Correlation functions of the open XXZ chain II

LPENSL-TH-03/08 Correlatio n functions of t h e op en XXZ chain I I N. Kitanine 1 , K. K. Kozlo wski 2 , J. M. Maillet 3 , G. Niccoli 4 , N. A. Sla vno v 5 , V. T erras 6 Abstract W e derive compact multiple integral formulas for several physical spin cor- relation functions in the semi-infinite XXZ chain with a longitudinal boundar y magnetic field. Our form ulas follow from several effective re-summations of the m ultiple integral repres e n tation for the elementary blo cks obtaine d in our pre - vious article (I). In the free fer mion po int we c ompute the lo cal mag netization as well as the density of energ y profiles. These qua n tities, in addition to their bulk b ehavior, exhibit F rie de l type oscilla tions induced by the b oundar y; their amplitudes depend on the b oundary magnetic field and decay alg ebraically in terms of the distance to the bo undary . 1 In tro duction The Hamilto nian of the Heisen b erg XXZ sp in -1 / 2 fin ite c hain [1] with d iagonal b ound ary conditions (namely with longitudinal b ou n dary m agnetic fields ) is d efi ned as [2, 3] H = M − 1 X m =1  σ x m σ x m +1 + σ y m σ y m +1 + ∆  σ z m σ z m +1 − 1  + h − σ z 1 + h + σ z M . (1.1) 1 LPTM, Unive rsit´ e de Cergy-P ontois e et CNRS, F r ance, k itanine@ptm.u-cergy . fr 2 Lab oratoire de Ph ysique, Univ ersit ´ e de Lyo n, ENS Lyon et CNRS, F rance, k aro l.kozl ow ski@ens- lyo n.fr 3 Lab oratoire de Physique, Universit ´ e de Lyon, ENS Lyon et CNR S, F rance, maillet@ens-lyon.fr 4 DESY Theory group, Hamburg, German y , Giuliano.Niccoli@ens-ly on.fr 5 Steklov Mathematical Institute, Mosco w, Russia, n sla vnov@mi.ras. ru 6 Lab oratoire de Physique, Univ ersit ´ e de Lyon, ENS Lyon et CNRS, F rance, veronique.terras@ens- lyo n.fr, on leav e of absence from LPT A, Universit ´ e Montpellier I I et CNRS , F rance 1 This is a linear op erator acting in the q u an tum space H = M ⊗ m =1 H m , H m ≃ C 2 , of dimension 2 M of the c hain. In this expression, σ ± m , σ z m denote local spin op erators (acting as Pa uli matrices) at site m , ∆ is the anisotropy parameter and h ± are the b ound ary (longitudinal) magnetic fields. W e hav e recentl y d ev elop ed a metho d to compute the so-called elemen tary blo c ks of correlation functions for this mo del (see [4], th at w e refer to as Pap er I in the follo wing) in the fr amew ork of the (algebraic) Bethe an s atz [5–18] for b oundary in- tegrable s y s tems [2, 3, 1 9– 31]. The results essentia lly agree with pr evious expressions deriv ed fr om the v ertex op erato r approac h [32, 33]. The pur p ose of the present pap er is to obtain th e p hysical spin correlation fu nctions for th is m o del, in particular, the one p oin t functions for the lo cal spin op erators at distance m from the b oundary as w ell as sev eral tw o p oint fun ctions (like b oun dary-bulk correlation functions). There are n um erous p h ysical interests in suc h quan tities that can b e measured in actual exp eriments [34–46]. In muc h the same wa y as in the bulk case [47–52], the computation of the physical correlation fu nctions amoun ts to obtain effecti v e re-summ ations of the m ultiple inte- gral represen tations deriv ed for th e elemen tary blo cks. F or example, the one p oin t functions at distance m f rom the b oundary , s uc h as the lo cal m agnetiza tion h σ z m i , can b e written as a sum of 2 m elemen tary blo c ks. W e will sho w ho w to obtain com- pact expressions for suc h ob jects, typicall y in v olving th e sum of only m terms, eac h con taining multiple integ rals whose int egran ts h a ve a s tr ucture similar to the one of the elemen tary blo cks. I n the free fer m ion p oin t we are able to compute these m u lti- ple integrals (and hence the corresp ond ing correlations functions) almost completely b y reducing them to single in tegrals. F or instance th e lo cal magnetization and the densit y of energy profi les (a quantit y of inte rest in th e study and the understanding of th e in terpla y b et we en qu an tu m en tanglemen t and qu an tum criticalit y [53–61]) are expressed as single in tegrals. Hence, their asymp totic b eha vior at long distance m from the b oundary can b e explicitly ev aluated. In addition to the bulk constant v alue they exh ib it F riedel t yp e oscillations [44–46, 60, 61], algebraically d eca ying with the distance m , their amplitud es b eing rational functions of the b oundary magnetic field, in agreemen t with fi eld theory predictions [38–42, 62–70]. W e start this p ap er with a short tec hnical int ro du ction concerning th e algebraic Bethe ans atz appr oac h to the op en XXZ spin -1 / 2 c h ain sub ject to diagonal b oundary magnetic fi elds. This pr eliminary section is follo wed in Section 3 by a reminder of the metho d prop osed in [4] to compute correlation functions of op en integrable mo dels in the framew ork of algebraic Bethe ansatz. In Section 4 we obtain form ulae for the action of lo cal op erators on arbitrary b ou n dary states in a form suitable for taking later on the thermo dynamic limit. Using these results, w e deriv e a ser ies represent ation for th e generating fu nction hQ m ( κ ) i of b ulk-b ound ary σ z correlation functions in Secti on 5. This form ula is the b ound ary analogue of the original series [49] in the bulk case. In Section 6, w e obtain a f orm ula for hQ m ( κ ) i alternativ e to the one inferred in Section 5. W e also giv e m ultiple in tegral repr esen tations for h σ + m +1 σ 1 1 i and for the lo cal densit y of energy . These formulae are obtained by a d irect resu mmation 2 of the corresp onding elemen tary blo c ks. It is worth stressing th at we actually h a ve t wo represent ations f or their in tegrand. The fi rst one is in the spirit of the b ulk case [52] and inv olv es the Izergin determinant representa tion [71] for the partition function of the six v ertex mo del with domain wall b ound ary conditions. The second one in v olv es the Tsuc h iy a [72] determinant representa tion for the p artition fu n ction of the six v ertex mo del with reflecting ends. The next section is dev oted to th e free ferm ion p oint . F or that case are able to reduce the m ultiple integrals to one d imensional ones. This allo ws us to write the leading asymptotics of the local magnetization and of the d ensit y of energy profiles as we ll as of the h σ + m +1 σ − 1 i correlation fun ction. Our conclusions are presented in the last section. 