Form factor approach to the asymptotic behavior of correlation functions in critical models

LPENSL-TH-10/11 F orm factor approa c h to the a symptotic b eha vior of correlation functions in critical mo dels N. Kitanine 1 , K. K. Kozlowski 2 , J. M. Maillet 3 , N. A. Sla vno v 4 , V. T erras 5 Abstract W e prop ose a form factor approach f or the computation of the large distance asymptotic behavior o f cor r elation functions in qua n tum cr itical (integrable) mo dels. In the large distance r egime we reduce the summation over all exc ited states to one ov er the particle/ hole excitations lying on the F ermi surface in the ther mo dynamic limit. W e compute these sums, ov er the so-called c r iti- cal form factor s, exactly . Thu s we obtain the leading la r ge distance b ehavior of ea ch oscillating harmonic of the corr elation function asymptotic ex pansion, including the co rresp onding a mplitudes. Our metho d is a pplicable to a wide v a r iety of in teg rable mo dels and yields precise ly the re s ults s temming from the Luttinger-liquid a pproach, the confor mal field theory predictions and our pr e- vious analysis of the correlation functions from their m ultiple-integral represen- tations. W e argue that our scheme a pplies to a general cla ss of non-integrable quantum critical mo dels as well. 1 IMB, UMR 5584 du CNR S, Universit ´ e de Bourgogne, F rance, Nikolai.Kita n ine@u-b ourgogne.fr 2 IUPUI, Department of Mathematical Sciences, Indianap olis, USA, k akozlo w@iupui.edu 3 Lab oratoire de Physique, UMR 5672 du CNRS, ENS Lyon, F rance, maillet@ens-lyo n.fr 4 Steklov Mathematical Institute, Moscow, Russia, n sla vn ov@mi.ras.ru 5 Lab oratoire de Physique, UMR 5672 du CNRS, ENS Lyon, F rance, veronique.terras@ens-ly on.fr 1 1 In tro d uction F orm factors and correlation functions are central ob jects for the stud y of dynamical prop erties of mo dels in quan tu m field theory and statistical mec hanics. F or general interac ting systems, their calculation remains a fan tastic c hallenge. In th e in tegrable mo del situation in lo w d imen- sion, see e.g. [1–14] and references therein, considerable progress has b een m ade d uring the last fort y yea r s to wa r ds the exact calculatio n of these d ynamical qu an tities [14–46]. Of particular in terest are the s o-called critical mo dels w here the correlation functions of lo cal op erators are b eliev ed, from Lu ttinger liquid approac h and conformal field th eory (CFT) predictions, to d eca y as p o we r laws in th e distance [47–57]. There are several wa ys to approac h the computation of correlation fu nctions in su c h mo dels, either f rom their massiv e field theories counterparts (in their sh ort distance b eha vior) or from critical lattice mo dels (lo oking at their long distance prop erties). In su c h limits one should reco v er the picture predicted by CFT, with the p ossi- ble additional insight int o the explicit corresp on d ence with th e p h ys ical lo cal op erators of the microscopic initial theories. The latter in formation is crucial for obtaining full control of th eir dynamics. In deed, w hile C FT is able to p r edict for example the algebraic co efficien ts in the op erator pr o duct expansion of lo cal op erators together with the v alues of the v arious critical exp onents driving the p o wer la w b ehavior of the correlat ion functions, it fails to d etermin e the (non u niv ers al) corresp onding amplitudes (v acuum exp ectati on v alues and form factors) whic h are in fact deeply ro oted in the detailed microscopic in teractions of the physical mo dels at h and. The pur p ose of the present article is to sh o w that un der quite general assumptions, whic h are v alid in the integ r ab le mo del situation, it is p ossible to determin e exactly the asymptotic b ehavi or of correlatio n fu nctions in critical mo dels starting from their expansion in terms of form factors. The u se of f orm factors in this context migh t app ear to b e a very strange str ategy (see ho wev er [58, 59]). Indeed, as has b een demonstrated [60–62], form factors of lo cal op erators in critical mo dels scale to zero with a non trivial p ow er la w in the system size (or equiv alently in the mass scale). This is not su rprising as it just r efl ects the corresp onding critical dimensions of the lo cal op erators (together with those of the creation op erators of the excited states) considered. Hence th e form factor expansion of correlatio n functions of local op erators in critical mod els is a large (infi nite) su m of small (v anishing) terms in the infin ite v olume limit. T h ere it should b e the collectiv e (v olume) effect of the sum th at comp en s ates the v anishing b eh a vior of eac h term of this su m. Hence, extracting this collectiv e effect directly from the form factor series seems a priori to b e a quite inv olv ed task. Ho wev er, a ve r y in teresting effect reve aled itself in our recen t d eriv ation of the leading asymptotic b ehavio r of correlation f unctions of the X X Z c hain obtained fr om their re-su m med multiple-in tegral repr esen tations [38]. Un exp ectedly , it app eared there that the exactly computed amplitudes of the leading p o wer law deca y of the spin–spin correlatio n fu nction w ere in fact g iv en by prop erly renormalized (by a p ositive p o wer of the s ystem s ize) form factors of the lo cal spin op erator. The very app ealing form of this answer w as a strong motiv ation for us to dev elop a metho d for stu dying such correlation functions from their form factor expansion. 