Thermodynamic limit of particle-hole form factors in the massless XXZ Heisenberg chain

LPENSL-TH-03/10 DESY 11-03 7 Thermo dynamic limit of particle-h ole form facto rs in th e massle ss XX Z Heis en b erg c hain N. Kitanine 1 , K. K. Kozlo wski 2 , J. M. Maillet 3 , N. A. Sla vno v 4 , V. T erras 5 Abstract W e study the ther mo dynamic lim it of the particle- ho le form factors of the XXZ Heisen b erg chain in the massles s regime. W e show that, in this limit, such form factor s decrease as a n ex plic itly co mputed power-law in the s ystem- size. Moreov er, the corr esp onding amplitudes can b e obtained a s a pro duct of a “ smo oth” and a “discrete” part: the for mer dep ends contin uous ly on the rapidities of the particles a nd holes, whereas the latter has an a dditio nal explicit depe ndence o n the set of in teger n um ber s that label each excited state in the asso ciated loga r ithmic B ethe equations . W e also show tha t sp ecial for m facto rs corres p o nding to zero-e ne r gy excitations lying on the F e r mi surfac e decrea se as a p ow er-law in the system size with the same critical exp onents as in the long- distance as y mptotic b ehavior of the related tw o- p o int cor relation functions. The metho ds we dev elop in this article are rather general and can be applied to o ther ma ssless integrable mo dels asso cia ted to the six-vertex R -matrix and having determinant repre s entations for their form factors. 1 In tro duction This article is a conti n uation of our study of form factors in massless quantum in tegrable mo dels. W e ha v e recen tly [1] inv estigated the thermody n amic limit of a sp ecial, Umklapp- t yp e, f orm f actor of the X X Z Heisen b erg spin c hain [2, 3, 4, 5, 6, 7, 8, 9] starting from its 1 IMB, UMR 5584 du CNRS, Un iversit ´ e de Bourgogne, F rance, Nicolai.Kitanine@u-b ourgogne.fr 2 DESY, Hamburg, Deu t sc hland, k arol.k a jetan.kozlo wski@desy .de 3 Lab oratoire de Physique, UMR 5672 du CNRS, ENS Lyon, F rance, maillet@ens-lyon.fr 4 Steklov Mathematical Institut e, Mosco w, Russia, n sla vno v@mi.ras.ru 5 Lab oratoire de Physique, UMR 5672 du CNRS, ENS Lyon, F rance, veronique.terras@ens-lyo n.fr 1 determinan t represen tatio n [10] in the algebraic Bet he Ansatz f ramew ork [11, 12, 13]. W e n o w apply this m etho d to general, particle-hole type, form factors. The m ain goal of our analysis is to pav e a wa y for the calculation of the asymptotic b eha vior of correlation fun ctions in massless in tegrable mo dels thr ough their form factor expansion. O u r results should also b e useful in the numerical computation o f dynamical structure factors throu gh their (exact) f orm factor expressions starting from fi nite size sys tems [14, 15, 16, 17, 18] as they allo w a precise a priori con trol of the dep end ence of eac h form f actor in terms of the system size. F or definiteness, w e c hose to f o cus on a particular mo del, th e XXZ spin-1 / 2 Heisen b erg c h ain [2] in the massless r egime and in an external magnetic field h > 0 for whic h determinan t represent ations of general form factors were obtained in [10]. Ho wev er, up to minor mo difica- tions sp ecific to the c hoice of th e mo del, our m etho d and r esu lts app ly as w ell to other massless, algebraic Bethe Ansatz-solv able mo dels with kno wn determinan t represen tations of their form factors lik e the non-linear Sc h r ¨ odinger mod el [19, 20] or the higher s p in XXX [2 1] and XXZ [22] c hains. The Hamiltonian of the XXZ chain is giv en b y , H = M X k =1  σ x k σ x k +1 + σ y k σ y k +1 + ∆( σ z k σ z k +1 − 1)  − h 2 M X k =1 σ z k , (1.1) where σ x,y ,z k are th e spin op erators (P auli matrices) acting on the k th site of the chain. The main pur p ose of this article is to study th e asymptotic b eha v ior of the form factors in the thermo dyn amic limit M → ∞ of the c hain and in the massless regime | ∆ | < 1. W e will tak e this limit starting from a chain of finite ev en size and sub ject to p erio dic b oundary conditions. Considering the fi nite c h ain allo ws u s to define rigorously the form factors of lo cal spin op erators as the norm alized matrix elemen ts F ( s ) ψ ψ ′ ( m ) = h ψ | σ s m | ψ ′ i k ψ k · k ψ ′ k , s = x, y , z , (1.2) where | ψ i an d | ψ ′ i are t w o eigenstates of the Hamilto nian (1.1 ). Such matrix element s can b e computed in a systemat ic wa y by using the solution to the q u an tum inv erse scattering problem [10, 23] together with the explicit f orm ulae for scalar pr o ducts [24, 25, 10] in the algebraic Bethe Ansatz framework [11, 12, 13, 26]. Using the closure r elation, one can express an y zero temp erature t w o-p oin t correlation function of spin op erators as a sum o ver the f orm facto rs of the co rresp on d ing local op erators, h ψ g | σ s m σ s ′ m ′ | ψ g i h ψ g | ψ g i = X | ψ ′ i F ( s ) ψ g ψ ′ ( m ) F ( s ′ ) ψ ′ ψ g ( m ′ ) . (1.3) Here | ψ g i denotes the ground s tate of the Hamiltonian (1.1), and th e summation is tak en with resp ect to all the eigenstates | ψ ′ i of (1.1). One p ossible w a y of d ealing with (1.3) wa s prop osed in [27, 28], where the sum o ver the form factors for the fi nite c hain was recast into a multiple con tour integral , the so-called master equation. This metho d then leads to v arious m ultiple integral r epresen tations f or the correlation functions in the thermo dynamic limit [27 , 28, 29] th at are in f act non-trivial summations of 2 previously obtained m ultiple in tegral representat ions for the correlation functions [30, 31, 32, 33, 34, 35]. Moreo ve r, in [36] we hav e sh o wn ho w to derive long-dista nce asymptotic b eha vior of certain t w o-p oin t correlation functions using suc h master equation representa tion. It agrees with pr ed ictions arising fr om the Luttinger-liquid and Conformal Field Theories appr oac hes [37, 38, 39, 40, 41, 42, 43, 44]. It is the v ery remark able structur e of the r esults obtained in [36] (the amplitudes of the p o wer law decrease of the long distance b eha vior of the t w o-p oin t correlation f unction are related to the mo dulu s squ ared of some prop er ly normalized f orm factors) that strongly suggested th at another p ossible w ay to analyze the asymptotic b ehavio r of the t wo-point correlation fu nctions w ould b e to tak e the thermo d ynamic limit directly in (1.3). It seems indeed qu ite natural that, in this limit, the main con tribution to the sum (1. 3) should s tem from excitatio ns ab ov e the ground state ha ving a finite energy . Su c h excited states can b e c h aracterized in terms of particles, h oles and/or string configur ations. In this framew ork, the sum o ver the complete set of states in (1.3) b ecomes a sum o ver all the p ossible particle- hole-string type excit ations ab ov e th e ground state which, after some p ossible regularizations, should b e rep laceable b y in tegrals in the thermo dyn amic limit. Another imp ortan t motiv ation to learn h o w to deal d irectly with the form factor expansion of the correlatio n fun ctions concerns the time dep endent case, as in that situation the asym p totic analysis throu gh the dynamical master equation [28] p oses y et unsolve d problems. The first step to carry out this pr ogram is therefore to determine the leading asymptotic b eha vior of the f orm factors of the mod el in the thermo dyn amic limit. It is the p urp ose of the present article to solve this p roblem for the so-call ed particle/hole t yp e form factors (corresp ond ing to diffusion states). The case of states asso ciated w ith complex solutions of the Bethe equ ations suc h as strin g typ e solutions (corresp onding to b ound states) requires the use of additional tec h niques and will b e considered in a separate pub lication. In [1], we ha v e sho wn that, in the M → ∞ limit, a sp ecial Umklapp-type form factor of the σ z op erator decreases as some negativ e p o w er of the system size M . The s triking feature of th is analysis is th at th is p o w er coincides with one of the critical exp onent s app earing in th e long- distance asymptotic b eha vior of the zero temp erature tw o-p oint correlation function h σ z 1 σ z m +1 i . In the present article, w e will show that ge neral form factors exhib it similar prop erties: they all deca y as some negativ e p o w er of the size M ; moreo ver, for s p ecial t yp es of form factors corresp ondin g, in th e th ermo dynamic limit, to p article-hole excitations lying on the F ermi surface, these exp onen ts coincide exactly with the critical exp onents describing the p ow er- la w deca y of the long-distance asymptotic b eha vior of the co rresp ond ing t wo -p oin t correlation functions. More precisely , in the thermo dynamic limit, the pro duct of tw o form factors app earing in (1.3) can b e pr esen ted in the follo wing form F ( s ) ψ g ψ ′ ( m ′ ) · F ( s ′ ) ψ ′ ψ g ( m ) = M − θ ss ′ e i P ex ( m − m ′ ) S ss ′ D ss ′ . (1.4) All the dep endence on the lattice distance m − m ′ is con tained in an ob vious p h ase fac tor 1 . There P ex is the excitatio n momen tum. The fi nite part of the form factors pro duct can b e separated in to t wo parts S ss ′ and D ss ′ . W e call them smo oth and discrete parts r esp ectiv ely . 