2 The op en XXZ spin- 1 / 2 c hain The sp ectrum of H can b e obtained by algebraic Bethe ansatz (ABA) [3]. The cen tr al to ol of this m etho d is the b oun d ary mono dromy matrix, whic h will b e defined after we int ro du ce some n ecessary notations. Here and in the follo win g w e adopt the stand ard parameterizat ions ∆ = cosh η and h ± = sinh η co th ξ ± . Let R : C → En d( V ⊗ V ), V ≃ C 2 , b e the R -matrix of the six-v ertex mo d el, obtained as the trigonometric solution of the Y ang-Baxter equation: R ( u ) = sinh ( u + η ) b R ( u ) , with b R ( u ) =     1 0 0 0 0 b ( u ) c ( u ) 0 0 c ( u ) b ( u ) 0 0 0 0 1     , (2.1) and b ( u ) = sinh u sinh( u + η ) , c ( u ) = sinh η sinh( u + η ) . (2.2) The bulk mono dr omy matrix T ( λ ) ∈ End( V 0 ⊗ H ), V 0 ≃ C 2 , is d efined as an ordered pro du ct of R matrices: T 0 ( λ ) = R 0 M ( λ − ξ M ) . . . R 01 ( λ − ξ 1 ) =  A ( λ ) B ( λ ) C ( λ ) D ( λ )  [0] . (2.3) The subscript 0 labels here the t w o-dimensional auxiliary space V 0 , whereas subscripts m runn in g from 1 to M refer to the quantum sp aces H m of th e c hain. Be sides, w e attac h an inhomogeneit y parameter ξ m to eac h site m of the c h ain. W e recall that T ( λ ) satisfies the Y ang-Baxt er algebra, on V 0 ⊗ V 0 ′ ⊗ H : R 00 ′ ( λ − µ ) T 0 ( λ ) T 0 ′ ( µ ) = T 0 ( λ ) T 0 ′ ( µ ) R 00 ′ ( λ − µ ) . (2.4) Let us also introdu ce the t w o b oundary matrices, K ± ( λ ) = K ( λ ± η / 2; ξ ± ), where K ( λ ; ξ ) is the 2 × 2 matrix acting on the auxiliary sp ace: K ( λ ; ξ ) =  sinh ( λ + ξ ) 0 0 sin h ( ξ − λ )  [0] . (2.5) 3 The b oundary mon o drom y matrix U ( λ ) [3] is built out of a pro duct of T ( λ ) and K + ( λ ), namely 1 U t 0 0 = T t 0 0 ( λ ) K t 0 + ( λ ) b T t 0 0 ( λ ) =  A ( λ ) B ( λ ) C ( λ ) D ( λ )  t 0 [0] , (2.6) where b T 0 ( λ ) = R 10 ( λ + ξ 1 − η ) . . . R M 0 ( λ + ξ M − η ) = ( − 1) M M Y j =1 [sinh ( λ + ξ j ) sin h ( λ + ξ j − η )] T − 1 0 ( − λ + η ) . (2.7) This b oundary mono drom y matrix satisfies the reflection algebra fir st in tro duced in [20]: R 00 ′ ( − λ + µ ) U t 0 0 ( λ ) R 00 ′ ( − λ − µ − η ) U t 0 ′ 0 ′ ( µ ) = U t 0 ′ 0 ′ ( µ ) R 00 ′ ( − λ − µ − η ) U t 0 0 ( λ ) R 00 ′ ( − λ + µ ) . (2.8) The comm uting charges of the XXZ spin-1 / 2 c hain with diagonal b ound ary con- ditions are r ealized by the one-parameter family of transfer matrices: T ( λ ) = tr 0 [ U 0 ( λ ) K − ( λ )] , (2.9) and th e Hamiltonian (1.1) is obtained in terms of the d eriv ativ e d T ( λ ) d λ | λ = η/ 2 in the homogeneous case ξ i = η / 2, i = 1 , . . . , M . Common eigenstate s of all transfer matrices (and th us of the Hamiltonian (1.1) in the h omogeneous case) can b e constructed by successive actions of B ( λ ) op erators on the r eference state | 0 i which is th e ferr omagnetic state with all the s pins up. More precisely , the state 2 | { λ } n 1 i b ≡ B ( λ 1 ) . . . B ( λ n ) | 0 i (2.10) is a common eigenstate of the transfer matrices if th e set of sp ectral parameters { λ } n 1 ≡ { λ j } 1 ≤ j ≤ n is a solution of the Bethe equations y j ( λ j ; { λ } n 1 ) = y j ( − λ j ; { λ } n 1 ) , j = 1 , . . . , n, (2.11) where y j ( x ; { λ } n 1 ) = ˆ y ( x ; { λ } n 1 ) s ( λ j , x − η ) , ˆ y ( x ; { λ } n 1 ) = − a ( x ) d ( − x ) sinh ( x + ξ + − η / 2) sinh ( x + ξ − − η / 2) × n Y l =1 s ( x − η , λ l ) . (2.12) 1 Note that it corresp onds to the matrix U + of our previous article (I). Since we consider only the ‘+’ case in th e present article, we do not sp ecify it in the notations. 2 In order to lig hten the formulae w e ha ve slightly c hanged the notation with respect to the one in P ap er I. Namely , the vector | { λ } n 1 i b corresponds to | ψ + ( { λ } ) i in [4]. Su c h a bound ary state should in p articular b e distinguished from the corresp onding bulk state that w e merely denote | { λ } n 1 i . 4 Here and in the follo wing, s ( λ, µ ) denotes the function s ( λ, µ ) = sinh ( λ + µ ) s in h ( λ − µ ) , (2.13) and the f unctions a ( λ ) and d ( λ ) stand resp ectiv ely for the eigen v alues of the bulk op erators A ( λ ) and D ( λ ) on the pseudo-v acuum | 0 i : a ( λ ) = M Y i =1 sinh ( λ − ξ i + η ) , d ( λ ) = M Y i =1 sinh ( λ − ξ i ) . (2.14) Of course it is also p ossible to implement the Bethe ansatz starting f r om the dual state h 0 | and acting on it with C ( λ ) op erators: b h{ λ } n 1 | ≡ h 0 |C ( λ 1 ) . . . C ( λ n ) . (2.15) The d escription of the groun d state of H in the h alf-infinite c hain dep ends on the regime. One should distinguish the t wo domains − 1 < ∆ ≤ 1 (massless regime) and ∆ > 1 (massiv e regime): α j = λ j , ζ = iη > 0 , ξ − = − i ˜ ξ − , with − π 2 < ˜ ξ − ≤ π 2 , for − 1 < ∆ ≤ 1 , α j = iλ j , ζ = − η > 0 , ξ − = − ˜ ξ − + iδ π 2 , with ˜ ξ − ∈ R , for ∆ > 1 , where δ = 1 f or | h − | < sinh ζ and δ = 0 otherwise. Thus, to a giv en set of ro ots { λ j } corresp onds a set of v ariables { α j } giv en b y the previous c hange of v ariables. Note that the nature of the ground s tate r ap id ities dep ends on the v alue of the b oundary field h − . Indeed, when ˜ ξ − < 0 or ˜ ξ − > ζ / 2, the groun d state of the Hamilto nian (1.1) is giv en in b oth r egimes by th e maxim um num b er N of ro ots λ j corresp onding to real (p ositiv e) α j suc h that cos p ( λ j ) < ∆. In the thermo dynamic limit M → ∞ , these ro ots λ j form a dense distribution on an in terv al [0 , Λ] of the real or imaginary axis. Their d ensit y ρ ( λ j ) = lim M →∞ [ M ( λ j +1 − λ j )] − 1 (2.16) satisfies th e int egral equation 2 π ρ ( λ ) + Λ Z − Λ i sin h(2 η ) s ( λ − µ, η ) ρ ( λ ) d λ = 2 i sinh η s ( λ, η / 2) , (2.17) with Λ = + ∞ in th e massless regime, and Λ = − iπ / 2 in the massive one. The densit y can b e expressed in terms of usual fu nctions: ρ ( λ ) =        1 ζ cosh ( π λ/ζ ) , − 1 < ∆ < 1; i π Q n ≥ 1  sinh nζ cosh n ζ  2 θ 3 ( iλ ; − ζ ) θ 4 ( iλ ; − ζ ) , 1 < ∆ . (2.18) 5 Ho wev er, wh en 0 < ˜ ξ − < ζ / 2, the g round state a lso admits a root ˇ λ (corresp onding to a complex ˇ α ) which tend s to η / 2 − ξ − with exp onen tially small corrections in the large M limit. I n that case, th e density of r eal ro ots is s till giv en by the solution of (2.17). 3 The ABA approac h to correlat ion functions A zero temp erature correlation function is the normalized exp ectation v alue, in the groun d state of the Hamilto nian (1.1), of some lo c al 3 op erator O m , hO m i = b h{ λ } N 1 |O m | { λ } N 1 i b b h{ λ } N 1 | | { λ } N 1 i b , (3.1) where the parameters λ are the solutions of the groun d state Bethe equations. In order to compute suc h a correlation function, one sh ould first d eriv e the ac- tion of the corresp ondin g local op erator on the b oundary state | { λ } N 1 i b , and then ev aluate the resulting s calar pro d ucts. W e ha v e constructed in [4] a metho d to solv e this problem. This metho d is based on a revisited v ersion of the quan tum inv erse problem, fir st introdu ced in [47, 73] for the XXZ spin c hain w ith p er io d ic b oundary conditions. Once a local op erator is reconstructed in terms of the ent ries of the bulk mono dromy matrix, its actio n on b oundary s tates can then b e computed thanks to the decomp osition of b ound ary states in terms of b ulk states and to the Y ang-Baxter comm u tation r elations. W e shall no w recall the main p oin ts of our metho d. 3.1 The bulk in verse problem revisited Prop osition 3.1 (Solution of the bulk in v erse problem) [47, 73] L et E ij m b e an elementary matrix acting non-trivial ly only on the m th site of the chain, then E ij m = m − 1 Y k =1 ( A + D ) ( ξ k ) tr  T 0 ( ξ m ) E ij 0  m Y k =1 ( A + D ) − 1 ( ξ k ) . (3.2) Note that, th anks to the crossing symmetry of the R -matrix, one can r ecast the in v erse of th e bulk tran s fer matrix at inhomogeneit y parameter ( A + D ) − 1 ( ξ k ) in terms of the transfer matrix at shifted p arameter ( A + D ) ( ξ k − η ), n amely ( A + D ) − 1 ( ξ k ) = ( A + D ) ( ξ k − η ) a ( ξ k ) d ( ξ k − η ) . (3.3) It is worth p ointing out that the p r o ducts of elementa ry matrices on the fi rst m sites of the c h ain define a b asis in the space of lo cal op erators O m , so th at (3.2) allo ws one to defin e a reconstruction for all su c h op erators. Ho wev er, this reconstruction is 3 i.e . acting non - trivially only in m ⊗ k =1 H k . 