2 The metho d that we p rop ose in the present pap er op ens up a w ay to analyze the large distance asymptotic b ehavio r of correlation functions of critical mo dels from their form factor expansion. Namely , it enables us to determine the leading b ehavior of eac h oscillating h armonic in the asymp totic expans ion of the correlation functions, includ ing n ot only th e exact p ow er la w deca y s bu t also the asso ciated amplitud es. T o describ e our metho d , w e first study the correlation fu nctions of quantum in tegrable mo dels that are solv able b y using the Bethe ansatz in their critical regime. The typica l example considered in this p ap er is the integrable Heisen b erg spin c hain. Ho w ever, the metho d that w e describ e in this pap er is also applicable to the one- dimensional Bose gas (or qu antum n on-linear S c hr¨ odinger mo d el), and more generally to an y quan tu m int egrable mo d el p ossessing d eterminan t repr esen tations f or the f orm factors of lo cal op erators [33, 63–65]. In fact we will f urther argue that, un d er qu ite natural assump tions, it applies to a general class of non-in tegrable critical m o dels as w ell. Generally sp eaking, indep end en tly on the integrabilit y of a quan tum mo del, th e zero tem- p erature limit of a correlation fu nction of lo cal op erators lo cated at t wo different space p oints x ′ and x ′ + x , sa y O 1 ( x ′ ) and O 2 ( x ′ + x ), redu ces to the groun d state exp ectation v alue of th eir pro du ct: hO 1 ( x ′ ) O 2 ( x ′ + x ) i ≡ h ψ g | O 1 ( x ′ ) O 2 ( x + x ′ ) | ψ g i h ψ g | ψ g i . (1.1) Ab o ve, | ψ g i denotes the ground state and th e co ordinate x can b e either a con tinuous parameter (for one dimensional field theories) or a discrete one (for lattice mo dels). The form factor expansion of the zero temp eratur e correlation fu nction is obtained by in- serting a complete set of eigenstates | ψ ′ i of the Hamilto n ian b et we en the lo cal op erators: hO 1 ( x ′ ) O 2 ( x ′ + x ) i = X | ψ ′ i F (1) ψ g ψ ′ ( x ′ ) F (2) ψ ′ ψ g ( x + x ′ ) . (1.2) Th us the correlation function is represented as a su m of matrix elemen ts (the s o-calle d form factors) F ( j ) ψ g ψ ′ ( x ) of the lo cal op er ators O j tak en b et ween the groun d state | ψ g i and an y excited state | ψ ′ i : F (1) ψ g ψ ′ ( x ′ ) = h ψ g | O 1 ( x ′ ) | ψ ′ i p h ψ g | ψ g ih ψ ′ | ψ ′ i , F (2) ψ ′ ψ g ( x + x ′ ) = h ψ ′ |O 2 ( x + x ′ ) | ψ g i p h ψ g | ψ g ih ψ ′ | ψ ′ i . (1.3) What m ak es the representati on (1.2) v ery app ealing is its straigh tforward physical inte rpreta- tion. Mo reo ve r , it pro v ed to p ro vide a very efficien t w ay to ev aluate the correlation fun ctions n u merically as so on as exact represen tations for the form factors are kno wn [46, 66–69]. The form factor expansions are well un dersto o d in the case of massive mo dels [19–25]. Actually , in these mo dels they are extremely effectiv e in computing the long distance exp onen tial deca y of the asso ciated correlation function. In particular, the leading long distance b eha vior of the correlation fun ctions generally stems there from a few classes of excited states only . Ho we ver, as we already discussed, the structur e and prop er ties of form factor expan s ions in massless (critical) mo dels are more inv olved, as an infinite to wer of classes of excited states 3 con tributes to the leading asymptotic b eha vior of the correlation fu nction, in con trast to the massiv e regime case. There h as r ecen tly b een imp ortant p rogress in computing the asymptotic b eha vior of correla- tion functions f or qu an tum in tegrable mo dels; these app roac hes rely basically on the Riemann– Hilb ert analysis of r elated F redholm determinan ts, an d are quite tec h n ical [38, 70, 72, 73]. W e b eliev e that the metho d that we prop ose in this pap er is simpler as it follo ws a p ath closer to the physica l intuition and hence sh ou ld b e of a wider use (in particular b eyond the int egrable case). As w e just men tioned, for critical mo dels, eac h term of the form factor s eries scales to zero as some n on-trivial p o w er θ (dep ending on the excited state | ψ ′ i selected by the lo cal op erators considered) of the system size L [60–62] for L → ∞ . In trans lation in v ariant systems w e ha v e F (1) ψ g ψ ′ ( x ′ ) · F (2) ψ ′ ψ g ( x + x ′ ) = L − θ e ix P ex A. (1.4) Here the dep endence on the distance of the f orm f actor is contai ned in the phase factor e ix P ex , where P ex is the m omen tum of the excited state | ψ ′ i relativ e to the ground state. Th e amplitude A do es not dep end on x and remains finite in the thermo dyn amic limit. Our strategy for the analysis of the sum (1.2) is as follo ws. Initially , the sum (1.2) ru ns through all the eigenstates and f or a finite v alue of the distance x all eigenstates sh ou ld indeed b e consid er ed . Ho wev er, in the long d istance regime, as follo ws from (1.4 ), eac h term in the form factor series is quickl y oscillating with the distance. Th is means that, similarly to what happ ens for oscillatory int egrals, the leading b eha vior of the form factor series stems from the lo calization of the sum around sp ecific critical p oint s (like saddle p oint s or edges of the summation int er v al). T his f act allo ws u s to carry out sev eral reasonable simplifications of the summation range that do not affect the leading asymp totic b eh avior of the correlation function. First of all, w e restrict the form factor sum (1.2) to th e summ ation ov er excitations of one sp ecific t yp e only , namely , the particle–hole excitations 1 . Second, in the large distance regime w e tak e in to accoun t only con tr ibutions coming from excitations close to the endp oin ts of the F ermi zone, the so-called critical excited s tates [38, 62]. In this wa y , w e naturally restrict ourselv es to the excitations app earing in th e conformal part of the s p ectrum of the mo del. W e will sho w that the r esulting restricted s u m o ve r critical excited states can b e computed exactly . The asymptotic expansion f or th e t wo- p oint functions of the X X Z mo del that we obtain in this w ay confir ms the predictions stemming from a corresp ondence with CFT/Luttinger liquid approac hes an d agrees with the part of the asymptotic b eha vior obtained through the exact Riemann–Hilb ert metho ds [38, 70–73]. The pap er is organized as follo ws. In Section 2 w e d efine the critical form factors and form u late our main strategy f or ev aluating their su ms in the large distance limit. W e giv e s ome 1 Hence, in particular, b oun d states will not contribute in th is approach, although we b elieve that they could app ear in th e analysis of dyn amical correlation functions. Of course, for some mo dels such as the non-linear Schr¨ odinger model, all the excited states with fi nite ex citation energy can b e describ ed in terms of particles and holes. 