1 Some form factors (see e.g. (3.1)) ma y also have an additional factor ( − 1) m − m ′ , whic h can b e remov ed b y the re-defi n ition of the Hamiltonian (1.1) 3 W e sho w that the s mo oth part S ss ′ dep end s contin uously on the r ap id ities of the particles and holes. On the con trary , it is only when the particles (holes) rapidities are s ep arated from the F ermi b ound ary that the discrete p art D ss ′ do es also dep end smo othly on these quan tities. As so on as the latte r app roac h the F ermi surface, the discrete s tr ucture of the form factors rev eals itself in D ss ′ . Th is means that a microscopic (of ord er 1 / M ) deviation of a particle (hole)’ s rapidit y leads to a macroscopic c hange of D ss ′ . W e believe that one of the consequen ces of such a structure is the p articular role pla y ed b y the F ermi b oun dary in b osonization tec hniqu es. The article is organized as follo w s. In section 2 w e giv e a description of th e low-l ying excited states of the mod el. F ollo w ing the ideas of [1], we consider excitations abov e th e ground state of the t wisted transfer matrix. Th is enab les us to form ulate p recisely our results in section 3, i.e. the structur e of th e large size asymptotic b eha vior of general particle-hole f orm factors in the XXZ sp in-1 / 2 c h ain. The remaining p art of th e article is dev oted to th e pro of of this resu lt. In sect ion 4 we sho w how form facto rs can b e redu ced to scala r pro du cts of a sp ecial t yp e that w e can repr esen t in terms of determin ants for c hains of finite length M . W e then calculate the thermo d ynamic limit of the form factors of the σ z op erator in section 5. In particular, we sho w that if the rap id ities of the particles and holes collapse on the F ermi surface then the form f actors hav e a discrete structure. In secti on 6 we obtain analogous results for the form factors of the σ ± op erators 2 . W e would lik e to stress that several methods and results used in the present artic le are b orro w ed fr om our previous wo rk [1]. W e conclude b y discussing the p ossibilit y of applying our resu lts to the calculation of the long-distance asymp totic b eha vior of the tw o-p oint functions at zero temp erature. S ev eral auxiliary Lemmas and pro ofs of tec hnical c h aracter are pr esen ted in th e t w o app endices. 2 Space of stat es of the mo del Since our ultimate goa l is to analyze the t w o-point correlation functions at zero temp eratur e through (1.3), the form factors we consider here corresp on d to matrix elements of a lo cal op erator b et ween the ground state and an excite d eigenstate of the Hamilt onian (1.1). In this section we recal l ho w such stat es can b e describ ed in the large M limit. In the algebraic Bethe Ansatz framew ork [11, 12, 13], the space of states of the mo d el (in finite volume) is constru cted by means of the Y ang-Baxter algebra realized by the op erator en tries of the mono dr om y matrix. In the XXZ case, and more generally for the class of mod els asso ciated with the s ix-vertex R-matrix, suc h mono dromy matrice s tak e the form T ( λ ) = A ( λ ) B ( λ ) C ( λ ) D ( λ ) ! , (2.1) where A , B , C , D are quan tum op erators d ep endin g on some sp ectral parameter λ . It satisfies Y ang-Baxter comm utation relations giv en by the six-v ertex R-matrix. In this framew ork, th e eigenstates | ψ i of the Hamilto nian (1.1) coincide with the ones of the transfer matrix T ( λ ) = tr T ( λ ) = A ( λ ) + D ( λ ), and can b e parameterized as | ψ i = | ψ ( λ 1 , . . . , λ N ) i , N = 0 , 1 , . . . , M / 2, 2 Recall that σ ± = 1 2 ( σ x ± iσ y ) 4 in terms of a set { λ j } N j =1 of solutions to the logarithmic Bethe equations M p 0 ( λ j ) − N X k =1 ϑ ( λ j − λ k ) = 2 π n j , j = 1 , . . . , N . (2.2) Here the fun ctions p 0 ( λ ) and ϑ ( λ ) are the bare momentum and phase, p 0 ( λ ) = i log sinh( iζ 2 + λ ) sinh( iζ 2 − λ ) ! , ϑ ( λ ) = i log  sinh( iζ + λ ) sinh( iζ − λ )  , (2.3) where 0 < ζ < π and cos ζ = ∆. Th e num b ers n j , − M / 2 < n j ≤ M / 2, are in tegers (for N o dd) or half-int egers (for N ev en ). 2.1 Thermo dynamic limit of the ground state The Bethe r o ots λ j , j = 1 , . . . , N , describing the ground state corresp ond to th e solution of th e system of logarithmic Bethe Ansatz equations (2.2) with the sp ecial c h oice n j = j − ( N + 1) / 2, where N is the num b er of do wn spins in the ground state 1 [4, 45, 9, 8] . Th us, for giv en M and h , these parameters λ j are fixed quant ities. In order to d escrib e ho w they b eh a ve in the thermo dyn amic limit ( N , M → ∞ su c h that N / M tends to some fixed a verag e densit y D ), we in tro duce the ground state counting function, b ξ ( ω ) = 1 2 π p 0 ( ω ) − 1 2 π M N X k =1 ϑ ( ω − λ k ) + N + 1 2 M , (2.4) whic h is build so that b ξ ( λ j ) = j / M for j = 1 , . . . , N . This function defin es the discrete densit y of the minimal energy state in th e N sector by b ρ ( ω ) = b ξ ′ ( ω ). It is p ossible to argue that, in the thermod ynamic limit, the parameters λ j fill densely a finite in terv al [ − q , q ] of th e real axis (the F ermi zone), and that the d iscrete densit y goes to a smo oth function ρ ( ω ) that solv es th e follo wing in tegral equation ρ ( λ ) + 1 2 π q Z − q K ( λ − µ ) ρ ( µ ) dµ = 1 2 π p ′ 0 ( λ ) , with K ( λ ) = ϑ ′ ( λ ) . (2.5) The v alue of the endp oint q of the F ermi zone is fixed by requ iring that R q − q ρ ( λ ) = D . N ote that, sin ce lim M →∞ b ξ ( λ 1 ) = 0, the thermo dyn amic limit ξ ( λ ) of the counting function is the an tideriv ativ e of ρ ( λ ) that v anishes at − q . Therefore ξ ( λ ) can b e expressed in terms of the dressed momentum p ( λ ), the F ermi momentum b eing defined b y k F = p ( q ) p ( λ ) = 2 π λ Z 0 ρ ( µ ) dµ, ξ ( λ ) = [ p ( λ ) + p ( q )] / 2 π, π D = p ( q ) ≡ k F . (2.6) 1 The num b er N dep ends on the magnetization of th e ground state, whic h is fixed b y the o verall magnetic field h . Hereafter, we call N sector the subspace of t h e space of states h a ving N spins down. 5 Sev eral other p rop erties of the ground state can b e describ ed through the solutions of other linear int egral equations. W e in tro d uce t w o suc h functions that will p la y an imp ortan t r ole in our analysis, namely the dr essed c h arge Z ( λ ) and the dr essed p hase φ ( λ, ν ), wh ic h satisfy Z ( λ ) + 1 2 π q Z − q K ( λ − µ ) Z ( µ ) dµ = 1 , (2.7) and φ ( λ, ν ) + 1 2 π q Z − q K ( λ − µ ) φ ( µ, ν ) dµ = 1 2 π ϑ ( λ − ν ) . (2.8) In fact, these t w o functions are not indep end en t. Using that K ( λ ) = ϑ ′ ( λ ), one easily obtains that Z ( λ ) = 1 + φ ( λ, q ) − φ ( λ, − q ) . (2.9) Another non-trivial relationship b et ween Z and φ has b een established in [46, 47]: 1 + φ ( q , q ) − φ ( − q , q ) = Z − 1 ( q ) . (2.10) 2.2 Thermo dynamic limit of the excited states T o d escrib e th e other eigenstates of the transf er matrix T ( λ ), it is conv enien t for further purp oses to consider in stead the t wisted transfer m atrix T κ ( λ ) = A ( λ ) + κD ( λ ), where κ is some complex (t w ist) parameter. Hereb y the excited states of the Hamiltonian (1.1) are obtained as the κ → 1 limit of the eigenstates of the t wisted transfer matrix T κ ( λ ) [27, 1]. In co mplete analogy w ith the standard algebraic Bethe Ansatz considerations, the eig en- states | ψ κ ( { µ } ) i of T κ ( λ ) can b e parameterized b y sets of solutions of the t wisted Bethe equa- tions M p 0 ( µ ℓ j ) − N κ X k =1 ϑ ( µ ℓ j − µ ℓ k ) = 2 π  ℓ j − N κ + 1 2  + 2 π α, j = 1 , . . . , N κ . (2.11) Suc h state s | ψ κ ( { µ } ) i dep end on N κ parameters { µ ℓ k } N κ k =1 , where N κ is the n u m b er of sp ins do wn and we hav e set κ = e 2 π iα . In this article we consider α to b e a r eal num b er. This restriction ℑ ( α ) = 0 is n ot crucial, but con venien t, since then h ψ κ ( { µ } ) | = ( − 1) N κ | ψ κ ( { µ } ) i † . R emark 2.1 . O ne could of cour se from the v ery b eginnin g set α = 0 and d eal d irectly with the standard excited states of the XXZ Hamiltonian (1.1). Ho wev er, we c hose to introdu ce this extra parameter and k eep it throughout our computations sin ce, as w e will sho w later on, an imp ortant cla ss of form factors can b e obtained by a mere shift of α by some in teger v alue. In the follo wing, w e will consider t wo case s of in terest: N κ = N for form factors of σ z , and N κ = N + 1 for the σ + form factors 1 . It f ollo ws from [8] that the set of real ro ots of th e t w isted Bethe equations is completely defined b y the c hoice of the set of in tegers ℓ j in the r.h.s. of (2.11). Ther efore, w e ha v e lab eled the ro ots of (2.11) by subscripts ℓ j . 1 W e do not consider the case N κ = N − 1 corresponding to form factors of σ − , since the last on es can b e obtained from the form factors of σ + . 6 Definition 2.1. The state p ar ameterize d by the r e al solutions to (2.11) with ℓ j = j , j = 1 , . . . , N κ , i s c al le d α -twiste d gr ound state in the N κ se ctor. In the thermo d ynamic limit, excitations corresp onding to real set of solutions to the twiste d Bethe equations ab o ve this α -t wisted ground state can b e describ ed in terms of particles and holes. Note that the states asso ciated to complex ro ots of the Bethe equations (like string solutions) are n ot considered in the p resen t article. They corresp ond in particular to form factors of b ound s tates [5, 6, 7, 48] that will b e dealt w ith in a separate publication. All this means that one deals with solutions to the t wisted Bethe equations where most of the ℓ j coincide with their v alue for the ground state : ℓ j = j except for n in tegers for whic h ℓ j 6 = j , with n remaining fi nite in the N κ → ∞ limit. T o b e m ore precise, we fi x 2 n d istinct in tegers h 1 , . . . , h n and p 1 , . . . , p n suc h that h k ∈ { 1 , . . . , N κ } and p k / ∈ { 1 , . . . , N κ } . The in tegers h k represent ”holes” in resp ect to the distributions of integ ers f or the ground state in the N κ sector, whereas the integ ers p k represent ”particles”. In other words, the excited state is describ ed by a set of in tegers { ℓ j } N κ j =1 suc h that ℓ j = j for j 6 = h 1 , . . . , h n , and ℓ h k = p k . The terminology of particles and holes tak es a clearer in terpretation in terms of the counting function b ξ κ asso ciated with the N κ -excited state: b ξ κ ( ω ) = 1 2 π p 0 ( ω ) − 1 2 π M N κ X k =1 ϑ ( ω − µ ℓ k ) + N κ + 1 2 M − α M . (2.12) One can argue that the coun ting function is m on otonously increasing on th e real axis. Thus it is p ossible to defin e unam biguous ly a set of real p arameters µ j as the u nique solutions to b ξ κ ( µ j ) = j / M . One can id en tify , among th is set of p arameters, the N κ solutions of the t wisted Bethe equations with in tegers ℓ j as b eing µ ℓ j as it s h ould b e. In this picture, an excited state no-longer corresp onds to the set of N κ consecutiv e solutions µ j to the equation b ξ κ ( µ j ) = j / M , but is rather obtained b y removi ng the solutions b ξ κ ( µ h a ) = h a / M and replacing th em by the solutions b ξ κ ( µ p a ) = p a / M . It is in this resp ect that µ h a stands for the rapidities of the holes and th at µ p a stands for those of the particles. F rom now on, we agree up on µ j b eing the solution to b ξ κ ( µ j ) = j / M with b ξ κ b eing giv en b y (2.12). Similarly as for the groun d state in the N sector, one can also