6 esp ecially con v enien t when one w ants to obtain the action on a bulk Bethe state; indeed, in such a case, the pr o duct of bu lk transfer matrices merely pr o duces a nu- merical factor. This is no longer the case w hen one acts on a b oundary Bethe state. The theorem b elo w allo ws one to reconstruct lo cal op erators in a wa y adapted to an action on b ound ary states. Theorem 3.1 [4] F or any set of inhomo g e neity p ar ameters { ξ i 1 , . . . , ξ i n } , the pr o d- uct of bulk op er ators T ǫ i n ǫ ′ i n ( ξ i n ) . . . T ǫ i 1 ǫ ′ i 1 ( ξ i 1 ) T ¯ ǫ i 1 ¯ ǫ ′ i 1 ( ξ i 1 − η ) . . . T ¯ ǫ i n ¯ ǫ ′ i n ( ξ i n − η ) (3.4) vanishes if, f or some k ∈ { i 1 , . . . , i n } , ǫ k = ¯ ǫ k . Th us w e ha ve : Corollary 3.1 A pr o duct of elementary matric es acting on the first m sites of the chain c an b e expr esse d as a single monomial in the entries of the bulk mon o dr omy matrix: E ǫ 1 ǫ ′ 1 1 . . . E ǫ m ǫ ′ m m = m Y i =1  a ( ξ i ) d ( ξ i − η )  − 1 × T ǫ ′ 1 ǫ 1 ( ξ 1 ) . . . T ǫ ′ m ǫ m ( ξ m ) T ¯ ǫ m ¯ ǫ m ( ξ m − η ) . . . T ¯ ǫ 1 ¯ ǫ 1 ( ξ 1 − η ) (3.5) with ¯ ǫ i = ǫ ′ i + 1 (mo d 2) . This result r ep resen ts a strong simplification. Indeed, it means that, ov er the 2 m monomials app earing in the reconstruction of a lo cal op erator (3.2), only one is non-zero. W e shall no w explain h o w to compute the action of this non-v anishing monomial on an arb itrary (bulk or b oun dary) state. 3.2 Action on bulk and b oundary states Before stating the lemma wh ich explains h o w to deriv e the action of the former monomial on a bulk state, we r ecall that the action of A ( µ ) or D ( µ ) on a bulk s tate | { λ } N 1 i ≡ Q N j =1 B ( λ j ) | 0 i pro duces t wo kinds of terms: th e dir e ct term , where all rapidities remain unchanged, and indir e ct terms where one λ j is replaced b y µ . Lemma 1 ( Action on a bulk state) [4] The action on a b u lk state | { λ } N 1 i of a string of op er ators O ǫ ′ i 1 ,...,ǫ ′ i n ǫ i 1 ,...,ǫ i n = T ǫ ′ i n ǫ i n ( ξ i n ) . . . T ǫ ′ i 1 ǫ i 1 ( ξ i 1 ) | {z } (1) T ¯ ǫ i 1 ¯ ǫ i 1 ( ξ i 1 − η ) . . . T ¯ ǫ i n ¯ ǫ i n ( ξ i n − η ) | {z } (2) (3.6) with ¯ ǫ l = ǫ ′ l + 1 (mo d 2) , satisfies the r estrictions: • The only non-zer o c ontributions of the tail op er ators (2) c ome fr om 7 (i) the indir e ct action of al l A ( ξ l − η ) op er ators; (ii) the dir e ct action of al l D ( ξ l − η ) op er ators. • In what c onc erns the he ad op er ators (1) , (iii) if ǫ ′ l = 1 , the action of the op er ator T ǫ ′ l ǫ l ( ξ l ) (i.e. A ( ξ l ) or B ( ξ l ) ) do e s not r e su lt in any substitution of a p ar ameter ξ i − η ; (iv) if ǫ ′ l = 2 , the action of the op e r ator T ǫ ′ l ǫ l ( ξ l ) (i.e. D ( ξ l ) or C ( ξ l ) ) substitutes ξ l − η with ξ l ; mor e over, if ther e wer e others p ar ameters ξ j − η , j 6 = l , in the initial state, they ar e stil l pr esent in the r esulting state. This lemma enab les us to compute the action of lo cal op erato rs on any (arbitrary) bulk state. In order to compu te the actio n on a b oundary state, we use the fact that the latter can b e decomp osed in terms of bulk states: Prop osition 3.2 (Boundary-bulk decomp osition) [4], [74] L et | { λ } n 1 i b b e an arbitr ary b oundary state, then i t c an b e expr esse d i n terms of bulk states as | { λ } n 1 i b = X σ i = ± i =1 ,...,n H B { σ i } ( { λ } n 1 ) | { λ σ } n 1 i , (3.7) with H B { σ i } ( { λ } n 1 ) = n Y j =1 H B σ j ( λ j ) · Y 1 ≤ r< s ≤ n sinh( λ σ r s − η ) sinh( λ σ r s ) . ( 3.8) In this expr e ssion, H B σ ( λ ) denotes the “one-p article” b oundary-bulk c o efficie nt, which c an b e written as H B σ ( λ ) = σ ( − 1) M d ( − λ σ ) sinh (2 λ + η ) sinh 2 λ sinh ( λ σ + ξ + − η / 2) . (3.9) Her e we have u se d the notations: λ r s = λ r − λ s , λ r s = λ r + λ s , (3.10) λ σ j j = σ j λ j , an d more generally { λ σ } n 1 = n λ σ j j o n 1 . (3.11) It is remark able that, b y using this decomp osition and the previous lemma, we are able to expr ess th e action of a lo cal op erator O m on an arbitrary b oundary state as a linear com bination of suc h b oun d ary states: O m | { λ } N 1 i b = X α m C α m ( { λ } ; { ξ } ) | { µ i } i ∈ α m i b , (3.12) where the su m mation is tak en ov er certain subsets { µ i } i ∈ α m of { λ } N 1 ∪ { ξ } m 1 , and where C α m are co efficient s which can b e computed generically 4 . 4 See section 5.3 of [4] for th e explicit expression in the case of a prod uct of elementary matrices. 8 3.3 F rom scalar pro ducts to correlation functions It no w remains, in ord er to obtain the correlation fun ction (3.1), to tak e the scalar pro du ct of th is r esulting combinatio n of states with the ground state b h{ λ } N 1 | . T his can b e done by using the trigonometric generaliz ation [4] of th e rational [74] form ula for the scalar pr o duct b et we en a b oun dary Bethe state and an arbitrary b oundary state. In particular, w e h a ve to ev aluate the follo wing t yp e of renormalized scalar pro du ct: S ( { λ } , { µ } ) = b h{ λ } | | { µ } i b b h{ λ } | | { λ } i b , (3.13) where the sets { λ } and { µ } are partitioned according to: { λ } = { λ a } a ∈ α − ∪ { λ b } b ∈ α + , { µ } = { λ a } a ∈ α − ∪ { ξ b } b ∈ γ + , (3.14) with | α + | = | γ + | . Here the parameters λ are the solutions of th e ground state b ound - ary Bet he equations, { ξ b } b ∈ γ + are a rbitrary inhomogeneities and α + ∪ α − is a partition of { 1 , . . . , N } . Since we are esp ecially intereste d in th e thermo d ynamic limit M → + ∞ of the correlation fun ction (3.1), w e only recall the explicit formula for the leading asymp- totic contribution of the renormalized scalar p ro duct [4]: S ( { λ } , { µ } ) =  1 M  | α + | S ( { λ } α + , { ξ } γ + ; { λ } α − ) det a ∈ α + b ∈ γ +  e S ab  . (3.15) The coefficient S ( { λ } α + , { ξ } γ + ; { λ } α − ) has b een computed in [4]. Not e that, since { λ } is a s olution of the b oundary Bethe equations, ther e is some sign arbitrariness in the exp ression of th is co efficient: indeed, it is in fact equal to S σ ( { λ } α + , { ξ } γ + ; { λ } α − ) = Q a,b ∈ α + a>b s ( λ b , λ a ) Q a,b ∈ γ + a>b s ( ξ b , ξ a ) Y a ∈ α − Q b ∈ α + s ( λ b , λ a ) Q b ∈ γ + s ( ξ b , λ a ) × Y b ∈ γ + ˆ y ( ξ b ; { λ } α + ∪ α − ) sinh(2 ξ b + η ) sinh(2 ξ b ) Y a ∈ α + sinh(2 λ σ a − η ) sinh(2 λ a ) ˆ y ( λ σ a ; { λ } α + ∪ α − ) sin h(2 λ a + η ) , (3. 16) for any v alue of σ a ∈ { + , −} , a ∈ α + , where ˆ y is the fu nction defin ed in (2.12). When M is large, the matrix elemen ts of e S reduce to e S ab ∼ M →∞ ( 2 iπ M sinh ( λ a − ξ − + η / 2) Ψ ( λ a , ξ b ) if λ a = ˇ λ , ρ − 1 ( λ a ) Ψ ( λ a , ξ b ) if λ a 6 = ˇ λ , (3.17) the corrections b eing of order O (1 / M ), and Ψ ( λ, ξ ) = ρ ( λ − ξ ) − ρ ( λ − η + ξ ) 2 sin h ( 2 ξ − η ) . (3.18) 9 The determinan t structur e of the scala r pro duct as well as p eculiarities of the co efficien ts C α m enable us to write: hO m i = 1 M m X { ν } I ⊂{ λ } N 1 ∪{ ξ } m 1 | I | = m H m ( { ν } I , { ξ } m 1 ) (1 + O (1 / M )) , (3.19) in which the co efficien t H m ( { ν } I , { ξ } m 1 ) can b e computed generically . T aking the thermo dynamic limit M → + ∞ we r ecast the sums ov er rep laced r apidities λ into in tegrals: 1 M N X i =1 X σ i = ± σ i f ( λ σ i ) − → M → + ∞ Z C d λ ρ ( λ ) f ( λ ) , ∀ f ∈ C 0 ( C ) . (3.20 ) In the b oundary mo del the con tour of in tegration C d ep ends on the anisotrop y pa- rameter ∆ and on the b oundary field h − . 