4 argumen ts confi rming th is app r oac h. In section 3 we p erform exactly the summation o ver th e critical form factors. T h is compu tation is giv en in a general, mo d el ind ep endent fr amew ork. It giv es the leading asymptotic b eha vior of all the oscillating harm onics. In section 4 w e apply this general result to compute the t wo-point functions for the Heisen b er g X X Z c h ain. W e repro du ce the leading asymptotic b eha vior of the t wo- p oin t functions as predicted by C FT and th e correlation amplitudes obtained b y the Riemann–Hilb ert approac h. Finally , in the conclusion, we argue that the ab o v e sc h eme app lies to a large class of critical (integrable and non-in tegrable) m o dels. In App endix A we pro ve the summation form u la w hic h p ermits us to compute all the ab o ve critical sums exactly . 2 F orm factor s eries in the large distance limit Let u s consider th e case of quantum integ r able mo dels for wh ic h the ground state solution of the Bethe equations can b e describ ed in terms of real rapidities λ j densely filling (with a densit y ρ ( λ )) the F erm i zone [ − q , q ]. In su ch mo dels, the logarithmic Bethe ansatz equations for the ground state Bethe ro ots λ j tak e the form [2, 3, 5–7]: Lp 0 ( λ j ) − N X k =1 ϑ ( λ j − λ k ) = 2 π  j − N + 1 2  , j = 1 , . . . , N . (2.1) Here L is the size of the mo del, and N is the num b er of quasi-particles in the groun d state. The fun ctions p 0 ( λ ) and ϑ ( λ ) are the bare momentum and bare scattering p hase of the quasi- particles in the corresp on d ing mo d el. T h e v alue N d ep ends on the parameters of the mo del (suc h as the c hemical p oten tial or external magnetic field). When w e take the thermo dynamic limit L → ∞ , the ratio N /L has a finite limit which we call the total densit y D = lim L →∞ N L . In order to describ e the form factor sum one needs to hav e a charact er ization of all the excited states as w ell. In the Bethe ansatz framewo rk, the excited states are parameterized b y rapidities { µ ℓ a } N ′ 1 solving a set of equations similar to (2.1) but inv olving other choice s of in tegers ℓ 1 , . . . , ℓ N ′ in the r hs : Lp 0 ( µ ℓ j ) − N ′ X k =1 ϑ ( µ ℓ j − µ ℓ k ) = 2 π  ℓ j − N ′ + 1 2  , j = 1 , . . . , N ′ . (2.2) Ab o ve, N ′ corresp onds to the n u m b er of quasi-particles con tained in th e giv en excited state. The v alues of N ′ to b e considered for carrying out the form f actor sums dep end on the t yp e of lo cal op erator O that one deals with. A lo cal op erator O only connects states h aving a ‘small’ difference in the n u m b er of qu asi-particles. In other words, for a giv en lo cal op erator, there exists an integ er k su c h that only states with | N − N ′ | 6 k giv e r ise to non-zero form factors. F or in stance, later on, w e will fo cu s on the X X Z sp in-1 / 2 c h ain. There, w e will consider the lo cal spin op erators, where either N ′ = N or N ′ = N ± 1. W e do how ev er stress that our 5 metho d allo ws one to consider more complicated cases of lo cal op erators that connect more distan t v alues of N and N ′ . F or excited states of the t yp e that w e are intereste d in, it is conv enient to describ e th e solutions of th e Bethe equations in terms of a set of in tegers alternativ e to ℓ 1 , . . . , ℓ N ′ . Namely one characte r izes the excitations in terms of holes in the Dirac sea of intege r s for the ground state ℓ j = j and additional, p article-lik e inte gers outside of the zone { 1 , . . . , N ′ } . In other w ord s , the in tegers ℓ j describing an excited n -particle–hole state tak e the form ℓ a = a , a ∈ { 1 , . . . , N ′ } \ { h 1 , . . . , h n } ℓ h a = p a , p a ∈ Z \ { 1 , . . . , N ′ } . (2.3) Eac h choice of pairwise distinct q u an tum num b ers { p a } and { h a } c haracterizes an excited state asso ciated with a configuration of the particle and the hole r apidities { µ p a } and { µ h a } , see e.g., [11, 12]. The particle/hole excitation con tribu tion to the t wo-point function has the form hO 1 ( x ′ ) O 2 ( x + x ′ ) i ph = lim L →∞ X { p } , { h } L − θ e ix P ex A ( { µ p } , { µ h }|{ p } , { h } ) . (2.4) Here we ha ve explicitly ind icated that the amplitud e A (1.4) dep ends on the particle/hole rapidities { µ p } and { µ h } as well as on the corresp ondin g in teger quan tum num b ers { p } and { h } . Let us comment briefly on this dual dep endence (see [62] f or more details). If all particle/hole rapidities are separated f rom the F er m i b oundaries, then the amplitude smo othly dep ends only on these quan tities A = A ( { µ p } , { µ h } ). How ever, as so on as these rapidities approac h the F ermi b ound aries, the amplitud e shows a discrete structur e: a microscopic (of order 1 /L ) d eviation of a particle (hole) rapidit y , corresp onding to a fin ite shift of th e in teger qu antum num b ers { p } or { h } , leads to a macroscopic c han ge of A . Hence, in this case, one should tak e int o account the discrete structure of the amplitude, and therefore we wr ite A = A ( { µ p } , { µ h }|{ p } , { h } ). In the follo win g, we will argue that the oscillatory charact er of the sum in (2.4) lo calizes it, in the absence of an y saddle p oints of the oscillating exp onen t 2 , in the vicinit y of the F ermi b ound aries ± q . F or this p u rp ose, w e introdu ce sev eral defin itions [62]. Definition 2.1. An n p article–hole e xc i te d state { µ ℓ a } is c al le d a critic al e xcite d state if the r apidities { µ p a } and { µ h a } define d by such a state ac cumulate on the two endp oints of the F ermi zone in the thermo dynamic limit. F orm factors c orr esp onding to any such a state ar e c al le d critic al form factors. The critical excite d states are c haracterized by the sp ecific d istribution of p articles and holes on the F erm i b ound aries, ± q . Namely , assume th at, in the thermo d ynamic limit, there are n ± p particles whose rapidities are equal to ± q and n ± h holes w hose r apidities are equal to ± q . Th en, ob viously , n + p + n − p = n + h + n − h = n. (2.5) 2 Such saddle p oints could app ear in th e t ime dep end ent case and will b e considered in a forthcoming pu bli- cation. 6 Definition 2.2. A given c ritic al excite d state b elongs to the P ℓ class if the distribution of p articles and holes on the F ermi b oundaries is such that n + p − n + h = n − h − n − p = ℓ, − n 6 ℓ 6 n. (2.6) Then such an excite d state has momentum 2 ℓk F in the thermo dynamic limit, wher e k F = π D is the F ermi momentum. Its asso ciate d critic al form factor wil l also b e said to b elong to the P ℓ class. No w w e would lik e to argue in fa v or of the lo calization of the form factor sum on the F ermi b ound aries in the large distance limit. F or this purp ose we w ill use an analogy with oscillatory in tegrals. Hence, supp ose that we deal with some multiple in tegral of the t yp e I n ( x ) = Z R \ [ − q ,q ] d n µ p Z [ − q ,q ] d n µ h f ( { µ p } , { µ h } ) n Y j =1 e ix ( p ( µ p j ) − p ( µ h j )) , x → ∞ , (2.7) where f ( { µ p } , { µ h } ) is a holomorphic function in a neigh b orh o o d of the real axis and p ( µ ) has no saddle p oints on th e in tegration con tours. By deformin g the integrat ion conto u rs int o the complex plane, one mak es the in tegrand exp onenti ally small ev erywh ere except in the vicinity of the endp oin ts ± q . Then the large x asymptotic analysis of (2.7) reduces to the calculatio n of the integ r al in small vicinities of the endp oints, where f ( { µ p } , { µ h } ) can b e replaced by f ( {± q } , {± q } ). Indeed, suc h r ep lacemen t do es not alter th e leading asym p totic b eha vior of I n ( x ). Now, if f ( { µ p } , { µ h } ) has integ r able singularities at ± q , for example f ( { µ p } , { µ h } ) = ( q − µ h 1 ) ν + ( µ h 1 + q ) ν − f r eg ( { µ p } , { µ h } ), then , when carrying out the asymptotic analysis in the vicinities of ± q , one has to ke ep the singular factors ( q ∓ µ h 1 ) ν ± as th ey are, but we can rep lace the regular part f r eg ( { µ p } , { µ h } ) b y an appropriate constan t f r eg ( {± q } , {± q } ). Again, this appro ximation d o es not alter the leading asymptotic b eh avior of I n ( x ). Th us, wo r king b y analogy , we ma y fairly exp ect th at, as so on as P ex has no sadd le p oin t, th e sum (2.4) w ill lo calize on the endp oin ts of the summation for the int egers p a and h a , n amely , in the language of rapidities, on the tw o endp oin ts of the F ermi zone. In other words, the sum will lo calize on the cr itical excited states 3 . T he part of the amplitud e A ( { µ p } , { µ h }|{ p } , { h } ) smo othly dep ending on the rapidities pla ys th e role of the regular part f r eg in the multiple in tegrals (2.7). There one can set { µ p } and { µ h } equal to their v alues in the given P ℓ class: A ( { µ p } , { µ h }|{ p } , { h } ) − → A ( { q } n + p ∪ {− q } n − p , { q } n + h ∪ {− q } n − h |{ p } , { h } ) = A ( ℓ ) ( { p } , { h } ) . (2.8) Suc h replacemen t should not c hange the leading asymptotic b eha vior o ver x of the form factor series. On the other hand the remaining part of the amplitude A ( ℓ ) ( { p } , { h } ) explicitly d ep endin g on ℓ and on the quant u m num b ers p and h p la ys the role of the singular factors ( q ∓ µ h 1 ) ν ± 3 Note that for discrete sums, the integr ation by part p rocedure could b e carried out, leading in simple cases to such a result in a p erfectly controlled wa y . 7 in the integ r al (2.7) sin ce, for large s ystem size L , it v aries quic kly in the vicinity of the F ermi b ound aries. In fact, in th e thermo dynamic limit, this rapid v ariation of the amplitude as a function of the rapidities m anifests itself as a kinematical p ole of the form f actor whenever one particle and one hole are app roac hing sim u ltaneously the same F ermi b oundary . In terms of the quantum num b ers { p } and { h } ho wev er, there is no singularity and the co efficien ts A ( ℓ ) are alw ays well defined. So, fixing the P ℓ class of critical form factors, we should take the sum in volving A ( ℓ ) ( { p } , { h } ) o v er all the excited s tates (n amely ov er all the p ossible qu an tum n u m b ers { p } and { h } ) within this class b efor e taking the thermo dynamic limit . This p icture fits p erfectly into th e sc heme of approximat ions u sed in th e app lication of CFT (or Luttinger liquid) metho d s to the computation of the asymptotic b eha vior of correlation functions. Moreo ve r , it giv es a clear in terpr etation of why these predictions actually work and are so univ ersal. Let u s p oin t h ere an imp ortan t r emark. Although we just used the m u ltiple-in tegral asymp- totic b ehavi or analogy , the sum that w e consid er here cannot b e replaced straight f orw ardly by an integral sum ev en for large s y s tem size L . Th e non-intege r nature of th e critica l exp onent θ w ould mak e suc h a corresp ond ence quite subtle, even tually pro ducing d ivergen t in tegrals times remaining v anishin g facto r s of L . Th is feature explains the difficulties already noted in the lit- erature, see e.g. [74–76], while trying to use th e form f actor app roac h directly in the con tinuum limit for massless mo dels. Th e main ad v anta ge of our approac h (in comparison to a fi eld theory framew ork) is in computing exactly the form factor sum for L large but finite, hence pro du cing an explicit comp ens ation for the v anishin g factor L − θ and making the th ermo dynamic limit L → ∞ straigh tforward, as w e will s h o w in section 3. 