Long-time and large-distance asymptotic behavior of the current-current correlators in the non-linear Schr"{o}dinger model

LPENSL-T H-01/11 DESY 10-253 Long-time and large-distance asymptotic b eha vior of the curren t-curren t correlators in the non-linear Sc hrö dinger mo del K. K. K ozlowski 1 , V. T erras 2 October 31, 2021 Abstract W e present a new method allowing us to derive the long -time and la rge-distance asymp- totic behavior of the correla tio ns functions of qua ntum integrable mo dels from their exa ct representations. Starting from the form factor expansio n o f the cor relation functions in ﬁnite volume, w e explain how to reduce the complexity o f the computation in the so- called in teracting integrable mo dels to the o ne app earing in free fermion equiv alent mo dels. W e apply our metho d to the time-dependent zero -temperature current-curren t correlation f unc- tion in the non-linear Schrödinger mo del and compute the ﬁrst few terms in its asymptotic expansion. Our result go es b ey ond the confor mal ﬁeld theory based predictions: in the time- dependent case, other t yp es of excitations than the o nes on the F ermi surface con tribute to the leading orders of the asymptotics. 1 In tro duction The successf ul resolution of man y -bo dy quantum in tegrable mo dels through Bethe Ansatz [4, 78, 83 , 68 ] naturally raised the question of the computation of their correlation f unct ions. In its full generalit y , this problem app eared ﬁrst as extremely complex due to the complica ted com binatorial represent ation of the eigenfunctions. Ho wev er, f or the particular v alues of the coupling constan ts for whic h the mo dels are equiv alen t to fr ee fermion ic ones, the use of sp eciﬁc methods (such as the Jordan-Wigner transformation) led to numerous explicit results [66, 64, 65, 75, 70, 3, 76, 88, 74 , 85, 40 , 73, 72, 57 , 33, 32, 34 , 17, 35 , 1 8, 19]. In this con text, diﬀeren t t yp es of eﬀective represen tations hav e b een obtained for the correlation f unct ions, notably in terms of determ inan ts. It has b een p ossible in particular to express tw o-p oin t functions in terms of F redholm determinan ts of compact op erators I + V , where the in tegral k ernel V b elongs to the class of in tegrable in tegral op erators [64, 65 , 57 , 17, 16]. These determinan ts represen tations ha v e been us ed to compute the a symptotic b ehavio r of the correlations functions at large distances [40, 71, 33, 32, 34, 35, 18 ]. 1 DESY, Hamburg, Deut sc hland, k arol.k a jetan.kozlo wski@desy .de 2 Laboratoire d e Physique, U MR 5 672 du CNRS, ENS Lyon, F rance, veronique.terras @ens-lyon.f r 1 It w as the understanding of the underlying algebra ic structures in truly in teracting in tegrable mo dels whic h ﬁnally op ened the w a y to the computation of their correlation functions. In particular, the algebraic v ersion of the Bethe Ansatz (ABA) [24] enabled Izergin, K orepin and their collab orators in Le ningrad to obtain the ﬁrst exact represen tations for the correlati on functions of the XXZ and the non-linear Schrödinger mo dels aw a y from their free fermion p oin ts (see [56] and references therein). These represen tations, based on ﬁnite-size determinan t represen tations for the norms of Bethe eigenstates [25, 26, 55] and for more general s calar pro duct s [82, 56], wer e ho w ev er not completely explicit due to the introduction of dual ﬁelds to handle the comb inatorial diﬃculties of the ABA approac h. T he ﬁrst explicit represen tations (m ultiple in tegral represen tations for elemen tary building blo c ks of the XXZ chain at zero temp erature) were obtained in the 90’ by Jim b o, Miki, Miw a and Nak a yashiki through q - v ertex op erators and solutions of the q -KZ equation [37, 39, 38]. A few y ears later, these represen tations, together w ith t heir generalization to the case of an extern al m agnetic ﬁeld, w ere repro duced by the Ly on group in the ABA framew ork thanks to the resolution of the so-called quan tum in v erse scattering problem [53, 54]. The Ly on group w as a lso able to pro vide ﬁnite-size determinan t represen tations for the form factors of the ﬁnite X X Z c hain [53]. Deve lopmen ts of the latter approac h led to in tegral represen tations for the tw o-p oin t function [50, 52], to their generalizatio ns in the tem p erature case [27, 28], in the t ime-dep ende n t case [51], and in all these cases tak en together [80]. On the other hand, new interesting algebraic represen tations f or the correlation functio ns w ere also obtained r ecen tly using reduc ed q -KZ equations [9, 10, 11], whic h led to the discov ery of a hidden Grassmann structure in the XX Z mo del [12, 8, 13, 41, 6, 7]. In this framewo rk, one of the most c hallenging problem is the deriv ation, f rom these ex- act represen tations, of the long-distance as y mpt otic b eha vior of the correlation functions. In general, zero-temp erature correlation functions of mo dels with a gapless sp ectrum are exp ected to deca y algebraically at large distances. The exp one n ts go v erning this p o w er-la w b eha vior, called critical ex p onen ts, are b eliev ed to b e universal, i.e. not dep ending on the microscopic details of th e model, but only on its ov erall symmetries. Predictions concerning these exp onen ts w ere obtained v ia the Luttinger liquid approac h [69, 29, 30, 31 ] or the connection to Conformal Field Theory [79, 14, 1, 15, 5], together with exact computations of ﬁnite size corrections to the sp ectrum [20, 86, 21, 87, 22]. Ho w ev er, un til very recen tly , it wa s not p ossible to conﬁrm these (indirect) predictions via some direct com putations based on exact results f or an in tegrable mo del out of its free fermion p oin t. T o tac kle this problem, one ma y notice that, in the ABA approac h, the correlation func- tions can b e written in terms of s erie s of m ultiple integr als that can b e though of as some m ultidimensional generalizations of the F redholm determinan ts whic h app ear in the study of the free fermionic mo dels. One ma y therefore attempt to extend and adapt the large arsenal of tec hniques a v ailable for free fermionic correlators to mo dels a w ay from their free fermion p oin ts. The main obstacle here comes from the highly coupled nature of the v arious series of m ultiple in tegrals represen tations for the correlation functions. A ctually , un til tw o years ago, there w ere only few cases out of the free fermion p oin t where the analysis could ha ve b een p erformed to some extend [48, 49, 61]. A ma jor breakthrough w as ac hieved in [45], whic h prop osed a system- atic metho d to p erform the asy mptotic analysis of the long-distance asymptotic b eha vior of the correlation functions of densities in in tegrable mo dels describ ed by a six-vertex t yp e R -m atrix and solv able b y the ABA. T here, the ide a w as to extract the asymptotic b eha v ior of the s eries of m ultiple in tegrals represen ting the correlator by making s ome connections with the asymptotes of F re dholm determinan ts. A lthough f ully functional, the metho d in volv ed sev eral subtle steps, and in particular a quite complex summation pro cess. The ﬁnal result itself, nev ertheless, hap- 2 p ened to b e q uite simply given in terms of a prop erly normalized form factor [45, 46]. This fact naturally suggests that the metho d prop osed in [45] migh t b e simpliﬁed if one w orks directly at the level of the form factor expansion of the correlation functions. It is the aim of the presen t article to exp ound suc h a simpler and more direct approac h, based on the f orm factor expansion of the correlati on functions in the ABA framew ork, on the ﬁnite-size determinan t represen tation f or the form factors in ﬁnite volume and on s ome of their main features in the thermodynamic limit. The main adv an tage compared to [45] is that it relies on the s tudy of more natural physical ob jects: whereas in [45] the correlation f unct ion w as written as a series o v er some elemen tary ob jects expressed in terms of bare quan tities (suc h as energy , momen tum), the so-called cy cle in tegrals, here w e c hose to pursue our s tudy of form factors a step further, and to use some of their thermodynamic prop erties (and in particular their represen tation in terms of dressed ph ysical quan tities) fr om the v ery beginning. This re sults into a ma jor simpliﬁcation of the summation pro cess. Indeed, in [ 45], the asymptotic s umm ation of the series of cycle in tegrals was quite subtle. In particular, it inv olv ed a summation of a whole class of corrections, sub-leading to eac h term of the s eries, but nev ertheless con tributing to the leading term in the ﬁnal result: they happ ened to b e resp onsible, once up on summation, for the dressing of bare physical quan tities. Since in our present approac h the series o ver form factors is already written in terms of the dressed quan tities, the s umm ation is muc h simpler: the series can b e connected to a F redholm determinan t and the le ading asymptotic b ehavior of the c orr elation function i s dir e ctly given by the le ading asymptotic b ehavior of this F r e dholm determinant . No subtle iden tiﬁcation of sub-leading terms is needed for the result. Let us b e sligh tly more precise ab out the diﬀerent steps of our metho d. Its s tarti ng p oin t is the form f acto r expansion of the correlation function in ﬁnite v olume. Using the determinan t represen tation of the form factors in ﬁnite vo lume [81 , 53], one can analyze their thermo dynamic limit in the particle/hole picture [82, 46, 43], and decomp ose their ﬁnite-size expression in to some (universal) “singular” or “discrete” part, whic h is es s en tially f ree-fermion-lik e and con tains the whole non-trivial scaling b eha v ior of the form factor in the thermo dynamic limit, and s ome (mo del-d ep enden t) “regular” part, whic h admits a smo oth thermo dynamic limit [43]. The idea is to reduce t he summ ation o v er form factors to a summation o ver their ﬁnite-size discrete parts, the (thermo dynamic limit of the) regular parts b eing treated as some dressing functionals. The main problem is that, for intera cting mo dels, the series w e consider is highly coupled. The origin of this coupling is tw ofold: on the one hand, the regular part is a complicated function of the particle/hole Bethe ro ots; on the other hand, the discrete part can b e seen as a certain functional of the shift function b et w een the considered exc ited state and the ground state, and hence introduces non-trivial couplings b et w een the particle and hole-t y p e Bethe ro ots. It is p ossible to s ligh tly simplify the structure of the diﬀeren t terms b y neglecting, on the lev el of the ﬁnite-size series, some con tributions that should v anish in the thermodynamic limit. W e th us obtain an eﬀectiv e (ﬁnite-size) form factor series with a simpler particle/hole dep enden ce than in the original series. This series can b e linke d to a decoupled one, reduced to the sum of the discrete parts of the form factors with a shift function that do es not dep end on the excited state, exactly as in a (generalized) f ree fermionic mo del. The sum can then b e computed and recast in terms of a determinan t whic h tends to a F red holm determinan t in the thermo dynamic limit. Th e leading asy mpto tic b eha vior of the correlation function then follo ws directly from the leading as y mpt otic b eha vior of the F redholm determinan t [60]. W e stress that, within this metho d, the computation of the asymptotic expansion of the correlation functions for "truly in teracting" models and for their f ree f erm ionic limi ts carry almost the same complexit y , in as m uc h as w e oﬀer a w ay of understanding the int eracting case 3 as a smo oth deformation of the free fermionic one. This joins in s ome sense the spirit of the w orks [12, 8, 13 , 41, 6, 7] on the hidden Grassmann structure i n the XXZ mo del with the main diﬀerence that, in our metho d, the f ree fermionic structure is only used as an in termediate step so as to carry out certain algebraic manipulations in a simpler w ay and deform the range of summations to a kind of analog of the steep es t descen t cont our. W e ch o ose to exp ose our metho d in the framew ork of the q uan tum non-linear Sch rö dinger mo del, whic h is probably the b est understo od and simplest in tegrable interac ting mo del. Note ho w ev er that, with minor mo diﬁcations sp eciﬁc to the structure of the mo del under in vestiga- tion (mor e complex s truc ture of the conﬁguration of the p ertinen t Bethe roots ab ov e the groun d state, the app earance of trigonometric functions . . . ), this metho d is in principle applicable to a ve ry wide class of int egrable mo dels whose f orm f acto rs are k no wn and admit a ﬁnite-size determinan t represen tation suc h as those obtained in [81, 5 3, 77]. W e apply our metho d to the deriv ation of the ﬁrst few terms in the long-distance and large-time asymptotic expansion of the correlation function of curren ts at zero temp eratu re (s ee Section 3 ). The case of the re- duced densit y matrix, together with some more precise mathematic al details about the metho d, will b e considered in a further publication [59 ]. W e will see that the asym ptotes in the time- dep endent c ase ar e not only issuing fr om excitations in the vicinities of the F ermi b oundaries (whic h correspond to the region of the sp ectru m tak en into accoun t b y the CFT/Luttin ger liquid approac h), but also fr om the excitations ar ound the sadd le-p oint λ 0 of the “plane-wave” c ombination xp ( λ ) − tε ( λ ) , wher e p and ε ar e r esp e c tively t he dr esse d momentum and ener gy of the excitations . More precisely , the exp onen ts go vernin g the p o w er-law deca y of the correlation function are given in terms of the diﬀeren t shift f unct ions b et ween the ground state and excited states ha ving one particle and one hole either on the F ermi b oundaries ± q or at the s addle- p oin t λ 0 . The asso ciated amplitudes are given b y the corresp onding, prop erly normalized, form factors of curren ts. The pap er is organized as follo ws. In Section 2, w e in tro duce the non-linear Sc hrö dinger (NLS) mo del and the diﬀeren t notations, relativ e to the description of the s pace of states and the thermo dynamic limit of the mo del, that will b e used throughout the article. This enables us, in Sec tion 3, to p resen t our main result: the leading lar ge-distance and long -time asymptotic b eha vior of the (zero-temp erature) correlation function of curren ts. The remaining part of the article is devoted to the deriv ation of this result. W e establish in Section 4 a (ﬁnite-size) form factor series represen tation for the correlation function (or more precisely for its generating function). In Section 5, w e explain ho w to relate our highly coupled series to a decoupled, f ree- fermion t y p e series that can b e summed up in to a ﬁnite-size determinan t. Finally , in Section 6, w e take the thermodynamic limit of the previous result, hence represen ting the form factor series in terms of a F redholm determinan t, and explain ho w to deriv e the ﬁrst leading terms of the long-time/large-distance asymptotic b eha vior of our series from those of the F redholm determinan t. T ec hnical details are gathered in a set of A ppendices: explicit represen tations for form factors are given in App endix A; the summation of the free-fermion t yp e series is p erformed in App endix B; the notion of functional translation op erator, whic h is used as a tec hnical to ol to link our series to a decoupled one, is in tro duce d in App endix C; in App endix D, w e sho w ho w to relate the summe d-up form factor series to the t yp e of series th at w as obtained in [45] by expanding the master equation; ﬁnally , in App endix E, w e explain ho w to contro l sub-leading corrections in the as y mpt otic series using the so-called Natte series in tro duce d in [60]. 