4 Action of lo cal op erators on b oundary states Using Corollary 3.1, Lemma 1 and the b ound ary -b ulk decomp osition of Prop o- sition 3.2, it is easy to compu te the action of a pro d uct of elemen tary matrices of the form (3.5 ) on an arbitrary b ound ary state. This compu tation w as explicitely p erformed in [4], and enabled us there to obtain some exp ressions for th e elementa ry building blo cks of correlation functions. The aim of the present article is to obtain suc h expressions for physical correlati on functions, and in p articular for one-p oin t functions. Therefore, if w e wan t to us e the metho d recalled in Section 3, the main p r oblem is to obtain some resu mmed formulas directly for the action of the lo cal spin op erators we consider. This is the pu rp ose of the p resen t section. In the first p art of this section, w e derive the action of the op erator Q m ( κ ) ≡ m Y i =1  E 11 i + κE 22 i  = m Y i =1 ( A + κD ) ( ξ i ) m Y i =1 ( A + D ) − 1 ( ξ i ) (4.1) on arbitrary b oundary states. hQ m ( κ ) i can b e in terpreted in the b oun d ary mo del as the generating function of the magnetiza tion at a distance m from the b ound ary: h 1 − σ z m 2 i = D m ∂ κ hQ m ( κ ) i| κ =1 , (4.2) where D m is th e lattice deriv ativ e : D m u m ≡ u m +1 − u m . Then, in the second p art of this section, we giv e the formulas for th e action of the lo cal spin op erato rs E 22 m = 1 − σ z m 2 , E 12 m = σ + m and E 21 m = σ − m on arb itrary b oundary states. Note that the action of E 11 m follo ws from the fact that E 11 m = 1 − E 22 m . 10 4.1 Action of Q m ( κ ) W e start by compu ting th e action of Q m ( κ ) on an arbitrary bu lk state, and then infer fr om this form u la its action on arbitrary b oundary states. Prop osition 4.1 The action of Q m ( κ ) on an arbitr ary bulk state | { λ } N 1 i c an b e expr esse d as Q m ( κ ) | { λ } N 1 i = m X n =0 X P λ ; P ξ R κ n ( P λ , P ξ ) | { ξ } γ + ∪ { λ } α − i . (4.3) In the ab ove formula, we sum over al l p ossible p artitions P λ and P ξ of the sets { λ } N 1 and { ξ } m 1 into subsets { λ } α + ∪ { λ } α − and { ξ } γ + ∪ { ξ } γ − r e sp e c tiv ely, satisfying the c onstr aint on the c ar dinality | α + | = | γ + | = n : P λ : { λ } N 1 = { λ } α + ∪ { λ } α − , | α + | = n , (4.4) P ξ : { ξ } m 1 = { ξ } γ + ∪ { ξ } γ − , | γ + | = n . (4.5) The c o efficient R κ n ( P λ , P ξ ) splits i nto two p arts, R κ n ( P λ , P ξ ) = R ( P λ , P ξ ) S κ n  { ξ } γ + , { λ } α +  , (4.6) the first one having a pr o duct structur e, R ( P λ , P ξ ) = Q a ∈ α +  a ( λ a ) Q b ∈ α − f ( λ b , λ a )  Q a ∈ γ +  a ( ξ a ) Q b ∈ α − ∪ α + f ( λ b , ξ a )  Y a ∈ γ − Q b ∈ γ + f ( ξ b , ξ a ) Q b ∈ α + f ( λ b , ξ a ) , (4.7 ) and the se c ond one, which dep ends her e only on the subsets { λ } α + and { ξ } γ + , b eing given as a r atio of two determinants, S κ n ( { ν } n 1 , { µ } n 1 ) = det n  M κ  { µ } n 1 , { ν } n 1  det − 1 n  1 sinh ( ν k − µ j + η )  . (4.8) The entries of the matrix M κ r e ad  M κ  { µ } n 1 , { ν } n 1  j k = t ( ν k , µ j ) − κ t ( µ j , ν k ) n Y a =1 a 6 = j f ( µ a , µ j ) f ( µ j , µ a ) n Y a =1 f ( µ j , ν a ) f ( ν a , µ j ) , (4.9) and the func tions f and t stand for t ( λ, µ ) = sinh η sinh ( λ − µ ) sinh ( λ − µ + η ) , f ( λ, µ ) = sinh ( λ − µ + η ) sinh ( λ − µ ) . (4.10) 11 The ab o ve theorem app ears as a non-trivial generalization of the action of Q m ( κ ) on bu lk Bethe eige nv ectors [49]. Indeed, when | { λ } i is n ot an eigenstate of the bu lk transfer m atrix, then Q m i =1 ( A + D ) ( ξ i ) d o es not act by m ultiplication an y more. Of course ou r result rep r o duces the p r evious case when we send the parameters λ to a solution of the b u lk Bethe equations. Pr o of — The pro of go es b y ind uction on m . Prop erty (4.3) is ob vious for m = 1. Assume that it holds f or some m . T o pro v e its v alidity for m + 1 w e hav e to compu te Q m +1 ( κ ) | { λ } N 1 i = ( A + κD ) ( ξ m +1 ) Q m ( κ ) ( A + D ) ( ξ m +1 − η ) a ( ξ m +1 ) d ( ξ m +1 − η ) | { λ } N 1 i . (4.11) Let u s firs t repro d uce the co efficient R κ n ( P λ , P ξ ) in the case when the partition P ξ is suc h that ξ m +1 6∈ { ξ } γ + . The corresp ond ing state | { λ } α − ∪ { ξ } γ + i can only b e obtained by the d ir ect action of ( A + κD ) ( ξ m +1 ). In order to repro d uce th e claimed form of the co efficien t R κ n it is enou gh to pro v e that ( A + κD ) ( ξ m +1 − η ) acts directly . Supp ose that this is n ot the case. Th en Q m ( κ ) acts on a s tate con taining ξ m +1 − η . In vir tu e of Lemma 1, the action of Q m ( κ ) on these s tates cannot replace ξ m +1 − η . Th us ( A + D ) ( ξ m +1 ) exc hanges ξ m +1 − η with ξ m +1 , which leads to a con tradiction. W e still hav e to repro duce the co efficien t R κ n ( P λ , P ξ ) corresp onding to states | { λ } α − ∪ { ξ } γ + i su c h that ξ m +1 ∈ { ξ } γ + . Theorem 3.1 yields the decomp osition: Q m +1 ( κ ) = A ( ξ m +1 ) Q m ( κ ) a ( ξ m +1 ) d ( ξ m +1 − η ) D ( ξ m +1 − η ) | {z } (1) + κD ( ξ m +1 ) Q m ( κ ) a ( ξ m +1 ) d ( ξ m +1 − η ) A ( ξ m +1 − η ) | {z } (2) , whereas Lemma 1 en s ures that • (1) only acts directly; ind eed A ( ξ m +1 ) cannot replace ξ m +1 − η by ξ m +1 ; • (2) acts indirectly and thus D ( ξ m +1 ) only acts by su bstitution. The form ula for R κ n (4.6) f ollo ws after c omputing the resu lting actions and rearranging the s u ms thanks to the re-summation formula pro vided b y the contour inte gral: 0 = I R ∪ R + iπ d z sinh ( z − ξ n +1 ) r Y a =1 f ( z , x a ) f ( ξ n +1 , x a ) S κ n +1  { ξ } n 1 ∪ { z } , { λ } n +1 1  . (4 .12) Note that the parameters x a app earing in the con tour in tegral (4.1 2) are generic.  Using th e b oundary-bulk decomp osition of P r op osition 3.2, one can n o w d educe from Pr op osition 4.1 the action of Q m ( κ ) on arbitrary b ou n dary states. Corollary 4.1 The action of Q m ( κ ) on an arbitr ary b oundary state | { λ } N 1 i b r e ads: Q m ( κ ) | { λ } N 1 i b = m X n =0 X P λ ; P ξ R κ n ( P λ , P ξ ) | { ξ } γ + ∪ { λ } α − i b . (4.13) 12 The sum over p artitions is define d as in The or em 4.1, and the c o efficient R κ n c an b e expr esse d as R κ n ( P λ , P ξ ) = X σ i = ± i ∈ α + R σ ( P λ , P ξ ) S κ n ( { ξ } γ + , { λ σ } α + ) , (4.14) wher e S κ n ( { ν } n 1 , { µ } n 1 ) is the bulk f unction define d in (4.8) , while R σ ( P λ , P ξ ) is the b oundary dr essing of (4.7) : R σ ( P λ , P ξ ) = Q a ∈ α +  a ( λ σ a ) Q b ∈ α −  f ( λ b , λ σ a ) f ( − λ b , λ σ a )   Q a ∈ γ +  a ( ξ a ) Q b ∈ α + f ( λ σ b , ξ a ) Q b ∈ α −  f ( λ b , ξ a ) f ( − λ b , ξ a )   × Y a ∈ γ − Q b ∈ γ + f ( ξ b , ξ a ) Q b ∈ α + f ( λ σ b , ξ a ) H B { σ } α + ( { λ } α + ) H B ( { ξ } γ + ) . (4. 