define the discrete densit y of the excit ed s tate b y b ρ κ ( ω ) = b ξ ′ κ ( ω ). It is ea sy to see that the thermo dynamic limits of this new coun ting fun ction and its asso ciated density coincide with the limits defin ed ab o v e in (2.5), (2.6). An excited state is most con venien tly c haracterized in terms of the s hift function b F ( ω ) = b F ( ω |{ µ p }|{ µ h } ) defined by b F ( ω ) = M  b ξ ( ω ) − b ξ κ ( ω )  . (2.13) The shift function describ es the sp acing b et wee n th e ro ot λ j for the groun d state in the N sector and the parameters µ j defined b y b ξ κ ( µ j ) = j / M : µ j − λ j = F ( λ j ) ρ ( λ j ) M + O ( M − 2 ) , (2.14) where F ( λ ) is the th er m o dynamic limit of the shift fu nction. Reca ll that w e consider the excited states in the N κ sector with N κ = N (for form f actors of σ z ) and N κ = N + 1 (for 7 form factors of σ + ). Resp ectiv ely we should distinguish b et ween t wo shift fu nctions F ( z ) ( λ ) and F (+) ( λ ) corresp onding to these t wo cases. Generically the s hift f u nction F ( λ ) satisfies the in tegral equation F ( λ ) + q Z − q K ( λ − µ ) F ( µ ) dµ 2 π = α + δ N 2  1 − ϑ ( λ − q ) π  + 1 2 π n X k =1  ϑ ( λ − µ p k ) − ϑ ( λ − µ h k )  . (2.15) where δ N = N − N κ . Using (2.7 ) and (2.8) we conclud e that F ( z ) ( λ ) = αZ ( λ ) + n X k =1 φ ( λ, µ p k ) − n X k =1 φ ( λ, µ h k ) . (2.16) and F (+) ( λ ) =  α − 1 2  Z ( λ ) + n X k =1 φ ( λ, µ p k ) − n X k =1 φ ( λ, µ h k ) + φ ( λ, q ) . (2.17) 3 The main results W e no w formulate a pr ecise statemen t of th e resu lts obtained in this article concerning the thermo dyn amic b eha vior of f orm factors. In fact, for later use, we will giv e this thermo d ynamic b ehavio r directly for the pro du cts of form factors app earing in the sp ectral expansion of the correlation fu nction (1.3). Ho wev er, it will b e made clear that our metho d allo ws also to obtain the b eha vior of eac h in dividual form factor. The pro of of the results we present below will b e giv en in the next sections. W e consider the form factors (1.2) of the lo cal spin op erators σ s m , s = + , − , z , b etw een the ground state | ψ g i and some (differen t) excit ed stat e | ψ ′ i of the Hamiltonian 1 . Here and in the follo wing, w e set | ψ g i ≡ | ψ ( { λ } ) i , { λ 1 , . . . , λ N } b eing the s olution of the Bethe equations (2.2) describing the groun d state of the Hamiltonian, and | ψ ′ i = lim κ → 1 | ψ κ ( { µ } ) i for { µ ℓ 1 , . . . , µ ℓ N κ } b eing a solution of the κ -t wisted Bethe equ ations (2.11) with n particles and n h oles. W e will b e sp ecially interested in the limiting case where, in the th ermo dynamic limit, all r apidities { µ p j } n j =1 and { µ h j } n j =1 of particles and holes condensate on the F ermi b ound aries. Su c h excit ed states h a ve a zero excitation energy in the thermod ynamic limit; w e exp ect that they will pro du ce the main con trib u tion to the asymptotic b eha vior of correlat ion functions (this fac t is already apparent in the predictions based on the Luttinger liquid theory and Conf ormal Field Theory for the long-distance asymptotic b eha vior of t wo-point fun ctions [37, 38, 39, 40 , 41, 42, 43, 44]). W e will also consider the opp osite limiting case, namely when all particle/hole rapidities r emain at finite distance fr om the F ermi b oundaries. In the thermo d ynamic limit, up to uniformly M − 1 · log M corrections, the p ro ducts of suc h form factors b ehav e as F ( z ) ψ g ψ ′ ( m ′ ) · F ( z ) ψ ′ ψ g ( m ) ∼ δ N ,N κ M − θ z z e i P ex ( m − m ′ ) ∂ 2 α S z z D z z    α =0 , F (+) ψ g ψ ′ ( m ′ ) · F ( − ) ψ ′ ψ g ( m ) ∼ δ N +1 ,N κ M − θ + − ( − 1) m − m ′ e i P ex ( m − m ′ ) S + − D + −    α =0 . (3.1) 1 The case when | ψ ′ i = | ψ g i is trivial: F ( ± ) ψ g ψ g ( m ) = 0, F ( z ) ψ g ψ g ( m ) = 2 D − 1. 8 Let us explain these formulas. These pro d ucts d ecrease as some n egativ e p ow er of the size M of the s y s tem, the exp onents θ z z , θ + − b eing sp ecified b elo w. Due to the translation in v ariance of the mod el, all the dep endence on the lattice s p acing m − m ′ is absorb ed into a phase factor, with P ex = lim M ,N → + ∞ n N κ X j =1 p 0 ( µ ℓ j ) − N X j =1 p 0 ( λ j ) o = 2 π αD + n X j =1  p ( µ p j ) − p ( µ h j )  (3.2) b eing the momentum of the excited state relativ e to the ground state. W e hav e split the constan t in f ron t of th e M and ( m − m ′ ) d ep endence in tw o parts, S ss ′ and D ss ′ , that w e call smo oth and d iscrete parts resp ectiv ely . The reason f or this denominatio n is that the smo oth part S ss ′ dep end s contin uously on the r apidities µ p j and µ h j of the p articles and holes, whereas the discrete part D ss ′ also dep ends on the set of in tegers app earing in the logarithmic Bethe Ansatz equations (2.11) for th e excited state. Such a discrete structur e can b e f airly we ll appro ximated by a smo oth function of the rapidities µ p j and µ h j as long as th ey are located at a fi nite d istance fr om the F ermi b ound aries. Ho wev er, as so on as th e rapidities of the particles or holes approac h the F ermi surface, the d iscr ete structure of the form facto rs can no longer b e neglected: a microscopic (of order 1 / M ) deviation of a particle (or hole) r apidit y leads to a macroscopic c hange in D ss ′ . 3.1 Smo oth part s The smo oth parts S z z and S + − are decomp osed into S z z = 2 π 2 sin 2  P ex 2  · A ( z ) n · e C ( z ) n , S + − = A (+) n · e C (+) n . (3.3) Here C ( z ) n , A ( z ) n are functionals acting on the shift function F ( z ) ( λ ) given in (2.16), while C (+) n , A (+) n are functionals acting on the shift function F (+) ( λ ) giv en in (2.17). T his action is su c h that the result d ep ends smo othly on the particle/hole r apidities { µ p } and { µ h } of the state | ψ κ ( { µ } ) i . Before giving explicit expressions for the quan tities in (3.3) we in tro duce the iπ -p erio dic Cauc h y transform of the shift fun ctions F ( z / +) ( λ ) on [ − q , q ], F ( z / +) ( w ) = 1 2 π i q Z − q F ( z / +) ( λ ) coth( λ − w ) dλ. (3.4) Then the co efficien t C ( z ) n = C ( z ) n ( { µ p } , { µ h } ) has the form C ( z ) n [ F ( z ) ] = C 0 [ F ( z ) ] + 2 π i n X j =1  F ( z ) ( µ h j − iζ ) + F ( z ) ( µ h j + iζ ) − F ( z ) ( µ p j − iζ ) − F ( z ) ( µ p j + iζ )  + n X j,k =1 log sinh( µ h j − µ p k − iζ ) sinh( µ p k − µ h j − iζ ) sinh( µ p j − µ p k − iζ ) sinh( µ h j − µ h k − iζ ) , (3.5) 9 where the fun ctional C 0 [ F ] reads C 0 [ F ] = − q Z − q F ( λ ) F ( µ ) sinh 2 ( λ − µ − iζ ) dλ dµ. (3.6) The co efficien t C (+) n = C (+) n ( { µ p } , { µ h } ) has similar representa tion, but it conta ins sev eral additional terms C (+) n [ F (+) ] = C ( z ) n [ F (+) ] − 2 π i  F (+) ( q + iζ ) + F (+) ( q − iζ )  + n X j =1 log sinh( µ h j − q − iζ ) sinh( µ h j − q + iζ ) sinh( µ p j − q − iζ ) sinh( µ p j − q + iζ ) . (3.7) The functional A ( z ) n = A ( z ) n ( { µ p } , { µ h } ) has the form A ( z ) n [ F ( z ) ] =  sin π α sin π F ( z ) ( − q )  2      e − 2 π i F ( z ) ( − q + iζ ) n Y k =1 sinh( q + µ h k + iζ ) sinh( q + µ p k + iζ )      2 ×      det  I + 1 2 π i U ( z ) ( w, w ′ )  det  I + 1 2 π K       2 . (3.8) The equation (3.8) con tains a ratio of F redholm determinants. T he in tegral operator I + 1 2 π K acts on the interv al [ − q , q ], whereas the integral op erators I + 1 2 π i U ( z ) acts on a countercl o c kw ise orien ted con tour Γ q surroun ding [ − q , q ]. This contour is su c h that it con tains all the ground state ro ots and no other sin gularit y of the k ern el. Th e in tegral k ernel is defined b y U ( z ) ( w, w ′ ) = − Φ( w ) e 2 π i  F ( z ) ( w ) − F ( z ) ( w + iζ )  K κ ( w − w ′ ) − K κ ( − q − w ′ ) 1 − e 2 π iF ( z ) ( w ) , (3.9) with K κ ( w ) = coth( w + iζ ) − κ coth( w − iζ ) , Φ( w ) = n Y k =1 sinh( w − µ p k ) sinh( w − µ h k + iζ ) sinh( w − µ h k ) sinh( w − µ p k + iζ ) . (3.10) The coefficien t A (+) n = A (+) n ( { µ p } , { µ h } ) also is prop ortional to a ratio of F redholm deter- minan ts A (+) n [ F (+) ] = sin ζ 2 π κ      e − 2 π i F (+) ( iζ 2 ) sinh( q − iζ 2 ) n Y k =1 sinh( µ h k − iζ 2 ) sinh( µ p k − iζ 2 ) · det  I + 1 2 π i U (+) ( w, w ′ )  det  I + 1 2 π K       2 . (3.11) Similarly to the inte gral op erator defin ed in (3.9 ), the op erator I + 1 2 π i U (+) acts on the con tour Γ q surroun ding [ − q , q ], with a ke rnel U (+) [ F ] ( w , w ′ ) = Φ( w ) sinh( w − q ) sinh( w − q + iζ ) e 2 π i  F (+) ( w ) − F (+) ( w + iζ )  K κ ( w, w ′ ) 1 − e 2 π iF (+) ( w ) , (3.12) where K κ ( w, w ′ ) = sinh( w ′ + 3 iζ 2 ) sinh( w ′ − iζ 2 ) sinh( w − w ′ − iζ ) − κ sinh( w ′ − 3 iζ 2 ) sinh( w ′ + iζ 2 ) sinh( w − w ′ + iζ ) , (3.13) and Φ( w ) is give n by (3.10). 10 3.2 Discrete parts and exp onents The v alues of the d iscrete parts D z z and D + − , as well as of th e exp onents θ z z and θ + − , dep end on whether particle s and holes are on th e F ermi surface or separated from it. W e sp ecify their expressions in the tw o particular cases w e mentio ned ab ov e: when all particle/hole r apidities remain at fin ite distance from the F ermi b oun daries, or w hen all particle/hole rapidities collapse on the F ermi b oundaries. Other inte rmediate configurations can b e obtained along similar tec hn iqu es. 3.2.1 P articles and holes a w a y from the F ermi b oundaries In the first case, the exp onen ts in (3.1) giving the algebraic deca y of the form factor with the system size read: θ z z = 2 n +  F ( z ) +  2 +  F ( z ) −  2 , θ + − = 2 n +  F (+) + + 1  2 +  F (+) −  2 , (3.14) where F ( z / +) ± = F ( z / +) ( ± q ) are giv en in terms of the sh ift fun ctions defined in (2.16) and (2.17). The discrete part D z z is giv en b y D z z ( { µ p } , { µ h } ) =  det n 1 sinh( µ p j − µ h k )  2 · D ( z ) [ F ( z ) ] n Y k =1 sin 2 ( π F ( z ) ( µ h k )) π 2 ρ ( µ h k ) ρ ( µ p k ) × n Y k =1 exp    2 q Z − q  F ( z ) ( λ )  coth( λ − µ p k ) − coth( λ − µ h k )  dλ    . (3.15) Here a fun ctional D ( z ) [ F ] h as the follo wing form: D ( z ) [ F ] = G 2 (1 − F − ) G 2 (1 + F + ) (2 π ) F − − F + [ ρ ( q ) sinh(2 q )] ( F − ) 2 +( F + ) 2 e C 1 [ F ] . (3.16) where G ( z ) is the Barnes function satisfying G ( z + 1) = Γ( z ) G ( z ), and w e ha ve set C 1 [ F ] = q Z − q F ′ ( λ ) F ( µ ) − F ( λ ) F ′ ( µ ) 2 tanh( λ − µ ) dλ dµ + F + q Z − q F + − F ( λ ) tanh( q − λ ) dλ + F − q Z − q F − − F ( λ ) tanh( q + λ ) dλ . (3.17) The discrete part D + − has the form D + − ( { µ p } , { µ h } ) =  det n 1 sinh( µ p j − µ h k )  2 · D (+) [ F (+) ] n Y k =1 sin 2 ( π F (+) ( µ h k )) π 2 ρ ( µ h k ) ρ ( µ p k ) × n Y k =1  sinh( µ p k − q ) sinh( µ h k − q )  2 exp    2 q Z − q  F (+) ( λ )  coth( λ − µ p k ) − coth( λ − µ h k )  dλ    , (3.18) 11 where the fun ctional D (+) [ F ] reads D (+) [ F ] = D ( z ) [ F ] · sinh(2 q )Γ 2 (1 + F + ) [ ρ ( q ) sinh(2 q )] 1+2 F + exp    2 q Z − q F + − F ( λ ) tanh( q − λ ) dλ    . ( 3.19) 3.2.2 P articles and holes on the F ermi b oundaries Definition 3.1. L et an excite d state c ontain n p articles and n holes. It is c al le d critic al excite d state, i f al l the r a pidities µ p , µ h = ± q in the thermo dynamic limit. We also say that this excite d state b elongs to the P r class if it c ontains n ± p p articles, r esp. n ± h holes, with r apidities e qual to ± q such that n + p − n + h = n − h − n − p = r, r ∈ Z . (3.20) The c orr esp onding form factors ar e c al le d critic al form factors of the P r class. The rapidities of the particles and holes in an excited state b elonging to the P r class are all lo cated in a close neighborh o o d of ± q . As a consequence, it is useful to re-parameterize the in tegers describing the p osition of p articles and holes according to p j = p + j + N κ , if µ p j = q , p j = 1 − p − j , if µ p j = − q , h j = N κ + 1 − h + j , if µ h j = q , h j = h − j , if µ h j = − q . (3.21) All the inte gers { p ± } and { h ± } in tro duced ab ov e are p ositiv e an d v ary in a range suc h that lim N →∞ P p ± j N = lim N →∞ P h ± j N = 0 , (3.22) whic h means that µ p and µ h indeed collapse to the F er m i b oun dary in the therm o dynamic limit. F or σ z and σ ± form factors b elonging to the P r class, the discrete parts are given b y D z z = D ( z ) [ F ( z ) r ] G 2 (1 + F ( z ) + ) G 2 (1 − F ( z ) − ) G 2 (1 + F ( z ) r, + ) G 2 (1 − F ( z ) r, − ) sin( π F ( z ) r, + ) π ! 2 n + h sin( π F ( z ) r, − ) π ! 2 n − h × R n + p ,n + h ( { p + } , { h + }| F ( z ) + ) R n − p ,n − h ( { p − } , { h − }| − F ( z ) − ) , (3.23 ) and D + − = D (+) [ F (+) r ] G 2 (2 + F (+) + ) G 2 (1 − F (+) − ) G 2 (2 + F (+) r, + ) G 2 (1 − F (+) r, − ) sin( π F (+) r, + ) π ! 2 n + h sin( π F (+) r, − ) π ! 2 n − h × R n + p ,n + h ( { p + } , { h + }| 1 + F (+) + ) R n − p ,n − h ( { p − } , { h − }| − F (+) − ) . (3.24 ) Here we agree u p on F ( z / +) r ( λ ) = F ( z / +) ( λ ) + r and F ( z / +) r, ± = F ( z / +) r ( ± q ). Th e fun ctionals D ( z ) and D (+) are giv en resp ectiv ely in (3. 16) and (3.1 9). The coefficient R n,m ( { p } , { h } | F ) is 12 defined as R n,m ( { p } , { h } | F ) = n Q j >k ( p j − p k ) 2 m Q j >k ( h j − h k ) 2 n Q j =1 m Q k =1 ( p j + h k − 1) 2 Γ 2 { p k + F } , { h k − F } { p k } , { h k } ! , (3.2 5) where w e ha v e used the standard h yp ergeometric type notation f or r atios of Γ functions: Γ a 1 , . . . , a ℓ b 1 , . . . , b j ! = ℓ Y k =1 Γ( a k ) · j Y k =1 Γ( b k ) − 1 . (3.26) The exp onents θ in this case are θ z z ( r ) =  F ( z ) r, +  2 +  F ( z ) r, −  2 , θ + − ( r ) =  1 + F (+) r, +  2 +  F (+) r, −  2 . (3.27) Note that th e shift f unctions F ( z / +) ( λ ) ent er the exp onents (3.27) only in the com bination F ( z / +) r ( λ ) = F ( z / +) ( λ ) + r . It is easy to see that F ( z / +) r ( λ ) are nothing else bu t the sh ift functions of the ( α + r )-t wisted groun d state s (see Definition 2.1) in the N and N + 1 sectors resp ectiv ely . R emark 3.1 . One can wr ite do wn the results of this section in a more symmetric form. F or this w e in tro duce t w o constan ts f ∓ describing the shift of the utmost parameters µ 1 and µ N κ of the ( α + r )-t wisted ground state w ith resp ect to the utmost parameters λ 1 and λ N of the ground state: f − = lim N ,M →∞ M b ρ ( λ 1 )( µ 1 − λ 1 ) , f + = lim N ,M →∞ M b ρ ( λ N )( µ N κ − λ N ) . (3.28) Then it is easy to see that f ± = F ( z ) r, ± for the σ z form factors. Ho wev er for the σ + form factors (where N κ = N + 1) one has f − = F (+) r, − , bu t f + = 1 + F (+) r, + . Then in b oth cases the exp onen ts (3.27) can b e written as θ = f 2 + + f 2 − . It is crucial to compare th e r esults (3.27) with the cr itical exp onen ts app earing in the asymptotic b eha vior of the t wo -p oin t correlat ion fu nctions predicted in [37, 38, 39, 40, 41, 42, 43, 44, 49, 50, 36]. In the case of θ z z ( r ), it follo ws from (2.9), (2.16) that F ( z ) r ( λ ) = ( α + r ) Z ( λ ). Therefore, for α = 0, we obtain that θ z z ( r ) = 2 r 2 Z 2 ( q ), | r | = 1 , 2 , . . . . These n um b ers coincide with the critical exp onents app earing in the asymptotic b eha vior of the h σ z 1 σ z m +1 i correlation fun ction and are asso ciated to th e oscilla ting term with momen tum 2 r k F . The exp onen t θ + − ( r ) h as similar prop erties. In this case, F (+) r ( λ ) = ( α + r − 1 / 2) Z ( λ ) + φ ( λ, q ). It follo ws from equations (2.9) and (2.10) that 1 + φ ( q , q ) = Z ( q ) + Z − 1 ( q ) 2 , φ ( − q , q ) = Z ( q ) − Z − 1 ( q ) 2 . (3.29) 13 F rom this we find F r, + + 1 = ( α + r ) Z ( q ) + Z − 1 ( q ) 2 , F r, − = ( α + r ) Z ( q ) − Z − 1 ( q ) 2 . (3.30) Th us, at α = 0 we h a ve θ + − ( r ) = Z − 2 ( q ) / 2 + 2 r 2 Z 2 ( q ), | r | = 0 , 1 , . . . . Again, these n um b ers coincide with the critical exp onen ts app earing in the asymp totic b ehavio r of the h σ + 1 σ − m +1 i correlation function and asso ciated to the oscillating term with momentum 2 r k F . Thus, we see that the b eha vior of critical f orm factors with resp ect to the size of the system coincides with the asymptotic b eha vior of t w o-p oin t correlat ion functions with r esp ect to the lattice distance. 4 F orm factors and scalar pro ducts The solution of the quan tum in v erse scattering problem enables us to express the lo cal s pin op erators in terms of the en tries of th e mono drom y matrix [10, 23]: σ s m = T m − 1 ( − iζ / 2 ) · tr  T ( − iζ / 2) σ s  · T − m ( − iζ / 2 ) . (4.1) In the l.h.s. of th is expression, the sy mb ol σ s m ( s = ± , z ) d enotes the corresp on d ing lo cal spin op erator at site m , whereas the sy mb ol σ s app earing in the r.h.s. should b e understo o d as a 2 × 2 P auli matrix m ultiplying the 2 × 2 m on o drom y matrix (2.1). Using (4.1), one can reduce the computation of the form factors of lo cal spin op erators in the finite XXZ c hain to the one of the scala r pr o ducts [10]. W e ha ve F ( z ) ψ ′ ψ g ( m ) = − e i ( m − 1) N P j =1 [ p 0 ( µ ℓ j ) − p 0 ( λ j )]  e i N P j =1 [ p 0 ( µ ℓ j ) − p 0 ( λ j )] − 1  × ∂ ∂ α h ψ κ ( { µ } ) | ψ ( { λ } ) i π i k ψ κ ( { µ } ) k · k ψ ( { λ } ) k    α =0 , (4.2) whic h is nonzero only if N κ = N . The form-factor F ( z ) ψ g ψ ′ is obtained b y complex co njugation F ( z ) ψ g ψ ′ ( m ) =  F ( z ) ψ ′ ψ g ( m )  ∗ . Similarly , one obta ins F ( − ) ψ ′ ψ g ( m ) = ( − 1) N + m +1 e i ( m − 1) N κ P j =1 p 0 ( µ ℓ j ) − im N P j =1 p 0 ( λ j ) h ψ κ ( { µ } ) | B  − iζ 2  | ψ ( { λ } ) i a ( − i ζ 2 ) k ψ κ ( { µ } ) k · k ψ ( { λ } ) k    α =0 , (4.3) F (+) ψ g ψ ′ ( m ) = ( − 1) N + m e i ( m − 1) N P j =1 p 0 ( λ j ) − im N κ P j =1 p 0 ( µ ℓ j ) h ψ ( { λ } ) | C  − iζ 2  | ψ κ ( { µ } ) i a ( − i ζ 2 ) k ψ κ ( { µ } ) k · k ψ ( { λ } ) k    α =0 , (4.4) whic h are nonzero only if N κ = N + 1. I n (4.3) and (4.4 ), a ( ν ) = sinh M ( ν − iζ / 2) d enotes the eigen v alue of the op er ator A ( ν ) on the reference state whic h, in the case of the spin-1/2 c hain, is the ferromagnetic state w ith all spins up . In the follo w ing, we will also u se the notation d ( ν ) = sinh M ( ν + iζ / 2) for the eigen v alue of D ( ν ) on th is reference state. R emark 4.1 . Although it is p ossible to set α = 0 directly in (4.3) an d (4.4), we prefer to tak e the limit α = 0 only in the v ery end of the calculatio ns to study the prop erties of th e scalar pro du cts for ge neral α . 14 In all the ab o v e scalar p ro ducts, one of the states is an eigenstate of the t wisted tran s fer matrix. T here exists an explicit determinan t representat ions for su c h scalar p ro ducts and the asso ciated norms: Prop osition 4.1. L et { λ } N 1 satisfy the gr ound state Bethe e quations (2.2) and { µ } N κ 1 solve the system of α -twiste d Bethe A nsatz e qu ations (2.11) . Then the fol lowing r epr esentations for the sc alar pr o ducts hold: h ψ κ ( { µ } ) | ψ ( { λ } ) i = δ N κ ,N · N Y a,b =1 sinh( µ ℓ a − λ b − iζ ) sinh( λ a − µ ℓ b ) · N Y j =1 n d ( µ ℓ j ) d ( λ j ) h e 2 iπ b F ( λ j ) − 1 io × 1 − κ 1 − e 2 π i b F ( − q ) · N Y a =1 sinh( q + λ a − iζ ) sinh( q − µ ℓ a − iζ ) · det Γ q  I + 1 2 π i b U ( z ) ( w, w ′ )  , (4.5) h ψ κ ( { µ } ) | B ( − iζ / 2) | ψ ( { λ } ) i = δ N κ ,N +1 · a ( − iζ / 2) sinh ( − iζ ) N Y j =1 n a ( λ j ) h 1 − e 2 iπ b F ( λ j ) io × N Q b =1 sinh( λ b − iζ / 2) N +1 Q b =1 sinh( µ ℓ b + iζ / 2) N +1 Y a =1 ( d ( µ ℓ a ) N Y b =1 sinh( µ ℓ a − λ b − iζ ) sinh( µ ℓ a − λ b ) ) · det Γ q  I + 1 2 π i b U (+)  w, w ′   . (4.6) In these expr essions b F ( λ ) (2.13) dep ends on whether we c onsider the N se ctor or the ( N + 1) se ctor (se e (2.12) ). The inte gr al op er ators I + 1 2 π i b U ( z ) and I + 1 2 π i b U (+) act on a close d anti- clo ckwise oriente d c ontour Γ q surr ounding the interval [ − q , q ] wher e the gr ound state r o ots { λ a } N a =1 c ondensate and c ontaining no other singularity of the kernels. The last ones have the form b U ( z ) ( w, w ′ ) = − N Y a =1 sinh( w − µ ℓ a ) sinh( w − λ a + iζ ) sinh( w − λ a ) sinh( w − µ ℓ a + iζ ) · K κ ( w − w ′ ) − K κ ( − q − w ′ ) 1 − e 2 π i b F ( w ) , (4.7) b U (+) ( w, w ′ ) = N +1 Y a =1 sinh( w − µ ℓ a ) sinh( w − µ ℓ a + iζ ) N Y a =1 sinh( w − λ a + iζ ) sinh( w − λ a ) · K κ ( w, w ′ ) 1 − e 2 π i b F ( w ) , (4.8) wher e K κ and K κ ar e given r esp e ctively by (3.10) and (3.13) . The representa tion (4.5) was obtained in [24, 25, 10, 36]. T he represent ation (4.6 ) is derive d in app end ix B. W e also r ecall the finite size determinant represen tations for the norms of Bethe states: 15 Prop osition 4.2. [51, 52, 53] L et µ ℓ 1 , . . . , µ ℓ N κ satisfy the system (2.12) . Then h ψ κ ( { µ } ) | ψ κ ( { µ } ) i = ( − 1) N κ N κ Y j =1  2 π iM b ρ κ ( µ ℓ j ) a ( µ ℓ j ) d ( µ ℓ j )  N κ Q a,b =1 sinh( µ ℓ a − µ ℓ b − iζ ) N κ Q a,b =1 a 6 = b sinh( µ ℓ a − µ ℓ b ) × det N κ Θ ( µ ) j k , (4.9) wher e Θ ( µ ) j k = δ j k + K ( µ ℓ j − µ ℓ k ) 2 π M b ρ κ ( µ ℓ k ) . (4.10) One can see from another repr esen tatio n for the scalar pro duct (B.1) that, when ℑ α = 0, h ψ κ ( { µ } ) | ψ ( { λ } ) i =  h ψ κ ( { µ } ) | ψ ( { λ } ) i  ∗ , (4.11) and h ψ ( { λ } ) | C ( − iζ 2 ) | ψ κ ( { µ } ) i = κ − 1 e i P N κ j =1 p 0 ( µ ℓ j )+ i P N j =1 p 0 ( λ j )  h ψ κ ( { µ } ) | B ( − iζ 2 ) | ψ ( { λ } ) i  ∗ . (4.12) F or th e calculation of the t wo-p oin t correlation fu nctions h σ z m σ z m ′ i (resp. h σ + m σ − m ′ i ), w e need actuall y to sum up the pro ducts F ( z ) ψ g ψ ′ ( m ) F ( z ) ψ ′ ψ g ( m ′ ) (resp. F (+) ψ g ψ ′ ( m ) F ( − ) ψ ′ ψ g ( m ′ )) o v er all eigenstates | ψ ′ i . W e ha ve the follo wing result: Prop osition 4.3. The pr o ducts of two form factors c an b e written in terms of the former sc alar pr o ducts as F ( z ) ψ g ψ ′ ( m ′ ) · F ( z ) ψ ′ ψ g ( m ) = e i ( m − m ′ ) b P ex 2 sin 2  b P ex 2  π 2 · ∂ 2 ∂ α 2 S z N     α =0 , (4.13) F (+) ψ g ψ ′ ( m ′ ) · F ( − ) ψ ′ ψ g ( m ) = ( − 1) m − m ′ e i ( m − m ′ ) b P ex · S + N    α =0 , (4.14) wher e S z N =     h ψ κ ( { µ } ) | ψ ( { λ } ) i k ψ κ ( { µ } ) k · k ψ ( { λ } ) k     2 , (4.15) S + N = − e − 2 π iα a 2 ( − iζ / 2 )      h ψ κ ( { µ } ) | B  − iζ 2  | ψ ( { λ } ) i k ψ κ ( { µ } ) k · k ψ ( { λ } ) k      2 . (4 .16) In (4.13) and (4.14 ) , b P ex denotes the r elative excitation momentum of the excite d states in the N κ se ctor in r esp e ct to the gr ound stat e in the N se ctor, b P ex = N κ X j =1 p 0 ( µ ℓ j ) − N X j =1 p 0 ( λ j ) . (4.1 7) 16 R emark 4.2 . Here, w e did not consider the pro duct F ( − ) ψ g ψ ′ ( m ) F (+) ψ ′ ψ g ( m ′ ), since the equal-time correlation function h σ − m σ + m ′ i can b e obtained fr om h σ + m σ − m ′ i b y the replacemen t m → m ′ . The calc ulation of the th ermo dynamic limit of b P ex is a relativ ely simple problem. Indeed, it follo ws from (2.4), (2.12) that b P ex 2 π = α N κ M + N κ X j =1 ˆ ξ κ ( µ ℓ j ) − N X j =1 ˆ ξ ( λ j ) + N 2 + N − N 2 κ − N κ 2 M = α N κ M + n X k =1  ˆ ξ κ ( µ p k ) − ˆ ξ κ ( µ h k )  . (4.18) Here w e ha ve used the an tisymmetry of the bare phase ϑ ( − λ ) = − ϑ ( λ ) and ˆ ξ κ ( µ ℓ j ) = ℓ j / M , ˆ ξ ( λ j ) = j / M . Th en, using (2. 6), w e obtain that the thermod ynamic limit P ex of b P ex is giv en b y the expression (3.2). The n on-trivial prob lem is the computatio n of the th ermo dynamic limit of the scala r pro d- ucts S z N and S + N . This problem will b e treated in the remaining p art of the article. 5 Thermo dynamic limit of S z N In the pap er [1] w e studied the th ermo dynamic limit of th e scalar pro duct S z N in the particular case where the state | ψ κ ( { µ } ) i w as the α -t wisted ground state in the N sector. In the general case, wh en the excited state con tains n particles and n holes, the result also d ep ends on their rapidities { µ p } and { µ h } . As we ha ve already announ ced, su c h d ep enden ce is not sufficien t to charac terize completely the th ermo dynamic limit of th e form factor. In deed, the latter decomp oses in to a pr o duct of a smo oth and a d iscr ete part. The s m o oth part can actually b e completely describ ed in terms of the r ap id ities of th e particles and h oles. Ho w ever, for the description of the d iscrete part one should use also the integ er num b ers { p a } and { h a } . In the first part of this section, we explicitly factorize S z N in to the aforemen tioned pr o d- uct. Th en , w e inv estigate the ther m o dynamic limit of the s mo oth part, p ostp oning the m ore complicated analysis of the discr ete p art until su bsection 5.3. Note that in this section we d eal only with the excited states in the N sec tor. Therefore the thermo dyn amic limit of the shift function b F ( λ ) is equal to F ( z ) ( λ ) giv en by (2.1 6). Ho wev er, in order to light en notations w e omit the sup erscript ( z ) throughout this sect ion, denoting the limiting v alue of the s hift function simply by F ( λ ). 5.1 Represen t ation of the scalar pro duct S z N Assume that the excited state | ψ κ ( { µ } ) i is an excited state in the N sector conta ining n particles with rapid ities { µ p a } n a =1 and n holes with rapidities { µ h a } n a =1 . One can fairly exp ect that the limiting v alue of S z N dep end s on the thermod ynamic limit of these rapidities { µ p } and { µ h } , i.e. lim S z N = S z n ( { µ p } , { µ h } ). Using Prop ositions 4.1 and 4.2 as w ell as (4.15), it is readily seen that S z N has the follo wing represent ation: S z N ( { µ p } , { µ h } ) = A ( z ) N ( { µ p } , { µ h } ) · D ( z ) N ( { µ p } , { µ h } ) · exp n C ( z ) N ( { µ p } , { µ h } ) o , (5.1) 17 where C ( z ) N ( { µ p } , { µ h } ) = N X a,b =1 log sinh( λ a − µ ℓ b − iζ ) sinh( µ ℓ b − λ a − iζ ) sinh( λ a − λ b − iζ ) sinh( µ ℓ a − µ ℓ b − iζ ) , (5.2) D ( z ) N ( { µ p } , { µ h } ) =  det N 1 sinh( µ ℓ j − λ k )  2 · N Y j =1 sin 2 π b F ( λ j ) π 2 M 2 b ρ ( λ j ) b ρ κ ( µ ℓ j ) , (5.3) and A ( z ) N ( { µ p } , { µ h } ) =      sin π α sin π b F ( − q ) N Y a =1 sinh( q + λ a − iζ ) sinh( q − µ ℓ a − iζ )      2    det Γ q h I + 1 2 π i b U ( z ) ( w, w ′ ) i    2 det N Θ ( λ ) j k · d et N Θ ( µ ) j k . (5.4) W e r emin d here that b U ( z ) has b een defined in (4.7) and that the N × N matrices Θ ( λ ) j k and Θ ( µ ) j k are Θ ( λ ) j k = δ j k + K ( λ j − λ k ) 2 π M b ρ ( λ k ) , Θ ( µ ) j k = δ j k + K ( µ ℓ j − µ ℓ k ) 2 π M b ρ κ ( µ ℓ k ) . ( 5.5) 5.2 Thermo dynamic limit of the smo oth part The smo oth p art of the scalar pro du ct consists of fac tors A ( z ) N and C ( z ) N . T heir limits can be computed exactly in th e same m anner as in [1]. W e set C ( z ) n ( { µ p } , { µ h } ) = lim N ,M →∞ C ( z ) N ( { µ p } , { µ h } ) , A ( z ) n ( { µ p } , { µ h } ) = lim N ,M →∞ A ( z ) N ( { µ p } , { µ h } ) . (5.6) W e illustrate the main idea b ehin d the calculation of C ( z ) n and A ( z ) n on the follo wing to y- example. Assume that f ( λ ) ∈ C 1 ( R ), and consider the compu tation of the thermo dynamic limit of the sum S f = N X j =1  f ( µ ℓ j ) − f ( λ j )  . (5.7) T o compu te this limit one can decomp ose the sum as follo ws S f = N X j =1  f ( µ j ) − f ( λ j )  + n X j =1  f ( µ p j ) − f ( µ h j )  → n X j =1  f ( µ p j ) − f ( µ h j )  + q Z − q F ( λ ) f ′ ( λ ) dλ. (5 .8) There w e ha v e used the leading b eha vior in M − 1 (2.14) for the sp acing b et ween µ j and λ j . It is con v enient to in tro duce a mo d ified shift function by F mod ( λ ) = F ( λ ) χ [ − q ,q ] ( λ ) + n X j =1  χ ] −∞ ,µ p j ] ( λ ) − χ ] −∞ ,µ h j ] ( λ )  , (5.9) where χ [ a,b ] is th e charac teristic fu nction of the interv al [ a, b ]. Th en equation (5.8) can b e recast simply as S f → ∞ Z −∞ F mod ( λ ) f ′ ( λ ) dλ. (5.10) 18 Therefore, pr o vided that one extends the integ ration con tours to the whole r eal axis and r eplaces F with F mod , one formally r educes the computations to th e case considered in [1] wh er e on e w as dealing with an exci ted state | ψ κ ( { µ }i ha ving no particles or holes. F rom this observ ation, the results of section 3 follo w straight forw ardly (by a mer e replace men t of the shift fun ction b y F mod ) from those of [1]. In this w a y , starting from the equations (5.2), (5. 4) w e arrive at (3.5), (3.8). Th us, the limits A ( z ) n and C ( z ) n are well defined for arbitrary p ositions of particles and holes. They dep end on { µ p } and { µ h } only . In particular, they do n ot dep end on the underlying in teger n um b ers. A t this stage it is inte resting to consider the limiting case where, in the thermo dynamic limit, all rapidities of p articles and holes condensate on th e F e rmi b oundaries. Corollary 5.1. L et | ψ κ ( { µ } ) i b elongs to the P r class as i n (3.20) , and denote C ( z ) n,r = C ( z ) n ( { + q } n + p ∪ {− q } n − p , { + q } n + h ∪ {− q } n − h ) , A ( z ) n,r = A ( z ) n ( { + q } n + p ∪ {− q } n − p , { + q } n + h ∪ {− q } n − h ) , (5.11) wher e subscripts show the numb er of elements in the c orr esp onding subsets. L e t F r ( λ ) = F ( λ )+ r and F r ( w ) b e the iπ - p erio dic Cauchy tr ansfo rm of F r ( λ ) . Then the c o efficient C ( z ) n,r takes the form C ( z ) n,r = C 0 [ F r ] , (5.12) wher e the f unctional C 0 [ F ] is define d in (3.6) . The c o efficient A ( z ) n,r b e c omes A ( z ) n,r =  sin π α sin π F ( − q )  2    e − 2 π i F r ( − q + iζ )    2      det  I + 1 2 π i U ( z ) r ( w, w ′ )  det  I + 1 2 π K       2 , (5.13) wher e U ( z ) r ( w, w ′ ) = − e 2 π i  F r ( w ) − F r ( w + iζ )  K κ ( w − w ′ ) − K κ ( − q − w ′ ) 1 − e 2 π iF ( w ) . (5.14) Pr o of. This follo ws straigh tforw ardly from F mod ( λ ) = F r ( λ ) χ [ − q ,q ] ( λ ).  Using (2.9 ), (2.16) w e find that F r ( λ ) = ( r + α ) Z ( λ ) , (5.15) and w e see that up to a replacemen t α + r → α our results coincide with the ones obtained in [1]. Th is coi ncidence is not acc iden tal. In the thermo d ynamic limit, the class P r con tains an infin ite num b er of states. In deed, kno wing that a p article or hole’s rapidit y equals to ± q in this limit does n ot allo w to fix the corresp ondin g in tegers p (r esp. h ) un am biguously . F or instance, knowing that µ p = q only allo ws one to sa y that p/ M → D . The latter is s atisfied as long as one chooses p = N + u M with u M = o ( M ) , but arbitrary otherwise. It is clear that qu antitie s h a vin g w ell defined thermo d ynamic limits (suc h as C ( z ) n,r and A ( z ) n,r ), m ust only dep end on the macrosco pic realization { µ p } and { µ h } of the excited state and n ot on the m icroscopic quantum num b ers { p } and { h } leading to such a macrosco pic configur ation 19 of the r apidities. Therefore, the calculation of their thermo dynamic limit in the case of an excited state b elonging to the P r class can b e done by choosing any of its representa tiv e. On e of these representa tiv es is giv en by the ( α + r )-t wisted ground state. I n deed, set n − p = n + h = 0 and c ho ose the inte gers describ ing the holes and particles according to h k = k , p k = N + k, k = 1 , . . . , r . (5.16) The excited state is thus describ ed b y the set of in tegers, ℓ j = j for j = r + 1 , . . . , N , and ℓ j = N + j for j = 1 , . . . , r . (5.17) Observe no w that if w e replace α b y α + r in the r.h.s. of (2.11), then up to a re-ordering of the equations, w e obtain the same set of in tegers (5.17). Thus, the ab o v e repr esen tativ e of the class P r coincides with the ( α + r )-t wisted ground state. Th er efore the smo oth part of all the critical form factors b elonging to the P r class coincides with the one of the form factor of the σ z op erator tak en b et ween the ground state and the ( α + r )-twiste d ground state. 