4 2 The NLS mo del: notations and deﬁniti ons The quan tum non-linear Sc hrö dinger mo del i s describ ed in terms of quan tum Bose ﬁelds Ψ( x, t ) and Ψ † ( x, t ) ob eying to canonical equal-time comm utation relations, with Hamiltonian H = L Z 0 n ∂ x Ψ † ∂ x Ψ + c Ψ † Ψ † ΨΨ − h Ψ † Ψ o d x. (2.1) The mo del is deﬁned on a ﬁnite in terv al of length L (w e tak e the thermodynamic limit L → ∞ later on) and is sub ject to perio dic b oundary cond itions. In (2.1), c denotes the coupling constan t and h the c hemical p oten tial. In this article, w e fo cus on the repulsive regime c > 0 in presence of a p ositiv e c hemical p oten tial h > 0 . Due to the conserv ation of the n um b er of particl es, this quan tum ﬁeld theory mo del is equiv alen t to a (quan tum mec hanical) one-dim ensional ma n y -bo dy gas of b osons sub ject to p oin t-lik e interac tions. Suc h a mo del w as ﬁrst in tro duced a nd solv ed by Lieb and Liniger in 1963 [68, 67], where it w as us ed as a test for Bogoliubov’s theory . It w as then thoroughly in vestigat ed. In particular, in 1982, Izergin and K orepin [36 ] (see also [56]) prop osed a lattice regularizatio n of the mo del allo wing to implemen t the ABA sc h eme [24] in ﬁnite vo lume. In this article, w e consider the zero-temp eratur e time-dep endent correlation function of cur- ren ts, i.e. the ground s tate exp ectation v alue h j ( x, t ) j (0 , 0) i , where j ( x, t ) = Ψ † ( x, t ) Ψ( x, t ) . Our aim is to ev aluate, in the thermo dynamic limit L → ∞ of the mo del, its large-distance and long-time asymptotic b eha vior. Prior to writing do wn our res ult, we in tro duce some necessary notations. Name ly , w e describe the space of states of the mo del and provide deﬁnitions of sev eral thermo dynamic quan tities whic h app ear in our resul t. In the algebraic Bethe ansatz framew ork, the eigenstates | ψ ( { λ } ) i of the H amiltonian (2.1) are parametrized b y a set of sp ectral parameters λ 1 , . . . , λ N solution to the system of Bethe equations. In their logarithm ic form, these Bethe equations read Lp 0 ( λ j ) + N X k =1 θ ( λ j − λ k ) = 2 π n j , j = 1 , . . . , N , (2.2) in terms of a set of in teger (for N o dd) or half-in teger (for N ev en) num b ers n 1 , . . . , n N . The bare momen tum p 0 ( λ ) and the bare phase θ ( λ ) app earin g ab o v e are giv en by p 0 ( λ ) = λ, θ ( λ ) = i log  ic + λ ic − λ  . (2.3) The energy of the state | ψ ( { λ } ) i is E ( { λ } ) = N X j =1 ε 0 ( λ j ) , with ε 0 ( λ ) = λ 2 − h. (2.4) The s olutions to the system of Bethe equations ha v e b een studied b y Y ang and Y a ng [89]. In particular, it has b een prov en that all solutions are sets of real n umbers and that a s et of solutions { λ j } can b e uniquely parametrized by a set of (half-)in tegers { n j } . In the follo wing, λ j , j = 1 , . . . , N , will denote the Bethe roots describing the ground state of the H ami ltonian (2.1). They corresp ond to the solution to (2.2) asso ciated with the follo wing c hoice of (half )-in tegers: n j = j − ( N + 1) / 2 , where the num b er of particles N is ﬁxed b y the 5 v alue of the c hemical p oten tial h . In the thermo dynami c limit ( L → ∞ , N → ∞ with N /L tending to some ﬁnite av erage density D ), the parameters λ j densely ﬁll a symmetric interv al [ − q , q ] of the real axis (the F ermi zone) with some densit y distribu tion ρ ( λ ) solving the follow ing linear in tegral equation ρ ( λ ) − 1 2 π q Z − q K ( λ − µ ) ρ ( µ ) d µ = p ′ 0 ( λ ) 2 π , (2.5) where K denotes the Lieb k ernel K ( λ ) = θ ′ ( λ ) = 2 c λ 2 + c 2 . (2.6) Excited states of (2.1) corresp ond to other solutions of the Bethe equations. F or tec hnical purp ose, it is actually con venien t to consider the twi ste d Bethe s tate s | ψ κ ( { µ } ) i instead. These are parameteriz ed by solutions µ ℓ 1 , . . . , µ ℓ N κ of the twi sted Bethe equations, Lp 0 ( µ ℓ j ) + N κ X k =1 θ ( µ ℓ j − µ ℓ k ) = 2 π  ℓ j − N κ + 1 2  + iβ , j = 1 , . . . , N κ . (2.7) Here β is some purely imaginary parameter, κ = e β , and ℓ 1 < ℓ 2 < · · · < ℓ N κ are s ome in tegers. Note that, the set of ro ots of (2.7) b eing completely deﬁned b y the s et of integ ers ℓ j , w e ha ve c hosen to lab el them accordingly . Ei genstates of the Hamiltonian (2.1) are obtained as limits when κ → 1 of the t wisted Bethe states | ψ κ ( { µ } ) i . F or purely imaginary β , the ro ots of (2.7) are real. The state param eterized b y the solutions of (2. 7) with ℓ j = j , j = 1 , . . . , N κ , is called the κ -t wisted ground state in the N κ -sector. In the thermo dynamic limit, excitations ab o v e this κ -t wisted ground state corresp ond to s olut ions suc h that most of the ℓ j ’s coincide with their v alue f or the ground state: ℓ j = j except for n in tegers (with n remaining ﬁnite in the N κ → ∞ limit) h 1 , . . . , h n ∈ { 1 , . . . , N κ } f or whic h ℓ h k = p k / ∈ { 1 , . . . , N κ } . The in tegers h k represen t "holes" with resp ect to the distributions of in tegers for the ground state in the N κ -sector, whereas the in tegers p k represen t "particle s". F or suc h an excited state | ψ κ ( { µ ℓ j } ) i , the coun ting function b ξ κ ( ω ) ≡ b ξ κ ( ω | { µ ℓ j } ) = 1 2 π p 0 ( ω ) + 1 2 π L N κ X k =1 θ ( ω − µ ℓ k ) + N κ + 1 2 L − iβ 2 π L , (2.8) whic h is mon otonously increa sing, deﬁnes unamb iguously a set of real "bac k grou nd" parameters µ j , j ∈ Z , as the unique solutions to b ξ κ ( µ j ) = j /L . Among this s et of paramete rs, the N κ solutions of the t wisted Bethe equations with in tegers ℓ j corresp ond to the solutions b ξ κ ( µ ℓ j ) = ℓ j /L , j = 1 , . . . , N κ , and the particle rapidities µ p a (resp ectiv ely the hole rapidities µ h a ), a = 1 , . . . , n , corresp ond to the solutions b ξ κ ( µ p a ) = p a /L (resp ectiv ely b ξ κ ( µ h a ) = h a /L ). Due to the partic ular t yp e of c orrelation function w e consider in this article (curren t-curren t correlation function), w e will restrict our study to states ha ving the same num b er of parameters as the ground state, i.e. N κ = N . T o describ e the p osition of the ro ots of an excited state | ψ κ ( { µ ℓ j } ) i with resp ect to the ground s tate roots { λ j } , it is conv enient to deﬁne the (ﬁnite- size) shift function b F ( ω ) ≡ b F ( ω |{ µ ℓ j } ) = L  b ξ ( ω ) − b ξ κ ( ω )  = 1 2 π N X k =1  θ ( ω − λ k ) − θ ( ω − µ ℓ k )  + iβ 2 π , (2.9) 6 in whic h b ξ κ is the excited state’s countin g f unct ion (2.8 ) whereas b ξ denotes the ground state coun ting function (at κ = 1 ). The spacing b et ween the ro ot λ j for the ground state and the bac kground parameters µ j deﬁned 3 b y b ξ κ ( µ j ) = j /L is then giv en b y µ j − λ j = F ( λ j ) Lρ ( λ j ) + O  L − 2  , j = 1 , . . . , N , (2.10) in whic h F ( λ ) is the thermo dynamic limit of the shift function, solution of the in tegral equation F ( λ ) − 1 2 π q Z − q K ( λ − µ ) F ( µ ) d µ = iβ 2 π − 1 2 π n X k =1  θ ( λ − µ p k ) − θ ( λ − µ h k )  . (2.11) In tro ducing the dressed phase φ ( λ, µ ) and the dressed c harge Z ( λ ) as the resp ectiv e solutions of the linear integr al equations φ ( λ, µ ) − 1 2 π q Z − q K ( λ − ω ) φ ( ω , µ ) d ω = θ ( λ − µ ) 2 π , (2.12) Z ( λ ) − 1 2 π q Z − q K ( λ − µ ) Z ( µ ) d µ = 1 , (2.13) w e get F ( λ ) = iβ 2 π Z ( λ ) − n X a =1  φ ( λ, µ p a ) − φ ( λ, µ h a )  . (2.14) Other imp ortan t thermo dynamic quan tities that w e need to introduce in order to formu late our result are the dressed energy ε ( λ ) and the dressed momen tum p ( λ ) , deﬁned as ε ( λ ) − 1 2 π q Z − q K ( λ − µ ) ε ( µ ) d µ = ε 0 ( λ ) , with ε ( ± q ) = 0 , (2.15) p ( λ ) = p 0 ( λ ) + q Z − q θ ( λ − µ ) ρ ( µ ) d µ = 2 π λ Z 0 ρ ( µ ) d µ. (2.16) Note that expression (2.16) enables us to express, in the large L limit, the coun ting function (2.8) of an y excited state w ith a ﬁnite n um b er o f particles and holes ab o ve the N -parti cle ground state in the f ollo wing form: b ξ κ ( ω | { µ ℓ j } ) = ξ ( ω ) + O( L − 1 ) , with ξ ( ω ) = 1 2 π p ( ω ) + D 2 , (2.17) D b eing the a verag e densit y in the thermo dynamic limit ( N /L → D ). In particular this means that, in the thermodynamic limit and up to O(1 /L ) corrections, the coun ting f unct ion do es not dep end on the particular lo caliz ation of the corresp onding Bethe ro ots { µ ℓ j } : it is the same for all particle/hole -t yp e excited states. 3 Since the counting function dep ends on the particular excited state we consider, so does th e set of b ac kground parameters: these are therefore deﬁned for e ach excited state { µ ℓ j } separately . 7 3 The main result: la rge-distance/long-time asymptotic b ehav - ior of the correlation function of curren ts W e are no w in p osition to form ulate our main result, namel y the leading asymptotic b eha vior of the zero-temp erat ure correlation function of curren ts h j ( x, t ) j (0 , 0) i in the large-distance and long-time regime, with x/t = const. Let u ( λ ) = p ( λ ) − t x ε ( λ ) , where p is the dressed momen tum (2.1 6) and ε the dressed energy (2.15), the ratio t/x b eing ﬁxed (and non-zero). W e assume that this function has a unique saddle-p oin t 4 λ 0 on R ( i. e. u ′ admits a unique zero λ 0 on R , whic h moreo v er satisﬁes u ′′ ( λ 0 ) < 0 ). One distinguishes t wo diﬀeren t regimes 5 according to whether λ 0 ∈ ] − q , q [ (time- lik e regime) o r λ 0 / ∈ [ − q , q ] (space-lik e regim e). Then, at large-distance and long-time ( x → + ∞ , t → + ∞ , x/t = const) and in the space-lik e regime, h j ( x, t ) j (0 , 0) i =  p F π  2 − Z 2 2 π 2 x 2 + t 2 v 2 F  x 2 − t 2 v 2 F  2 (1 + o(1)) + 2 cos(2 xp F ) ·   F q − q   2 [ − i ( x − v F t )] Z 2 [ i ( x + v F t )] Z 2 (1 + o(1)) + √ 2 π e − i π 4 p ′ ( λ 0 ) p tε ′′ ( λ 0 ) − xp ′′ ( λ 0 ) e ix [ p ( λ 0 ) − p F ] − itε ( λ 0 ) ·   F λ 0 q   2 [ − i ( x − v F t )] [ F λ 0 q ( q ) − 1] 2 [ i ( x + v F t )] [ F λ 0 q ( − q )] 2 (1 + o(1)) + √ 2 π e − i π 4 p ′ ( λ 0 ) p tε ′′ ( λ 0 ) − xp ′′ ( λ 0 ) e ix [ p ( λ 0 )+ p F ] − itε ( λ 0 ) ·   F λ 0 − q   2 [ − i ( x − v F t )] [ F λ 0 − q ( q ) ] 2 [ i ( x + v F t )] [ F λ 0 − q ( − q )+1 ] 2 (1 + o(1 )) , (3.1) whereas in the time-lik e regime, h j ( x, t ) j (0 , 0) i =  p F π  2 − Z 2 2 π 2 x 2 + t 2 v 2 F  x 2 − t 2 v 2 F  2 (1 + o(1)) + 2 cos(2 xp F ) ·   F q − q   2 [ − i ( x − v F t )] Z 2 [ i ( x + v F t )] Z 2 (1 + o(1)) + √ 2 π e i π 4 p ′ ( λ 0 ) p tε ′′ ( λ 0 ) − xp ′′ ( λ 0 ) e − ix [ p ( λ 0 ) − p F ]+ itε ( λ 0 ) ·   F q λ 0   2 [ − i ( x − v F t )] [ F q λ 0 ( q ) +1] 2 [ i ( x + v F t )] [ F q λ 0 ( − q )] 2 (1 + o(1)) + √ 2 π e i π 4 p ′ ( λ 0 ) p tε ′′ ( λ 0 ) − xp ′′ ( λ 0 ) e − ix [ p ( λ 0 )+ p F ]+ itε ( λ 0 ) ·   F − q λ 0   2 [ − i ( x − v F t )] [ F − q λ 0 ( q ) ] 2 [ i ( x + v F t )] [ F − q λ 0 ( − q ) − 1 ] 2 (1 + o(1 )) . (3.2) In these expressions, p F = ± p ( ± q ) = π D is the F ermi momen tum, v F = ε ′ ( q ) /p ′ ( q ) is the F ermi velocity , whereas Z = Z ( ± q ) is the v alue of the dressed charg e on the F ermi surface. The exp onen ts of the last t w o terms in (3.1)-(3.2) are giv en in terms of the v alues at the F ermi b oundar ies ± q of some sp eciﬁc s hift functions (2.14) F µ p µ h (at β = 0 ) b et ween the ground state and an excited state with one particle at µ p and one hole at µ h : the shift functions F λ 0 ± q ( λ ) = − [ φ ( λ, λ 0 ) − φ ( λ, ± q )] (3.3) 4 It is clear that the bare counterpart u 0 = p 0 ( λ ) − t x ε 0 ( λ ) of this function fulﬁlls this prop erty: u ′ 0 admits a unique zero λ 0 on R , and more o ver u ′′ 0 ( λ 0 ) < 0 . Considering the f orm of th e in tegral eq uatio ns (2.15) and (2.16 ) , one easily sees that u ′ admits at least one zero. The case of sev eral saddle-points can in principle b e treated similarly , and will give rise to several co ntributions. 5 W e do not consider here the case λ 0 = ± q . Indeed, this sp eciﬁc case w ould demand some further analysis. 8 b et w een the ground state and an excited state with one particle at λ 0 and one hole at ± q in (3.1) (space-lik e regime), and the shift functions F ± q λ 0 ( λ ) = − [ φ ( λ, ± q ) − φ ( λ, λ 0 )] (3.4) b et w een the ground state and an excited state with one particle at ± q and one hole at λ 0 in (3.2) (time-lik e regime). The corresponding amplitudes are giv en in terms of some prop erly normalized form factors F µ p µ h of the curren t op erator b et w een the ground state and an excited state con taining one particle at µ p and one hole at µ h . The connexion   F q − q   2 = l im L → + ∞  L 2 π  2 Z 2     h ψ q − q | j (0 , 0) | ψ g i || ψ q − q || · || ψ g ||     2 , (3.5) b et w een the amplitude   F q − q   2 and the square of the norm of an Umklapp form factor of the densit y operator b et we en the ground s tate | ψ g i and an excited state | ψ q − q i con taining one particle and one hole at the opp osite ends of the F ermi b ound aries w as already noticed in [45, 46]. A similar phenomenon happ ens for the other terms. M ore precisely , the amplitudes   F λ 0 ± q   2 app earin g in (3.1) (space-lik e regime) corresp ond to the thermo dynamic limit of the prop erly normalized norm squared of the form f acto r of the curren t op erat or take n b et w een the ground state | ψ g i and an eigenstate | ψ λ 0 ± q i with a particle at λ 0 and a hole at ± q :   F λ 0 q   2 = l im L → + ∞  L 2 π  1+[ F λ 0 q ( q ) − 1] 2 +[ F λ 0 q ( − q )] 2     h ψ λ 0 q | j (0 , 0) | ψ g i || ψ λ 0 q || · || ψ g ||     2 , (3.6)   F λ 0 − q   2 = l im L → + ∞  L 2 π  1+[ F λ 0 − q ( q ) ] 2 +[ F λ 0 − q ( − q )+1 ] 2     h ψ λ 0 − q | j (0 , 0) | ψ g i || ψ λ 0 − q || · || ψ g ||     2 . (3.7) Similarly , the amplitudes   F ± q λ 0   2 app earin g in (3.2) (time-lik e regime) corresp ond to the ther- mo dynamic limit of the prop erly normalized norm squared of the f orm factor of the curren t op erator b et wee n the ground state | ψ g i and an eigenstate | ψ ± q λ 0 i with a particle at ± q and a hole at λ 0 :   F q λ 0   2 = l im L → + ∞  L 2 π  1+[ F q λ 0 ( q ) +1] 2 +[ F q λ 0 ( − q )] 2     h ψ q λ 0 | j (0 , 0) | ψ g i || ψ q λ 0 || · || ψ g ||     2 , (3.8)   F − q λ 0   2 = l im L → + ∞  L 2 π  1+[ F − q λ 0 ( q ) ] 2 +[ F − q λ 0 ( − q ) − 1 ] 2     h ψ − q λ 0 | j (0 , 0) | ψ g i || ψ − q λ 0 || · || ψ g ||     2 . (3.9) The explicit expressions of (3.5)-(3.9) are giv en in App endix A.3. W e now commen t this result. First of all, the ﬁrst t wo lines of each expression app ear as a direct generalization, in volving the relativistic com binations x ± v F t , of the large-distance asymptotic b eha v ior of the corresp onding static correlation function. They are in agreemen t (up to the amplitude of the third term, computed in [45, 46]) w ith the pred ictions comin g from the CFT/Luttinger liquid appro x imations. This is indeed not surprising, since suc h terms in volv e particle/hole excitations on the F ermi b oundary and are therefore correctly taken in to accoun t b y a linearization of the sp ectrum around this p oin t . Ho wev er, when time is of the same order of magnitude as distance ( i .e. x/t ∼ O(1) ), other types of excitations also con tribute to 9 the asy mpt otics, namely those in v olving particles (for the space-like regime) or holes (for the time-lik e regime) with rapidities lo cated at the saddle-p oin t λ 0 . These con tributions app ear in the last tw o lines of (3.1), (3.2). R emark 3.1 . This result is also v alid when x/ t → 0 ( t >> x ). In the t / x → 0 limit ( i .e. t << x ), one has λ 0 → + ∞ , and the last t wo terms of (3.1)-(3.2) exhibit a very quick oscillation except around x = 0 , whic h reconstructs the δ ( x ) -part of the equal-time correlation function. Hence, as exp ected , the con tribution of these terms v anishes at large distances. R emark 3.2 . It is easy to see that our result coincides, at the free f ermi on p oin t c = + ∞ , with the leading asymptotic b eha v ior obtained through a direct computation of the correlation function. In that case λ 0 = x/ (2 t ) and the com bination of the last tw o terms of (3.1)-(3.2) reduces to: i r π t e iα π 4 e iα ( x 2 4 t + th )  e − iα [ xq − t ( q 2 − h )] x − 2 q t − e iα [ xq + t ( q 2 − h )] x + 2 q t  , (3.10) where α = 1 in the space-lik e regime and α = − 1 in the time-lik e regime. Note that this con tribution is dominan t with resp ect to the other p o wer -la w decaying terms. 