15) Her e H B { σ } α + ( { λ } α + ) and H B ( { ξ } γ + ) stand for the b oundary-bulk c o efficients (3.8) asso ciate d r esp e ctively to { λ } α + , { σ } α + , and to { ξ } γ + , { σ } γ + = { 1 , . . . , 1 } . Pr o of — T he p ro of is a str aightforw ard consequence of the b oun dary-bulk decomp o- sition (3.7) applied to Prop osition 4.1. More pr ecisely , expressing the b oundary state |{ λ } N 1 i b in terms of the bulk states |{ λ σ } N 1 i , and using (4.3), we get Q m ( κ ) | { λ } N 1 i b = m X n =0 X P λ ; P ξ X σ i = ± 1 ≤ i ≤ N H B { σ } ( { λ } N 1 ) R κ n ( P λ σ , P ξ ) | { ξ } γ + ∪ { λ σ } α − i . W e now u se the fact that H B { σ } ( { λ } N 1 ) = Y b ∈ α − Q a ∈ α + f ( − λ σ b , λ σ a ) Q a ∈ γ + f ( − λ σ b , ξ a ) H B { σ } α + ( { λ } α + ) H B ( { ξ } γ + ) H B 1 , { σ } α − ( { ξ } γ + ∪ { λ } α − ) , where H B 1 , { σ } α − ( { ξ } γ + ∪ { λ } α − ) is the b oundary -b ulk co efficien t of |{ ξ } γ + ∪ { λ } α − i b in term s of |{ ξ } γ + ∪ { λ σ } α − i . Not e that the first factor of this p ro duct combines with the pro du cts o ver b ∈ α − in the expression (4.7) of R ( P λ σ , P ξ ), and that the resulting factor, Y b ∈ α − Q a ∈ α +  f ( λ σ b , λ σ a ) f ( − λ σ b , λ σ a )  Q a ∈ γ +  f ( λ σ b , ξ a ) f ( − λ σ b , ξ a )  , is actually indep endan t of the v alue of σ i for i ∈ α − . It enables u s to r econstruct the b ound ary state |{ ξ } γ + ∪ { λ } α − i b , w ith a co efficien t wh ic h reduces to (4.14).  13 4.2 Action of lo cal spin op er at or s W e list here the action of the lo cal spin op erators σ − m , σ + m and E 22 m on bulk and b ound ary states. W e omit the pro ofs since, although a little more tec hnical, they parallel the one concernin g the actio n of Q m ( κ ). Prop osition 4.2 The action of σ − m , E 22 m and σ + m on an arbitr ary bulk state |{ λ } N 1 i c an b e expr esse d as σ − m |{ λ } N 1 i = m − 1 X n =0 X P − λ , P ξ R − n ( P − λ , P ξ ) |{ ξ } γ + ∪ { λ } α − i , E 22 m |{ λ } N 1 i = m − 1 X n =0 N X c 1 =1 X P 22 λ , P ξ R 22 n ( P 22 λ , P ξ ) |{ ξ } γ + ∪ { λ } α − i , σ + m |{ λ } N 1 i = lim λ N +1 → ξ m m − 1 X n =0 N X c 1 =1 N +1 X c 2 =1 c 2 6 = c 1 X P + λ , P ξ R + n ( P + λ , P ξ ) | { ξ } γ + ∪ { λ } N 1 \ { λ } e α + i , in which the sums run over the fol lowing p artitions P ξ : { ξ } m 1 = { ξ } γ + ∪ { ξ } γ − , with | γ + | = n + 1 , (4.16) P − λ : { λ } N 1 = { λ } α + ∪ { λ } α − , with | α + | = n, (4.17) P 22 λ : { λ k } 1 ≤ k ≤ N k 6 = c 1 = { λ } α + ∪ { λ } α − , with | α + | = n, (4.18) P + λ : { λ k } 1 ≤ k ≤ N k 6 = c 1 ,c 2 = { λ } α + ∪ { λ } α − , with | α + | = n. (4.19) We also define the fol lowing p artitions, asso ciate d r esp e ctively to (4.18) and to (4.19) , e P 22 λ : { λ } N 1 = { λ } e α + ∪ { λ } α − , with e α + = α + ∪ { c 1 } , (4.20) e P + λ : { λ } N +1 1 = { λ } e α + ∪ { λ } e α − , with e α + = α + ∪ { c 1 , c 2 } . (4.21) The c o efficients R − n ( P − λ , P ξ ) , R 22 n ( P 22 λ , P ξ ) and R + n ( P + λ , P ξ ) ar e given as R − n ( P − λ , P ξ ) = R ( P − λ , P ξ ) lim ξ → ξ m Q a ∈ γ + sinh( ξ a − ξ ) Q a ∈ α + sinh( λ a − ξ ) b S n ( { λ } α + , { ξ } γ + ; ξ , ∅ ) , ( 4.22) R 22 n ( P 22 λ , P ξ ) = R ( e P 22 λ , P ξ ) sinh η Y a ∈ α + f ( λ a , λ c 1 ) × lim ξ → ξ m Q a ∈ γ + sinh( ξ a − ξ ) Q a ∈ e α + sinh( λ a − ξ ) b S n ( { λ } α + , { ξ } γ + ; ξ , { λ c 1 } ) , (4.23) 14 R + n  P + λ , P ξ  = R  e P + λ , P ξ  f ( λ c 2 , λ c 1 ) 2 Y i =1  sinh η Y a ∈ α + f ( λ a , λ c i )  × Q a ∈ γ + sinh( λ N +1 − ξ a + η ) Q a ∈ e α + sinh( λ N +1 − λ a + η ) b S n  { ξ } γ + , { λ } α + ; λ N +1 , { λ c 1 , λ c 2 }  . (4.24) Her e R ( P λ , P ξ ) is given by (4.7) , and the structur e of the factor b S n ( { ξ } n +1 1 , { λ } n 1 ; ξ , { µ } p 1 ) is similar to (4.8) : b S n ( { ξ } n +1 1 , { λ } n 1 ; ξ , { µ } p 1 ) = n Q a =1 n +1 Q b =1 sinh( ξ b − λ a + η ) Q a>b sinh( ξ a − ξ b ) Q a>b sinh( λ b − λ a ) × det n +1 h c M ( { λ } n 1 , { ξ } n +1 1 ; ξ , { µ } p 1 ) i , (4.25) wher e the matrix elements of c M ar e obtaine d as  c M  { λ } n 1 , { ξ } n +1 1 ; λ n +1 , { µ } p 1  j k =  M κ ( λ j , { µ } )  { λ } n +1 1 , { ξ } n +1 1  j k , (4.26) with κ ( λ j , { µ } ) = (1 − δ j,n +1 ) n ′ Y i =1 f ( µ i , λ j ) f ( λ j , µ i ) , in which δ ij denotes the Kr one cker symb ol and M κ is define d as i n (4.9) . Using again th e b oundary-bulk decomp ositio n, we are now in p osition to list the action of local spin op erators on b oundary s tates. Corollary 4.2 With the same notations as in Pr op osition 4.2, the action of σ − m , E 22 m and σ + m on an arbitr ary b oundary state |{ λ } N 1 i b takes the form σ − m |{ λ } N 1 i b = m − 1 X n =0 X P − λ , P ξ R − n ( P − λ , P ξ ) |{ ξ } γ + ∪ { λ } α − i b , E 22 m |{ λ } N 1 i b = m − 1 X n =0 N X c 1 =1 X P 22 λ , P ξ R 22 n ( P 22 λ , P ξ ) |{ ξ } γ + ∪ { λ } α − i b , σ + m |{ λ } N 1 i b = l im λ N +1 → ξ m m − 1 X n =0 N X c 1 =1 N +1 X c 2 =1 c 2 6 = c 1 X P + λ , P ξ R + n ( P + λ , P ξ ) | { ξ } γ + ∪ { λ } N 1 \ { λ } e α + i b . The b oundary c o effici e nts R − , R 22 and R + have a structur e similar to their c orr e- sp onding bu lk c ounterp arts: R − n ( P − λ , P ξ ) = X σ i = ± i ∈ α + R σ ( P − λ , P ξ ) lim ξ → ξ m Q a ∈ γ + sinh( ξ a − ξ ) Q a ∈ α + sinh( λ a − ξ ) b S n ( { λ } α + , { ξ } γ + ; ξ , ∅ ) , 15 R 22 n ( P 22 λ , P ξ ) = X σ i = ± i ∈ e α + R σ ( e P 22 λ , P ξ ) sinh η Y a ∈ α + f ( λ a , λ c 1 ) × lim ξ → ξ m Q a ∈ γ + sinh( ξ a − ξ ) Q a ∈ e α + sinh( λ a − ξ ) b S n ( { λ } α + , { ξ } γ + ; ξ , { λ c 1 } ) , R + n  P + λ , P ξ  = X σ i = ± i ∈ e α + R σ  e P + λ , P ξ  f  λ σ c 2 , λ σ c 1  2 Y i =1  sinh η Y a ∈ α + f  λ σ a , λ σ c i   × Q a ∈ γ +  f ( − λ N +1 , ξ a ) sin h( λ N +1 − ξ a + η )  Q a ∈ e α +  f ( − λ N +1 , λ σ a ) sinh( λ N +1 − λ σ a + η )  b S n  { ξ } γ + , { λ σ } α + ; λ N +1 , { λ σ c i }  , wher e R σ is define d as in (4.1 5) and b S n is the bulk quantity (4.25) . 5 Correlation fu n ctions in the half-infinite c hain W e apply the r esults of the p revious section to deriv e the exp ecta tion v alues of the generating function hQ m ( κ ) i of h σ z m i , and of h σ + 1 σ − m +1 i in th e ground state of the half-infinite chain. Th ese are the b ound ary analogues of the r esults pu blished in [49]. 5.1 The generating function hQ m ( κ ) i Prop osition 5.1 The gener ating fu nction hQ m ( κ ) i i s obtaine d, in the thermo dy- namic limit M → + ∞ , as the homo gene ous limit of the qu antity hQ m ( κ ) i = m X n =0 1 ( n !) 2 I Γ + ( { ξ } m 1 ) d n z (2 iπ ) n Z C D d n λ m Y a =1 n Y b =1 f ( z b , ξ a ) f ( λ b , ξ a ) W − ( { λ } n 1 , { z } n 1 ) × det n [ M κ ( { λ } , { z } )] d et n [Ψ ( λ j , z k )] , (5.1) in which M κ is given by (4.9) , and W − is the b oundary dr essing, W −  { λ } n 1 1 , { z } n 2 1  = n 2 Q j =1 sinh ( z j + ξ − − η / 2) n 1 Q j =1 sinh ( λ j + ξ − − η / 2) × n 1 Q a =1 n 2 Q b =1 sinh ( z b + λ a − η ) n 2 Q a 1 . Pr o of — Corollary 4.1 yields the action of Q m ( κ ) on a b oundary state. It is con v e- nien t to note th at the coefficient R σ  P λ , P ξ  (4.15) can b e rewr itten as R σ  P λ , P ξ  =  Y a ∈ α + σ a  (sinh η ) | γ + | Y b ∈ γ + ∪ γ − Q a ∈ γ + a 6 = b f ( ξ a , ξ b ) Q a ∈ α + f ( λ σ a , ξ b ) W − ( { λ σ } α + , { ξ } γ + ) × Q a>b sinh( ξ a − ξ b ) Q a>b sinh( λ b − λ a ) Q a ∈ α + Q b ∈ γ + sinh( ξ b − λ a + η ) S σ ( { λ } α + , { ξ } γ + ; { λ } α − ) − 1 , (5.5 ) in whic h S σ ( { λ } α + , { ξ } γ + ; { λ } α − ) is the fun ction defi ned in (3.16). Then , using the reduced scalar p ro duct formula (3.15) and ab s orbing th e s u ms o v er p artitions P ξ in to auxiliary z in tegrals 5 , we ob tain the former repr esen tation. Note that the con tour conta ins Γ +  ˇ λ  for large p ositiv e b ound ary fi eld since we ha v e to absorb the con tribu tion coming from the rep lacemen t of the complex ro ot ˇ λ as explained in [4].  5 W e refer the reader to [49] for technical details. 17 5.2 The ground state exp ect at ion v alue h σ + 1 σ − m +1 i Using the same metho d as for th e generating fun ction hQ m ( κ ) i , we can also com- pute the ground state exp ectation v alue h σ + 1 σ − m +1 i . It giv es h σ + 1 σ − m +1 i = m − 1 X n =0 sinh( ξ 1 + ξ − − η / 2) n !