Large-distance and long-time asymptotic behavior of the reduced density matrix in the non-linear Schrödinger model
aa r X i v : . [ m a t h - ph ] M a y DESY 10-230
Large-distance and long-time asymptotic behavior of the reduceddensity matrix in the non-linear Schrödinger model.
K. K. Kozlowski Abstract
Starting from the form factor expansion in finite volume, we derive the multidimen-sional generalization of the so-called Natte series for the time and distance dependentreduced density matrix at zero-temperature in the non-linear Schrödinger model. Thisrepresentation allows one to read-o ff straightforwardly the long-time / large-distanceasymptotic behavior of this correlator. This method of analysis reduces the complexityof the computation of the asymptotic behavior of correlation functions in the so-calledinteracting integrable models, to the one appearing in free fermion equivalent models.We compute explicitly the first few terms appearing in the asymptotic expansion. Partof these terms stems from excitations lying away from the Fermi boundary, and hencego beyond what can be obtained by using the CFT / Luttinger liquid based predictions.
One-dimensional quantum models with a gapless spectrum are believed to be critical at zero temperature. Inother words, in these models, the ground state expectation values of products of local operators should decay,for large distances of separation between the operators, as some power-law in the distance. It is also believedthat, for a generic class of Hamiltonians, the actual value of the exponents governing this power-law decay, theso-called critical exponents, does not depend on the microscopic details of the interactions in the model, but onlyon its overall symmetries [21, 24]. Therefore two models belonging to the same universality class should becharacterized by the same critical exponents.It has been argued that the equal-time correlation functions in quantum critical one-dimensional models exhibitconformal invariance in the large-distance regime [74]. Hence, its appears plausible to infer their large-distanceasymptotics from those of the associated conformal field theory (CFT). The central charge of the CFT lying in theuniversality class of the model can be deduced from the finite-size corrections to the ground state energy [3, 11].The possibility to compute such finite-size corrections for many integrable models allowed the identification of thecentral charge and scaling dimensions leading to the predictions for the critical exponents [5, 17, 18, 54, 55, 84] DESY, Hamburg, Deutschland, [email protected]
1f the long-distance asymptotics. We remind that it is also sometimes possible to give predictions for the criticalexponents [25, 26, 67] by putting the model in correspondence with a Luttinger liquid [27].Due to their wide applicability and relative simplicity, it is more than desirable to test these CFT / Luttingerliquid based predictions versus some exact calculations carried out on such models; this starting from first principleand in such a way that no approximation (apart from assuming a large distance) is made to the very end. Suchexact computations have been carried out in the 70’s and 80’s on various two-point functions appearing in freefermion equivalent models such as the Ising [13, 70, 71], the XY model at the critical magnetic field [68, 69] or theimpenetrable Bose gas [83]. The latter approaches were then made much more systematic (and also simplified)with the occurrence of a Riemann–Hilbert based approach to free-fermion models’ asymptotics [30] together withthe development of the non-linear steepest-descent method [16]. Indeed, the latter constitutes a relatively simpleand systematic tool for carrying out the asymptotic analysis [29, 31, 32] of Riemann–Hilbert problems associatedwith Fredholm determinants representing the correlators in free fermionic models.However, obtaining long-distance asymptotic expansions of two-point functions for models not equivalent tofree fermions faced several additional problems of technical nature. This fact takes its roots in that even obtainingexplicit expressions for the correlation functions in the so-called interacting integrable models demands to over-come new types of combinatorial intricacy that disappears when dealing with free fermion equivalent models. Thefirst approach to the problem of computing correlators out of the free fermion point can be attributed to Izerginand Korepin [35, 36]. These authors managed to construct certain series representations for the correlation func-tions of the non-linear Schrödinger model and the XXZ spin-1 / n th summand appearingin these series was only defined implicitly by induction. Low- n calculation allowed them for an e ff ective per-turbative characterization of a vicinity of the free fermion point. First manageable expressions for correlators atzero temperature in an interacting integrable model were obtained by Jimbo, Miki, Miwa, Nakayashiki throughthe vertex operator approach. They have provided multiple integral representations for the matrix elements ofthe so-called elementary blocks † in the massive [39] regime of the infinite XXZ chain. These results where laterextended to the massless regime of this chain [40] or to a half-infinite chain subject to a longitudinal magneticfield acting on one of its ends [38]. The multiple integral representations were then reproduced, in the frameworkof the algebraic Bethe Ansatz by Kitanine, Maillet and Terras for the massive and massless regime of the periodicXXZ chain [52]. These two series of works opened a way towards a systematic and e ff ective computation ofvarious types of multiple integral and / or combinatorial representations for the correlation functions in numerousintegrable models. In particular, it was possible to derive e ff