4 F orm factor expansio n for the correlation function in ﬁni te v olume The deriv ation of the result we ha v e just announced is based on the form factor expansion for the correlation function. In this section, we write down the f orm factor series represent ation for the curren t-curren t correlation function in ﬁnite v olume, or more precisely for its generating function. Eac h term of the series can b e ex press ed by using th e ﬁnite-size determinan t repre- sen tation f or the form factors of the mo del and their relation with the o verlap scalar pro ducts. Suc h a series will b e the starting p oin t f or our study . Let us consider the zero-temperature correlation function of curren ts h j ( x, t ) j (0 , 0) i , with j ( x, t ) = e itH j ( x, 0) e − itH . Inserting the sum o v er the complete (see [23]) set of Bethe states | ψ ′ i b et w een the t wo op erators w e obtain h j ( x, t ) j (0 , 0) i = X ψ ′ e − it E ex h ψ g | j ( x, 0) | ψ ′ i h ψ ′ | j (0 , 0) | ψ g i || ψ g || 2 · || ψ ′ || 2 , (4.1) in whic h E ex denotes the diﬀerence of energies b et ween the excited state | ψ ′ i and the ground state | ψ g i . The matrix elemen ts of j ( x, 0) ca n b e easily obtained in terms of the matrix elemen ts of the op era tor e β Q x , with Q x = R x 0 j ( y , 0)d y : h ψ g | j ( x, 0) | ψ ′ i || ψ g || · || ψ ′ || = ∂ x ∂ β h ψ ( { λ j } ) | e β Q x | ψ κ ( { µ ℓ j } ) i k ψ ( { λ j } ) k · || ψ κ ( { µ ℓ j } ) ||      β =0 . (4.2) In their turn, the latter can b e computed from the results of [44 , 77, 58] in terms of the scalar pro duct s: h ψ ( { λ j } ) | e β Q x | ψ κ ( { µ ℓ j } ) i = e ix P κ ex h ψ ( { λ j } ) | ψ κ ( { µ ℓ j } ) i , (4.3) with P κ ex = P N j =1 [ p 0 ( µ ℓ j ) − p 0 ( λ j )] . In these expressions, { λ } parametrizes the ground state whereas { µ ℓ j } is a s et of solutions to the twisted Bethe equations (2.7) asso ciated with the set 10 of inte gers { ℓ j } , s uch that | ψ κ ( µ ℓ j ) i → | ψ ′ i when κ → 1 (we recall that κ = e β ). Note that it is enough to consider s tate s with N κ = N , as otherwise, the corresponding matrix elemen t is zero. Therefore, (4.1) can b e rewritten as h j ( x, t ) j (0 , 0) i = X ℓ 1 <...<ℓ N e ix P ex − it E ex     ∂ y ∂ β  e iy P κ ex h ψ ( { λ j } ) | ψ κ ( { µ ℓ j } ) i k ψ ( { λ j } ) k · || ψ κ ( { µ ℓ j } ) ||  y ,β =0     2 , (4.4) in which we ha v e set P ex = lim κ → 1 P κ ex . Using the fact that P ex = 0 if | ψ ( { λ j } ) i = | ψ ( { µ ℓ j } ) i and that h ψ ( { λ j } ) | ψ κ ( { µ ℓ j } ) i = 0 if | ψ ( { λ j } ) i 6 = | ψ ( { µ ℓ j } ) i in virtue of the orthogonali t y of Bethe states [23 ], we obtain h j ( x, t ) j (0 , 0) i = − [ ∂ β P κ ex | β =0 ] 2 − X ψ ′ 6 = ψ g e ix P ex − it E ex P 2 ex ∂ β h ψ ( { λ j } ) | ψ κ ( { µ ℓ j } ) i β =0 · ∂ β h ψ κ ( { µ ℓ j } ) | ψ ( { λ j } ) i β =0 || ψ g || 2 · || ψ ′ || 2 . (4.5) On the other hand, s etting E κ ex = P N a =1 [ ε 0 ( µ ℓ a ) − ε 0 ( λ a )] and using similar argumen ts, ∂ 2 x ∂ 2 β ( e ix P κ ex − it E κ ex ·     h ψ ( { λ j } ) | ψ κ ( { µ ℓ j } ) i || ψ κ ( { µ ℓ j } ) || · || ψ ( { λ j } ) ||     2 ) β =0 = − 2 δ ψ ′ ,ψ g [ ∂ β P κ ex | β =0 ] 2 − 2(1 − δ ψ ′ ,ψ g ) e ix P ex − it E ex P 2 ex ∂ β h ψ ( { λ j } ) | ψ κ ( { µ ℓ j } ) i β =0 · ∂ β h ψ κ ( { µ ℓ j } ) | ψ ( { λ j } ) i β =0 || ψ g || 2 · || ψ ′ || 2 . (4.6) Ab o ve w e hav e used the fact that, for β ∈ i R , the solutions of the κ -t wisted Bethe equations (2.7) are real, whic h means that h ψ ( { λ } ) | ψ κ ( { µ ℓ j } ) i = h ψ κ ( { µ ℓ j } ) | ψ ( { λ } ) i . (4.7) Let us deﬁne Q κ N ( x, t ) = X ℓ 1 < ··· <ℓ N e ix P N a =1 [ u 0 ( µ ℓ a ) − u 0 ( λ a )]      h ψ ( { λ j } ) | ψ κ ( { µ ℓ j } ) i   ψ κ ( { µ ℓ j } )   · k ψ ( { λ j } ) k      2 , (4.8) with u 0 ( λ ) = p 0 ( λ ) − t x ε 0 ( λ ) . Compari ng (4.5) and (4.6), we get that Q κ N ( x, t ) is the generating function of the time and space -dep ende n t zero-temperature curren t-curren t correlation function of the NLS mo del in ﬁnite v olume: h j ( x, t ) j (0 , 0) i = 1 2 ∂ 2 x ∂ 2 β Q κ N ( x, t )   β =0 . (4.9) W e recall that, in (4.8), the set of parameters { λ j } denotes the solution of the Bethe equations (2.2) parametrizing the ground state of the Hamiltonian (2.1), whereas { µ ℓ j } stands for the unique solutio n of the κ -t wisted logarithmic Bethe equations (2.7) deﬁned by the c hoice of in tegers ℓ 1 < · · · < ℓ N . The sum runs here through all the p ossible cho ices of ordered N -tuples of in tegers ℓ 1 < · · · < ℓ N , and ther efore throug h all the excited states | ψ κ ( { µ ℓ j } ) i in the N -particle s ect or [23]. 11 Eac h term of the f orm factor series (4.8) inv olves a normalized scalar pro duct b et w een a Bethe s tate and a t wisted Bethe s tate. F or the mo del in ﬁnite v olume, such scalar products (and form factors) admit ﬁnite-size determinan t represen tations [81, 58]. When the size L b ecomes large, these scalar pro ducts (and form f acto rs) exhibit a non-trivial b eha vior with resp ect to the size of the system [82, 2]. F or scalar pro ducts b et w een the ground s tate and an ex cited state with a ﬁnite num b er of particles and holes suc h as those describ ed in Section 2, it w as sho wn in [46, 43] ho w to extract the leading large L asy mpto tic b eha v ior from their ﬁnite s ize determinan t represent ations. The detailed represen tation f or the pro ducts of form f acto r that app ear in eac h term of the series (4.8), as w ell as its leading b eha v ior in the thermo dynamic limit, is recalled in App endix A. In ﬁnite volume, the corresp onding normalized scalar pro du ct takes the follo wing form:      h ψ ( { λ j } ) | ψ κ ( { µ ℓ j } ) i   ψ κ ( { µ ℓ j } )   · k ψ ( { λ j } ) k      2 = b D N ( { λ j } , { µ ℓ j } ) · b G N ( { λ j } , { µ ℓ j } ) , (4.10) in whic h b D N ( { λ j } , { µ ℓ j } ) = N Y j =1 sin 2 [ π b F ( λ j )] π 2 L 2 b ξ ′ ( λ j ) b ξ ′ κ ( µ ℓ j ) ·  det N 1 λ j − µ ℓ k  2 (4.11) is the so-called discrete part of the form f actor, whic h contain s the whole non-trivial singu- lar part of the form factor with resp ect to the sy s tem- size (and is quite univ ersal), whereas b G N ( { λ j } , { µ ℓ j } ) is a dressing function whic h, for a n -particle/ n -hole excited state as deﬁned in Section 2, admits a smo oth thermody namic limit: lim L,N →∞ b G N ( { λ j } , { µ ℓ j } ) = G n  { µ p a } { µ h a }  . (4.12) Here G n is a holomorphic function of the particle/hol e rapidities { µ p a } and { µ h a } . The mi- croscopic details of this dressing function (see App endix A ) dep end on the mo del and do not really matter for our study . W e will only use the fact that G n can b e represen ted in terms of a functional G n  { µ p a } { µ h a }  ≡ G   n  ·     { µ p a } { µ h a }  (4.13) of the function  n  λ     { µ p a } { µ h a }  = n X a =1  1 λ − µ p a − 1 λ − µ h a  . (4.14) The behavior of the thermo dynamic limit of the discrete part is more diﬃcult to obtain (see [46, 43] and App endix A). I ts kno wledge is ho wev er unnecessary for our purp ose, since our study will directly rely on the ﬁnite-size form ula (4.11). W e will simply use the fact that (4.11) can b e understo od as a f unc tional of the ﬁnite-size s hift function b F , dep ending moreo ver on the extra sets of particle/hole integ ers { p a } and { h a } : b D N ( { λ j } , { µ ℓ j } ) ≡ b D N ,n  { p a } { h a }   b F  . (4.15) More precisely , the function b ξ b eing ﬁxed and the set of parameters λ j b eing deﬁned as the pre-image of the set { j /L : j = 1 , . . . , N } by this function, it means that: 12 • the function b ξ κ in (4.11) should b e understo o d as a functional of the shift function b F through the relation b ξ κ = b ξ − L − 1 b F ; • the set of in tegers ℓ j b eing ﬁxed b y the c hoice of { p a } and { h a } (and vic e-versa ), the parameters µ ℓ j are obtained as their pre-image by the function L b ξ κ . 5 Eﬀectiv e decoupl i ng of the ﬁnite- s i ze form factor serie s The structure of the summations in (4.8) is extremely in tricate. Indeed, the sets { µ ℓ j } of Bethe parameters o ver whic h we sum up are implicit functions of the N in tegers ℓ j lab elling the corresp onding excited states. A ltho ugh one can build on suc h a description so as to us e the m ultidimensional residue theory to recast this complex sum into a con tour integra l (the so- called master e quation [52, 51, 44]), the later still remains diﬃcult to handle. In particular, one cannot p erform the thermody namic limit directly on the level of the master e quation. Therefore, one has to expand the cont our in tegral int o yet another series (the so-called m ultidimensional F redholm series) that has a presumably w ell deﬁned thermo dynamic limit. The asymptotic analysis of the obtained s erie s, decomposed in terms of cycle in tegrals, then relies on a non- trivial s ummation pro cess, resulting in particular int o the dressing of bare quan tities (energy , momen tum) in to dressed ones. Although suc h an approac h w as s ucce ssfully applied in [45] to pro duce the leading as y mpt otic b eha v ior of the time-indep enden t correlation functions of the XXZ chain and of the NLS mo del (see also [63] for the temp erature-dependen t case), we wish here, so as to b ypass some of the subtleties of [45], to prop ose an alternativ e line of though by pursuing our study directly on the form factor series. The idea of our metho d is the follo wing: the s eries (4.8) can b e related to an essen tially free fermion t yp e series (see App endix B), this via some reasonable ph ys ical assumptions con- cerning the con tribution of eac h of its terms in the thermo dynamic limit, and via some f ormal manipulation s. Indeed , it is sho wn in App endix B that if, f or eac h excited state, • the rap idities of the particles and holes are decoupled, i.e. they are determin ed b y a coun ting function that do es not dep end on the p osition of the ro ots of the corresp onding excited state, • the dressing part b G N of the form factors is decoupled, then the series in ﬁnite volu me can b e summed up in to a ﬁnite-size determinan t. W e explain in this section ho w to reduce the study of our highly coupled series to this simple case. It means that, in our setting, the k ey-role in the summation is pla y ed b y the univ ersal discrete part b D N of the form factor s exactly as in the free fermion case. This f act should b e put in parallel with the key-role pla yed, in the master equation-based asymptotic analysis [45], b y the Cauc h y determinan t part of the form factors. 5.1 An eﬀectiv e form factor series Considering that w e will ﬁnally b e in terested in the thermo dynamic limit L → + ∞ of the generating function (4.8), w e no w sligh tly simplify its form factor series. Our simpliﬁcations rely on t w o kinds of assumptions: (i) the con tribution to the sum (4.8) of a state ha ving a macroscopically (with resp ect to L ) diﬀeren t energy and momen tum f rom the ground state should v a nish in the thermo dy- namic limit; 13 (ii) for each term of the series, "non-discrete" quant ities ( i.e. parts of the form factor expres- sion b eha ving smo othly at the thermo dynamic limit) should con tribute to the thermo dy- namic limit of the sum (4.8) only through their leading order in L . Assumption (i) can b e attributed in particular to the extremely q uic k oscillation of the phase factors for states having large excitation momen ta and energies. It means notably that: • w e can use the particle/hole picture to describ e the large- L b eha vior of the form factors, k eeping in mind that the relev an t ( i .e. the one not v anishing in the L → + ∞ limit) part of the form factor expansion corresp onds to a summation o ver states with a ﬁnite (or at least gro wing m uch less than L ) num b er of particle/hole excitations ab o ve the ground state; • w e can in tro duce a "cut-oﬀ" (with resp ect to L ) of the range of in tegers on which w e sum up in ( 4.8) since, for v ery large in tegers, momen tum and energy of the co rresp onding state b ecome also very large. Doing this, we roughly sp eaking neglect correcting terms in the lattice size L . It is th us reason- able to assume that, on the same ground, one can also neglect s ome ﬁnite-size corrections to the leading thermo dynamic b eha vior of the form factors, as stated in (ii). More precisely , w e mak e the followi ng simpliﬁcations in the series (4.8): • w e re place the ﬁnite -size shift function b F , the smo oth part b G N and the oscillatin g exp onen t P N a =1 [ u 0 ( µ ℓ a ) − u 0 ( λ a )] b y their leading con tribution in the thermo dynamic limit; • in the obtained expression, we no w understand the rapidities of the particles µ p a (resp. of the holes µ h a ), corresp ondin g to a particular ch oice of in tegers ℓ 1 < · · · < ℓ N , as b eing deﬁned in terms of the thermo dynamic limit F of the shift function (and no longer in terms of its ﬁnite-size coun terpart (2.9)) as the pre-images of p a /L (resp. h a /L ) b y the coun ting function b ξ F = b ξ − F /L ; in other w ords, the 2 n parameters { µ p a } and { µ h a } are obtained as the unique solutions to the system of 2 n equations b ξ ( µ p a ) − 1 L F  µ p a     { µ p a } { µ h a }  = p a L and b ξ ( µ h a ) − 1 L F  µ h a     { µ p a } { µ h a }  = h a L , (5.1) with a = 1 , . . . , n . These simpliﬁcations result into an eﬀectiv e series that w e assume to ha ve the same value at the thermo dynamic limit as the original series. In other w ords, w e conjecture that, lim L,N →∞ Q κ N ( x, t ) eﬀ = lim L,N →∞ Q κ N ( x, t ) ≡ Q κ ( x, t ) , (5.2) with Q κ N ( x, t ) eﬀ = e − xβ p F π X ℓ 1 < ··· <ℓ N ℓ a ∈B L e ix n P a =1 [ u ( µ p a ) − u ( µ h a )] · b D N ,n  { p a } { h a }  F  ·     { µ p a } { µ h a }  × G   n  ·     { µ p a } { µ h a }  . (5.3) In (5.3), the sums are restricted to the set B L =  ℓ ∈ Z | − w L ≤ ℓ ≤ w L  , where w L is a cut-oﬀ gro wing as L 1+ ǫ , for some ǫ > 0 . F or a given set of int egers { ℓ j } , the parameters 14 p a and h a deﬁne the p osition of particles and holes as explained in Section 2. The (ﬁnite- size) represen tation for the form factor has b een partly replaced by its thermo dynamic limit. More precisely , the dressing function and the phase factor, whic h b oth b eha ve smo othly at the thermo dynamic limit, ha ve b een replaced b y their lead ing equiv alen ts. Note that, as a consequence, it is now the dressed energy and momen tum that a ppear in the phase factor ( cf App endix A): u ( λ ) = p ( λ ) − t x ε ( λ ) . In (5.3), w e hav e k ept the ﬁnite-size expression of the discrete part, but w e ha ve nev ertheless replaced the express ion of t he ﬁnite-size shift function b F b y its limiting v alue F (2.14). Finally , the particles and holes’ rapidities ass ociated to a giv en set of intege rs ℓ 1 < · · · < ℓ N are obtained (in terms of F ) via the system of 2 n equations (5.1). R emark 5.1 . W e stress that, in (5.3), we are still in "ﬁnite volum e" ( i .e. f or the momen t, L and N are kep t ﬁnite). W e ha ve only mo diﬁed the original series (4.8 ) arguing that it should not aﬀect the v alue of its thermo dynam ic limit. R emark 5.2 . The introduction of the cut-oﬀ w L is conv enient to a v oid problems of con vergenc e in our further manipulations of the series. Indeed, as far as w e remain in ﬁnite volume , w e now deal with ﬁnite s um s only . 