3 Summation of critical form factors W e hav e already insisted that critical states are gathered into v arious classes h a ving distinct v alues of their excitation momen tum in the thermo dyn amic limit. M ore precisely , all states b elonging to the P ℓ class h a ve momentum 2 ℓk F . Also, w e stress that all critical states ha ve, in the thermo d ynamic limit, a v anishing excitation energy (the latter go es to zero as 1 /L ), this indep end en tly of their class. Actually , similarly to w hat happ ens for the excitation energy , for L large but fin ite, the momen tum of any critical state deviates from 2 ℓk F b y terms of order 1 /L . Namely , it follo ws from the Bethe equations (2.1) and (2.2) that P ex ≡ N ′ X j =1 p 0 ( µ ℓ j ) − N X j =1 p 0 ( λ j ) = 2 π L n X k =1 ( p k − h k ) , (3.1) where p and h are the in teger qu an tum n umb ers of particles and holes. It is conv enien t to 8 re-parameterize these in tegers for the critical excited states as follo ws p j = p + j + N ′ , if µ p j = q , p j = 1 − p − j , if µ p j = − q , h j = N ′ + 1 − h + j , if µ h j = q , h j = h − j , if µ h j = − q . (3.2) All the in tegers { p ± } and { h ± } defined ab o ve are p ositiv e and v ary in a range such that lim N →∞ 1 N X p ± j = lim N →∞ 1 N X h ± j = 0 . (3.3) It means that all th e particles and holes collapse to ± q in the thermo dynamic limit. Then the expression for the excitation momen tum of critical states b elonging to the P ℓ class b ecomes P ex = 2 π L ℓN ′ + 2 π L P ( d ) ex . (3.4) Ab o ve, we ha ve set P ( d ) ex = n + p X j =1 p + j − n − p X j =1 p − j + n + h X j =1 h + j − n − h X j =1 h − j + n − p − n + h . (3.5) W e recall that in the therm o dynamic limit N ′ /L → D with D = k F /π . It is customary in th e Bethe ansatz framew ork to describ e excited states in terms of th eir shift fu nction, see e.g. [12]. Th e latter c haracterizes the w a y in which the Bethe ro ots (rapidities of quasi-particles) of a giv en excited state are shifted with resp ect to the ground state ones. It is in fact the result of the non trivial interact ions that hold in the mo del, making the quasi- particles inside the F ermi zone mov e sligh tly due to the cr eation of p article/hole excitat ions. In the case of a generic particle/hole excited state , this shift function dep ends on the r apidities of the particles { µ p } and holes { µ h } o ccurrin g in the give n state. Ho w ever, in the case of critical excited states, this shift fun ction is the same f or all the states b elonging to the same P ℓ class, and is denoted by F ℓ ( λ ). Of course, the f unction F ℓ ( λ ) d ep ends on the mo del, and on th e excited state selected by the lo cal op erator un der consider ation. F or in tegrable mo dels, it is obtained as a solution of a linear integ r al equation 4 . It was sho wn in [60, 62] that the exp onents θ can b e expressed in terms of the F ermi b oundary v alues of this fun ction: F − ℓ ≡ F ℓ ( − q ) , F + ℓ ≡ F ℓ ( q ) + N ′ − N . In particular th e exp onen ts θ for the form facto r s of the class P ℓ are all equ al to a same v alue, θ ℓ , whic h can b e written as [62] θ ℓ = ( F − ℓ + ℓ ) 2 + ( F + ℓ + ℓ ) 2 . (3.6) 4 The linear character of the in tegral equation satisfied by the shift function implies that F ℓ ( λ ) linearly dep ends on the integer ℓ (see section 4). 9 As w as mentio n ed ab ov e the v alues of th e shift function F ± ℓ dep end on the sp ecific mo del and on the op er ators O j that w e deal with. How ever the functional expression (3.6) for θ ℓ in terms of the constan ts F ± ℓ is univ ersal and mo del indep end en t. Th us, the sum o ve r form factors (2.4) b eing restricted to the critical ones tak es th e form hO 1 ( x ′ ) O 2 ( x + x ′ ) i cr = lim L →∞ ∞ X ℓ = −∞ L − θ ℓ e 2 ixℓk F X { p ± } , { h ± } n + p − n + h = n − h − n − p = ℓ e 2 πix L P ( d ) ex A ( ℓ ) ( { p ± } , { h ± } ) . (3.7) Observe that this su m is separated into t wo parts: th e external su m ov er d ifferen t P ℓ classes, and the sums ov er the quant u m num b ers { p ± } , { h ± } within a fixed P ℓ class. W e call the last ones critic al sums of or der ℓ . Calculating these critical sums we d eal only with the d iscrete amplitude A ( ℓ ) ( { p ± } , { h ± } ) w eigh ted with the exp onents exp ( 2 π ix L P ( d ) ex ). It is remark able that A ( ℓ ) has a pu rely kinematical in terpr etation and that, up to a constant factor, its fun ctional form is univ ers al and mo del indep end en t. Consider the ℓ -shifted state | ψ ′ ℓ i with the Bethe ro ots giv en by the follo win g equations: Lp 0 ( µ j ) − N ′ X k =1 ϑ ( µ j − µ k ) = 2 π  j + ℓ − N ′ + 1 2  , j = 1 , . . . , N ′ . (3.8) This excited s tate b elongs to the P ℓ class. Since all the form factors of the P ℓ class scale as L − θ ℓ , we defin e the ren orm alized amplitud e F (1) ℓ F (2) ℓ whic h corresp ond s to matrix elemen ts of the op erators O 1 and O 2 b et ween the ground state and the ℓ -shifted state | ψ ′ ℓ i : F (1) ℓ F (2) ℓ = lim L →∞ L θ ℓ h ψ g |O 1 ( x ′ ) | ψ ′ ℓ ih ψ ′ ℓ |O 2 ( x ′ ) | ψ g i h ψ g | ψ g ih ψ ′ ℓ | ψ ′ ℓ i . (3.9) This quantit y is fin ite in the thermo dynamic limit [61, 62] and, due to the translation inv ariance, it do es not dep en d on x ′ . The discrete amplitudes corresp onding to a giv en P ℓ class can b e expr essed in terms of th e ab o ve quan tity . Namely , on e has [62] A ( ℓ ) ( { p ± } , { h ± } ) = F (1) ℓ F (2) ℓ G 2 (1 + F + ℓ ) G 2 (1 − F − ℓ ) G 2 (1 + ℓ + F + ℓ ) G 2 (1 − ℓ − F − ℓ )  sin( π F + ℓ ) π  2 n + h  sin( π F − ℓ ) π  2 n − h × R n + p ,n + h ( { p + } , { h + }| F + ℓ ) R n − p ,n − h ( { p − } , { h − }| − F − ℓ ) , (3.10 ) where G ( z ) is the Barnes function satisfying G ( z + 1) = Γ( z ) G ( z ), and R n,n ′ ( { p } , { h }| F ) = n Q j >k ( p j − p k ) 2 n ′ Q j >k ( h j − h k ) 2 n Q j =1 n ′ Q k =1 ( p j + h k − 1) 2 n Y k =1 Γ 2 ( p k + F ) Γ 2 ( p k ) n ′ Y k =1 Γ 2 ( h k − F ) Γ 2 ( h k ) . (3.11) 10 Let us commen t the formulas ab o ve . The representa tion (3.1 0 ) was derived straightforw ardly for the p article-hole f orm factors in the X X Z Heisen b erg c hain [62] an d quan tum one-dimen s ional Bose gas [72]. It represents the thermo dynamic limit of known d eterminan t formulas for form factors for fi nite N and L [33, 77]. In this representat ion, only the constant F (1) ℓ F (2) ℓ dep end s on the mo del and on the op erators considered. The rest of the equation (3.10) is univ ersal. Th is o ccurs b ecause the particle–hole form factors for b oth mo dels are prop ortional to generalized C auc hy determinan ts, namely , F ψ g ψ ′ ∼ N Q j >k ( λ j − λ k ) N ′ Q j >k ( µ ℓ j − µ ℓ k ) N Q j =1 N ′ Q k =1 ( λ j − µ ℓ k ) . (3.12) W e call the rhs of (3.12) the generalized C au ch y determinan t b ecause for N ′ = N it giv es known expression for the standard C auc hy determinan t det[1 / ( λ j − µ ℓ k )] in terms of the rap id ities λ j of the quasi-particles in the ground state and µ ℓ k in the excited state. T he app earance of suc h an ob ject in th e form u las for form factors h as a clear physica l inte r pretation. The fact that F ψ g ψ ′ = 0 as so on as λ j = λ k or µ ℓ j = µ ℓ k simply tak es in to account the f ermionic str ucture of the particle/hole description of the ground state and of the excitations. The singularities at λ j = µ ℓ k pro du ce, in the th ermo dynamic limit, p oles o ccurrin g wheneve r the rapidities of particles and h oles coincide at the F ermi b oundaries. S uc h a b eh a vior near the F ermi b oundary has a pur ely kinematical origin and is quite unive r sal. It app ears for example in the form factor b o otstrap program. Th ere th e residues of the n -particle form facto r at th e particle/an tiparticle p oles are giv en in terms of ( n − 1)-particle form factors [20]. On the other hand it is not difficu lt to c h ec k (see e.g. [62]) that the com bination of the generalized Cauc hy d eterminan ts corresp onding to the ratios ( F (1) ψ g ψ ′ F (2) ψ ′ ψ g ) / ( F (1) ℓ F (2) ℓ ) of form factors giv es, in the thermo dy n amic limit, exactly the unive r sal p art of th e amplitud e (3.10 ), pro vid ed F (1) ψ g ψ ′ and F (2) ψ ′ ψ g are critical form factors of the P ℓ class. Thus, a rather complicated, at fi rst sight , dep endence of the f orm factors on the q u an tum num b ers suc h as in (3.10) h as a simple ph ysical origin. One can observe that the expr ession (3.10) is factorize d into the pr o duct of t w o parts corre- sp ondin g to the left and righ t F ermi b ound aries. Such facto r ization o ccurs in th e thermo dy n amic limit, when we neglect all corrections of order o (1) at L → ∞ . If we do not neglect these correc- tions, then w e evidently pro duce subleading con tr ib utions to the form factors of order L − θ ℓ − 1 , L − θ ℓ − 2 etc. On the other hand in this case w e do n ot ha ve a complete factorization of th e fin al answ er, lik e in (3.10), and thus, we can say that higher order corrections describ e an effectiv e coupling b etw een the t w o F ermi b oundaries. Using the r epresen tation (3.10) w e can present the su m of the critical form factors in the 11 follo wing f orm: hO 1 ( x ′ ) O 2 ( x + x ′ ) i cr = lim L →∞ ∞ X ℓ = −∞ L − θ ℓ e 2 ixℓk F F (1) ℓ F (2) ℓ G 2 (1 + F + ℓ ) G 2 (1 − F − ℓ ) G 2 (1 + ℓ + F + ℓ ) G 2 (1 − ℓ − F − ℓ ) × X { p ± } , { h ± } n + p − n + h = n − h − n − p = ℓ exp  2 π ix L P ( d ) ex   sin( π F + ℓ ) π  2 n + h  sin( π F − ℓ ) π  2 n − h × R n + p ,n + h ( { p + } , { h + }| F + ℓ ) R n − p ,n − h ( { p − } , { h − }| − F − ℓ ) . (3.13 ) W e would lik e to men tion th at this equation giv es explicitly the con tribu tion of the critical form factors in the thermo dyn amic limit. It is imp ortan t to notice that the critica l sum s lik e in (3.13) are univ er s al and hence app ear already in the mo dels equiv alen t to free ferm ions . There, of course, the sh ift function is a trivial constan t. I n deed, in free mo d els one cannot obtain a n on-trivial shift fun ction b ecause of the absence of in teraction. Therefore in suc h mo dels a shift b et wee n the ground state quasi-particles and the ones describ ing the excited state either o ccurs due to non-v anishing difference N − N ′ , or it can b e created artificially b y considerin g twiste d excite d states (see section 4). I n con trast, for the interact in g mo dels w e deal with, a non-trivial shift fu nction arises automatically . I t is remark able, ho wev er, that b eing restricted to the states b elonging to a fixed P ℓ class, th is non- trivial sh ift fun ction enters the critical su ms only through its v alues at the F ermi b oundaries, namely th rough the tw o constan ts F ± ℓ . In this sense one can s ay that we reduced our m o del to an effectiv e (d eform ed) free f ermion one in ev ery P ℓ class. W e stress ho wev er, that for int eracting systems suc h a reduction cann ot b e carried out un iformly for all critical form factors, in con trast to wh at happ ens for a free theory: ev ery P ℓ class of form factors should b e d escrib ed by its o wn (deformed b y the v alues F ± ℓ ) free fermion theory . The most imp ortant p rop erty of the critical sum s is that they can b e exp licitly compu ted due to the follo w ing su m mation form ula: X n,n ′ > 0 n − n ′ = ℓ X p 1 < ···