( n + 1)! I Γ + ( { ξ } m +1 1 ) n +1 Y k =1 d z k 2 iπ Z C D n +1 Y k =1 d λ k Z C A d λ n +2 × m +1 Y a =2 n +1 Q b =1 f ( z b , ξ a ) n Q b =1 f ( λ b , ξ a ) n Y b =1 sinh( λ b − ξ 1 ) sinh( λ b − ξ m +1 ) n +1 Y b =1 sinh( z b − ξ m +1 ) sinh( z b − ξ 1 ) × n +1 Q b =1 sinh( λ b − λ n +1 + η ) n +2 Q b =1 sinh( λ n +2 − λ b + η ) n +1 Q b =1  sinh( z b − λ n +1 + η ) sinh( λ n +2 − z b + η )  W −  { λ } n +2 1 , { z } n +1 1  × n +2 Q b =1 sinh( ξ 1 + λ b − η ) n +1 Q b =1 sinh( ξ 1 + z b − η ) det n +1 h c M  { λ } n 1 , { z } n +1 1 ; ξ m +1  i × det n +2 [Ψ( λ j , ξ 1 ) , Ψ( λ j , z 1 ) , . . . , Ψ ( λ j , z n +1 )] . (5.6) In this expression, W − denotes the b oundary quan tity (5.2 ), c M  { λ } n 1 , { z } n +1 1 ; ξ m +1  is a s implified notation for the matrix c M  { λ } n 1 , { z } n +1 1 ; ξ m +1 , ∅  defined in (4.2 6) , C D is th e con tour (5.4), and C A denotes th e follo wing con tour ( A -t yp e con tour): C A =  ] − Λ + η ; Λ + η [ ∪ Γ −  ˇ λ  , if − ζ / 2 < ˜ ξ − < 0 , ] − Λ + η ; Λ + η [ , otherwise. (5.7) In the homogeneous limit, this r esults simplifies in to h σ + 1 σ − m +1 i = m − 1 X n =0 sinh ξ − n !( n + 1)! I Γ + ( η/ 2) n +1 Y k =1 d z k 2 iπ Z C D n +1 Y k =1 d λ k Z C A d λ n +2 × n +1 Y a =1  sinh( z a + η / 2) sinh( z a − η / 2)  m n Y a =1  sinh( λ a − η / 2) sinh( λ a + η / 2)  m n +2 Q b =1 sinh( λ b − η / 2) n +1 Q b =1 sinh( z b − η / 2) × n +1 Q b =1 sinh( λ b − λ n +1 + η ) n +2 Q b =1 sinh( λ n +2 − λ b + η ) n +1 Q b =1  sinh( z b − λ n +1 + η ) sinh( λ n +2 − z b + η )  W −  { λ } n +2 1 , { z } n +1 1  × d et n +1 h c M ( { λ } n 1 , { z } n +1 1 ; η / 2) i det n +2 [Ψ( λ j , η / 2) , Ψ( λ j , z 1 ) , . . . , Ψ ( λ j , z n +1 )] , 18 6 An alternativ e resummation 6.1 Bulk type resumations W e ha ve obtained in the p revious sections a series representat ion for the g enerating function. It happ ens, just as in the bulk case [52], that it is also p ossible to derive a totally different representat ion for hQ m ( κ ) i . Th e latte r is based on a r e-summation of its expansion with resp ect to elemen tary b lo c ks : hQ m ( κ ) i = h m Y i =1  E 11 i + κE 22 i  i = m X s =0 κ s F s , (6.1) where F s = 1 s ! ( m − s )! X π ∈ Σ m h E ǫ π ( 1) ǫ π (1) 1 . . . E ǫ π ( m ) ǫ π ( m ) m i , ǫ i =  2 , i = 1 ...s, 1 , i = s + 1 ...m, (6.2) and Σ m is th e group of p ermutations of m elements. These elemen tary blo c ks we re computed in [4]. Th ey can b e written as multiple in tegrals in the h alf-infinite size limit: h E ǫ 1 ǫ ′ 1 1 . . . E ǫ m ǫ ′ m m i = ( − 1) m − s Z C D s Y i =1 d λ i Z C A m Y i = s +1 d λ i det m [Ψ ( λ i , ξ j )] m Q ij sinh ( λ ij − η ) sinh  λ ij − η  s Y p =1   i p − 1 Y j =1 sinh ( ξ j − λ p ) m Y j = i p +1 sinh ( ξ j − λ p − η )   × m Y i =1 sinh ( ξ i + ξ − − η / 2) sinh ( λ i + ξ − − η / 2) m Y p = s +1   i p − 1 Y j =1 sinh ( ξ j − λ p ) m Y j = i p +1 sinh ( ξ j − λ p + η )   . (6.3) The ind ices i p are d efined by  { i : 1 ≤ i ≤ m, ǫ ′ i = 2 } = { i 1 < · · · < i s } , { i : 1 ≤ i ≤ m, ǫ i = 1 } = { i s +1 > · · · > i m } . (6.4) F or simplicit y , we consider f rom no w on th e massless regime (although all wh at follo ws can b e p erformed in th e massive regime as w ell). In that case, η = − iζ and the contours of integrati on C D and C A dep end on the b oundary magnetic field h − as follo ws: range of ξ − D − con tour A − contour ζ / 2 < | ˜ ξ − | < π / 2 C D = R C A = R − iζ ζ / 2 > ˜ ξ − > 0 C D = R S Γ +  ˇ λ  C A = R − iζ − ζ / 2 < ˜ ξ − < 0 C D = R C A = { R − iζ } S Γ −  ˇ λ  (6.5) W e recall that ˇ λ = η / 2 − ξ − , and that Γ ± ( z ) is a small lo op around z of index ± 1. 19 W e now p erform a change of v ariables in th e A -t yp e con tours : λ ′ A = λ A − iζ . Moreo ve r, we shift the inhomogeneities around zero δ i = ξ i + iζ / 2, and defin e a i = 3 / 2 − ǫ i =  1 / 2 ( ǫ i = 1) for A -t yp e , − 1 / 2 ( ǫ i = 2) for D -type . (6.6) This gives h E ǫ π ( 1) ǫ π (1) 1 . . . E ǫ π ( m ) ǫ π ( m ) m i = ( − 1) [ π ] Z C D d s λ Z e C A d m − s λ det m h e Ψ ( λ i , δ j ) i Q ik sinh  δ k − λ π ( j ) + ia π ( j ) ζ  sinh  δ j − λ π ( k ) − ia π ( k ) ζ  sinh  λ π ( j ) π ( k ) − ia π ( j ) π ( k ) ζ  sinh  λ π ( j ) π ( k ) − ia π ( j ) π ( k ) ζ  × m Y j,k =1 sinh ( λ j + δ k − ia j ζ ) m Y j =1 sinh ( ξ − + δ j ) sinh ( λ j + ξ − − ia j ζ ) . (6.7) Here e Ψ ( λ, δ ) = Ψ ( λ, δ − iζ / 2) = ρ ( λ − δ ) − ρ ( λ + δ ) 2 sin h 2 δ , (6.8) e C A =  ] − Λ ; Λ [ ∪ Γ −  ˇ λ − η  if − ζ / 2 < ˜ ξ − < 0 , ] − Λ ; Λ [ otherwise. (6.9) and ( − 1) [ π ] is th e signature of the p erm utation. One can compute the su m o ver p erm utations (6.2) just as in th e bu lk case [52]. It leads to th e follo win g in tegral representat ion for F s : Prop osition 6.1 (Bulk-t yp e resummation) The gener ating function of the spin c orr elation function hQ m ( κ ) i c an b e e xpr esse d as hQ m ( κ ) i = m X s =0 κ s F s (6.10) with F s = 1 s ! ( m − s )! Z C D d s λ Z e C A d m − s λ det m h e Ψ ( λ i , δ j ) i Q ik sinh λ j k s ( λ j k , ia j k ζ ) sinh  λ j k − ia j k ζ  . (6.13) Similar repr esen tations can b e obtained for other correlation fun ctions. Here we giv e only t w o imp ortan t examples: the lo cal densit y of energy and the h σ + m +1 σ − 1 i t wo-point fu nction. The lo cal density of energy E m = h σ x m σ x m +1 + σ y m σ y m +1 + ∆( σ z m σ z m +1 − 1) i , (6. 14) can b e written as a sum of m term s E m = m − 1 X s =0 ˜ E s , (6.15) eac h of them conta ining m + 1 in tegrals ˜ E s = 1 s ! ( m − 1 − s )! Z C D d s λ Z e C A d m − s − 1 λ Z C D d λ m Z e C A d λ m +1 det m +1 h e Ψ ( λ i , δ j ) i Q i ˜ ξ − > 0 C = R S Γ + ( ˇ λ ) S Γ − ( − ˇ λ ) − ζ / 2 < ˜ ξ − < 0 C = R S Γ + ( iζ + ˇ λ ) S Γ − ( − iζ − ˇ λ ) (6.19) W e extract the totally ev en p art of the int egrand app earing in (6.7) according to Z C d xf ( x ) = 1 2 X σ = ± Z C d xf ( x σ ) , x σ = σ x . ( 6.20) W e get F s = 1 s !( m − s )!2 m Z C d m λ det m h e Ψ ( λ i , δ j ) i Q ik s ( λ j , λ k ) s ( λ j k , ia j k ζ ) s  λ j k , ia j k ζ  , (6.22) and the sums ov er negations h a ve b een absorb ed into H s ( { λ } , { δ } ): H s ( { λ } , { δ } ) = X σ i = ± m Y j =1  σ j sinh  λ σ j − ξ − − ia j ζ   Y j >k sinh  λ σ j k + ia j k ζ  sinh  λ σ j k  × m Y j,k sinh  λ σ j + δ k − ia j ζ  Z m ( { λ σ } , { δ } ) . (6.23) Equation (6.23) implies that H s ( { λ } , { δ } ) is a symmetric fu nction of the parameters λ and of the parameters δ . Moreo v er, e 2( m − 1) λ j H s ( { λ } , { δ } ) is a p olynomial in eac h of the e 2 λ j v ariables of degree 2( m − 1). Finally , it is a matter of str aigh tforward computations to c h ec k that H s satisfies th e reduction prop erties: H s | λ 1 = ± ( δ 1 − ia 1 ζ ) ( { λ i } m i =1 ; { δ k } m k =1 ) = ± H s ( { λ i } m i =2 ; { δ k } m k =2 ) × sinh (2 ( δ 1 − ia 1 ζ )) s in h ( δ 1 − ξ − ) m Y j =2 s ( λ j , δ 1 + ia 1 ζ ) s ( δ 1 − 2 ia 1 ζ , δ k ) . (6.24) These are th e r eduction p r op erties of Z m ( { λ } , { δ } ), the p artition fu nction of the six- v ertex mo d el with reflecting ends [72]. Supplement ing th is result with the equalit y of the t wo functions at m = 1, we obtain that H s ( { λ } ; { δ } ) is s -in d ep endent and equal to Z m ( { λ } , { δ } ). Hence, we ha v e the follo w ing result: Prop osition 6.2 (Boundary-t yp e resummation) The gener ating function of the spin c orr elation function hQ m ( κ ) i c an b e e xpr esse d as hQ m ( κ ) i = m X s =0 κ s F s (6.25) with F s = 1 s ! ( m − s )!2 m Z C d m λ det m h e Ψ ( λ i , δ j ) i Q ik s ( λ j , λ k ) s ( λ j k , ia j k ζ ) s  λ j k , i a j k ζ  , (6.27) 23 and Z m ( { λ } , { δ } ) = m Q j,k =1  s ( λ j , δ k + iζ / 2) s ( λ j , δ k − iζ / 2)  Q ik  s ( λ j , λ k ) s ( λ j k , iζ ) s  λ j k , iζ   × m Y k >l s ( λ k l , z k l ) s  λ k l , z k l  s ( δ l , δ k ) m Y p =1 ϕ ( z p ) sin h ( ξ − + ξ p ) s ( λ p , ξ − + z p ) , (6.29) where ϕ ( z ) = sinh 2 z κ − i ( z + iζ / 2) /ζ s ( z , iζ / 2) . (6.30) This b ound ary-t yp e resummation y ields an in tegrand n ot only symm etric in { λ } but also in v arian t un der a rev ersal of an y inte gration v ariable λ . These prop erties allo w to compute completely the so called emp tiness formation p robabilit y at ∆ = 1 / 2 and f or v anishing b oundary magnetic fields. It also allo ws to obtain th e leading asymptotics of th is quan tit y at the free fermion p oin t [75]. 