5.2 T o wards a free-fermion type series Although the eﬀectiv e series (5.3) is already simpler that the original one it is still highly coupled: • the thermody namic limit of the shift function (2.14) still dep ends on the p osition of particles and holes, • the rapidities of the particles and holes ha ve to b e computed for eac h excited state s epa- rately ( i.e. they are excited s tate dep enden t), • the expression of the functional G is extremely in tricate. Our aim is no w to relate Q κ N ( x, t ) eﬀ to a decoupled series, so as to reduce its analysis to the one of the generalized free-fermion case that is carried out in App endix B. This can b e done b y understanding the coupling b et w een v ariables as the result of the action of some functional translation op erator s 6 (see App endix C). The function  n (4.14) depends line arly on a function of the rapidities of the particles and of the holes. This means that one can formally express this dep endence b y means of the action of a functional translation op erator, as ex plained in App endix C: for an y suﬃcien tly s mooth functional F s upported on a neigh b orhoo d U of the real axis, F   n  ·     { µ p a } { µ h a }  = n Y a =1 exp ( Z R d λ  1 λ − µ p a − 1 λ − µ h a  δ δ ω ( λ ) ) · F [ ω ]     ω =0 . (5.4) The thermo dynami c limit of the shift function (2.14) also dep ends line arly on a function of the rapidities of the particles and holes. Ho wev er, the situation is sligh tly more complicated than in (5.4) since, in virtue of (5.1), the parameters { µ p a } and { µ h a } are themselv es functionals of the shift function. As explained in App endix C, one can still represen t an y smo oth functional of 6 The use of functional translation op erators is simply a con venien t and compact wa y to manipulate generalized Lagrange series (se e App endix C of [4 5]). In fact, the whol e reasoning that f ollows can instead b e performed b y writing explicitly the corresponding series. 15 this shift function in terms of translation op erators, pro v ided that one imp oses some op erator ordering : · : and that one takes in to accoun t the con tribution of the Jacobian coming from the s ummation of the corresp onding mul ti-dimensional Lagrange series (see form ulae (C.5 ) and (C.11)): F  F  ·     { ¯ µ p a } { ¯ µ h a }  = : n Y a =1 exp ( Z R d λ [ φ ( λ, ¯ µ h a ) − φ ( λ, ¯ µ p a )] δ δ τ ( λ ) ) · F [ ν τ ] :     τ =0 J. (5.5) Ab o ve the op erator ordering : · : is suc h that, in the formal series expansion in p o wers of δ /δτ ( λ ) , all functional deriv ative operators are lo cated on the left. ν τ is the function ν τ ( λ ) = i β Z ( λ ) 2 π + τ ( λ ) , (5.6) and J is the Jacobian J = d et R  I − δ Γ[ ν τ ]( λ ) /δτ ( µ )  | τ = F of the functional Γ[ ν τ ]( λ ) = n X a =1 [ φ ( λ, ¯ µ h a ) − φ ( λ, ¯ µ p a )] , where ¯ µ ℓ ≡ b ξ − 1 ν τ  ℓ L  with ℓ ∈ Z . (5.7) This Jacobian is ev aluated at τ = F , where F coincides with the shift function o ccuring in the l.h.s. of (5.5). W e s tress that the bar o ccuring in the parameters ¯ µ p a (resp. ¯ µ h a ) app earing in the r. h. s. of (5.5) indicates that these are to b e understo o d as the pre-images of p a /L (resp. h a /L ) b y the coun ting function b ξ ν τ , as in (5.7). It is easy to con vince oneself that, provided that the nu m b er of particle/hole excitations is ﬁnite, J = 1 + O( L − 1 ) . In the light of our previous argumen ts, only this sector of excitations is exp ected to con tribute to the L → + ∞ limit of the form factor expansion . As a consequence, in order to b e consisten t with our previous appro ximations, w e also drop the ﬁnite-size corrections due to this Jacobian b y considering f rom now on that our eﬀective series (still abusiv ely denoted b y the same symbol Q κ N ( x, t ) eﬀ ) is in fact give n by the expression (5.3 ) in whic h eac h term is m ultiplied b y the in v erse of this Jacobian. Hence, the use of the functional deriv ativ e leads to the follow ing represen tation: b D N ,n  { p a } { h a }  F  ·     { µ p a } { µ h a }  · G   n  ·     { µ p a } { µ h a }  · J − 1 = : n Y a =1 e R R d λ [ φ ( λ, ¯ µ h a ) − φ ( λ, ¯ µ p a )] δ δτ ( λ ) n Y a =1 e R R d λ  1 λ − ¯ µ p a − 1 λ − ¯ µ h a  δ δω ( λ ) · b D N ,n  { p a } { h a }  [ ν τ ( · )] · G [ ω ( · )] :     τ =0 ω =0 . (5.8) Here w e hav e insisted on the f act that b D N ,n and G are functionals acting on the argumen t · of ν τ ( · ) and ω ( · ) . Als o, on the l.h.s. of (5.8), we ha ve explicitly p oin ted out the parametric dep ende nce of F and  n on the parameters { µ p a } and { µ h a } . In the r.h.s. of (5.8), the functional b D N ,n acts on a shift function ν τ that do es not dep end an ymore on the particle/hole rapidities. In other wo rds, the eﬀectiv e s hift function ν τ b ecomes independent of the summation v ariables and hence mimics the one app earing in the generalized free fermion mo del studied in A ppendix B. Moreo ver , the equations deﬁning the p osition of the particle’s/hole’ s rapidities also b ecome decoupled: the rapidit y ¯ µ ℓ is no w the unique solution 16 to b ξ ν τ ( ¯ µ ℓ ) = ℓ/L , and its v alue do es not dep end anymore on the choi ce of the other inte gers describing the excited state, but only on the function τ . It is con venien t to express, on a formal lev el, eac h term of the pre-factor in (5.8) as a ratio of tw o exp onen ts e b g ( ¯ µ p a ) / e b g ( ¯ µ h a ) , where b g is an op erat or v alued function (w e ha v e used the hat so as to insist on this prop ert y): b g = b g τ + b g ω , with b g τ ( λ ) = − Z R d µ φ ( µ, λ ) δ δ τ ( µ ) , b g ω ( λ ) = Z R d µ µ − λ δ δ ω ( λ ) . ( 5.9) In the follo wing, we set b E 2 − = e − ixu − b g . (5.10) The op erator order : · : b eing linear, w e can exc hance it with a ﬁnite summation symbol suc h as the one app earing in the expression (5.3) for Q κ N ( x, t ) eﬀ . Hence, w e obtain Q κ N ( x, t ) eﬀ = e − β xp F π : X ℓ 1 < ··· <ℓ N ℓ j ∈B L n Y a =1 b E 2 − ( ¯ µ h a ) b E 2 − ( ¯ µ p a ) · b D N ,n  { p a } { h a }  [ ν τ ] · G [ ω ] :     τ =0 ω =0 = e − β xp F π : N Y a =1 b E 2 − ( ¯ µ a ) b E 2 − ( λ a ) · X N  ν τ , b E 2 −  · G [ ω ] :     τ =0 ω =0 , (5.11) in whic h X N is the generalize d free-fermionic functional (B.1) studied in App endix B. 5.3 Represen tation in terms of a ﬁnite-size determinan t In App endix B we hav e recast the generalized free-fermionic functional X N [ ν, E 2 − ] in to a ﬁnite- size determinan t (B.2), this without an y further appro ximations. Suc h a represen tation is obtained through pur ely algebr aic manipulations on the initial deﬁnition (B.1) for X N [ ν, E 2 − ] . These t wo represent ations a re still eq ual when applied to the (non-comm utativ e) op erator v alued functions ν τ and b E − : it is indeed not a problem to implemen t the appropriate op erator order at an y step of the computation p erformed in App endix B. One ma y therefore use the determinan t represen tation (B.2) to obtain new represen tations for our series (5.11), pro v ided that one imp oses an op erator order on the entr ies of the columns of the determinan t 7 [84]. This leads to Q κ N ( x, t ) eﬀ = e − β xp F π : N Y a =1 b E 2 − ( ¯ µ a ) b E 2 − ( λ a ) · det N  δ j k + b V ( L ) ( λ j , λ k ) L b ξ ′ ( λ k )  · G [ ω ] :     τ =0 ω =0 , (5.12) where the (operator v alued) k ernel is giv en as b V ( L ) ( λ k , λ j ) = 4 sin[ π ν τ ( λ k )] sin[ π ν τ ( λ j )] 2 iπ ( λ k − λ j ) n b E ( L ) + ( λ k ) b E − ( λ j ) − b E ( L ) + ( λ j ) b E − ( λ k ) o , (5.13) with b E ( L ) + ( λ ) = i b E − ( λ )      ¯ B L Z ¯ A L  d µ 2 π b E − 2 − ( µ ) µ − λ + b E − 2 − ( λ ) 2 cot[ π ν τ ( λ )] + I L  ν τ , b E − 2 −  ( λ )      . (5.14) 7 Such an order can for instance b e implemented explicitly by expanding the determinant in to a sum ov er p erm utations. 17 The expression of the functional I L can b e found in (B.5). Suc h an expression pro vides a formal re-summation of the eﬀectiv e form factor series (5.3). In the next section, we show ho w to tak e the thermo dynamic limit of (5.12 ). 6 Thermo dynamic lim it and asymptotic analysis W e no w build on the results of the previous s ecti on so as to derive the leading large-d istance/long- time asymptotic b eha v ior of the thermo dynami c limit Q κ ( x, t ) of the generating function Q κ N ( x, t ) . 6.1 Represen tation in terms of a F redholm determinan t in the thermo dy- namic limit The next step o f our study is to obtain a con v enien t represen tation f or the thermo dynami c limit N , L → + ∞ of the generating function (4.8 ). Recall at this p oin t that the eﬀectiv e series (5.3 ) w as built s o as to approac h, in this limit, the s ame v alue Q κ ( x, t ) as the original form factor series (4.8). W e will therefore use the represen tation (5.12) to derive a s uita ble expression for Q κ ( x, t ) . In principle, prior to taking the thermo dynamic limit of a formal represen tation such as (5.12), one should ﬁrst compute the eﬀect of the translation op erat ors: the op erati on of taking the thermo dynam ic limit is indeed a priori only allo wed on express ions deﬁned in terms of explicit holomorphic functions ( i .e. whic h do not cont ain any op erator v alued f unct ions). In App endix D, such a pro cedur e is explicitly p erformed so as to obtain, starting from (5.12), a particular s eries re presen tation of the therm o dynamic limit Q κ ( x, t ) of Q κ N ( x, t ) . More precisely , in App endix D, w e expand the ﬁnite-size determinan t in (5.12) in to a sum o ver determinan ts of all s ub-m atrices of V ( λ j , λ k ) , f acto rize the Cauc hy part of each of these determinan ts, compute the eﬀects of the translation op erators, and then take the thermo dy namic limit. This enables us to express Q κ ( x, t ) as a series of m ultiple in tegrals coinciding, in the equal-time case, with the series expansion derived from the master equation represen tation in [45]. In fact, it is easy to con v ince oneself, b y carrying out the computation in the reverse order, that the operation of taking the thermo dynamic limit N , L → + ∞ and the one of compu ting the eﬀect of the translation op erators do actually comm ute. I ndeed, let us consider the expression (5.12) in whic h w e ha ve sent directly N , L → + ∞ , hence replacing sums by in tegrals in the prefactor and the ﬁnite-size determinan t represen tation for X N b y the F redholm determinan t (B.11) : e − β xp F π : e − q R − q ν τ ( λ ) [ ix u ′ ( λ )+ b g ′ ( λ )] d λ det  I + V [ ν τ , u, b g ]  · G [ ω ] :    τ =0 ω =0 , (6.1) where V [ ν, u, g ]( λ, µ ) = 4 sin[ π ν ( λ )] sin[ π ν ( µ )] 2 iπ ( λ − µ ) n E + ( λ ) E − ( µ ) − E + ( µ ) E − ( λ ) o (6.2) with E − ( λ ) = e − ix u ( λ ) 2 − g ( λ ) 2 , (6.3) E + ( λ ) = iE − ( λ )    Z R  d µ 2 π E − 2 − ( µ ) µ − λ + E − 2 − ( λ ) 2 cot[ π ν ( λ )]    . (6.4) 18 Then, if we decomp ose the F redholm determinan t in (6.1) in to its F redholm series and subse- quen tly compute the eﬀect of the function al translation op erators, w e obtain the same r epr esen- tation (D.24 ) as b y p erforming all these operations in the opp osite order. This means (pro vided that the series (D.24) is con v ergent, whic h is not completely ob vious but seems neverthel ess a reasonable assumption) that the thermo dynamic limit of the eﬀectiv e series (5.12), whic h is supp osed to giv e the thermo dynami c limit of the original form factor series (4.8), is eﬀectiv ely giv en by the expression (6.1), i.e. Q κ ( x, t ) = e − β xp F π : e − q R − q ν τ ( λ ) [ ix u ′ ( λ )+ b g ′ ( λ )] d λ det  I + V [ ν τ , u, b g ]  · G [ ω ] :    τ =0 ω =0 , (6.5) with a kernel V giv en b y (6.2)-(6.4). It also means that w e can no w use any other existing represen tation for the F redholm determinan t so as to compute the eﬀect of the translation op erator s and reco ver standard scalar-v alued f unc tions. I n f act, it is not v ery con venien t for our purp ose to expand the determinan t in to its F redholm series like in App endix D s ince the latter do es not pro vide an y information on its large- x asymptotic b eha v ior. This large- x asymptotic b eha vior w as studied in [60] precisely with the goal of computing the asy mpt otic b eha vior of (6.5), and a muc h more con v enien t (with resp ect to the x → + ∞ limit) represen tation for the F redholm determinan t w as obtained there. 6.2 Large- x asymptotic b eha vior of the F redholm determinan t The large x asymptotic analysis of the F redholm determinan t with kernel (6.2 )-(6. 