N or h k > N . Thus, the series in eac h h j in (A.4) b ecomes truncated at h j = N , and we obtain the determinan t of an N × N matrix: f 0 ( ν, w ) = det j =1 ,...,N k =1 ,...,N [ δ j k + V ( j, k )] . (A.7) Setting ν = N in (A.5 ) w e get, after simp le alge b ra, V ( j, k ) = w ( j + k ) / 2 N Q m =1 m 6 = j ( j − m ) N Q m =1 m 6 = k ( k − m ) ∞ X p =0 w p N Q m =1 ( p + m ) 2 ( p + k )( p + j ) . (A.8) Let us transform the det( I + V ) obtained as det( I + V ) = det N ( AA T + AV A T ) (det N A ) 2 , (A.9 ) with A j k = w − k / 2 k j − 1 , j, k = 1 , . . . , N . (A.10) Ob viously , (det N A ) 2 = w − N ( N +1) / 2 N Y j >k ( j − k ) 2 = w − N ( N +1) / 2 N − 1 Y k =0 ( k !) 2 . (A.1 1) T o compute the pr o duct AV A T one can use the follo wing id en tit y: N X ℓ =1 ℓ s − 1 ( ℓ + p ) N Q m =1 m 6 = ℓ ( ℓ − m ) = ( − 1) N + s p s − 1 N Q m =1 ( p + m ) , s = 1 , . . . , N , (A.12) whic h follo w s fr om I | z | = R z s − 1 dz ( z + p ) N Q m =1 ( z − m ) = 0 , s = 1 , . . . , N , with R > max( N , p ) . (A.13) Then w e obtain f 0 ( ν, w ) = w N ( N +1) / 2 N − 1 Q k =0 ( k !) 2 det N  N X p =1 w − p p j + k − 2 + ( − 1) j + k ∞ X p =0 w p p j + k − 2  . (A.14) 20 Setting here w = e − t , w e get f 0 ( ν, w ) = e − tN ( N +1) / 2 N − 1 Q k =0 ( k !) 2 det N  ∂ j + k − 2 t  N X p =1 e pt + ∞ X p =0 e − pt   = e − tN ( N +1) / 2 N − 1 Q k =0 ( k !) 2 det N  ∂ j + k − 2 t e N t 1 − e − t  . (A.15) W e n o w use that lim u 1 ,...,u N → u 0 v 1 ,...,v N → v 0 det N Φ( u j , v k ) ∆( u )∆( v ) = N − 1 Y k =0 1 ( k !) 2 det N  ∂ j + k − 2 Φ( u 0 , v 0 ) ∂ u j − 1 0 ∂ v k − 1 0  , ( A.16) where Φ( u, v ) is an y tw o-v ariable function such that all deriv ativ es entering (A.16) exist. The notation ∆ means the V andermonde determinants of the corresp onding v ariables. Clearly the equation (A.15) can b e written as a homogeneous limit of the t yp e (A.16): f 0 ( ν, e − t ) = e − tN ( N +1) / 2 lim u 1 ,...,u N → 0 v 1 ,...,v N → 0 det N  e N ( t + u j + v k ) 1 − e − t − u j − v k  ∆ − 1 ( u )∆ − 1 ( v ) . (A.17) It is a Cauc hy d eterminan t; hen ce, f 0 ( ν, e − t ) = lim u 1 ,...,u N → 0 v 1 ,...,v N → 0 ∆( e − u )∆( e − v ) ∆( u )∆( v ) N Y j =1 e u j + v j N Y j,k =1  1 − e − t − u j − v k  − 1 . (A.18) The calculatio n of th e homogeneous limit b ecomes no w trivial and w e obtain f 0 ( ν, w ) = (1 − w ) − N 2 . (A.19) A.2 Non-in teger ν Here we only giv e a sk etc h of the pro of. L et us hav e | w | = 1 but w 6 = 1. Let also ν satisfy − 1 / 2 ≤ ℜ ( ν ) ≤ 1 / 2. T h e general v alue of ν can b e reac hed b y analytic conti n u ation. It is readily seen that the infinite matrix V in (A.6 ) can b e factorized as a pro du ct of t wo matrices: f 0 ( ν, w ) = det [ I − U ( ν ) U ( − ν )] , (A.20) where U j k ( ν ) = sin π ν π ( j + k − 1) w ( j + k − 1) / 2 Γ( j − ν )Γ( k + ν ) Γ( j )Γ( k ) , j, k = 1 , 2 , . . . . (A.21) Let T ( ν ) and e T ( ν ) b e the follo wing T o eplitz matrices: T j k ( ν ) = w ( k − j ) / 2 j − k + ν and e T j k ( ν ) = w j − k T j k ( ν ) . (A.22) 21 Using the follo wing id en tit y for the Γ-functions [91], ∞ X j =0 Γ( j − ν + 1) j !( j + p ) = Γ( p )Γ( ν )Γ(1 − ν ) Γ( p + ν ) , (A.23) it is easy to sho w th at U ( − ν ) T ( − ν ) = e T ( ν ) U ( ν ) = H ( ν ) , (A.24) where H ( ν ) is a Hanke l matrix: H j k ( ν ) = w ( k + j − 1) / 2 j + k − 1 + ν . (A.25) This means that th e d eterminan t repr esentati on (A.20) can b e rewritten in terms of T o eplitz and Hank el matrices: f 0 ( ν, w ) = det h I − e T − 1 ( ν ) H ( ν ) H ( ν ) T − 1 ( − ν ) i . (A.26) No w w e ha v e to recall some general results for T o eplitz and Hankel matrices. L et a ( z ) b e a smo oth function on a unit circle. Its F our ier co efficien ts are [ a ] n = 1 2 π 2 π Z 0 dθ e − inθ a  e iθ  . W e also define the fun ction ˜ a such th at [˜ a ] n = [ a ] − n . The Hank el and T o eplitz op erators asso ciated with such a smo oth fu n ction can b e d efined as H j k [ a ] = [ a ] j + k − 1 , T j k [ a ] = [ a ] j − k , with j, k = 1 , 2 , . . . . W e will use also th e f ollo w ing iden tit y f or the Hankel and T o eplitz matrices: for any t wo functions smo oth on the u nit circle, it is easy to d emonstrate, using basic prop erties of the F ourier co efficient s , that T [ ab ] = T [ a ] T [ b ] + H [ a ] H [ ˜ b ] . (A.27) Assume that the fu nction a adm its the Wiener–Hopf factorization a ( z ) = a + ( z ) a − ( z ) with unit constan t term, where a + ( z ) = exp ( ∞ X k =1 z k [ln a ] k ) , a − ( z ) = exp ( ∞ X k =1 z − k [ln a ] − k ) . Then the corresp ond ing T o eplitz matrix can also b e factorized as T [ a ] = T [ a − ] T [ a + ]. More- o v er, it can b e easily shown that, for any function b , the follo w ing identitie s hold: T [ ab ] = T [ a − ] T [ a + b ] = T [ a − b ] T [ a + ]. 22 The last result we need w as obtained b y H. Widom [89] for the determinan t of pr o ducts of T o eplitz matrices. If a and b are smo oth and adm it the Wiener–Hopf factorization, the follo wing determinan t can b e computed: det  T − 1 [ a + ] T [ a + b − ] T − 1 [ b − ]  = exp ( ∞ X k =1 k [ln a ] k [ln b ] − k ) . (A.28) No w we can apply these results to compute th e determinant (A.26 ). W e d efine the follo w ing functions: a ( z ) = ∞ X n = −∞ w n/ 2 z n n + ν , b ( z ) = ∞ X n = −∞ w − n/ 2 z n n − ν . (A.29) The corresp ondin g T o eplitz and Hank el op erators are exactly the ones app earing in (A.26): H [ a ] = − H [ ˜ b ] = H ( ν ) , T [ b ] = T ( − ν ) , T [ a ] = e T ( ν ) . (A.30) It is easy to write an explicit form of these functions. Setting w 1 / 2 = e iψ , w e obtain a ( e iθ ) = 2 iπ e − iν ( θ + ψ ) 1 − e 2 iπ ν , with θ ∈ [ − ψ , 2 π − ψ ] , (A.31) b ( e iθ ) = 2 iπ e iν ( θ +2 π − ψ ) 1 − e 2 iπ ν , with θ ∈ [ − 2 π + ψ , ψ ] . (A.32) It is imp ortant to n ote that, although th ese fun ctions are not smo oth on the unit circle, all the results for the T o eplitz and Hanke l matrices men tioned ab ov e can nev ertheless b e used also for suc h fu nctions [90]. Both functions admit the Wiener–Hopf factorizatio n ; it is easy to see that [ln a ] n = ln  π sin π ν  δ n, 0 + ν n w n/ 2 (1 − δ n, 0 ) , [ln b ] n = ln  − π sin π ν  δ n, 0 − ν n