7 The free fermion p oin t 7.1 Lo cal magnetization at distance m The first (and the most imp ortant) application of the r e-su m mation metho ds giv en ab o v e is the magnetiza tion profi le. This one-p oin t fun ction h σ z m i = 1 − 2 D m ∂ κ hQ m ( κ ) i | κ =1 , can b e computed at the free fermion p oin t by using the tw o different t yp es of r e- summations for the generation fun ction. W e giv e here b oth deriv ations. 7.1.1 First met ho d In the fr ee ferm ion p oin t, the n th term of the series (5.1) b eh a ves as ( κ − 1) n . Th us, after taking the κ d eriv ativ e and sending κ to 1, only the n = 1 term su r viv es. 24 A t ζ = π / 2, we ha v e hQ m ( κ ) i = m X n =0 ( κ − 1) n (2 π ) 2 n ( n !) 2 Z C D d n λ I Γ + ( { ξ } ) d n z m Y a =1 n Y b =1 tanh ( λ b − ξ a ) tanh ( z b − ξ a ) × n Y j =1  sinh ( z j + ξ − + iπ / 4) sinh ( λ j + ξ − + iπ / 4) sinh 2 λ j  × det n  1 sinh ( λ j − z k )  det n  1 s ( λ j , z k )  . (7.1) Th us 6 , for m ≥ 2, h σ z m i = ( − 1) m π Z C D d λ sinh ( λ − ξ − − iπ / 4) sinh ( λ + ξ − + iπ / 4) [tanh ( λ + iπ / 4)] 2( m − 1) cosh 2 ( λ + iπ / 4) . (7.2) Computing, if it exists, the r esidue at ˇ λ we get, f or m ≥ 2, h σ z m i = − 2Θ ( h − − 1) h 2 − − 1 h 2 m − + ( − 1) m π Z R d λ h − + i tanh ( λ + iπ / 4) 1 + ih − tanh ( λ + iπ / 4) [tanh ( λ + iπ / 4)] 2( m − 1) cosh 2 ( λ + iπ / 4) , (7.3) where Θ ( x ) is the Hea viside step function. The s tandard ∆ = 0 change of v ariables, e ip = − tanh ( λ − iπ / 4) , (7.4) yields h σ z m i = − 2Θ ( h − − 1) h 2 − − 1 h 2 m − + ( − 1) m π π Z 0 d p e − 2 i ( m − 1) p e − ip + ih − e ip − ih − . (7.5) Th us h σ z m i displa ys F riedel t yp e oscilla tions in duced by the b oun dary . Moreo v er it deca ys as 1 /m when m → + ∞ : h σ z m i = 2 ( − 1) m π m h − h 2 − + 1 + O  1 m 2  , m >> 1 . (7.6) Here we reco ve r the r esu lts of [69], since we h a ve h − = √ 2 α − in Bilstein’s nota- tions. When | h − |→ ∞ we conclude from (7.5) th at the first site is totally d ecoupled from the others as h σ z m i m ≥ 2 go es to its bulk a verage v alue 0. Ac tually in this limit the mo del is in corresp ondence with a Kondo mo d el with a spin 1 / 2 impurity [29]. 6 W e do not give h σ z 1 i as it corresp onds to ∂ κ hQ 1 ( κ ) i without taking the lattice deriv ative. 25 But in this case th e imp u rit y is completely screened, and the o verall magnetization in zero. W e also reco ve r from (7.5), just as exp ected f r om the spin r ev ersal symmetry , that h σ z m i = 0 when th e b oun dary field v anish es. Actually this obs er v ation holds f or all ∆ as inferr ed from th e structure of the mon o drom y matrix (2.6) on the first site. When one sets ξ 1 = η / 2 and ξ − = 0 then it acts as a diagonal matrix on the first site, a sign of th e claimed d ecoupling. 7.1.2 Second met ho d Starting from the re-summation form ula (6.11 ) of Prop osition 6.1, we implement the s im p lification du e to ζ = π / 2. If we p erform the c hange of v ariables e ip = − tanh ( λ − iπ / 4) (7.7) in (6.11) at ζ = π / 2 then we arr iv e at F s = (2 i ) m ( m − 1) 2 s ! ( m − s )! s Y j =1 Z C D dp j 2 π  e ip j + e − ip j e ip j − ih −  m Y j = s +1 Z C A dp j 2 π  e ip j + e − ip j e − ip j − ih −  × s Y k =1 m Y j = s +1   e − ip k − e ip j  (sin p j + sin p k )  s Y j,k =1 j >k   e − ip j − e − ip k  (sin p j − sin p k )  × m Y j,k = s +1 j >k   e ip j − e ip k  (sin p k − sin p j )  . (7.8) The conto urs of integ ration are C A = ] 0 ; π [ , h − ≥ − 1 , C A = ] 0 ; π [ ∪ Γ −  e − ip = − ih −  , h − < − 1 , (7.9) C A = ] − π ; 0 [ , h − ≥ − 1 , C A = ] − π ; 0 [ ∪ Γ +  e ip = − ih −  , h − < − 1 , (7.10) C D = ] 0 ; π [ , h − ≤ 1 , C D = ] 0 ; π [ ∪ Γ +  e ip = − ih −  , h − > 1 . (7.11) Once we in tro duce the fun ction θ κ ( p ) =  κ p ∈ C D , 1 p ∈ C A , (7.12) w e can re-sum the terms F s in to a single m -fold in tegral for h Q m ( κ ) i : hQ m ( κ ) i = (2 i ) m ( m − 1) 2 m ! Z C A ∪ C D m Y j =1  d p j 2 π θ κ ( p j ) e ip j + e − ip j e ip j − ih −  × m Y j >k ≥ 1  e − ip j − e − ip k  (sin p j − sin p k ) . (7.13) 26 One can then exp r ess the generating fun ction as a single determinant hQ m ( κ ) i = det m [ U ( κ )] , (7.14) U j k ( κ ) = 1 2 π Z C A ∪ C D d p θ κ ( p ) e − ip ( j − 1) e ipk − ( − 1) k e − ipk e ip − ih − . (7.15) T o simplify this resu lt w e add to eac h ro w of U ( κ ) the next one m ultiplied b y ih − : hQ m ( κ ) i = det m e U ( m ) ( κ ) , e U ( m ) j k ( κ ) = 1 2 π Z C A ∪ C D d p θ κ ( p )  e ip ( k − j ) − ( − 1) k e − ip ( j + k )  , j < m, e U ( m ) mk ( κ ) = U mk ( κ ) . It is easy to see that Q m (1) = 1. Computing the first deriv ative of the generating function one reco vers the result already obtained from the series (7.5): h σ z m i = ( − 1) m π Z C D d p e − 2 ip ( m − 1) e − ip + ih − e ip − ih − . = − 2 h 2 − − 1 h 2 m − Θ ( h − − 1) + ( − 1) m π π Z 0 d p e − 2 ip ( m − 1) e − ip + ih − e ip − ih − . (7.16) 7.2 Lo cal density of energy The lo cal d en sit y of energy is another interesting quanti t y [61] that one can ev al- uate for the XX0 c h ain: E m = h σ x m σ x m +1 + σ y m σ y m +1 i . (7.17 ) Starting from (6.16) and usin g the same tec hn ique as in the pr evious sub -section one easily obtains the f ollo wing representati on for the densit y of energy: E m = − 2 i Z C D d p 2 π Z C A d q 2 π  e i ( p + q ) + 1  det m +1 [ V ( p, q )] . (7.18) 27 The entries of V ( p, q ) read V j k ( p, q ) = 1 2 π Z C A ∪ C D d p ′  e ip ′ ( k − j ) − ( − 1) k e − ip ′ ( j + k )  = δ j k , j < m − 1 , V m − 1 k ( p, q ) = 1 2 π Z C A ∪ C D d p ′ e − ip ′ ( m − 2) e ip ′ k − ( − 1) k e − ip ′ k e ip ′ − ih − =  h − i  k − m +1 Θ ( k − m + 1) , V mk ( p, q ) = e − ipm e ipk − ( − 1) k e − ipk e ip − ih − , V m +1 k ( p, q ) = e − iq m e iq k − ( − 1) k e − iq k e iq − ih − . (7.19) Finally , the in tegrals o v er C A can b e represen ted as Z C A = Z C A ∪ C D − Z C D . (7.20) Accordingly , E m reduces to a su m of tw o 3 × 3 determin ants: E m = − 2 i       1 ih − − h 2 − F ( m − 1 , m − 1) F ( m − 1 , m ) F ( m − 1 , m + 1) 0 1 ih −       − 2 i       1 ih − − h 2 − F ( m, m − 1) F ( m, m ) F ( m, m + 1) 0 0 1       , (7.21 ) where F ( j, k ) = 1 2 π Z C D d p e − ipj e ipk − ( − 1) k e − ipk e ip − ih − . (7.22) The compu tation of th ese determinants yields E m = − 2 i ( F ( m, m ) − F ( m − 1 , m + 1)) − 2 h − ( F ( m, m − 1) − F ( m − 1 , m )) , (7.23) or, m