5.3 Thermo dynamic limit of the discrete part As we ha v e seen, the m o dified sh ift fun ction (5.9) can b e used for the calculation of the ther- mo dynamic limits of su ms (pro du cts, determinants) in the case w hen w e d eal w ith smo oth functions of the rapidities of the excited state . On e cannot apply this metho d in the case of the factor D ( z ) N ( { µ p } , { λ h } ) as it dep ends on the Cauc hy determinan t of { λ } and { µ } . In the thermo dyn amic limit, certain en tries of this matrix b ecome div ergen t and more ca re is needed for the calculation of this limit. Prop osition 5.1. The discr ete p ar t D ( z ) N b ehaves in the thermo dynamic as D ( z ) N ( { µ p } , { µ h } ) = M − F 2 + − F 2 − · E 0 ( { µ h , µ p } ; { h a , p a } ) · n Y k =1 H N ( µ h k , h k ) × n Y k =1 P N ( µ p k , p k ) · D ( z ) [ F ] ·  1 + O  log M M  . (5.18) The facto r E 0 dep ends explicitly on the i nte gers { h a } and { p a } which ar e the quantum numb ers defining the excite d state E 0 = n Q j,k =1 j 6 = k ( h j − h k )( p j − p k ) ϕ ( µ h j , µ h k ) ϕ ( µ p j , µ p k ) n Q j,k =1 ( p j − h k ) 2 ϕ 2 ( µ p j , µ h k ) , (5.19) wher e ϕ ( λ, µ ) = (2 π sinh( λ − µ )) / ( p ( λ ) − p ( µ )) and p ( λ ) is the dr esse d momentum (2.6) . The factors P N and H N also dep end on the quantum numb ers defining the holes and the p articles P N ( µ p k , p k ) = e J [ F ( z ) ] ( µ p k ) ρ ( µ p k ) Γ 2 p k , p k − N + F ( µ p k ) p k + F ( µ p k ) , p k − N ! , (5.20) 20 H N ( µ h k , h k ) = sin 2 ( π F ( µ h k )) e J [ F ( z ) ] ( µ h k ) π 2 ρ ( µ h k ) Γ 2 h k + F ( µ h k ) , N + 1 − h k − F ( µ h k ) h k , N + 1 − h k ! . (5.21) Ther e we have intr o duc e d the functional J which is define d in terms of the dr esse d momentum p ( λ ) and acts on the shift function F : J [ F ] ( ω ) = 2 q Z − q  F ( λ ) ∂ λ log ϕ ( λ, ω ) + F ( λ ) − F ( ω ) p ( λ ) − p ( ω ) p ′ ( λ )  dλ . (5.22) Final ly, we r emind that the functional D ( z ) is given i n (3.16) R emark 5.1 . The d efinition of P N ( µ p k , p k ) is giv en in the case w h ere the rapid ities of all particles are to the righ t of the F ermi zone ( ie p k > N ). If some particles ha ve their rapidities to the left of the F ermi zo ne ( p k ≤ 0), then the corr esp onding argumen ts of the Γ-functions in (5.20) b ecome negativ e inte gers. T h e aforemen tioned formula remains h o wev er v alid provided that the arguments of these Γ-function are und ersto o d as limits Γ( p k ) Γ( p k − N ) = lim ε → 0 Γ( p k + ε ) Γ( p k − N + ε ) = ( − 1) N Γ( N + 1 − p k ) Γ(1 − p k ) . (5.23) Pr o of. F ollo wing the strategy app lied in [1], w e multiply an d divide the original Cauc hy deter- minan t b y the Cauc h y determinan t of the coun ting fun ctions b ξ κ ( λ ). Let b ϕ ( λ, µ ) = sinh( λ − µ ) b ξ ( λ ) − b ξ ( µ ) , b ϕ κ ( λ, µ ) = sinh( λ − µ ) b ξ κ ( λ ) − b ξ κ ( µ ) . (5.24) After some algebra, we recast D ( z ) N ( { µ p } , { λ h } ) in to the follo win g pro du ct: D ( z ) N ( { µ p } , { λ h } ) = b E 0 · n Y k =1 b H k · n Y k =1 b P k · D N , 0 . ( 5.25) Here b E 0 dep end s only on the particle/hole rapidities and the corresp onding in tegers: b E 0 = n Q j,k =1 j 6 = k ( h j − h k )( p j − p k ) b ϕ κ ( µ h j , µ h k ) b ϕ κ ( µ p j , µ p k ) n Q j,k =1 ( p j − h k ) 2 b ϕ 2 κ ( µ p j , µ h k ) . (5.26) The factors b P k dep end on the r apidities µ p k , integ ers p k , the groun d state parameters { λ } , and the ro ots µ j suc h that b ξ κ ( µ j ) = j / M , j = 1 , . . . , N . Their explicit represen tatio ns are b P k = 1 b ρ κ ( µ p k ) N Y j =1 ( j − p k − b F ( µ p k ) j − p k − b F ( λ j ) · b ϕ κ ( µ p k , µ j ) b ϕ κ ( µ p k , λ j ) ) 2 Γ 2 p k , p k − N + b F ( µ p k ) p k + b F ( µ p k ) , p k − N ! . (5.27) 21 The representa tions for b H k are similar to (5.27), b ut they dep end on µ h k and h k : b H k = sin 2 ( π b F ( µ h k )) π 2 b ρ κ ( µ h k ) N Y j =1 ( j − h k − b F ( λ j ) j − h k − b F ( µ h k ) · b ϕ κ ( µ h k , λ j ) b ϕ κ ( µ h k , µ j ) ) 2 × Γ 2 h k + b F ( µ h k ) , N + 1 − h k − b F ( µ h k ) h k , N + 1 − h k ! . (5.28) Finally , D N , 0 do es not dep end on the particle/hole rapidities and the corresp onding in tege rs D N , 0 = N Y j =1 b ρ κ ( λ j ) b ρ ( λ j ) N Y j,k =1 b ϕ κ ( λ j , λ k ) b ϕ κ ( µ j , µ k ) b ϕ 2 κ ( µ j , λ k ) × N Y j >k 1 − b F ( λ j ) − b F ( λ k ) j − k ! 2 N Y j,k =1 j 6 = k 1 − b F ( λ j ) j − k ! − 2 N Y j =1 ( sin  π b F ( λ j )  π b F ( λ j ) ) 2 . (5.29) The fact that b E 0 → E 0 follo ws fr om b ϕ κ → ϕ . The calculation of the limit of the firs t pro du ct in (5 .29) is based on the definition of the shift function (2.13): N Y j =1 b ρ κ ( λ j ) b ρ ( λ j ) = N Y j =1 1 + ( b ξ ′ κ ( λ j ) − b ξ ′ ( λ j )) b ρ ( λ j ) ! = N Y j =1 1 − b F ′ ( λ j ) M b ρ ( λ j ) ! → exp   − q Z − q F ′ ( λ ) dλ   = e F − − F + . (5.30) The limit of the remaining p art of D N , 0 w as computed in [1], wh at giv es us D N , 0 = M − F 2 + − F 2 − · D ( z ) [ F ] ·  1 + O  log M M  . (5.31) Finally , th e limits b P k → P N ( µ p k , p k ) and b H k → H N ( µ h k , h k ) follo w from lim N ,M →∞ N Y j =1 ( j − k − b F ( µ k ) j − k − b F ( λ j ) b ϕ κ ( µ k , µ j ) b ϕ κ ( µ k , λ j ) ) 2 = e J [ F ] ( µ k ) . (5.32) This formula is pro ved in app endix A (see (A.8)).  Assuming general p ositions of the particles and holes, we can not mak e f u rther simplifica- tions in the f orm ula for D ( z ) N . Therefore w e no w study tw o limiting cases of inte rest, exactl y as w e did for the s m o oth p arts: first wh en all particles and holes are separated from the F ermi b ound aries, and then when all p articles and holes are in the vicinit y of the F e rmi zone (critica l form factor in the class P r ). Corollary 5.2. Supp o se that, in the thermo dyn amic limit, p articles and holes ar e sep ar ate d fr om the F ermi b oundaries: µ p a 6 = ± q and µ h a 6 = ± q . Then the thermo dynamic limit of the factor D ( z ) N b e c omes a smo ot h function of the r apidities { µ p } and { µ h } : D ( z ) N = M − 2 n − F 2 − − F 2 + · D ( z ) n ( { µ p } , { µ h } ) ·  1 + O  log M M  , (5.33) 22 wher e D ( z ) n ( { µ p } , { µ h } ) is given by the expr ession (3.15) for D z z . Pr o of. If the rap id ities of particles and holes are sep arated from the F ermi b oundaries, then all the argumen ts of the Γ-functions in (5.20) and (5.21) are large and w e can apply the Stirling form ula for their simplification. Then lim N ,M →∞ Γ 2 p k , p k − N + b F ( µ p k ) , N + 1 − h k − b F ( µ h k ) , h k + b F ( µ h k ) p k − N , p k + b F ( µ p k ) , N + 1 − h k , h k ! = lim N ,M →∞  p k − N p k  2 F ( µ p k )  N − h k h k  − 2 F ( µ h k ) =  p ( µ p k ) − p ( q ) p ( µ p k ) − p ( − q )  2 F ( µ p k )  p ( q ) − p ( µ h k ) p ( µ h k ) − p ( − q )  − 2 F ( µ h k ) , (5.34) where we hav e used (2.6). Combining (5.34) with the explicit form (5.22) of J ( ω ), we arrive at P N ( µ p k , p k ) H N ( µ h k , h k ) = sin 2 ( π F ( µ h k )) π 2 ρ ( µ h k ) ρ ( µ p k ) × exp    2 q Z − q  F ( λ )  coth( λ − µ p k ) − coth( λ − µ h k )  dλ    . (5.35) The limit of b E 0 is almost trivial: lim N ,M →∞ b E 0 M 2 n =  det n 1 sinh( µ p j − µ h k )  2 . (5.36) Gathering all these results, w e get the claim.  Hence, in the limit wh en the rapidities of particles and holes remain at finite distance from the F ermi b oundaries, the discrete part D ( z ) N , up to an M -dep end en t normalization, can b e appro ximated b y a smo oth function of these rapidities. Ho w ev er, as so on as some rapidities go to ± q , then certain int egrals in the second line of (3.15) b ecome ill-defined. There can also app ear singularities in the C auc hy determinan ts presen t in the first line of (3.15 ). Therefore, the discrete structure of D ( z ) N manifests itself when particles and holes condens ate on the F ermi b ound aries. W e no w d escrib e the asymptotic b eha vior of an excited s tate | ψ κ ( { µ ℓ j }i b elonging to the P r class. This means that the rapidities of all particles and h oles are located in vicinities of the F er m i b oundaries w ith the co ndition (3.20). Any s uc h state, can b e p arameterized by a s et of int egers as giv en in (3.21). Corollary 5.3. The asymptotic b eha vior of D ( z ) N for an excite d state in the P r class p ar ame- terize d as in (3.21) r e ads D ( z ) N ( { + q } n + p ∪ {− q } n − p , { + q } n + h ∪ {− q } n − h ) = M − F 2 r, + − F 2 r, − D ( z ) n,r  1 + O  log M M  , (5.37) wher e D ( z ) n,r is given b y the expr ession (3.23) for D z z . 23 R emark 5.2 . Thence, when particles and h oles are in the vicinity of the F ermi b oun d aries, the b eha vior of D ( z ) N cannot b e called a th er m o dynamic limit. Ind eed, the factors R n + p ,n + h and R n − p ,n − h , and thus the form factor, d ep end explicitly on the microscopic c haracteristics p ± and h ± of the excited state. Pr o of. In this case, the limits E 0 , P N ( µ p k , p k ) and H N ( µ h k , h k ) should b e re-calculated. Let u s first consider the co efficien t b E 0 . Usin g that ϕ ( q , q ) = ϕ ( − q , − q ) = ρ − 1 ( q ) and ϕ ( q , − q ) = ϕ ( − q , q ) = sinh(2 q ) /D , w e get lim N ,M →∞ b E 0 =  D ρ ( q ) sinh(2 q )  2 r 2 ρ 2 n ( q ) lim N ,M →∞  det n 1 p j − h k  2 . (5.38) Using the sp ecific parametrization (3. 