4) wa s p er- formed in [60] using Riemann-Hilbert problem-based techn iques. There, it w as pro ven that, under some h yp othesis ab out the regularit y and b eha vior of t he functions ν , u and g deﬁned on some op en neigh b orho od of the real axis (see [60] for more precisions), and pro vided that the function u has a unique saddle-poin t λ 0 on R (with λ 0 6 = ± q ), det [ − q ,q ]  I + V  [ ν, u, g ] = exp ( q Z − q  ixu ′ ( λ ) + g ′ ( λ )  ν ( λ ) d λ ) × ( B x [ ν, u ] + X ǫ = ± 1 e iǫx [ u ( q ) − u ( − q )]+ ǫ [ g ( q ) − g ( − q )] B x [ ν + ǫ, u ] + 1 x 3 2 X ǫ = ± 1 e iαx [ u ( λ 0 ) − u ( ǫq )]+ α [ g ( λ 0 ) − g ( ǫq )] b ( ǫ,α ) 1 [ ν, u ] B x [ ν, u ] + R x [ ν, u, g ] ) , (6.6) with α = +1 in the space-like regime 8 ( λ 0 > q ) and α = − 1 in the time-lik e regime ( λ 0 ∈ ] − q , q [ ). In (6.6), the f unc tional B x reads B x [ ν, u ] = e i π 2 [ ν 2 ( q ) − ν 2 ( − q )] e B [ ν ] [2 q x ( u ′ ( q ) + i 0 + )] ν 2 ( q ) [2 q x u ′ ( − q )] ν 2 ( − q ) , (6.7) 8 The case λ 0 < − q was not considered in [60]. 19 with e B [ ν ] given b y (A.22), and b ( ǫ,α ) 1 [ ν, u ] = e − iα π 4 [2 q x u ′ ( ǫq ) + i 0 + ] 2 ǫαν ( ǫq ) p − 2 π u ′′ ( λ 0 ) u ′ ( ǫq ) ( − ǫ ) ν ( ǫq ) ( λ 0 − ǫq ) 2  λ 0 + q λ 0 − q − i 0 +  − 2 αν ( λ 0 ) × (e − 2 iπ ν ( λ 0 ) − 1) 1 − α (e − 2 iπ ν ( ǫq ) − 1) α Γ(1 − ǫαν ( ǫq )) Γ(1 + ǫαν ( ǫq )) e α e J [ ν ]( λ 0 ) − α e J [ ν ]( ǫq ) , (6.8) where e J is given b y (A.25). R x [ ν, u, g ] is a remainder which is uniformly of order O  log x x  in what concerns the non-oscilla ting correction s, of order O  B x [ ν + ǫ, u ] log x x  in what concerns the oscillat ing correc tions at e iǫx [ u ( q ) − u ( − q )] , and of order O  B x [ ν, u ] b ( ǫ,α ) 1 [ ν, u ] log x x 5 / 2  in what concerns the oscillating corrections at e iαx [ u ( λ 0 ) − u ( ǫq )] . It w as sho wn in [60] how to obtain a series represen tation for this remainder, the so-called Natte s eries, whic h p ossesses the prop ert y of b eing we ll ordered with resp ect to its large- x b eha v ior (in can b e sho wn that its n -th term is (uniformly) a O ( x − na ) , for some 0 < a < 1 that dep ends on ν , u and g ). Hence, this series is w ell adapted for the study of the asy mpt otic b eha vior of the determinan t (on the con trary to its F redholm series). The form of suc h a Natte series is recalled in App endix E. R emark 6.1 . Note that for |ℜ ( ν ) | < 1 / 2 the ﬁrst (non-oscillating) term B x [ ν, u ] in (6.6) is alw ays leading, at large x , with resp ect to the other ones. This leading term will pro duce the leading asymptotic b eha vior of the generating f unc tion Q κ ( x, t ) , as w e will see in the next subsection. Ho wev er, recall that w e ha v e to diﬀeren tiate twice with resp ect to x and with resp ect to β at β = 0 in order to obtain the correlation function h j ( x, t ) j (0 , 0) i . By suc h a pro cess, the ﬁrst oscillating corrections ma y b ecome leading with resp ect to non-oscillating terms. This is the reason wh y we also consider these corrections in (6.6). 6.3 Asymptotic b eha vior of the correlation function So as to obtain the asymptotic b eha vior of the correlation function, it remains to compute the eﬀect of the f unct ional translation op erators on the representati on (6.6) of (6.5). First, we observ e that the exp onen tial pre-factor of (6.5) is canceled b y the one in (6.6): exp ( − q Z − q ν τ ( µ ) b g ′ ( µ ) d µ ) exp ( q Z − q ν τ ( µ ) b g ′ ( µ ) d µ ) = 1 . (6.9) Suc h a simpliﬁcatio n is justiﬁed b y the fact that, when applying (6.9) on an y f unct ionals of ν τ or ω , w e observe an algebr ai c cancellation for eac h term of the : · : ordered series expansion in δ /δτ and δ /δω for these exp onen ts 9 . Therefore, Q κ ( x, t ) = e − β xp F π : ( B x [ ν τ , u ] + X ǫ = ± 1 e iǫx [ u ( q ) − u ( − q )]+ ǫ [ b g ( q ) − b g ( − q )] B x [ ν τ + ǫ, u ] + 1 x 3 2 X ǫ = ± 1 e iαx [ u ( λ 0 ) − u ( ǫq )]+ α [ b g ( λ 0 ) − b g ( ǫq )] b ( ǫ,α ) 1 [ ν τ , u ] B x [ ν τ , u ] + R x [ ν τ , u, b g ] ) · G [ ω ] :     τ =0 ω =0 . (6.10) 9 Such an algebraic cancellation is very similar to the one o ccuring when computing ( √ I − A ) 2 through a T a ylor series expansion around zero. 20 It can b e pro ved that, by computing the eﬀect of the functional translation op erato rs o c- curing in the remainder R x [ ν τ , u, b g ] , one obtains corrections that are of the same order ( i .e. O( log x x ) corrections to each of the terms already presen t in the asymptotics) as originally in (6.6). This is explicitly done in App endix E b y using the so-called Natte series represen tation of the F redholm determinan t derived in [60 ]. More precisely , it is show n in App endix E that the eﬀect of the translation op erators do not mix the orders in x among the diﬀeren t terms of this (w ell-ordered) series. Therefore, the le ading asymptotic b ehavior of the gener ati ng function fol lows dir e ctly fr om t he ab ove le ading asymptotic b ehavior of the F r e dholm determinant . Since no translation op erator is app lied on the ﬁrst (non- oscillating) term of (6.10), w e simply need to set τ = 0 and ω = 0 into the corresponding expressions. The action of the translation op erator s e ǫ [ b g ( q ) − b g ( − q )] on the second term of (6.10) results in to replacing the function G [ ω ] b y G   1  ·     ǫq − ǫq  ≡ G 1  ǫq − ǫq  , (6.11) and the function ν τ b y the shift function F  ·     ǫq − ǫq  = iβ 2 π Z −  φ ( · , ǫq ) − φ ( · , − ǫq )  =  iβ 2 π + ǫ  Z − ǫ (6.12) asso ciated to a state with one particle and one hole lo cated on the opp osite ends of the F ermi zone. Similarly , the action of the translation operators e α [ b g ( λ 0 ) − b g ( ǫq )] results in to replacing G [ ω ] and ν τ resp ectiv ely b y G   1  ·     λ 0 ǫq  ≡ G 1  λ 0 ǫq  and F  ·     λ 0 ǫq  = iβ 2 π Z −  φ ( · , λ 0 ) − φ ( · , ǫq )  (6.13) in the s pac e-lik e regime α = +1 , and b y G   1  ·     ǫq λ 0  ≡ G 1  ǫq λ 0  and F  ·     ǫq λ 0  = iβ 2 π Z −  φ ( · , ǫq ) − φ ( · , λ 0 )  (6.14) in the time-like regime α = − 1 . Therefore, we get Q κ ( x, t ) = e − β xp F π  B x  iβ 2 π Z, u  G 0  1 + O  log x x  + X ǫ = ± 1 e iǫx [ u ( q ) − u ( − q )] B x  iβ 2 π + ǫ  Z, u  G 1  ǫq − ǫq   1 + O  log x x  + 1 x 3 2 X ǫ = ± 1 e ix [ u ( λ 0 ) − u ( ǫq )] b ( ǫ, +1) 1  F  ·     λ 0 ǫq  , u  B x  F  ·     λ 0 ǫq  , u  G 1  λ 0 ǫq   1 + O  log x x   (6.15) in the space-lik e regime, and similar expressions in the time-lik e regime. Namely , in the time- lik e regime, the exp onen t in the last term c hanges sign and b ( ǫ, +1) 1 , F  · | λ 0 ǫq  , G 1  λ 0 ǫq  are replac ed, resp ectiv ely , by b ( ǫ, − 1) 1 , F  · | ǫq λ 0  and G 1  ǫq λ 0  . In order to obtain the leading asymptotic b eha vior of the correlation function of curren ts, it remains ﬁnally to compute the s econ d x -deriv ativ e and second β -deriv ative at β = 0 of the previous result, cf (4. 9). The deriv ativ es of the ﬁrst term (the non-oscillating one) in (6.15) pro duce the constan t and the non-oscillating term app earing in (3.1 ) and (3.2 ) (w e ha ve used (A.20)). In their turns, the deriv ativ es of the t wo t yp es of oscillating corrections pro duce the corresp ond ing oscillating terms in (3.1), (3.2 ), with amplitudes given by (A.28)-(A.30). 21 7 Conclusion In this article, w e hav e propos ed a new metho d to deriv e, starting from ﬁrst principles, the leading asymptotic b eha vior of the t w o-p oin t correlation f unct ions of q uantum in tegrable sys- tems. T o explain the main steps of this method, w e c hose to f o cus on the example of the curren t-curren t correlation function of the quan tum non-linear Sc hrö dinger mo del. The case of the ﬁeld/conjugated ﬁeld correlator h Ψ † ( x, t ) Ψ(0 , 0) i , together with a rigorous setting for carrying out all the manipulations with op erator v alued determinan ts, will b e given in [59]. Our result go es b ey ond the CFT/Luttinger liquid based predictions: the saddle-p oin t con- tributions that app ear in the asymptotic b eha vior (3.1)-(3.2) inv olve excitations aw ay from the F ermi surface and cannot b e neglected for x/t ﬁnite. Compared to the approac h of [45], based on the master equation represen tation for the correlation functions, the presen t study relies directly on their form factor expansion. W e w ould lik e to conclude this article by making here a f ew commen ts ab out the s imi larities and diﬀerences b et ween the t w o metho ds. Of course, since the master equation can b e understoo d as the result of a summation o v er the form f actor s, the s piri t of the t w o approac hes is essen tially the same. In particular, in b oth approac hes, the asymptotic ana lysis relies essen tially on the singular p art of the f orm factor which is explicitly extracted (in the form of a Cauc hy determinan t squared), whereas the (mo del-d ep enden t) regular part is treated as a dressing part whic h is f ormally decoupled (or link ed to decoupled f unc tions) to mak e the analysis p ossible. This enables one to dra w a link b et w een the quant it y to estimate and the F redholm determinan t of a generalized s ine kerne l. The asy mptotic analysis of this F redholm determinan t then leads to the asy mptotic b eha v ior of the correlatio n function. Ho wev er, there exist some essential diﬀerences b et ween the t w o approac hes. In [45 ], the correlation function w as expanded into a series whose building blo c ks w ere the so-called cycle inte gr als . These cycle inte grals could then b e related to a F redholm determinan t, whic h allo wed one to access to their asymptotic b eha vior. The physical interpr etation of these ob j ects w as ho w ev er not clear, and they happened to b e q uite indirectly related to the correlation functions. In fact, once the asymptotic b eha vior of these cycle in tegrals established, one had to sum up the s erie s s o as to obtain the asymptotic b eha vior of the correlation function. In this pro cess, the main problem w as that not only the le ading asymptotic b ehavior of i ndividual cycle i nte gr als was c ontributing to the le ading or der for the c orr elation function s : one had to p erform a ﬁne and non-trivial study [47, 45, 62] of the asymptotic s eries s o as to gather terms at al l or der that ﬁnally , when summed up, were con tributing to the leading order. In fact, all these terms w ere rearranging themselv es into some generalization of a m ultiple Lagrange s eries pro ducing, when s umm ed up, a dr essing of b ar e quantities (energy and momen tum) into dr esse d ones . On the con trary , here, one deals from the very b eginning with ob jects ha v ing a clear ph ysical in terpretation, the form factors. In this con text, the dr esse d quantities app e ar natur al ly when c onsidering the thermo dynamic lim it of these form factors . Hence, p erforming the summation o ver the form factors, w e can connect the series to a F redholm determ inan t that is already expressed in terms of these dressed quant ities. Therefore, the asymptotic study is m uc h simpler: it happ ens that the le ading asymptotic b ehavior of the F r e dholm determinant gives dir e ctly the le ading asymptotic b ehavior of the c orr elation function . In other word s, there is no need to resort to highly non-trivial s umm ation as in [45]. In fact, we wou ld lik e to stress that the whole pro cess describ ed in the core of this article is quite simple and direct. Once the form factor series written do wn and the eﬀectiv e con- 22 tributing part of the form factor established, the result of the summation can b e expressed, in the thermo dynamic limit, in terms of a F redholm determinan t. The leading asy mp totic b e- ha vior of the correlation function follo ws directly from the leading as ympto tic b eha vior of this F redhom determinan t. Ev en if, in the cou rse of th e co mputations, we use some functiona l trans- lation op erators to relate our series to a decoupled one, whenev er the action of these functional translation op erator s has to b e computed, it is quite straigh tforward , i. e. it pro duces a simple translation of the functional on whic h it acts. On the contrary , if one w an ts to recov er, starting from the form f acto r expansion, a s erie s s imil ar to the one studied in [45 ], the computations are m uc h more in v olved (see App endix D). In particular, the action of the translation op erators on the series of A ppendix D produces non-trivial eﬀects: one has to deal with summations of generalized Lagrange series, whic h results into an undressing of the dress ed q uantiti es into bare ones. It is therefore clear that most of the mathematical complexit y corresp onding to this non-trivial summation has already b een tak en into accoun t by the fact that w e had some pre- cise description (the particle-hole picture) of our form f actors, whic h allow ed us f rom the v ery b eginnin g to deal with dressed quan tities instead of bare ones. Note that the simpler setting of our metho d enables us here to consider the time-dependen t case, which is not s o ob vious within the approac h of [45]. Note also that, in principle, there is no in trinsic obstruction preve n ting us from obtaining higher order terms in the asymptotic expans ion for the correlation f unct ions. F or this it is enough to reﬁne the asymptotic expansion of the F redholm determinan t. Of course, th ere is a pr ice to pay for this simplicit y: the need of som e clear picture to describ e the form factors. Whereas in [45] al l kinds of Beth e roots wer e automatically tak en into acco un t within the master equation framew ork (and this without an y precise s tud y of the sp ectrum), here w e s trong ly rely on the fact that the sp ectrum of the mo del w e consider is particularly simple: all excited states can b e describ ed in terms of particles and holes. In order to apply our metho d to the case of the X X Z spin c hain, for example, one has also to take in to accoun t the contri bution of complex solutions. A lthough it seems that, f or the time-indep enden t case, these s olutions do not contr ibute to the leading asymptotic b eha vior of correlation functions (see [42]), the question remains open in the time-dependen t case. This will b e the sub j ect of a further study . A c kno wledgem en ts V. T. is supp orted b y CNRS and by the ANR gran t “DIADEMS”. K. K. K. is supp orted b y the EU Marie-Curie Excellence Gran t MEXT-CT-2006- 042695. K. K. K. w ould lik e to thank the Theoretical Ph ysics group of the Lab ora tory of Ph ys ics at ENS Lyon for hospitalit y , whic h mak es this collab oration p ossible. V.T. w ould lik e to thank LPTHE (P aris VI Univ ersity) for hospitalit y . W e w ould lik e to thank N. Kitanine, J. M. Maillet and N. A. Slavno v for stim ulating discussions and commen ts . V. T. wo uld also lik e to thank M. Civelli and S. T eb er for their interest in this work. A The form factors and their thermo dynamic limi t In this app endix, w e presen t the explicit expression and the leading thermo dynamic b eha vior of the com binations of ﬁnite-v olume form factors whic h appe ar in eac h term of the series (4.1), 23 i.e. , with the notations of Section 4, of e − it E ex h ψ g | j ( x, 0) | ψ ′ i h ψ ′ | j (0 , 0) | ψ g i || ψ g || 2 · || ψ ′ || 2 = 1 2 ∂ 2 x ∂ 2 β  e ix P κ ex − it E κ ex     h ψ ( { λ j } ) | ψ κ ( { µ ℓ j } ) i   ψ κ ( { µ ℓ j } )   · k ψ ( { λ j } ) k     2  β =0 . (A.1) These results b eing the complete analogues, in the case of the NLS mo del, of those deriv ed in [46, 43] for the XXZ c hain, w e skip the details of