w n/ 2 (1 − δ n, 0 ) . (A.33) No w we can easily compute the d eterminan t (A.26 ): f 0 ( ν, w ) = det h I + T − 1 [ a ] H [ a ] H [ ˜ b ] T − 1 [ b ] i = det h T − 1 [ a ] T [ ab ]] T − 1 [ b ] i , (A.34) where w e u sed (A.27). Applying the Wiener–Hopf factorization we rewrite th e determinan t as f 0 ( ν, w ) = det h T − 1 [ a + ] T [ a + b − ] T − 1 [ b − ] i . Finally , u sing (A.28) and (A.33), we obtain f 0 ( ν, w ) = exp ( − ∞ X n =1 ν 2 n w n ) = (1 − w ) − ν 2 . (A.35) 23 A.3 Summation form ula for ℓ 6 = 0 The general ℓ case can b e reduced to the case ℓ = 0 by a simple com bin atorial tric k whic h w e call ‘b ac kground s hift’. It is conv en ient to consider the s u m in (3.14) as a sum o ve r all p ossible excitatio ns o v er a Dirac sea (all n on-p ositiv e in tegers) wh ich can b e constru cted as either particles (p ositiv e in tegers p j ) or holes (non-p ositiv e in tegers 1 − h j ) with a fi x ed difference b et ween num b ers of p articles and holes n p − n h = ℓ . Equiv alent ly , eve r y term of the sum can b e considered as an excitation ov er a shifted Dirac sea (all in tegers less than or equal to ℓ ) w ith particles in the p ositions ℓ + ˜ p k , ˜ p k = 1 , 2 , . . . and holes at 1 + ℓ − ˜ h k , ˜ h k = 1 , 2 , . . . . This second parametrization is con venien t b ecause there the num b er of p articles is equ al to the num b er of holes. Before giving the general case w e illustrate the bac kground shift in the simplest case ℓ = 1. Let us consider the follo wing su m: f 1 ( ν, w ) = ∞ X n =0 X p 1 < ···

k ( p j − p k ) 2 n Q j >k ( h j − h k ) 2 n +1 Q j =1 n Q k =1 ( p j + h k − 1) 2 n +1 Y k =1 Γ 2 ( p k + ν ) Γ 2 ( p k ) n Y k =1 Γ 2 ( h k − ν ) Γ 2 ( h k ) . (A.36) W e tak e an arb itrary term of this su m with a giv en n and giv en p ositions of particles and holes. W e can distinguish t wo p ossible situations: p 1 = 1 and p 1 > 1. In the first case there is no hole at p oint 1 and we d efine th e p ositions of particles and holes with resp ect to the shifted Dirac sea as follo ws: ˜ p j = p j +1 − 1 , j = 1 , . . . , n, ˜ h j = h j + 1 , j = 1 , . . . , n. Th us w e obtain a configuration with n particles and n holes. Sub stituting this in to (A.36), after elemen tary algebra w e arrive at n +1 Y j =1 w p j − 1 n Y j =1 w h j  sin π ν π  2 n n +1 Q j >k ( p j − p k ) 2 n Q j >k ( h j − h k ) 2 n +1 Q j =1 n Q k =1 ( p j + h k − 1) 2 n +1 Y k =1 Γ 2 ( p k + ν ) Γ 2 ( p k ) n Y k =1 Γ 2 ( h k − ν ) Γ 2 ( h k ) = Γ 2 (1 + ν ) n Y j =1 w ˜ p j − 1 n Y j =1 w ˜ h j  sin π ν π  2 n × det n 1 ˜ p j + ˜ h k − 1 ! 2 n Y k =1 Γ 2 ( ˜ p k + ν + 1)Γ 2 ( ˜ h k − ν − 1) Γ 2 ( ˜ p k )Γ 2 ( ˜ h k ) . 24 In the second case ( p 1 > 1) there is a hole at p oint 1. W e set ˜ p j = p j − 1 , j = 1 , . . . , n + 1 , ˜ h 1 = 1 , ˜ h j = h j − 1 + 1 , j = 2 , . . . , n + 1 , and w e obtain a configuration with n + 1 particles and n + 1 holes. S ubstituting this in to (A.36), w e obtain n +1 Y j =1 w p j − 1 n Y j =1 w h j  sin π ν π  2 n n +1 Q j >k ( p j − p k ) 2 n Q j >k ( h j − h k ) 2 n +1 Q j =1 n Q k =1 ( p j + h k − 1) 2 n +1 Y k =1 Γ 2 ( p k + ν ) Γ 2 ( p k ) n Y k =1 Γ 2 ( h k − ν ) Γ 2 ( h k ) = Γ 2 (1 + ν ) n +1 Y j =1 w ˜ p j − 1 n +1 Y j =1 w ˜ h j  sin π ν π  2( n +1) × det n +1 1 ˜ p j + ˜ h k − 1 ! 2 n +1 Y k =1 Γ 2 ( ˜ p k + ν + 1)Γ 2 ( ˜ h k − ν − 1) Γ 2 ( ˜ p k )Γ 2 ( ˜ h k ) . It is easy to see that, u p to a general co efficien t Γ 2 (1 + ν ), we retriev e exactly the same terms as in the ℓ = 0 case (A.1 ). It is also easy to observ e that the bac kgroun d s hift pro d uces a one to one corresp ondence b et w een the configurations of particles and holes in (A.36 ) and in (A.1), up to the replacemen t of ν b y ν + 1. This leads to a v ery simple resu lt: f 1 ( ν, w ) = Γ 2 (1 + ν ) f 0 ( ν + 1 , w ) . (A.37) W e can no w u s e this approac h in the case of general ℓ . Let for d efiniteness ℓ > 0. W e define f ℓ ( ν, w ) for | w | < 1 and arbitrary complex ν by the follo wing s er ies: f ℓ ( ν, w ) = ∞ X n =0 X p 1 < ···

k ( p j − p k ) 2 n Q j >k ( h j − h k ) 2 n + ℓ Q j =1 n Q k =1 ( p j + h k − 1) 2 n + ℓ Y k =1 Γ 2 ( p k + ν ) Γ 2 ( p k ) n Y k =1 Γ 2 ( h k − ν ) Γ 2 ( h k ) . (A.38) W e consider an arbitrary term of this sum with n + ℓ parameters p k and n parameters h j . W e in tro d uce a new equiv alen t set of parameters { ˜ p, ˜ h } as follo ws. • There is a v alue m , 0 6 m 6 ℓ su c h that for k 6 m all the parameters p k 6 ℓ , while for k > m all the p arameters p k > ℓ . Then for k = m + 1 , . . . , n + ℓ we define ˜ p k − m = p k − ℓ . • There are ℓ − m in tegers ℓ > j 1 > j 2 > · · · > j ℓ − m > 1 such that p k 6 = j i , ∀ k , i . Then w e define ˜ h i = ℓ − j i + 1, for i = 1 , . . . , ℓ − m . 25 • W e d efine ˜ h k + ℓ − m = h k + ℓ , for k = 1 , . . . , n . As in the case ℓ = 1 an adv an tage of this new set of parameters is the fact that there are the same num b ers ( n + ℓ − m ) of parameters { ˜ p } and { ˜ h } . After some compu tations (similar to those for the ℓ = 1 case but more cumbersome), we obtain n + ℓ Y j =1 w p j − 1 n Y j =1 w h j  sin π ν π  2 n n + ℓ Q j >k ( p j − p k ) 2 n Q j >k ( h j − h k ) 2 n + ℓ Q j =1 n Q k =1 ( p j + h k − 1) 2 n + ℓ Y k =1 Γ 2 ( p k + ν ) Γ 2 ( p k ) n Y k =1 Γ 2 ( h k − ν ) Γ 2 ( h k ) = w ℓ ( ℓ − 1) / 2 ℓ Y j =1 Γ 2 ( ν + j ) n + ℓ − m Y j =1 w ˜ p j + ˜ h j − 1  sin π ν π  2( n + ℓ − m ) × det n + ℓ − m 1 ˜ p j + ˜ h k − 1 ! 2 n + ℓ − m Y k =1 Γ 2 ( ˜ p k + ν + ℓ )Γ 2 ( ˜ h k − ν − ℓ ) Γ 2 ( ˜ p k )Γ 2 ( ˜ h k ) . It is easy to note th at there is a one to one corresp ond ence b etw een all the p ossible configurations { p, h } and { ˜ p, ˜ h } . Hence w e r etrieve the s um (A.1) in term s of the p arameters ˜ p and ˜ h with a simple prefactor: f ℓ ( ν, w ) = w ℓ ( ℓ − 1) / 2   ℓ Y j =1 Γ 2 ( ν + j )   f 0 ( ν + ℓ, w ) = w ℓ ( ℓ − 1) / 2   ℓ Y j =1 Γ 2 ( ν + j )   (1 − w ) − ( ν + ℓ ) 2 . 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