21) of the in tegers b elonging to the P r class w e see that the thermo d ynamic limit of the Cauc h y determinan t in (5.38) can b e factoriz ed in t w o parts:  det n 1 p j − h k  2 → N − 2 r 2 L n + p ,n + h ( { p + } , { h + } ) L n − p ,n − h ( { p − } , { h − } ) , (5.39) with L n,m ( { p } , { h } ) = n Q j >k ( p j − p k ) 2 m Q j >k ( h j − h k ) 2 n Q j =1 m Q k =1 ( p j + h k − 1) 2 . (5.40) This leads eve n tually to the follo wing estimate lim N ,M →∞ b E 0 ( M ρ ( q ) sinh(2 q )) 2 r 2 = ρ 2 n ( q ) L n + p ,n + h ( { p + } , { h + } ) L n − p ,n − h ( { p − } , { h − } ) . (5 .41) When computing the limits of b P k and b H k , some of th e argumen ts of th e Γ-fun ctions are not large an ymore. Th er efore, we can only use Stirling formula p artly . W e h a ve n Y k =1 Γ 2 p k , p k − N + b F ( µ p k ) , N + 1 − h k − b F ( µ h k ) , h k + b F ( µ h k ) p k − N , p k + b F ( µ p k ) , N + 1 − h k , h k ! ∼ N − 2 r ( F + + F − ) Γ 2 { p + + F + } , { h + − F + } , { h − + F − } , { p − − F − } { p + } , { h + } , { h − } , { p − } ! . (5.42) W e precise that the notation { p + + F + } means { p + j + F + } n + p j =1 and s imilarly f or other sets of parameters in (5.42). Com bining all these results we obtai n lim N ,M →∞ b E 0 ( M ρ ( q ) sinh(2 q )) 2 r 2 +2 r ( F + + F − ) n Y k =1 b P k b H k =  sin( π F + ) π  2 n + h  sin( π F − ) π  2 n − h R n + p ,n + h ( { p + } , { h + }| F + ) R n − p ,n − h ( { p − } , { h − }| − F − ) × exp    2 r q Z − q F + − F ( λ ) tanh( q − λ ) dλ + 2 r q Z − q F − − F ( λ ) tanh( q + λ ) dλ    . (5.43) After some simple algebra we get the claim.  24 T o conclude this section we w ould lik e to stress that the ev aluation of the S z N thermo dyn amic limit w as done w ithout u s ing the explicit form of the shift function F ( λ ) (2.16). W e h av e used this represen tatio n only in (5.15) in order to relate the sm o oth part of the critical form fact ors of P r class with the one corresp on d ing to the ( α + r )-t wisted groun d state. Ho wev er in all other resp ects F ( λ ) pla y ed the role of a free fun ctional parameter. 6 The scalar pro duct S + N W e no w study the s calar pro du ct S + N for | ψ κ ( { µ } ) i b eing an α -t wisted excited state with n particles an d n h oles in the N + 1 sector ( i.e. N κ = N + 1). As previously , { µ p a } n a =1 and { µ h a } n a =1 denote the rapidities of the p articles and holes resp ectiv ely . Th e notation b F ( λ ) no w means th e sh ift function in the N + 1 secto r, whose ther m o dynamic limit F (+) ( λ ) is giv en by (2.17). Ho w ever, like in the previous section, we omit the sup erscript (+) in order to ligh ten the notations. 6.1 Represen t ation for the scalar pro duct S + N In comparison with the S z N case, the excited state no w dep ends on N + 1 parameters µ . Using the d eterminan t represen tation giv en in Prop osition 4.1 as w ell as the n orm form ula of Prop osition 4.2, we get the follo wing represen tation for S + N (4.16): S + N ( { µ p } , { µ h } ) = A (+) N ( { µ p } , { µ h } ) · exp n C (+) N ( { µ p } , { µ h } ) o · D (+) N ( { µ p } , { µ h } ) . (6.1) Here D (+) N ( { µ p } , { µ h } ) = M π 2 b ρ ( λ N +1 ) sin 2 π b F ( λ N +1 ) N +1 Q a =1 sinh 2 ( λ N +1 − µ ℓ a ) N Q a =1 sinh 2 ( λ N +1 − λ a ) · D ( z ) N +1 ( { µ p } , { µ h } ) , (6.2) where D ( z ) N +1 is giv en b y (5.3) with N replaced b y N + 1. I t is expressed in term s of the N + 1 parameters µ ℓ a as w ell as the N + 1 parameters λ j defined b y b ξ ( λ j ) = j / M , b ξ ( λ ) b eing the coun ting fun ction (2.4) for the grou n d state. In other w ord s λ j , j = 1 , . . . , N , are Bethe ro ots for the ground state, whereas λ N +1 is the p oin t where M b ξ ( λ ) take s its next in teger v alue, i e M b ξ ( λ N +1 ) = N + 1. The co efficien t C (+) N is no w mo dified according to exp h C (+) N ( { µ p } , { µ h } ) i = N +1 Y a =1     sinh( λ a − λ N +1 − iζ ) sinh( µ ℓ a − λ N +1 − iζ )     2 · exp h C ( z ) N +1 ( { µ p } , { µ h } ) i , (6.3) where, up to the evident mo difications stemming from N → N + 1, C ( z ) N +1 is giv en b y (5.2). Finally the factor A (+) N has the follo w ing form : A (+) N ( { µ p } , { µ h } ) = sin ζ 2 π κ          N Q a =1 sinh( λ a − iζ 2 ) N +1 Q a =1 sinh( µ ℓ a − iζ 2 )          2 ·    det h I + 1 2 π i b U (+) ( w, w ′ ) i    2 det N +1 Θ ( µ ) j k · det N Θ ( λ ) j k , (6.4) 25 and w e refer to (4.8) for the d efinition of the k ernel b U (+) ( w, w ′ ). 6.2 Thermo dynamic limit of the smo oth part The smo oth part of S + N consists of the co efficients A (+) N and exp C (+) N . The computation of their therm o dynamic limits C (+) n and A (+) n do es not con tain an y subtleties comparing w ith the deriv ation describ ed in section 5.2. The u s e of the mo dified s h ift fun ction (5.9) formally reduces the compu tations to the case where the excited state d o es n ot con tain particles and h oles. In this w a y , starting from the r epresen tations (6.3), (6.4) w e arrive at the equations (3.7), (3.11). One can easily see that the coefficien ts C (+) n and A (+) n are w ell defined for general p ositions of particles and h oles. In p articular, for the critic al form facto rs of the P r class the smo oth part effectiv ely dep ends on the shif t fu nction F r ( λ ) = F ( λ ) + r . 6.3 Thermo dynamic limit of the discrete part Prop osition 6.1. The discr ete p ar t D (+) N b ehaves in the thermo dynamic as D (+) N ( { µ p } , { µ h } ) = M − F 2 − − ( F + +1) 2 E 0 ( { µ h , µ p } , { h a , p a } ) · n Y k =1 e H N +1 ( µ h k , h k ) × n Y k =1 e P N +1 ( µ p k , p k ) · D (+) [ F ] ·  1 + O  log M M  , (6.5) wher e e H N +1 ( µ h k , h k ) = H N +1 ( µ p k , p k ) ( N + 1 − h k − F ( µ h k )) − 2 ϕ − 2 ( q , µ h k ) , (6.6) e P N +1 ( µ p k , p k ) = P N +1 ( µ p k , p k ) ( N + 1 − p k − F ( µ p k )) 2 ϕ 2 ( q , µ p k ) . (6.7) The functions entering these definitions ar e the same as in P r op osition 5.1, bu t one should r eplac e N b y N + 1 for the c o efficients H N +1 and P N +1 . We also r emind that the functional D (+) is given i n (3.19) . Pr o of. As the limit of D ( z ) N +1 is already kno wn, it remains to ev aluate the limit of the pro duct P = N +1 Q a =1 sinh 2 ( λ N +1 − µ ℓ a ) N Q a =1 sinh 2 ( λ N +1 − λ a ) . (6.8) Harping on the steps in the analysis of D ( z ) N , we m ultiply and divide b y the coun ting function b ξ ( λ ) (2.4). W e obtain P = N +1 Q a =1  N + 1 − ℓ a − b F ( µ ℓ a )  2 M 2 N Q a =1 ( N + 1 − a ) 2 · N +1 Q a =1 b ϕ 2 ( λ N +1 , µ ℓ a ) N Q a =1 b ϕ 2 ( λ N +1 , λ a ) . ( 6.9) 26 Then using Corollary A.1 w e find that, in the thermod ynamic limit, the pro d uct P b ehav es as P → e J [ F ] ( q ) M 2 ρ 2 ( q ) Γ 2 N + 1 − F + N + 1 , − F + ! · n Y k =1  N + 1 − p k − F ( µ p k ) N + 1 − h k − F ( µ h k ) · ϕ ( q , µ p k ) ϕ ( q , µ h k )  2 , (6.10) with J [ F ] ( q ) defi n ed as in (5.22). Using now that N is large, w e ha v e e J [ F ] ( q ) Γ 2 N + 1 − F + N + 1 , − F + ! → Γ 2 (1 + F + ) sin 2 π F + π 2 ( ρ ( q ) sin h ( 2 q ) M ) 2 F + exp    2 q Z − q F ( λ ) − F + tanh ( λ − q )    . (6.11) T aking into accoun t (3. 16), (3.19) we find D ( z ) [ F ] · P · M π 2 b ρ ( λ N +1 ) sin 2 ( π b F ( λ N +1 )) → D (+) [ F ] M 2 F + +1 n Y k =1  N + 1 − p k − F ( µ p k ) N + 1 − h k − F ( µ h k ) · ϕ ( q , µ p k ) ϕ ( q , µ h k )  2 . (6.12) It remains to com bine the obtained result with the kn o wn limit of D ( z ) N +1 and to substitute it in to (6.2). A few algebraic manipulations lead then to the claim.  W e no w particularize this result to the t w o limiting cases w e considered previously , namely when all particles and holes are separated f rom the F ermi b oundaries, or when all particles and holes are on the F ermi b oundaries. Corollary 6.1. Supp o se that, in the thermo dyn amic limit, p articles and holes ar e sep ar ate d fr om the F ermi b ounda ries: µ p a 6 = ± q and µ h a 6 = ± q . Then the th ermo dynam ic limit of D (+) N b e c omes a smo ot h function of the r apidities { µ p } and { µ h } : D (+) N = M − 2 n − F 2 − − ( F + +1) 2 · D (+) n ( { µ p } , { µ h } ) ·  1 + O  log M M  , (6.13) wher e D (+) n ( { µ p } , { µ h } ) is giv e n by the expr ession (3.18) for D + − . Pr o of. If the rapidities of particles and holes are separated fr om the F ermi b oundaries, then n Y k =1  N + 1 − p k − F ( µ p k ) N + 1 − h k − F ( µ h k ) · ϕ ( q , µ p k ) ϕ ( q , µ h k )  2 → n Y k =1  sinh ( q − µ p k ) sinh ( q − µ h k )  2 . (6.14) Substituting this limit in to (6.5) and using the r esults of the pr evious section we arriv e at the statemen t.  Corollary 6.2. The asymptotic b ehavior of D (+) N for an excite d state in the P r class p ar am e- terize d as in (3.21) r e ads D (+) N ( { + q } n + p ∪ {− q } n − p , { + q } n + h ∪ {− q } n − h ) = M − ( F r, + +1) 2 − F 2 r, − D (+) n,r  1 + O  log M M  , (6.15) wher e D (+) n,r is given b y the expr ession (3.24) for D + − . 27 Pr o of. In this case n Y k =1  N + 1 − p k − F ( µ p k ) N + 1 − h k − F ( µ h k ) · ϕ ( q , µ p k ) ϕ ( q , µ h k )  2 →  M ρ ( q ) sinh(2 q )  − 2 r Q n + p k =1 ( p + k + F + ) 2 Q n + h k =1 ( h + k − 1 − F + ) 2 . (6.16) Substituting this limit in to (6.5) and using the r esults of the pr evious section we arriv e at the statemen t.  Conclusion In this article w e hav e pro vided the first steps to wa rds the study of co rrelation f unctions vi a the form facto r approac h. F or this purp ose w e ha ve calc ulated the thermo dynamic limits of the so-called particle/hole form factors. Although w e h a ve considered the sp ecific case of the XXZ chain, our metho d is straigh tforwardly app licable to other massless int egrable mo d els solv able by means of the algebraic Bethe An s atz and h a vin g determinant represen tatio ns of their form f actors. In particular, one can use it for the ca lculation of th e thermo dynamic limit of form fac tors in the mo del of one-dimensional b osons. Determinant representati ons for the form f actors of this mo del in the finite vo lume w ere giv en in [19, 20]. Their thermo dyn amic limit can b e directly obtained from the results giv en in the pr esen t article. Our results sho w that th e idea to replace, in the thermo dyn amic limit, the sum ov er excited states b y an in tegratio n with resp ect to particles and h oles sh ould b e essen tially mo difi ed. In particular, a form al replacemen t of th e d iscrete su ms o v er n -particle/hole excited s tates | ψ ′ ( { µ p } , { µ h } ) i by integrals as X | ψ ′ ( { µ p } , { µ h } ) i F ( s ) ψ g ψ ′ ( m ) F ( s ′ ) ψ ′ ψ g ( m ′ ) → M 2 n Z F ( s ) ψ g ψ ′ ( m ) F ( s ′ ) ψ ′ ψ g ( m ′ ) n Y j =1 ρ ( µ p j ) ρ ( µ h j ) dµ p j dµ h j , (6.17) leads to senseless results. First of all, ev en in the region where the form factors dep end smo othly on { µ p } and { µ h } , the coefficien t M 2 n do es not comp ensate the factors M − θ ss ′ , therefore in the ab o v e expression, the r.h.s. v anishes. On the other hand, the corresp onding integral b ecomes div ergen t wh en th e rapidities of particles and holes approac h the F ermi b oundaries (see (3.1 5), (3.18)). Lastly , due to the discrete structure of the form factors w h en some rapidities agglome rate on the F ermi b ou n daries, one cannot replace th e sum o v er the exc ited states in the P r class b y an int egral. F or suc h excit ed states, one has to tak e the microscopic stru cture of the excited state in to acco unt and p erform the discrete sums o v er the paramete rs p ± , h ± (3.21). In a forthcoming p ublication [54], we will describ e a w ay to o verco me these d ifficulties. W e will show that in the asymptotic regime (large lat tice distances m b et ween lo cal spin op ertors) only particles and holes ha ving their rapidities close to the F er m i b oundaries co n tribute to the form factor sums. In suc h a limit, one can send the rapidities of th e particles and holes to ± q in the smo oth part of the f orm f actor. Hence, it b ecomes a constan t that only dep ends on the P r class S ss ′ = S ( r ) ss ′ ( {± q } , {± q } ). In its turn the sum mation o ve r th e in tegers { p } and { h } presen t in the discrete part of the form factors leads to the natural re-scaling M → 2 π m of the s ystem 28 size M int o the d istance m b etw een the op erators. T his mec hanism explains the app earance of the critical exp onen ts θ ss ′ in the asymptotic b eha vior of the correlation functions. Ac kn o wledgemen ts J. M. M., N. S . and V. T. are supp orted b y CNRS. W e also ac kn o w ledge the supp ort from the GDRI-471 of CNRS ”F renc h-Rus sian net wo rk in Theoretical and Mathematica l Physic s” and RFBR-CNRS-09-01-931 06L-a. N. K., J. M. M. and V. T are also supp orted by the ANR gran t DIADEMS 10 BLAN 01200 4 an d N. S. by the P r ogram of R AS Mathematica l Metho ds of the Nonlinear Dynamics, RFBR-11-01-00 440-a . K. K. K. is supp orted b y the EU Marie-Curie Excellence Gran t MEXT-C T -2006- 04269 5. N. K., N. S . and K . K. K wo uld lik e to thank the Theoretical Ph ysics group of the L ab oratory of Physic s at ENS Ly on for h ospitalit y , w hic h mak es th is collab oration p ossible. N.K. and V. T. w ould like to thank LPTHE (P aris VI Univ ersit y) for hospitalit y . A Summation iden tities Lemma A.1. L et f ∈ C 1 ([0 , a ]) for some a > D . L et ( h M ) b e a se quenc e of inte gers such that h M M tends to some finite value x h ∈ [0 , a ] when M → + ∞ . Then, in the limit N , M → + ∞ , N/ M → D , the fol lowing su ms vanish: lim N ,M →∞ N X k =1 k 6 = h M     f ( k M ) − f ( h M M ) ( k − h M ) n     = 0 , n ≥ 2 . (A.1) Pr o of. Let n = 2. Then N X k =1 k 6 = h M | f ( k M ) − f ( h M M ) | ( k − h M ) 2 ≤ 1 M 2 N X k =1 k 6 = h M | f ( k M ) − f ( h M M ) − k − h M M f ′ ( h M M ) | ( k − h M M ) 2 + | f ′ ( h M M ) | M N X k =1 k 6 = h M 1 k − h M . (A.2) The second term v anishes in the limit, as it is of order log N M . T h e first term v anishes du e to the Euler–Maclaurin su m mation form u la: lim N ,M →∞ N/ M → D 1 M 2 N X k =1 k 6 = h M | f ( k M ) − f ( h M M ) − k − h M M f ′ ( h M M ) | ( k − h M M ) 2 = lim N ,M →∞ N/ M → D 1 M D Z 0 | f ( x ) − f ( x h ) − ( x − x h ) f ′ ( x h ) | ( x − x h ) 2 dx = 0 . (A.3) F or n > 2, we ha ve N X k =1 k 6 = h M     f ( k M ) − f ( h M M ) ( k − h M ) n     = N X k =1 k 6 = h M     1 ( k − h M ) n − 2 · f ( k M ) − f ( h M M ) ( k − h M ) 2     ≤ N X k =1 k 6 = h M | f ( k M ) − f ( h M M ) | ( k − h M ) 2 → 0 , (A.4) whic h ends the pro of.  29 Lemma A.2. With the hyp othesis of L emma A.1, let n 0 ∈ N b e such that sup x ∈ [0 ,D ] | f ( x ) | < n 0 and | f ( x h ) | < n 0 . Then S n 0 ; N ( f ) = N X k =1 | k − h M |≥ n 0 log k − h M + f ( k M ) k − h M + f ( h M M ) − → N ,M →∞ N/ M → D Z D 0 f ( x ) − f ( x h ) x − x h dx . (A.5) Pr o of. Expanding the logarithms in to their T aylor series w e obtain S n 0 ; N ( f ) = N X k =1 | k − h M |≥ n 0 ∞ X r =1 ( − 1) r +1 r · f r ( k M ) − f r ( h M M ) ( k − h M ) r . (A.6) The result of Lemma A.1 sho ws that only the terms corr esp onding to r = 1 giv e non-v anishing con trib utions. Th ese are computed as a Riemann su m: lim N ,M →∞ N/ M → D S n 0 ; N ( f ) = lim N ,M →∞ N/ M → D 1 M N X k =1 | k − h M |≥ n 0 f ( k M ) − f ( h M M ) k − h M M = Z D 0 f ( x ) − f ( x h ) x − x h dx. (A.7)  Corollary A.1. L et f and ( h M ) sa tisfy the hyp othesis of L emma A.1. L et ξ ( λ ) b e a strictly monotono us smo oth func tion on R such that ξ − 1 (0) = − q and ξ ( D ) = q . Define λ k = ξ − 1 ( k / M ) , k ∈ Z and ρ ( λ ) = ξ ′ ( λ ) . Then, indep endently whether the thermo dyna mic limit λ h of λ h M b elongs or not to the i nterval [ − q , q ] , one has lim N ,M →∞ N/ M → D N Y k =1 k − h M + f ( λ k ) k − h M + f ( λ h M ) = exp    q Z − q f ( λ ) − f ( λ h ) ξ ( λ ) − ξ ( λ h ) ρ ( λ ) dλ    . (A.8) Pr o of. Assume that h M ∈ { 1 , . . . , N } and set e f = f ◦ ξ − 1 . Cho ose n 0 > sup x ∈ [0 ,D ] | e f ( x ) | and decomp ose the original p ro duct in to N Y k =1 k − h M + f ( λ k ) k − h M + f ( λ h ) = Y | k − h M | b sinh( µ ℓ a − µ ℓ b ) sinh( ν b − ν a ) · d et N κ Ω κ ( { µ } , { ν }|{ µ } ) . (B.1) The N κ × N κ matrix Ω κ ( { µ } , { ν }|{ µ } ) is defined as (Ω κ ) j k ( { µ } , { ν }|{ µ } ) = a ( ν j ) t ( µ ℓ k , ν j ) N κ Y a =1 sinh( µ ℓ a − ν j − iζ ) − κ d ( ν j ) t ( ν j , µ ℓ k ) N κ Y a =1 sinh( µ ℓ a − ν j + iζ ) , (B.2) with t ( µ, ν ) = − i sin ζ sinh( µ − ν ) sinh( µ − ν − iζ ) and ( a ( µ ) = sinh M ( µ − iζ / 2 ) d ( µ ) = sinh M ( µ + iζ / 2) . (B.3) In order to obtain the scala r pro du ct (4.5) one should set here ν j = λ j for j = 1 , . . . , N , where λ j are the Bet he roots describing the groun d state. F o r the scalar prod uct (4.6 ) one should set in addition ν N +1 = − iζ / 2. T o obtain a F redholm determinant repr esen tation for the scalar p ro du cts w e should present the original determinant of the matrix Ω κ in the follo wing form det N κ Ω κ ( { µ } , { ν }|{ µ } )   ν j = λ j = H ( { µ } , { λ } ) d et N  δ j k + e Ω( λ j , λ k |{ µ } ) N Y a =1 , a 6 = j ( λ j − λ a ) − 1  . (B.4) Here e Ω( λ j , λ k |{ µ } ) is a new N × N matrix, H ( { µ } , { λ } ) an external coefficien t. If w e succeed to find a rep r esen tatio n of the t yp e (B.4 ), then w e can replace the determinan t of the N × N matrix by F redholm determinant of an integral op erator as det N  δ j k + e Ω( λ j , λ k |{ µ } ) N Y a =1 , a 6 = j ( λ j − λ a ) − 1  = det Γ q  I + 1 2 π i e Ω( w , w ′ |{ µ } ) N Y a =1 ( w − λ a ) − 1  . (B.5) Here the in tegral op erator in the r.h.s. of (B.5) acts on a coun terclo c kwise orien ted con tour Γ q surroun ding the F ermi zone [ − q , q ]. This conto ur is su c h that it con tains all the ground state ro ots λ j and no other singularit y of the kernel e Ω( w, w ′ |{ µ } ). 31 The pro of of (B.5) is quite ob vious. Ind eed, expanding the F redholm determinant in to th e series of multiple inte grals we see that eac h of these integrals redu ces to th e su m of the r esidues in the p oints λ j . Thus, the series of m u ltiple in tegrals turns in to the series of m ultiple sums. Then one can ea sily convince oneself that this series coincides with the expan s ion of th e N × N determinan t in the l.h.s. of (B.5). Th us, ou r goal is to pass from the original represen tation (B .1) to th e form (B.4). Th e w ay w e use here is b ased on the ext raction of a Cauc hy determinan t from the original determinan t defining the scalar pr o duct, an idea which w as fir st in tro duced in [55, 56]. I n th e case of the scalar pro du ct (4.5) this w as done in [36]. The same metho d with min or mo difications describ ed b elo w can b e used for the pr o of of the second determin ant iden tit y (4.6). F ollo win g [36], w e consider the matrix Ω κ ( { z } , { ν }|{ z } ) for t wo sets of N κ = N + 1 generic parameters { z } and { ν } . In this case w e w rite det N +1 [Ω κ ] = det N +1 (Ω κ A ) det N +1 A with A j k = N +1 Q a =1 sinh( z j − ν a ) N +1 Q a =1 a 6 = j sinh( z j − z a ) × ( coth( z j − ν k ) for k ≤ N , 1 for k = N + 1 . (B.6) The effect of multiplica tion b y the matrix A is computed similarly to the metho d presen ted in [36]. W e get det N +1 [Ω κ ( { z } , { ν }|{ z } )] = d et N +1  1 sinh ( z k − ν j )  · det N +1 S j k , (B.7) where S j k = δ j k Y κ ( ν j |{ z } ) + Q N +1 a =1 sinh( ν k − z a ) Q N +1 a =1 a 6 = k sinh( ν k − ν a ) · ∂ ∂ y k Y κ ( ν j |{ y } )     { y } = { ν } , k ≤ N , (B.8) S j,N +1 = Y κ ( ν j |{ ν } ) , (B.9) and w e ha v e set for arbitrary complex y 1 , . . . , y N +1 Y κ ( ν |{ y } ) = a ( ν ) N +1 Y k =1 sinh( y k − ν − iζ ) + κ d ( ν ) N +1 Y k =1 sinh( y k − ν + iζ ) . (B.10) Finally we r ed uce the size of det N +1 S by one b y p erforming linear com b inations of the lines det N +1 [ S j k ] = S N +1 ,N +1 det N  S j k − S j,N +1 S N +1 ,k S N +1 ,N +1  . (B.11) Th us, w e arr ive at the follo wing representa tion for d et N +1 Ω κ dep end ing on generic complex z 1 , . . . , z N +1 and ν 1 , . . . , ν N +1 : det N +1 [Ω κ ( { z } , { ν }|{ z } )] = S N +1 ,N +1 det N +1  1 sinh ( z k − ν j )  det N  S j k − S j,N +1 S N +1 ,k S N +1 ,N +1  , (B.12) where S j k are giv en b y (B.8), (B.9). 32 No w we set z a = { µ ℓ a } for a = 1 , . . . , N + 1, ν k = λ k for k = 1 , . . . , N and ν N +1 = − iζ / 2. W e assume that the p arameters { µ ℓ a } satisfy the sys tem (2.11), while the parameters { λ k } satisfy the system (2.2). Then we obta in for j ≤ N , Y κ ( ν j | { ν } ) a ( λ j ) Q N b =1 sinh( λ b − λ j − iζ ) = κ sinh( λ j − iζ 2 ) − s in h( λ j + 3 iζ 2 ) , ∂ Y κ ( ν j | { y } ) /∂ y k a ( λ j ) Q N b =1 sinh( λ b − λ j − iζ )      { y } = { ν } = κ sinh( λ j − iζ 2 ) tanh( λ k − λ j + iζ ) − sinh( λ j + 3 iζ 2 ) tanh( λ k − λ j − iζ ) , (B.13) and for j = N + 1, Y κ ( ν N +1 | { ν } ) = a ( − iζ / 2) sinh ( − iζ ) N Y b =1 sinh( λ b − iζ 2 ) , ∂ Y κ ( ν N +1 | { y } ) /∂ y k    { y } = { ν } = a ( − iζ / 2) sinh( − iζ ) coth ( λ k − iζ 2 ) N Y b =1 sinh( λ b − iζ 2 ) . (B.14) It also follo ws from (2.4), (2.12) and the d efinition of the shift fun ction (2.1 3) th at κ N +1 Y a =1 sinh( µ ℓ a − w + iζ ) sinh( µ ℓ a − w − iζ ) N Y a =1 sinh( λ a − w − iζ ) sinh( λ a − w + iζ ) = e 2 π i b F ( w ) . 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