the computations. A.1 Thermo dynam ic limit of the space and time-dependen t phase factor The thermo dynamic limit of e ix P κ ex − it E κ ex generates dressed quan tities. This limit b eing com- pletely smo oth, it is easy to s ee that e ix P κ ex − it E κ ex = exp ( ix q Z − q u ′ 0 ( λ ) F ( λ ) d λ + ix n X a =1  u 0 ( µ p a ) − u 0 ( µ h a )  ) (1 + O(1 /L )) = exp ( − xβ 2 π q Z − q u ′ 0 ( λ ) Z ( λ ) d λ + ix n X a =1  u ( µ p a ) − u ( µ h a )  ) (1 + O(1 /L )) , in whic h w e ha ve used the explicit form (2.14) of the shift function F , and where u is the follo wing com bination of dressed quan tities u ( λ ) = p ( λ ) − tε ( λ ) /x . This ident iﬁcation w as p ossible due to u ( λ ) = u 0 ( λ ) − q Z − q u ′ 0 ( µ ) φ ( µ, λ ) d µ. (A.2) Note that, due to the f act that ε ′ 0 is an o dd function whereas the dressed charge Z is even, one has q Z − q d λ Z ( λ ) u ′ 0 ( λ ) = 2 p F . (A.3) A.2 Represen tation of the normalized scalar pro duct Let { λ j } b e the solution of the sy stem of Bethe equations (2.2) parametrizing the ground state of (2.1) and let { µ ℓ j } b e a set of Bethe ro ots of (2.7) parametrizing an excited state ab o ve the κ -t wisted ground s tate in the N -particle sector. Then the normalized mo dulus squared of the corresp ond ing ov erlap scalar product in ﬁnite volum e can b e represent ed as      h ψ ( { λ j } ) | ψ κ ( { µ ℓ j } ) i   ψ κ ( { µ ℓ j } )   · k ψ ( { λ j } ) k      2 = b D N ( { λ j } , { µ ℓ j } ) · c W N ( { λ j } , { µ ℓ j } ) · b A N ( { λ j } , { µ ℓ j } ) , (A.4) where b D N , c W N and b A N are giv en by b D N ( { λ j } , { µ ℓ j } ) = N Y j =1 sin 2 [ π b F ( λ j )] π 2 L 2 b ξ ′ ( λ j ) b ξ ′ κ ( µ ℓ j ) ·  det N 1 λ j − µ ℓ k  2 , (A.5) 24 c W N ( { λ j } , { µ ℓ j } ) = N Y j,k =1 ( λ j − µ ℓ k − ic ) ( µ ℓ j − λ k − ic ) ( λ j − λ k − ic ) ( µ ℓ j − µ ℓ k − ic ) , (A.6) b A N ( { λ j } , { µ ℓ j } ) = (1 − κ ) 2 (1 − e − 2 iπ b F ( θ 1 ) )(1 − e − 2 iπ b F ( θ 2 ) ) N Y a =1 ( θ 1 − λ a + ic )( θ 2 − λ a + ic ) ( θ 1 − µ ℓ a + ic )( θ 2 − µ ℓ a + ic ) × det Γ  I + 1 2 iπ b U ( λ ) θ 1 ( ω , ω ′ )  · det Γ  I + 1 2 iπ b U ( λ ) θ 2 ( ω , ω ′ )  det N h δ j k − K ( λ j − λ k ) 2 π L b ξ ′ ( λ k ) i · d et N h δ j k − K ( µ ℓ j − µ ℓ k ) 2 π L b ξ ′ κ ( µ ℓ k ) i . (A.7) W e recall that b ξ κ and b ξ denote resp ectiv ely the ex cited state coun ting function (2.8) and the ground state coun ting f unction at κ = 1 , whereas b F is the ﬁnite-size shift function (2.9). Here θ 1 and θ 2 are some arbitrary real parameters. The in tegral op erator I + 1 2 iπ b U ( λ ) θ acts on the closed con tour Γ surrounding the ground state roots { λ j } and no othe r singularit y of the ker nel. This k ernel reads b U ( λ ) θ ( ω , ω ′ ) = − N Y a =1 ( ω − µ ℓ a )( ω − λ a + ic ) ( ω − λ a )( ω − µ ℓ a + ic ) · K κ ( ω − ω ′ ) − K κ ( θ − ω ′ ) 1 − e − 2 iπ b F ( ω ) , (A.8) with K κ ( ω ) = 1 ω + ic − κ ω − ic . (A.9) R emark A.1 . Although eac h individua l term, in (A.7), do es dep end on the set of auxiliary parameters θ k , the o ve rall com bination b A N do es not. This w as pro ven in [45]. The f acto r b D N ( { λ j } , { µ ℓ j } ) (A.5) is the so-called discrete par t of the f orm factor. It con tains all the non-trivial "singular part" of the form factor (see [43 ]). On the con trary , the factor b G N ( { λ j } , { µ ℓ j } ) ≡ c W N ( { λ j } , { µ ℓ j } ) · b A N ( { λ j } , { µ ℓ j } ) (A.10) admits a s mooth thermodynamic limit. It can b e though t of as a dressing function. It is equal to 1 at the free-fermion p oin t. In the case of an excited state | ψ κ ( { µ ℓ j } ) i with a ﬁnite num b er of particles and holes as describ ed in Section 2, it is easy to compute the thermo dynamic limit of the dressing function b G N (see [43]): lim L,N →∞ b G N ( { λ j } , { µ ℓ j } ) = G n  { µ p a } { µ h a }  ≡ W n  { µ p a } { µ h a }  · A n  { µ p a } { µ h a }  , (A.11) with W n  { µ p a } { µ h a }  = n Y a,b =1 ( µ p a − µ h b − ic )( µ h a − µ p b − ic ) ( µ p a − µ p b − ic )( µ h a − µ h b − ic ) × e − 2 iπ n P a =1 P ǫ = ± {C [ F ]( µ p a + iǫc ) −C [ F ]( µ h a + iǫc ) } + C 0 [ F ] , (A.12) A n  { µ p a } { µ h a }  = (1 − κ ) 2 e − 2 iπ {C [ F ]( θ 1 + ic )+ C [ F ]( θ 2 + ic ) } (1 − e − 2 iπ F ( θ 1 ) )(1 − e − 2 iπ F ( θ 2 ) ) n Y j =1 ( θ 1 − µ h j + ic )( θ 2 − µ h j + ic ) ( θ 1 − µ p j + ic )( θ 2 − µ p j + ic ) × det  I + 1 2 iπ U θ 1  det  I + 1 2 iπ U θ 2  det 2  I − 1 2 π K  . (A.13) 25 In these expressions, F denotes the shift function F (2.11), C [ ν ] is the rational Cauc hy transform of the function ν on [ − q , q ] , C [ ν ]( λ ) = q Z − q d µ 2 iπ ν ( µ ) µ − λ , (A.14) whereas C 0 is the follow ing functional: C 0 [ ν ] = − q Z − q d λ d µ ν ( λ ) ν ( µ ) ( λ − µ − ic ) 2 . (A.15) The in tegral kernel U θ tak es the form U θ ( ω , ω ′ ) = − n Y a =1 ( ω − µ p a )( ω − µ h a + ic ) ( ω − µ h a )( ω − µ p a + ic ) e 2 iπ C [ F ]( ω ) { K κ ( ω − ω ′ ) − K κ ( θ − ω ′ ) } e 2 iπ C [ F ]( ω + ic ) (1 − e − 2 iπ F ( ω ) ) . (A.16) R emark A.2 . The dep endence on the particle/ hole rapidities of G n is t w ofold: explicitly in the ab o v e expressions, and also in the shift f unc tion F . The latter is a holomorphic function of { µ p a } and { µ h a } , cf (2.14). Therefore, G n is itself a holomorphic f unction of { µ p a } and { µ h a } . This function can, in fact, b e understo od as a f unc tional G [  n ] of the function  n (4.14) (note that the dependence in n is exclusiv ely con tained in  n ). F or this it is enough to observe that F ( λ ) = iβ 2 π Z ( λ ) − Z Γ( R ) d µ 2 iπ φ ( λ, µ )  n  µ     { µ p a } { µ h a }  , (A.17) where the con tour Γ( R ) surrounds the real axis count erclo c k wise, and that n Y a =1 e f ( µ p a ) − f ( µ h a ) = exp ( Z Γ( R ) d λ 2 iπ f ( λ )  n  λ     { µ p a } { µ h a }  ) (A.18) for an y holomorphic functions f with a suﬃcien tly mild gro wth at inﬁnit y on Γ( R ) . In particular, w e hav e n Y a,b =1 ( µ p a − µ h b − ic )( µ h a − µ p b − ic ) ( µ p a − µ p b − ic )( µ h a − µ h b − ic ) = exp ( − Z Γ( R ) d λ d µ (2 iπ ) 2 log( λ − µ − ic )  n  λ     { µ p a } { µ h a }   n  µ     { µ p a } { µ h a }  ) . (A.19) R emark A.3 . The explicit factor (1 − κ ) 2 in (A.13) stresses that, for generic conﬁgurations of the param eters { µ p a } and { µ h a } , G n , considered as a function of β , has in ge neral a zero of order 2 at β = 0 . Note how ever that there exist some exceptions, in particular at the f ree fermion p oin t c = + ∞ , when G is identi cally 1 or, for general c , when the excited state coincides with the ground state in the limit β → 0 ( i.e. for n = 0 ). In the latter case, the limit can b e tak en as in [45] and we obtain lim β → 0 G 0 = 1 . (A.20) 26 The study of the thermo dynamic b eha v ior of the factor b D N (A.5) is slightly more tec hnical. Pro ceeding as in [43 ], one obtains that, for a n particle/hole ex cited state, b D N ( { λ j } , { µ ℓ j } ) =  L 2 π  − F 2 ( q ) − F 2 ( − q ) − 2 n [2 q p ′ ( q )] − F 2 ( q ) − F 2 ( − q ) e B [ F ] × Γ 2  { p j } , { p j − N + F ( µ p j ) } , { h j + F ( µ h j ) } , { N + 1 − h j − F ( µ h j ) } { p j − N } , { p j + F ( µ p j ) } , { h j } , { N + 1 − h j }  × det 2 n  1 µ p j − µ h k  n Y j =1 sin 2 [ π F ( µ h j )] e J [ F ]( µ p j ) − J [ F ]( µ h j ) π 2 p ′ ( µ p j ) p ′ ( µ h j )  1 + O  log L L  . (A.21) The expression (A.21 ) is giv en in terms of the b elo w f unctional of the shift function (2.14): e B [ ν ] = e C 1 [ ν ] G 2 (1 + ν ( q )) G 2 (1 − ν ( − q )) (2 π ) ν ( q ) − ν ( − q ) (A.22) where G is the Barnes G function and C 1 [ ν ] = 1 2 q Z − q d λ d µ ν ′ ( λ ) ν ( µ ) − ν ′ ( µ ) ν ( λ ) λ − µ + ν ( q ) q Z − q ν ( q ) − ν ( λ ) q − λ d λ + ν ( − q ) q Z − q ν ( − q ) − ν ( λ ) q + λ d λ. (A.23) W e ha v e also deﬁned J [ F ]( λ ) = 2 F ( λ ) log  λ − q λ + q p ( λ ) − p ( − q ) p ( λ ) − p ( q )  + e J [ F ]( λ ) , (A.24) with e J [ ν ]( λ ) = 2 q Z − q ν ( µ ) − ν ( λ ) µ − λ d µ. (A.25) Finally , Γ  { a k } { b k }  = n Y k =1 Γ( a k ) Γ( b k ) , (A.26) with the prescription that, should some of the particles in ( A .21) ha v e their rapidities to the left of the F ermi zone, the argumen ts of the Γ -functions ha ve to b e understo od as limits Γ( p k ) Γ( p k − N ) = lim ǫ → 0 Γ( p k + ǫ ) Γ( p k − N + ǫ ) . (A.27) Note that in (A.21) the thermo dynamic limit has only b een taken partly . Indeed, s ince the complete thermo dynami c b eha vior of (A.21) dep ends on whether the corresp ondin g particles and holes remain or not at a ﬁnite distance from the F ermi b oundaries. I n the next subsection, w e particularize these expressions to the s pecial form factors with one particle and one hole whic h app ear in the results (3.1 ), (3.2). 27 A.3 Explicit v alue of the amplitudes W e collect here the explicit v alues of the non-unive rsal amplitudes app earing in (3.1)-(3.2), whic h corresp ond to the normalized one particle/hole form f actors (3.5)-(3.9). They can b e easily obtained from the expressions (A.12)-(A.21) and read   F q − q   2 =   F − q q   2 = − 2 p 2 F π 2 e B  F q − q  e e J [ F q − q ]( q ) − e J [ F q − q ]( − q ) [2 q p ′ ( q )] [ F q − q ( q ) +1] 2 +[ F q − q ( − q )+1 ] 2 Γ 2  1 + F q − q ( q )  Γ 2  1 + F q − q ( − q )  × ∂ 2 β sin 2 h i β 2 Z + π F q − q ( − q ) i G 1  q − q      β =0 = − 2 p 2 F π 2 e B  F − q q  e e J [ F − q q ]( − q ) − e J [ F − q q ]( q ) [2 q p ′ ( q )] [ F q − q ( q ) − 1] 2 +[ F q − q ( − q ) − 1 ] 2 Γ 2  1 − F − q q ( q )  Γ 2  1 − F − q q ( − q )  × ∂ 2 β sin 2 h i β 2 Z + π F − q q ( q ) i G 1  − q q      β =0 , (A.28) in terms of the shift functions F ± q ∓ q ( λ ) = φ ( λ, ∓ q ) − φ ( λ, ± q ) = ± Z ( λ ) ∓ 1;   F λ 0 ǫq   2 = −  p ( λ 0 ) − ǫp F λ 0 − ǫq  2  λ 0 − q λ 0 + q  2 F λ 0 ǫq ( λ 0 ) [2 q p ′ ( q )] − [ F λ 0 ǫq ( q ) ] 2 − [ F λ 0 ǫq ( − q )] 2 +2 ǫF λ 0 ǫq ( ǫq ) 2 π 2 p ′ ( λ 0 ) p ′ ( q ) × e B  F λ 0 ǫq  e e J [ F λ 0 ǫq ]( λ 0 ) − e J [ F λ 0 ǫq ]( ǫq ) Γ 2  1 − ǫF λ 0 ǫq ( ǫq )  × ∂ 2 β sin 2 h i β 2 Z + πF λ 0 ǫq ( ǫq ) i G 1  λ 0 ǫq      β =0 , (A.29) in terms of the shift functions F λ 0 ǫq = φ ( λ, ǫq ) − φ ( λ, λ 0 ) , with ǫ = ± 1 ;   F ǫq λ 0   2 = −  p ( λ 0 ) − ǫp F λ 0 − ǫq  2  q − λ 0 q + λ 0  − 2 F ǫq λ 0 ( λ 0 ) [2 q p ′ ( q )] − [ F ǫq λ 0 ( q ) ] 2 − [ F ǫq λ 0 ( − q )] 2 − 2 ǫF ǫq λ 0 ( ǫq ) 2 π 2 p ′ ( λ 0 ) p ′ ( q ) × e B  F ǫq λ 0  e e J [ F ǫq λ 0 ]( ǫq ) − e J [ F ǫq λ 0 ]( λ 0 ) Γ 2  1 + ǫF ǫq λ 0 ( ǫq )  × ∂ 2 β sin 2 h i β 2 Z ( λ 0 ) + π F ǫq λ 0 ( λ 0 ) i G 1  ǫq λ 0      β =0 , (A.30) in terms of the shift f unctions F ǫq λ 0 = φ ( λ, λ 0 ) − φ ( λ, ǫq ) , with ǫ = ± 1 . W e recall that the functionals e B and e J are resp ectiv ely giv en b y (A.22) and (A.25), and that the expressions of the dressing f unctions G 1 are obtained through (A.12)-(A.13). R emark A.4 . F or generic parameters, the β -deriv ativ es will apply directly to the factor (1 − κ ) 2 of G 1 (see remark A.3). Ho wev er, in the free fermi on p oin t c = + ∞ , they will apply o n the sin us squared, s ince in that case the corresp onding shift functions v anish (and G 1 ≡ 1 ). There may exist other particular v alues of the parameters for whic h the shift function b ecomes an in teger but the o verall expressions are anyw a y smo oth (w e ma y then ha ve to apply the β -deriv ativ es to other factors suc h as the Barnes functions). 28 B Study of a generali zed free fermionic generating function In this app endix, w e consider the case of a generalized free-fermionic mo del for whic h the eﬀectiv e generating function (5.3) can b e represen ted as a ﬁnite-size determinan t. Namely , w e supp ose here that the shift f unction do es not dep end on the p osition of the ro ots parametrizin g the corresp onding excited state and that the dressing function G n is separated (whic h happ ens in particular at the f ree fermion p oin t c = + ∞ of the NLS mo del) . More precisely , for a giv en coun ting function b ξ (for example the ground state coun ting function of the NLS mo del), whic h deﬁnes a set of real parameters λ j , j = 1 , . . . , N b y b ξ ( λ j ) = j /L , w e consider the functional X N  ν, E 2 −  = X ℓ 1 < ··· <ℓ N ℓ j ∈B L N Y j =1 E 2 − ( λ j ) E 2 − ( µ ℓ j ) · N Y j =1 sin 2 [ π ν ( λ j )] π 2 L 2 b ξ ′ ( λ j ) b ξ ′ ν ( µ ℓ j ) ·  det N 1 λ j − µ ℓ k  2 , (B.1) where E − 1 − and ν are holomorphic functio ns on some op en neigh b orho od of R . The ab o v e expression should b e understo od as follo ws: • the function b ξ ν is deﬁned b y b ξ ν = b ξ − L − 1 ν ; • the multi ple sum runs through all the p ossible cho ices of N -tuples of i n tegers ℓ 1 < · · · < ℓ N b elongin g to the set B L = { j ∈ Z | − w L < j < w L } , where w L ∼ L 1+ ǫ , ǫ > 0 , is some cut-oﬀ whic h go es to + ∞ with L . • for a given set of intege rs ℓ j , the parameters µ ℓ j are obtained as the pre-image of ℓ j /L b y the function b ξ ν . This means, in particular, that they dep end on ν . Moreo ver, we restrict our study those functions ν that make b ξ ν a "go od" coun ting f unc tion ( i.e. ensuring a one-to-one correspondence b et w een the rapidities and the in tegers). Prop osition B.1. The functional X N  ν, E 2 −  c an b e r e c ast as the fol lowing ﬁ nite size determi- nant: X N  ν, E 2 −  = det N  δ j k + V ( L ) ( λ j , λ k ) L b ξ ′ ( λ k )  . (B.2) The c orr esp on ding ﬁ nite-size kernel is given as V ( L ) ( λ, µ ) = 4 sin[ π ν ( λ )] sin[ π ν ( µ )] 2 iπ ( λ − µ ) n E ( L ) + ( λ ) E − ( µ ) − E ( L ) + ( µ ) E − ( λ ) o , (B.3) wher e E ( L ) + ( λ ) = iE − ( λ )      B L Z A L  d µ 2 π E − 2 − ( µ ) µ − λ + E − 2 − ( λ ) 2 cot[ π ν ( λ )] + I L  ν, E − 2 −  ( λ )      , (B.4) In thi s expr ession, the last term I L  ν, E − 2 −  ( λ ) c orr esp onds to the inte gr al I L  ν, E − 2 −  ( λ ) = Z C ↑ d z 2 π E − 2 − ( z ) z − λ 1 1 − e − 2 iπ L b ξ ν ( z ) + Z C ↓ d z 2 π E − 2 − ( z ) z − λ 1 e 2 iπ L b ξ ν ( z ) − 1 . (B.5) 29 Final ly, the in te gr ation endp oints A L and B L ar e such that L b ξ ν ( A L ) = − w L − 1 / 2 and L b ξ ν ( B L ) = w L + 1 / 2 , and C ↑ / ↓ ar e some oriente d c ontours, include d i n the j oint domain of holomorphy of ν and E − 1 − . Theses c ontours join the p oints A L and B L thr ough the upp er/lower half plane r esp e ctively, it has b e en depicte d on Fi g. 1. b b A L B L C ↑ C ↓ Figure 1: Con tour C ↑ / ↓ . Pr o of — The summand in (B.1) is a symmetric function of the N summation v ariables µ ℓ j whic h v anishes on the diagonals ℓ j = ℓ k , j 6 = k . Therefore, we can replace the summation o ver the fundamen tal simplex ℓ 1 < · · · < ℓ N in the N th p o w er cartesian pro duct B N L b y a summation o ver the whole space B N L , provided that we divide the result b y N ! . The s umm ation domain b eing no w symmetric, w e can in v oke the an tisymmetry of the determinan ts so as to replace one of the Cauc hy determinan ts b y N ! times the pro duc t of its diagonal en tries. This last op eratio n pro duces a separation of v ariables, whic h enables us to recast the result into a single determinan t by int ro ducing the sum o ver ℓ j in to the j th line of the determin an t: X N  ν, E 2 −  = N Y j =1 4 sin 2 [ π ν ( λ j )] b ξ ′ ( λ j ) · det N M , (B.6) with M j k = δ j,k X ℓ ∈B L E 2 − ( λ j ) · E − 2 − ( µ ℓ ) 4 π 2 L 2 b ξ ′ ν ( µ ℓ ) ( µ ℓ − λ j ) 2 + (1 − δ j,k ) X ℓ ∈B L E 2 − ( λ j ) · E − 2 − ( µ ℓ ) 4 π 2 L 2 b ξ ′ ν ( µ ℓ ) ( λ j − λ k )  1 µ ℓ − λ j − 1 µ ℓ − λ k  . (B.7) The ab o ve discrete sums can b e expressed in terms of Hilb ert transforms (plus corrections that v anish in the L → + ∞ limit). Indeed, X ℓ ∈B L E − 2 − ( µ ℓ ) 2 π L b ξ ′ ν ( µ ℓ )( µ ℓ − λ ) = − iE − 2 − ( λ ) e 2 iπ L b ξ ν ( λ ) − 1 + Z C ↑ ∪ C ↓ E − 2 − ( z ) d z 2 π ( z − λ )(e 2 iπ L b ξ ν ( z ) − 1) = − iE − 2 − ( λ ) e 2 iπ L b ξ ν ( λ ) − 1 − Z C ↑ E − 2 − ( z ) d z 2 π ( z − λ ) + I L  ν, E − 2 −  ( λ ) = B L Z A L  E − 2 − ( z ) d z 2 π ( z − λ ) − E − 2 − ( λ ) 2 cot  π L b ξ ν ( λ )  + I L  ν, E − 2 −  ( λ ) . (B.8) 30 Diﬀeren tiating the ab o ve expression with resp ect to λ leads to X ℓ ∈B L E − 2 − ( µ ℓ ) 2 π L b ξ ′ ν ( µ ℓ )( µ ℓ − λ ) 2 = ∂ ∂ λ B L Z A L  E − 2 − ( z ) d z 2 π ( z − λ ) − ∂ λ E − 2 − ( λ ) 2 cot  π L b ξ ν ( λ )  + E − 2 − ( λ ) π L b ξ ′ ν ( λ ) 2 sin 2 [ π L b ξ ν ( λ )] + ∂ λ I L  ν, E − 2 −  ( λ ) . (B.9) By applying (B.8), (B.9) in (B.7 ), and using the fact that L b ξ ν ( λ j ) = L b ξ ( λ j ) − ν ( λ j ) = j − ν ( λ j ) , w e obtain M j k = δ j,k b ξ ′ ( λ j ) 4 sin 2 [ π ν ( λ j )] + E − ( λ j ) E − ( λ k ) E ( L ) + ( λ j ) E − ( λ k ) − E ( L ) + ( λ k ) E − ( λ j ) 2 iπ L ( λ j − λ k ) = E − ( λ j ) E − ( λ k ) · b ξ ′ ( λ k ) 4 sin[ π ν ( λ j )] sin[ π ν ( λ k )] ( δ j,k + V ( L ) ( λ j , λ k ) L b ξ ′ ( λ k ) ) , (B.10) in whic h E ( L ) + is given by (B.4). Note that w e recov ered the deriv ative of the λ -t yp e count ing function b ξ thanks to the iden tity b ξ ν = b ξ − L − 1 ν . This ends the proof of the prop osition. Let us no w supp ose that, in the thermodynamic limit L, N → + ∞ with N/L → D ( D ﬁnite), the ro ots λ j condensate on some sy mmetric in terv al [ − q , q ] of the real axis with the densit y ρ ( λ ) = lim L,N →∞ b ξ ′ ( λ ) of the NLS mo del. Let us moreo ver supp ose that, in this limit, the coun ting function b ξ ν tak es the form (2.17) in terms of the dressed momen tum p ( λ ) of the NLS mo del. W e demand in addition that log E − 2 − has at most a p olynomial gro wth in λ when ℜ ( λ ) → ±∞ . Then the thermody namic limit of X N  ν, E 2 −  is we ll deﬁned and giv en by the follo wing F redholm determinan t: lim M ,N →∞ X N  ν, E 2 −  = det [ − q ,q ] [ I + V ] (B.11) with k ernel V ( λ, µ ) = 4 sin[ π ν ( λ )] sin[ π ν ( µ )] 2 iπ ( λ − µ )  E + ( λ ) E − ( µ ) − E + ( µ ) E − ( λ )  . (B.12) Here, the f unc tion E + ( λ ) reads E + ( λ ) = iE − ( λ )    Z R  d µ 2 π E − 2 − ( µ ) µ − λ + E − 2 − ( λ ) 2 cot[ π ν ( λ )]    . (B.13) Indeed, it follo ws f rom the form (2.17) of the coun ting function b ξ ν that A L and B L tend resp ectiv ely to −∞ and + ∞ in the thermodynamic limit. Moreo ver, it can easily b e sho wn that I L [ ν, E 2 − ] = O ( L − 1 ) . The thermo dynamic limit of X N [ ν, E 2 − ] is therefore a direct consequence of the fact that det N [ δ k ℓ + o( L − 1 )] → 1 , whenev er the o( L − 1 ) s ym b ol is uniform in the en tries. C F unctional T ranslation op erator In this app endix, w e p ro vide a heuristic approac h to the notion of functional translation op erator that w e use in Section 5.2 to express o ur highly coupled form f actor series in terms of a decoupled 31 one. As we will s ee, such an ob ject is in fact a conv enien t to ol to manipulate generalized m ulti- dimensional Lagrange series (see App endix C of [45]). W e ref er to [59] for a more explicit and rigorous construction . The notion of one-dimension al translation op erator, whic h acts on some holomorphic func- tion g as e α∂ ω · g ( ω )   ω =0 = X n ≥ 0 α n n ! g ( n ) (0) = g ( α ) , (C.1) can easily b e extended to the multi dimensional case: s Y p =1 e α p ∂ ω p · g s ( ω 1 , . . . , ω s )   ω p =0 = g s ( α 1 , . . . , α s ) . (C.2) By analogy , w e can th us deﬁne a translation op erato r acting on functionals as T γ · U [ τ ]   τ =0 = exp    Z C d ω γ ( ω ) δ δ τ ( ω )    U [ τ ]    τ =0 = U [ γ ] . (C.3) Here, the in tegral is taken on a con tour C (t ypically , an inte rv al of the real axis ), and U is a functional acting on holomorphic f unc tions deﬁned on a neighborho o d U of C . F orm ula (C.3) can b e understo o d as the result of the action of ﬁnite dimensional transla- tion op erators on a ﬁnite dimensional appro ximation of U . More precisely , let us consider a discretization t 1 , . . . , t s of C and a collection of holomorphic functions U s ( { z i } s i =1 ) suc h that, for an y holomorphic function γ , U s ( { γ ( t i ) } s i =1 ) − → s → + ∞ U [ γ ] . Then, w e can represen t the translation op erator in terms of limits of the ﬁnite v ariable case: T γ · U [ τ ]   τ =0 = lim s → + ∞ s Y p =1 exp  γ ( t p ) ∂ ∂ τ ( t p )  U s  { τ ( t k ) } s 1     τ =0 . (C.4) Suc h discretizations allo w us to compute the action of more complicated translation opera- tors. Let us deﬁne, as in Section 5.2, the op erator ordered version : O : of s ome expression O con taining functional deriv ativ es of the t yp e δ /δ τ ( λ ) (suc h as in (C.3 )) as b eing the express ion where all function al deriv ativ e op erators are pla ced on the left (in eac h term of the T a ylor series expansion of O ). Then, let us consider, for some functionals Γ and U , the follo wing generalizatio n of (C.3): L = : exp ( Z C Γ[ τ ]( µ ) δ δ τ ( µ ) d µ ) U [ τ ] :    τ =0 (C.5) ≡ + ∞ X n =0 1 n ! Z C d n µ n Y i =1 δ δ τ ( µ i ) · ( n Y p =1 Γ[ τ ]( µ p ) U [ τ ] )      τ =0 . (C.6) After discretization one gets L = lim s → + ∞ L s (C.7) 32 with L s = : s Y p =1 exp  Γ s ( { τ k } s 1 )( τ p ) ∂ τ p  U s ( { τ k } s 1 )    τ k =0 : (C.8) ≡ + ∞ X n 1 ,...,n s =0 s Y ℓ =1 ∂ n ℓ τ ℓ n ℓ ! · ( s Y p =1 Γ n p s ( { τ k } )( τ p ) U s ( { τ k } ) )      τ k =0 , (C.9) in whic h we hav e set τ k ≡ τ ( t k ) . The last series is a m ulti-dimension al Lagrange series whi c h can b e computed as L s = U s ( { ν ( t k ) } ) det s  δ j k − ∂ τ k Γ s ( { τ ℓ } )( t j )  τ k = ν ( t k ) , (C.10) where ν ( t k ) , k = 1 , . . . , s , are obtained as the solutions of the system ν ( t k ) = Γ s ( { ν ( t ℓ ) } )( t k ) , k = 1 , . . . , s . The s → + ∞ limit can b e tak en, at least f orm ally . It yields L = U [ ν ] det C h I − δ δ τ ( µ ) Γ[ τ ]( λ ) i τ = ν , (C.11) in whic h ν is the solution to the equation ν ( µ ) = Γ[ ν ]( µ ) . (C.12) D The Master equation issued-li k e series represen tation In this app endi x w e obtain, starting from (5.12)-(5.14), an alternativ e series represen tation for Q κ N ( x, t ) eﬀ . This represen tation is in the spirit of the series obtained in [45 ] for the equal-time correlation functions. D.1 The ﬁnite-size series Expanding the determinan t in (5.12) in to its ﬁnite F redholm series (this ex pansion is an imme- diate consequence of the Laplace expansion for determinan ts), w e obtain: Q κ N ( x, t ) eﬀ = e − β p F π x : N Y j =1 b E 2 − ( ¯ µ j ) b E 2 − ( λ j ) N X n =0 1 n ! N X i 1 ,...,i n =1 n Y s =1 1 L b ξ ′ ( λ i s ) det n  b V ( L ) ( λ i j , λ i k )  G [ ω ] :    τ =0 ω =0 , (D.1) in whic h ˆ V ( L ) ( λ, µ ) is giv en b y (5.1 3). It is con v enient to use the analytic prop erties of the op erator v alued mappings b E ( L ) + and b E − so as to re-cast this ﬁnite-size ke rnel V ( L ) in to a more compact con tour integ ral represen tation: b V ( L ) ( λ, µ ) = 4 sin[ π ν τ ( λ )] sin [ πν τ ( µ )] b E − ( λ ) b E − ( µ ) I C q d z (2 iπ ) 2 b E ( L ) + ( z ) b E − 1 − ( z ) ( z − λ )( z − µ ) , (D.2) 33 where C q surrounds [ − q , q ] (see Fig. 2), λ, µ ∈ [ − q ; q ] and b E ( L ) + ( z ) b E − 1 − ( z ) = i Z C ( L ) d y 2 π f ( L ) ( y , ν τ ( y )) y − z b E − 2 − ( y ) . (D.3) In this last expression the con tour C ( L ) = C ( L ) E ∪ e C q ∪ C ↑ ∪ C ↓ consists of a union of four contou rs. T w o of them, C ↑ / ↓ , are as depicted in Fig. 1, and the re maining t wo, C ( L ) E ∪ e C q , app ear in Fig. 2 . In particular, the con tour e C q encircles the con tour C q of (D.2), whic h enables us to recast the in tegral k ernel b V ( L ) ( λ, µ ) in a more compact form. The in tegrand in (D.3) inv olv es the function f ( L ) ( y , ν τ ( y )) = 1 C ( L ) E ( y ) − 1 e C q ( y ) e − 2 iπ ν τ ( y ) − 1 + 1 C ↓ ( y ) e 2 iπ L b ξ ν τ ( y ) − 1 + 1 C ↑ ( y ) 1 − e − 2 iπ L b ξ ν τ ( y ) . (D.4) b b b b − q q b A L b B L e C q C q C ( L ) E Figure 2: Con tours C ( L ) E , C q and e C q . The endp oin ts b A L and b B L are suc h that L b ξ ν τ ( b A L ) = − w L − 1 / 2 and L b ξ ν τ ( b B L ) = w L + 1 / 2 . F actorizing the in tegrals o v er z (D.2) out of the determinan t in (D.1) and then using the symmetry of the summand in order to reconstruct a s econ d Cauc hy determinan t, we get Q κ N ( x, t ) eﬀ = e − β p F π x : N Y ℓ =1 b E 2 − ( ¯ µ ℓ ) b E 2 − ( λ ℓ ) N X n =0 1 ( n !) 2 N X i 1 ,...,i n =1 n Y s =1 ( 4 sin 2 [ π ν τ ( λ i s )] L b ξ ′ ( λ i s ) b E 2 − ( λ i s ) ) × I C q d n z (2 iπ ) 2 n Z C ( L ) d n y ( − 2 iπ ) n n Y s =1  f ( L ) ( y s , ν τ ( y s )) y s − z s b E − 2 − ( y s )  · det 2 n  1 z j − λ i k  G [ ω ] :     τ =0 ω =0 . (D.5) Since b E 2 − = e − ixu − b g τ − b g ω , (D.6) it remains to compute the eﬀect of the t w o t yp es of functional translation op erators o ccuring in (D.5). W e start b y taking the ω -translation in to accoun t, namely n Y a =1 e b g ω ( y a ) e b g ω ( λ i a ) N Y a =1 e b g ω ( λ a ) e b g ω ( ¯ µ a ) · G [ ω ]    ω =0 = G   N + n  ·     { λ j } N 1 ∪ { y s } n s =1 { ¯ µ j } N 1 ∪ { λ i s } n s =1   = G N  { λ j } N 1 \ { λ i s } n s =1 ∪ { y s } n s =1 { ¯ µ j } N 1  . (D.7) Ab o ve , we ha v e ﬁrst computed the action of the translation op erators according to (5.4), then used the equalit y (4.13). W e should no w compute the action of the second set of translation 34 op erator s in v olving b g τ . Since the parameters ¯ µ ℓ ≡ ¯ µ ℓ [ ν τ ] = b ξ − 1 ν τ ( ℓ/L ) , ℓ = 1 , . . . , N , at which the b g τ are ev aluated, are themselv es functionals of τ , w e should tak e care of the op erat or ordering. The expression we hav e to compute is of the t yp e: : N X n =0 1 ( n !) 2 N X i 1 ,...,i n =1 I C q d n z (2 iπ ) 2 n Z C ( L ) d n y ( − 2 iπ ) n exp  Z R Γ { i a } [ τ ]( ω ) δ δ τ ( ω ) d ω  · R ( n ) { i a }  { y s } n 1 { z s } n 1  [ ν τ ] :     τ =0 , (D.8) in whic h R ( n ) { i a } is a smo oth f unc tional of ν τ that, moreo ver, dep ends on the in tegration v ariables { y s } and { z s } , and the functional Γ { i a } [ τ ]( λ ) driving the functional translation reads Γ { i a } [ τ ]( λ ) = N X ℓ =1 [ φ ( λ, ¯ µ ℓ [ ν τ ]) − φ ( λ, λ ℓ )] + n X s =1 [ φ ( λ, λ i s ) − φ ( λ, y s )] . (D.9) A ccording to App endix C (see (C.5)), computing the action of a functional translation op erator with w eigths Γ { i a } [ τ ]( λ ) as ab o ve amoun ts to summing up a mult i-dimensional Lagrange s eries. In our case, the result is obtained b y replacing the functional arg umen t ν τ of R ( n ) { i a } b y the function ν b r , in whic h b r is the solution to the non-linear functional equation b r ( t ) = Γ { i a } [ b r ]( t ) , a nd then b y dividing the obtained expression by the corresp onding functional Jacobian J { i a } [ b r ] = det R  I − δ Γ { i a } [ τ ]( λ ) /δ τ ( λ )    τ = b r (see (C.11)). Finally , Q κ N ( x, t ) eﬀ admits the followi ng represen tation: Q κ N ( x, t ) eﬀ = e − β p F π x N X n =0 1 ( n !) 2 N X i 1 ,...,i n =1 I C q d n z (2 iπ ) 2 n Z C ( L ) d n y ( − 2 iπ ) n e ix U ( L ) { i a } ( { λ a } , { ¯ µ a } , { y s } ) J { i a } [ b r ] × n Y s =1 ( 4 sin 2 [ π ν b r ( λ i s )] L b ξ ′ ( λ i s ) ( y s − z s ) f ( L ) ( y s , ν b r ( y s )) ) × det 2 n  1 z j − λ i k  G N  { λ j } N 1 \ { λ i s } n s =1 ∪ { y s } n s =1 { ¯ µ j [ ν b r ] } N 1  . (D.10) In (D.10), we hav e set U ( L ) { i a } ( { λ a } , { ¯ µ a [ ν b r ] } , { y s } ) = N X ℓ =1 [ u ( λ ℓ ) − u ( ¯ µ ℓ [ ν b r ])] + n X s =1 [ u ( y s ) − u ( λ i s )] (D.11) and explicitly insisted on the fact that ¯ µ ℓ ≡ ¯ µ ℓ [ ν b r ] , ℓ = 1 , . . . , N , are now functionals of the solution b r to b r ( t ) = Γ { i a } [ b r ]( t ) . D.2 T aking the thermo dynamic limit F or ﬁnite N , (D.10) gives a rather implicit represen tation. W e are how e v er in terested in com- puting the thermo dynamic limit L, N → + ∞ of this expression (in fact it is only in this limit that the eﬀectiv e series (D.10) is supposed to coincide with the ori ginal form factor series (4.8)). In this limit of in terest (D.10) can b e considerably simpliﬁed. 35 In particular, the non-linear functional equation b r ( t ) = Γ { i a } [ b r ]( t ) for b r turns in to a linear in tegral equation for r ( t ) , with b r ( t ) − → N ,L →∞ r ( t ) : r ( t ) = ν r ( t ) − i β Z ( t ) 2 π = q Z − q ∂ 2 φ ( t, λ ) ν r ( λ ) d λ + n X s =1 [ φ ( t, λ i s ) − φ ( t, y s )] . (D.12) Indeed, in the thermo dynamic limit, the Bethe ro ots λ j for the ground s tate condensate on [ − q , q ] with the density ρ ( λ ) , whereas the parameters ¯ µ j [ ν b r ] (deﬁned in terms of the coun ting function b ξ ν b r ), are s hifted with resp ect to the λ j ’s according to: ¯ µ j [ ν b r ] − λ j = ν r ( λ j ) Lρ ( λ j ) + O( L − 2 ) , j = 1 , . . . , N . (D.13) Ab o ve , we ha v e denoted the thermo dynamic limit of the function ν b r b y ν r . The linear in tegral equation (D.12) is readily solv ed as so on as one observ es that the deriv ative ∂ 2 φ of the dressed phase (2.12) with resp ect to its second v ariable is actually related to the resolven t R of the Lieb k ernel ( ∂ 2 φ ( t, λ ) = − R ( t, λ ) , with ( I − K/ 2 π )( I + R ) = I ): ν r ( t ) = i β 2 π + 1 2 π n X s =1 [ θ ( t − λ i s ) − θ ( t − y s )] . (D.14) It also follows f rom (D.12) that the Jacobian of the transformation (see (C.11)) is simply giv en b y J = d et [ − q ,q ]  I − ∂ 2 φ  = det − 1 [ − q ,q ] h I − K 2 π i . (D.15) The expression (D.14) for the thermo dynami c limit of ν b r ( t ) along with the es timations for the shift (D.13) of the parameters ¯ µ j [ ν b r ] with resp ect to the λ j ’s allo ws one to compute the thermodynamic limit of U ( L ) { i a } ( { λ a } , { ¯ µ a } , { y s } ) . Namely , U ( L ) { i a } ( { λ a } , { ¯ µ a } , { y s } ) − → N ,L →∞ − q Z − q u ′ ( λ ) ν r ( λ ) d λ + n X a =1 u ( y a ) − u ( λ i a ) = − i β p F π + n X a =1 [ u 0 ( y a ) − u 0 ( λ i a )] . (D.16) T o obtain this limit, w e ha v e used the deﬁnitions (2.15)-(2.16) of ε and p so as to re-cast the expressions in vo lving int egrals of u = p − tε/x in terms of u 0 = p 0 − tε 0 /x . The thermodynamic limit of G N can b e computed along the lines of [43 ]. Indeed, one has F  λ     { λ j } N 1 ∪ { y s } n s =1 { ¯ µ j } N 1 ∪ { λ i s } n s =1  − → N ,L →∞ ν r ( λ ) . (D.17) The latter implies that N Y j =1 ω − ¯ µ j + iǫc ω − λ j + iǫc · e − 2 iπ C [ F ]( ω + iǫc ) − → N ,L →∞ 1 , for ǫ ∈ {± 1 , 0 } . (D.18) 36 By applying iden tity (D.18) to the v arious pro ducts ente ring in the deﬁnition of G N , one can con vince oneself that det [ − q ,q ]  I − K/ 2 π  N Y s =1 4 sin 2 [ π ν r ( λ i s )] G N  { λ j } N 1 \ { λ i s } n s =1 ∪ { y s } n s =1 { ¯ µ j } N 1  − → N ,L →∞ n Y s =1  1 − κ V − V +  λ i s     { λ i a } n 1 { y a } n 1  · F n  { λ i a } n 1 { y s } n 1  . (D.19) Ab o ve , w e hav e set V ± µ      { λ j } N j =1 { z j } N j =1 ! = N Y a =1 ic ∓ ( µ − λ a ) ic ∓ ( µ − z a ) , (D.20) and, agreeing that the auxiliary argumen ts of V ± are undercurren t by those of F n , F n  { λ s } n 1 { y s } n 1  = (1 − κ ) 2 det[ I − K/ 2 π ] n Y s =1  1 − V + ( λ s ) κV − ( λ s )  2 Y j =1  V − ( θ j ) det n  δ j k + b V ( θ j ) j k  1 − κV − ( θ j ) /V + ( θ j )  × n Y a,b =1 ( y a − λ b − ic ) ( y b − λ a − ic ) ( y a − y b − ic ) ( λ b − λ a − ic ) , (D.21) with b V ( θ ) k ℓ = − n Q s =1 ( λ k − y s ) n Q s =1 6 = k ( λ k − λ s ) K κ ( λ k − λ ℓ ) − K κ ( θ − λ ℓ ) V − 1 − ( λ k ) − κV − 1 + ( λ k ) . (D.22) It only remains to deal with the y -in tegrations o ver C ( L ) . In this limit, the in tegrals ov er C ↑ / ↓ giv e v anishing con tributions and hence the y -type int egration con tour b oils do wn to the con tour C = C ( ∞ ) E ∪ e C q with a weigh t function that no w reads f ( y , ν r ( y )) = 1 C ( ∞ ) E ( y ) − 1 e C q ( y ) e − 2 iπ ν r ( y ) − 1 . (D.23) The con tour C ( ∞ ) E corresp ond s to the L → + ∞ limit of the con tour C ( L ) E depicted on Fig. 2. Therefore, in the thermo dynamic limit, Q κ N ( x, t ) eﬀ − → N ,L →∞ Q κ ( x, t ) , with Q κ ( x, t ) = + ∞ X n =0 ( − 1) n ( n !) 