ore explicitly , E m = − 4 π + 2 i π ( − 1) m Z C D d p e − ip (2 m − 1) e − ip + ih − e ip − ih − . (7 .24) The constan t term repro d u ces the bulk result. Th e influence of the b ound ary app ears in the oscillating term. In the m → ∞ limit the lo cal densit y of en er gy b eha v es as E m = − 4 π + 2 π m ( − 1) m 1 − h 2 − 1 + h 2 − + O  1 m 2  . (7.25) 28 7.2.1 Tw o-p oin t function h σ + m +1 σ − 1 i The preceding metho d can b e successfully ap p lied to compute other typ es of t w o- p oint fun ctions (lik e b oundary-bu lk t w o p oin t functions), here w e giv e the example of h σ + m +1 σ − 1 i . In the free fermion p oint, after the u sual c hange of v ariables and some straigh tfor- w ard b ut tedious calculations, one obtains f rom (6.18) a s im p le determinant form ula for this ob ject. h σ + m +1 σ − 1 i = − i det m +1 V + − , (7.26) V + − j k = 1 2 π i   π Z 0 − 2 π Z π   dp  e ip ( k − j − 1) − ( − 1) k e − ip ( j +1+ k )  , j ≤ m − 1 , V + − mk = 1 2 π Z C D d p e − ipm e ipk − ( − 1) k e − ipk e ip − ih − , V + − m +1 k = 1 2 π Z C A dp e − ipm e ipk − ( − 1) k e − ipk e ip − ih − (7.27) Computing the in tegrals in the first m − 1 ro ws and using the fact that su m of the last t wo ro ws is δ k ,m +1 w e red uce this representa tion to a determinan t of a m × m matrix h σ + m +1 σ − 1 i = i ( − 1) m 2 π m det m ˜ V + − , (7.28) ˜ V + − j k =(1 + ( − 1) j − k ) ( j + 1)(1 − ( − 1) k ) + k (1 + ( − 1) k ) ( j + 1) 2 − k 2 , j ≤ m − 1 , ˜ V + − mk = Z C D d p e − ipm e ipk − ( − 1) k e − ipk e ip − ih − , (7.29) This determinant can b e computed for any v alue of m . Ho wev er the result is qu ite differen t for m o dd or ev en. The details of the computation are giv en in App endix 29 B, her e w e give only the final r esult for the tw o p oint fun ction h σ + 2 a σ − 1 i = − i 2 2 a − 1 π 2   2 a − 1 Y j =1 Γ( j ) Γ( j + 1 2 )   2 Γ( a − 1 2 )Γ 3 ( a + 1 2 ) Γ( a )Γ( a + 1) × Z C D d p P 2 a − 1  cos p  e − ip (2 a − 1) e ip − ih − (7.30) h σ + 2 a +1 σ − 1 i = − 2 2 a π 2   2 a Y j =1 Γ( j ) Γ( j + 1 2 )   2 Γ( a + 3 2 )Γ 3 ( a + 1 2 ) Γ( a )Γ( a + 1) × π Z 0 dq cos q Z C D d p P 2 a  cos( q − p )  e − 2 ipa e ip − ih − , (7.31) where P m ( x ) are Legendre p olynomials. Asymptotic analysis of these expression yields the follo win g leading b ehavio r of h σ + m +1 σ − 1 i h σ + m +1 σ − 1 i =( − 1) m A ( h − ) m − 3 4  1 + O ( 1 √ m )  (7.32) A ( h − ) = s 2 π (1 + h 2 − ) exp  1 4 Z ∞ 0 dt t  e − 4 t − 1 cosh 2 t  . (7.33) 8 Conclusion In this article we h a ve obtained different t yp es of ph ysical correlation fun ctions of th e op en X X Z c hain from r e-summations of the multiple integral s d er ived in [4] for the elemen tary blo c ks. At the f ree-fermion p oin t, we w ere able to use these represent ations to d er ive explicit results su c h as the formula for the density of en er gy profiles, a quantit y arising in th e stu dy of quant um enta nglemen t in spin c hains [61]. Just as in the b ulk case, the question concerning the asymptotic b eha vior of the correlation functions outside of the free-fermion p oin t naturally arises. The problem is of the same order of d ifficult y as in the bulk mod el. Indeed, the multiple int egrals differ from their bulk coun terparts only by factors d u e to the Z 2 symmetry λ → − λ and the presence of b oundary fields. One could also wonder if it wo uld b e p ossible to tell something ab out the d ynam- ical or temp erature correlation fun ctions. It seems th at this generalization is highly non-trivial. Finally , we would lik e to s tr ess that our expressions also s im p lify at other partic- ular p oin ts suc h as ∆ = 1 / 2. F or ins tance, when ∆ = 1 / 2, one can already compute completely the so-called emptiness formation probabilit y when h − = 0 [75]. 30 App endices A Asymptotic of the tw o-p oin t function h σ + m +1 σ − 1 i In the last sectio n w e obtained a d etermin ant r epresen tation (7.29) f or th e t wo- p oint function h σ + m +1 σ − 1 i . T his determinant can b e compu ted for any v alue of m . Ho wev er the results are quite d ifferen t for m o dd or eve n. If m is o d d: m = 2 a − 1, the r esult can b e written in the follo wing form det m ˜ V + − =2 m +1 π m − 3   m Y j =1 Γ( j ) Γ( j + 1 2 )   2 Γ( a − 1 2 )Γ 3 ( a + 1 2 ) Γ( a )Γ( a + 1) × a X b =1 Γ( a − b + 1 2 )Γ( a + b − 1 2 ) Γ( a − b + 1)Γ( a + b ) × π Z 0 dp e − ip (2 a − 1) e ip (2 b − 1) + e − ip (2 b − 1) e ip − ih − (A.1) If m is ev en: m = 2 a , the r esu lt is quite sim ilar but there is a v ery imp ortan t difference: det m ˜ V + − =2 m +1 π m − 3   m Y j =1 Γ( j ) Γ( j + 1 2 )   2 Γ( a + 3 2 )Γ 3 ( a + 1 2 ) Γ( a )Γ( a + 1) × a X b =1 b ( b + 1 2 )( b − 1 2 ) Γ( a − b + 1 2 )Γ( a + b + 1 2 ) Γ( a − b + 1)Γ( a + b + 1) × π Z 0 dp e − 2 ipa e 2 ipb − e − 2 ipb e ip − ih − (A.2) Asymptotic analysis of the prefactors in (A.1) and (A.2) is rather simp le, n amely:   2 a − 1 Y j =1 Γ( j ) Γ( j + 1 2 )   2 Γ( a − 1 2 )Γ 3 ( a + 1 2 ) Γ( a )Γ( a + 1) = √ 2 π 2 m m − 1 4 exp  1 4 Z ∞ 0 dt t  e − 4 t − 1 cosh 2 t   1 + O ( 1 m )  (A.3)   2 a Y j =1 Γ( j ) Γ( j + 1 2 )   2 Γ( a + 3 2 )Γ 3 ( a + 1 2 ) Γ( a )Γ( a + 1) = √ 2 π 2 m +1 m 3 4 exp  1 4 Z ∞ 0 dt t  e − 4 t − 1 cosh 2 t   1 + O ( 1 m )  (A.4) 31 F or m o d d the sum in (A.1) can b e rewr itten as f ollo ws: a X b =1 Γ( a − b + 1 2 )Γ( a + b − 1 2 ) Γ( a − b + 1)Γ( a + b )  e − 2 ip ( a − b ) + e − 2 ip ( a + b − 1)  = 2 a − 1 X l =0 Γ( l + 1 2 )Γ(2 a − l − 1 2 ) Γ( l + 1)Γ(2 a − l ) e − 2 ipl , (A.5) and can b e repr esen ted in terms of the Legendre p olynomials P m (cos p ) 2 a − 1 X l =0 Γ( l + 1 2 )Γ(2 a − l − 1 2 ) Γ( l + 1)Γ(2 a − l ) e − 2 ipl = Γ(2 a − 1 2 )Γ( 1 2 ) Γ(2 a ) 2 F 1 ( 1 2 , 1 − 2 a ; 3 2 − 2 a ; e − 2 ip ) = π e − ipm P m (cos p ) (A.6) Using L ap lace asymptotic f ormula, P m (cos p ) =  2 π m sin p  1 2 cos  p  m + 1 2  − π 4  + O  1 m 3 2  , (A.7) for the remaining integ ral w e obtain the follo wing leading term π Z 0 dp P m (cos p ) e − ipm e ip − ih − = − i r π m (1 + h 2 − )  1 + O ( 1 √ m )  (A.8) Assem bling all the co nt ributions w e obtain the follo wing lea ding term for th e t wo- p oint f unction (for m o dd): h σ + m +1 σ − 1 i = ( − 1) m s 2 π (1 + h 2 − ) exp  1 4 Z ∞ 0 dt t  e − 4 t − 1 cosh 2 t  m − 3 4  1 + O ( 1 √ m )  (A.9) The same result h olds for m ev en, bu t the deriv ation is a little bit m ore tricky . The su m in (A.2) can b e once again rewritten in a more simple wa y a X b =1 b ( b + 1 2 )( b − 1 2 ) Γ( a − b + 1 2 )( h − )Γ( a + b + 1 2 ) Γ( a − b + 1)Γ( a + b + 1) π Z 0 dp e − 2 ipa e 2 ipb − e − 2 ipb e ip − ih − = 1 2 a X b = − a 1 b + 1 2 + 1 b − 1 2 ! Γ( a − b + 1 2 )Γ( a + b + 1 2 ) Γ( a − b + 1)Γ( a + b + 1) π Z 0 dp e 2 ip ( b − a ) e ip − ih − = i a X b = − a Γ( a − b + 1 2 )Γ( a + b + 1 2 ) Γ( a − b + 1)Γ( a + b + 1) π Z 0 dq e − 2 iq b cos q π Z 0 dp e 2 ip ( b − a ) e ip − ih − = iπ π Z 0 dq cos q π Z 0 dp P m (cos( q − p )) e − imp e ip − ih − (A.10) 32 where w e introdu ced an additional integral to b e able to express the resu lt once again in terms of the Legendre p olynomials. Asymptotic analysis of these in tegrals giv es iπ π Z 0 dq cos q π Z 0 dp P m (cos( q − p )) e − imp e ip − ih − = −  π m  3 2 2 i q 1 + h 2 −  1 + O ( 1 √ m )  , (A.11) and it leads on ce again to the same leading term (A.9) for the tw o-p oint fun ction. Ac kn o wledgmen ts J.M. 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