in order to efficien tly describ e the p oin t lik e defect at x 0 . Here, we only provide a first glimpse on the con tin uum limit of the discrete NLS mo dels. T o obta in the con tin uum coun terparts of the higher in tegrals of motion a nd the asso ciated 14 Lax pa irs requires subtle manipulatio ns. Suc h an explicit construction is b ey ond the intended scop e of the presen t in v estigation, how eve r w e shall analyze this in triguing issue in full detail in future works [21]. 6 Discuss ion Let us briefly summarize the main findings of the presen t study . The main aim of this w ork w as the in ve stigation of the discrete NLS mo del in the presen t of an ultra lo cal in tegra ble defect. Based on purely algebraic considerations w e w ere able to extract the first c harges in in volution for the discrete mo del. The mo del is b y construction integrable given that the bulk L matr ices as w ell as the Lax matrix asso ciated to the defect are required to ob ey the same P oisson brac k et structure. Then, again b y exploiting the underly ing algebra, w e extracted the asso ciated Lax pairs. P articular a t ten tion w as giv en to the construction of the Lax pairs around the defect p oint. It tur ned out that the b eha vior of more and mor e p oints around t he defect is affected as w e mov e to higher order expressions. Ha ving this information at our disp osal w e w ere able to deriv e the sets of the difference equations of motion. Finally , we pro vided a first insigh t of the contin uum b ehavior of the system by considering the contin uum limit of the first t wo in tegrals of motion. This led to certain sew ing o r compatibilit y conditions t ha t ensure P oisson comm uta tivit y of the fir st t w o in tegra ls of motio n at the contin uum case as w ell. It is w orth noting that at t his stage it is difficult to conclude whether or not our r esults corresp ond to t he results of e.g. [8] or if any comparison whatso ev er can b e made, give n that the issue of in tegr a bilit y is still o p en in [8]. The sys tematic con t in uum limit of discrete mo dels in the presence of integrable defects in the spirit of [19] is our next t a rget [21], and it will pro vide a deep er understanding on the connection with earlier works. In fact, this systematic pro cess will lead to the deriv ation of the con tinuum limit of hig her integrals of motion, suc h as H (3) and t he corresp onding Lax pairs, as we ll as the asso ciated constraints (sewing conditio ns). Note that the entailed higher constrain ts will in v olv e as exp ected spa- tial deriv ativ es. Moreo ver, P oisson comm ut a tivit y of all the entailed c har g es needs to b e explicitly c hec ked so that we can claim that integrabilit y holds. Compatibility of the higher sewing conditions should b e a lso explic itly c heck ed. These are highly non-trivial tec hnical p oin ts, and will b e presen ted in full detail in future inv estigations. 15 Similar ideas ma y b e put forw ard in the case of o ther w ell known proto t yp e models suc h as the Heisen b erg mo del (see e.g. [1 1]) aiming also at t he in v estigation of the corresp onding con tinuum theories. In an y case, a detailed analysis on con tin uum in tegrable mo dels in the presence of defects turns out to b e a fundamental issue, whic h will b e addressed in forthcoming publications. Ac kno wledgemen t s I am indebted t o J. Av a n fo r illuminating discussions, us eful suggestions, and ongoing collab oration on this sub ject. References [1] G. D elfino, G. Mussardo and P . Simonetti, Ph ys. Lett. 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