2 q Z − q d n λ (2 iπ ) n I C q d n z (2 iπ ) n Z C d n y (2 iπ ) n n Y a =1 e ix [ u 0 ( y a ) − u 0 ( λ a )] × n Y s =1  f ( y s , ν r ( y s )) y s − z s  1 − κ V + V −  λ s     { λ a } { y a }   det 2 n  1 z j − λ k  F n  { λ a } { y a }  . (D.24) Here, w e remind that ν r is giv en by (D.14) and is a function of { y s } and { λ j } . 37 D.3 The equal-time case W e no w show that in the equal-time case, one reco v ers the series obtained from the master equation in [45]. This stems from the fact that, at t = 0 , the y -inte grals o ver C ( ∞ ) E do not con tribute. Indeed, when t = 0 , w e can deform the con tour C ( ∞ ) E in to R + iδ , with δ > 0 and small but s uch that the line R + iδ is ab o ve e C q . In order to prov e this ass ertion, w e ﬁrst build on the symmetries of the integra nd in (D.24 ) so as to split the y -in tegrations in to those along R + iδ ( y a , a = 1 , . . . , k ) and th ose along e C q ( y a , a = k + 1 , . . . , n ), with k = 1 , . . . , n . The integ rals o ver e C q can then b e computed by residues. One ev ent ually obtains Q κ ( x, 0) = + ∞ X n =0 n X k =0 ( − 1) k n ! k !( n − k )! q Z − q d n λ (2 iπ ) n I C q d n z (2 iπ ) n Z R + iδ d k y (2 iπ ) k n Y a = k + 1 e ix [ z a − λ a ] k Y a =1 e ix [ y a − λ a ] y a − z a × det 2 n  1 z j − λ k  n Q s =1  1 − κ V − V +  λ s     { λ a } n 1 { y a } k 1 ∪ { z a } n k +1  n Q a = k + 1  κ V − V +  z a     { λ a } n 1 { y a } k 1 ∪ { z a } n k +1  − 1  F n  { λ a } n 1 { y a } k 1 ∪ { z a } n k +1  . (D.25) When considered as a function of y 1 , . . . , y k , the in tegrand in (D.25) has no p oles (or even other singularities) in the half-planes ℑ ( y k ) ≥ δ . I ndee d, no p oles can arize from the θ j dep ende n t terms in F n in as m uc h as this function do es not dep end on θ j , cf [45]. Also the p oten tial singularities of the determin an ts at the zero es of 1 − κ ( V − /V + )( λ a ) are only apparen t due to the presence of the pre-factors Q s [1 − κ − 1 ( V + /V − )( λ s )][1 − κ ( V − /V + )( λ s )] distributed in b et w een (D.21) and (D.25). The p oten tial p oles at y a = λ b ± ic in tro duced b y these pre-factors are cancelled by the zero es of the double pro duct in the last line of (D.21). In its turn, the p oles in the upp er-hal f plane at y a = z s + ic , with a = 1 , . . . , k and s = k + 1 , . . . , n , that are in tro duced b y the double pro duct in (D.21), are comp ensated 10 b y the same p oles presen t in Q n s = k +1 [1 − κ ( V − /V + )( z s )] . Therefore the only singularities in the y v ariables corresp ond to the zero es of κ V + V −  z s     { λ a } n 1 { y a } k 1 ∪ { z a } n k +1  − 1 . (D.26) Ho wev er, one can alw a y s sq ueez e the z -in tegration con tours in (D.25) so that ℑ ( z a ) = 0 ± , a = 1 , . . . , n . In suc h a situation, it is readily seen that for y a ∈ R + iδ , δ > 0 , one has     κ V + V −  λ s     { λ a } n 1 { y a } k 1 ∪ { z a } n k +1      > 1 . (D.27) In other w ords, the aforemen tioned equation has no solutions in the upp er half-plane. This lac k of singularities allo ws one to sim ultaneously deform the y -in tegration con tours from R + iδ to R + i ∆ , with ∆ > 0 and as large as desired. Due to the presence of the oscillatory factors e ixy a , these inte grals will contri bute as e − ∆ x . Therefore, by sending ∆ → + ∞ , w e see that these con tributions v anish. 10 The p oles at y a = z s − ic lying in the low er half-plane are n ot explicitly compen sated but this is irrelev an t for our purp oses 38 W e ha ve thus pro ven that, in the t = 0 case, the s eries of m ultiple inte gral represen tation for Q κ ( x, 0) b oils dow n to Q κ ( x, 0) = N X n =0 ( − 1) n ( n !) 2 q Z − q d n λ (2 iπ ) n I C q d n z (2 iπ ) n n Y s =1 e ix ( z s − λ s ) det 2 n  1 z j − λ k  F n  { z a } { λ a }  × n Y s =1 1 − κ V − V +  λ s     { λ a } { z a }  1 − κ V − V +  z s     { λ a } { z a }  . (D.28) In order to pro v e our assertion, it remains to show that the last line in (D.28) do es not con tribute once that the z -in tegrals are taken . In virtue of the sy mm etry of the in tegrand, one can replace one of the Cauc hy determinan ts b y n ! times the pro ducts of its diagonal en tries. By expanding the rema ining Cauc h y determinan t in to a sum o ve r p erm utations, one sees that one has to compute double p oles at z p k = λ p k with p 1 < · · · < p ℓ with p i ∈ { 1 , . . . n } and 1 < ℓ < n . A ll other p oles will b e s imp le, leading to the equality b et wee n the tw o s ets { z a } a 6 = p k = { λ a } a 6 = p k . These double p oles will pro duc e ﬁrst order deriv atives with resp ect to z p k at z p k = λ p k . Therefore, the eﬀect of these double p oles can b e tak en into accoun t by setting 11 z a = λ a for a 6 = p k , k = 1 , . . . , ℓ , and z p k = λ p k + ǫ p k , and then taking the ﬁrst order ǫ p k -deriv ative s of the obtained expression at ǫ p k = 0 . As a consequence, only the linear in each ǫ p k order of the integra nd will con tribute. Ho wev er, under suc h a substitution, one can readily con vince oneself that n Y s =1 1 − κ V − V +  λ s     { λ a } { z a }  1 − κ V − V +  z s     { λ a } { z a }  = ℓ Y s =1  1 + iκ κ − 1 ℓ X k =1 ǫ p s ǫ p k K ′ ( λ p s − λ p k ) + O ( ǫ 2 p k )  = 1 + O( ǫ 2 p k ) . The linear order in the ǫ p k ’s v anishes. Therefore, we are led to Q κ ( x, 0) = N X n =0 ( − 1) n ( n !) 2 q Z − q d n λ (2 iπ ) n I C q d n z (2 iπ ) n n Y s =1 e ix ( z s − λ s ) det 2 n  1 z j − λ k  F n  { z a } { λ a }  . (D.29) Once up on taking the complex conjugate w e reco v er, w ord for wo rd, the series obtained in [45]. R emark D.1 . Note that our con ven tions corresp ond to x 7→ − x with resp ect to the work [45]. E Con trolli ng sub-leading corrections: the Natte seri e s repre- sen tation The Natte series represen tation for the F redholm determinan t det[ I + V ] is obtained [60] from a sp eciﬁc represen tation of the solution to the R iema nn–Hilb ert problem asso ciated with the in tegrable integ ral op erator I + V . As this particular represen tation for the solution of the Riemann–Hilbert problem is obtained b y a series of con tour deformations on the so-called init ial Riemann–Hilbert problem asso ciated with the in tegral op erator I + V , it can b e seen that the 11 The integ rand is symmetric with resp ect to the in tegration v ariables z a and λ a . 39 Natte series stems from a certain n umber of algebr aic tr ansformations 12 carried out on the initial F redholm series for the determinan t. This fact allo ws one, at least on the formal ground, to use this Natte series represen tation in (6.5). The rigorous justiﬁcation of the p ossibilit y to use the Natte series is given in [59]. E.1 The form ula for t he remainder In fact, the Natte series corresp onds to a represen tation of the remainder R x [ ν, u, g ] . The latter is expressed as a series of mul tiple in tegrals. The in tegrands app earing in this series hav e go od prop erti es with resp ect to the large- x limit. More precisely , the Natte series f or the sine ker nel (6.2) tak es the follo wing form: R x [ ν, u, g ] = X n ≥ 1 X K n X E n ( k ) Z C { ǫ t } d n z t (2 iπ ) n H n ; x ( { ǫ t } , { z t } )[ ν ] Y t ∈ J ( k ) e ǫ t g ( z t ) . ( E.1) The second sum app earing ab o v e runs through all the n -tuples k b elonging to K n =  k = ( k 1 , . . . , k n ) : k a ∈ N , a = 1 , . . . , n and suc h that n X s =1 sk s = n  . (E.2) F or eac h elemen t k of K n , one deﬁnes the ass ociated set of triplets J ( k ) (with cardinal n ), J ( k ) =  t = ( t 1 , t 2 , t 3 ) , t 1 ∈ [ [ 1 ; n ] ] , t 2 ∈ [ [ 1 ; k t 1 ] ] , t 3 ∈ [ [ 1 ; t 1 ] ]  , (E.3) whic h pro vides a con venien t w ay of lab elling s ets of n v ariables. The third sum runs through all the p ossible c hoices of elemen ts b elonging to the s et E n ( k ) =  { ǫ t } t ∈ J ( k ) : ǫ t ∈ {± 1 , 0 } and t 1 X t 3 =1 ǫ t = 0 ∀ ( t 1 , t 2 ) ∈ [ [ 1 ; n ] ] × [ [ 1 ; k t 1 ] ]  . In other words, E n ( k ) consists of sets of n parameters ǫ t ∈ {± 1 , 0 } , indexed b y triplets t = ( t 1 , t 2 , t 3 ) ∈ J ( k ) and sub ject to summation constrain ts. Finally , there is an n -fold integr al app earin g in the n th summand of (E.1). The in tegration v ariables z t are, again, indexed b y the triplets t = ( t 1 , t 2 , t 3 ) of J ( k ) . The con tours of inte gration C { ǫ t } dep end on the set { ǫ t } ∈ E n ( k ) . They are realized as n -f old Cartesian products of on e-dimensional compact curves corresp ond ing to v arious deformations of R . The in tegrand H n ; x ( { ǫ t } ; { z t } )[ ν ] is a smo oth functional of ν whic h is also a function of the integ ration v ariables z t . This f unc tional depends on the c hoice of the parameters { ǫ t } from E n ( k ) and on x . It is a holomorphic function of the in tegration v ariables { z t } b elongin g to some op en neigh b orhoo d of the in tegration con tour C { ǫ t } . The Natte series con v erges for x large enough in as muc h as, for n large,      Z C { ǫ t } d n z t (2 iπ ) n H n ; x ( { ǫ t } , { z t } )[ ν ] Y t ∈ J ( k ) e ǫ t g ( z t )      ≤ c 2  c 1 x  nc 3 , (E.4) 12 Such a s contour deforma tions or algebraic summations. 40 where c 1 and c 2 are s ome constan ts dep ending on the v alues taken b y u and g in some small neigh b orhoo d of R and on the v alues tak en b y ν on a small neigh b orho od of [ − q , q ] . The constan t c 3 > 0 only dep ends on ν . Finally , one has H 1; x = O( x −∞ ) and, f or n ≥ 2 , H n ; x ( { ǫ t } , { z t } )[ ν ] = O( x −∞ ) + [ n/ 2] X b =0 b X p =0 [ n/ 2] − b X m = b − [ n 2 ] e ix [ u ( λ 0 ) − u ( − q ) ] x 2[ ν ( − q )] ! αb e ix [ u ( q ) − u ( − q )] x 2[ ν ( q )+ ν ( − q )] ! m − αp × 1 x n − b 2 · H m,p,b n ; x ( { ǫ t } , { z t } )[ ν ] , (E.5) where, for all admissible v alues of m, b, p , H m,p,b n ; x ( { ε t } , { z t } )[ ν ] = O( x n e w − δ n, 2 ) (E.6) uniformly on the integ ration con tour. Ab o ve, w e ha v e s et e w = 2 su p  |ℜ [ ν ( z ) − ν ( τ q )] | : | z − τ q | ≤ δ , τ = ±  , (E.7) where δ > 0 can b e tak en as small as desired. These f unct ions H m,p,b n ; x ( { ε t } , { z t } ) [ ν ] are s uc h that their asympto tic expansion i n to in verse p o wers of x posses s es p oles that are encirc led b y the part of the con tour C { ǫ t } pro ducin g algebraic (in x large enough) con tribution to the in tegral. As a consequence, the n -f old in tegration o ccurring in the n th term of the series can pro duce z deriv atives. It was sho wn in [60] that the total order of these deriv ative is at most n . Once that these deriv ative s are taken, the es timates in (E.6) chan ge f rom O( x n e w − δ n, 2 ) to O( x − δ n, 2 log n x ) . E.2 The Natte series for the determinan t W e now use the Natte series represen tation (E.1) for the remainder R x [ ν, u, b g ] to pro v e that the s uble ading corrections app earing in the large- x asymptotic b eha vior of the (non op erator- dep ende n t) F redholm determinan t and included in R x [ ν, u, g ] still remain corrections with re- sp ect to the leading terms once that one computes the action of the functional translation op erator s. By expanding the F redholm determinan t in (6.5) in to its Natte s erie s, w e obtain Q κ ( x, t ) = e − β xp F π : ( B x [ ν τ , u ] + X ǫ = ± 1 e iǫx [ u ( q ) − u ( − q )]+ ǫ [ b g ( q ) − b g ( − q )] B x [ ν τ + ǫ, u ] + 1 x 3 2 X ǫ = ± 1 e iαx [ u ( λ 0 ) − u ( ǫq )]+ α [ b g ( λ 0 ) − b g ( ǫq )] b ( ǫ,α ) 1 [ ν τ , u ] B x [ ν τ , u ] + X n ≥ 1 X K n X E n ( k ) Z C { ǫ t } d n z t (2 iπ ) n Y t ∈ J ( k )  e ǫ t b g ( z t )  H n ; x ( { ǫ t } , { z t } )[ ν τ ] B x [ ν τ , u ] ) · G [ ω ] :      ω =0 τ =0 . (E.8) Note that in (E.8), w e ha ve already simpliﬁed the tw o exp onen ts. Also, w e remind that α = 1 in the space-lik e regime and α = − 1 in the time-lik e regime. The action of the translation op erator s o ccuring in the ﬁrst t w o lines has already b een computed in Section 6.3. In order to compute this action in the last line of (E.8), it is conv enien t to in tro duce the notations a { ǫ t } = # { t : ǫ t = 1 } and { z ± t } = { z t : t ∈ J ( k ) such th at ± ǫ t > 0 } . (E.9) 41 It then readily f ollo ws that Y t ∈ J ( k )  e ǫ t b g ω ( z t )  · G [ ω ]    ω =0 = G a { ǫ t }  { z + t } { z − t }  , (E.10) and : Y t ∈ J ( k )  e ǫ t b g τ ( z t )  · H n ; x ( { ǫ t } , { z t } )[ ν τ ] B x [ ν τ , u ] :    τ =0 = H n ; x ( { ǫ t } , { z t } )[ F { ǫ t } ] B x [ F { ǫ t } , u ] . (E.11) Note that, in order to obtain (E.10), w e ha ve used the fact that ǫ t ∈ {± 1 , 0 } for all t ∈ J ( k ) and that these parameters are sub ject to the condit ion P t ∈ J ( k ) ǫ t = 0 . As a consequence, # { z + t } = # { z − t } . Also we hav e set F { ǫ t } ( λ ; { z t } ) ≡ F { ǫ t } ( λ ) = ν ǫ t ( λ, { z t } ) = iβ Z ( λ ) 2 π − X t ∈ J ( k ) ǫ t φ ( λ, z t ) . (E.12) This leads to the follo wing represen tation for the generating function: Q κ ( x, t ) = Q κ ; α asym ( x, t ) + X n ≥ 1 X K n X E n ( k ) Z C { ǫ t } d n z t (2 iπ ) n × H n ; x ( { ǫ t } , { z t } )[ F { ǫ t } ] B x [ F { ǫ t } , u ] G a { ǫ t }  { z + t } { z − t }  . (E.13) Here, Q κ ; α asym ( x, t ) denotes the leading asymptotic part that is giv en in (6.15) or by its similar expression in the time-lik e regim e (without the O  (ln x ) / x  remainders since the se are no w explicitly tak en into accoun t in the rest of the ab o ve equation). Note that the n -fold integr ation o ccuring in the n th term of the series can pro duce z deriv a- tiv es whose total order is at most n . It thu s f ollo ws from the represen tation f or the functional H n ; x (E.5) and from the form of the estimates (E.6) that, indeed, the remainder produces corrections of the form written in (6.15). References [1] I . 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Long-time and large-distance asymptotic behavior of the current-current correlators in the non-linear Schr"{o}dinger model

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