ective representations in the case of finite temperature[23], non-equal times [49], models in finite volume [37], higher spin chains [15],... These results should be seenas of uttermost importance from the conceptual point of view: the multiple integral representations for the correla-tors of interacting integrable models naturally provides an interpretation for these objects as a new class of specialfunctions (of the distance, time, coupling constants, ...). However, the complexity of the integrands appearing insuch multiple integral representations makes the thorough description (computation at certain specific values ofthe distance / coupling or extracting their large-distance / long-time behavior etc ...) of these new special functions aquite challenging problem. Many investigations that followed where oriented towards a better understanding ofthese special functions. In particular, it was observed that the multiple integral representations for the elementaryblocks of the XXZ chain can be reduced to one dimensional integrals by a case-by-case analysis [10, 47, 50, 75].This observation led to the proof that it is possible to separate the multiple integrals representing the elementaryblocks of the XXZ chain on the algebraic level [7]. In its turn, this led to the discovery of a Grassmann structurein the XXZ chain [6, 8, 9]. Among many other developments such as the possibility to compute the one-pointfunctions of the sine-Gordon model [41, 42], the existence of such a Grassmann structure constitutes a promisingdirection towards bringing the complexity of the analysis of the correlation function in the XXZ chain to the oneof a free fermion problem. A completely other method for reducing the complexity of the multiple integral repre- † these constitute a basis on which it is possible to decompose all correlation functions of the model / long-time decay of theso-called one-particle reduced density matrix at finite temperature in the non-linear Schrödinger model (NLSM).They have been able to provide operator valued expressions for the correlation length. The dual field vacuum ex-pectation values where computed in [80], which led to a description of the correlation length in terms of a solutionto a non-linear integral equation. We would like to mention that until recently, although formal, the dual fieldapproach was the only approach alternative to a CFT / Luttinger liquid based correspondence that allowed one towrite down certain predictions for the critical exponents.There have also been developments aiming at obtaining alternative types of e ff ective series of multiple integralrepresentations for the distance dependent two-point functions [48, 51]. The guideline being a construction ofsuch a representation that would allow one to carry out a long-distance asymptotic analysis of the two-pointfunctions. This project has met a success in [45]. This article developed a first fairly rigorous method allowingone to compute, starting from "first principles", the long-distance asymptotic behavior of the spin-spin correlationfunction in the massless regime of the XXZ spin-1 / / Luttinger liquid-based predictions for thecritical exponents in this model but also provided explicit expressions for the amplitudes in front of the power-lawwhich, in their turn, cannot be predicted by universality arguments. These explicit formulae for the amplitudeswere then identified with certain, properly normalized in respect to the size of the system, form factors of thespin operators [44, 46]. This identification allowed one to point out the universality in the power of the system-size that one uses for normalizing the form factor associated with the amplitudes. The aforementioned methodof asymptotic analysis was applied recently to study the long-distance asymptotic behavior of the correlationfunctions at finite temperature in the NLSM [63].The large-distance / long-time asymptotic behavior of the correlation functions in massless one dimensionalquantum models goes beyond the predictions stemming from a correspondence with a CFT / Luttinger liquid.Hence, this constitutes a clear motivation for obtaining such an asymptotic behavior from exact considerationson some integrable model as this could help to understand their structure and origin in the general case whereexact computations are not feasible. We would like to mention that there already exists several exact results rel-ative to this regime of the asymptotics in the case of free fermion equivalent models [32, 68, 69, 72]. We alsowould like to remind that there has been proposed recently [28] a non-linear Luttinger liquid theory allowing oneto predict the leading power-law behavior near the edges of the purely hole or particle specta for dynamic struc-ture factors and spectral functions † at low energy and momentum. This approach has been combined with BetheAnsatz considerations to propose values for the amplitudes in front of this behavior [12].This article develops a method allowing one to compute the zero-temperature asymptotic behavior of thecorrelation functions in integrable models starting from the form factor expansion for two-point functions. Thefact that we build our method on the form factor expansion allows us to include the time-dependence and henceaccess to the large-distance and long-time asymptotic behavior. The method has been introduced recently onthe example of the current-current correlators [65]. Here, we provide many elements of rigor to the method andtreat the example of the one particle reduced density matrix in the non-linear Schrödinger model. We wouldlike to stress that this method of asymptotic analysis not only allows us to carry out the analysis in the large- † These quantities refer to space and time Fourier transforms of particular two-point functions / long-time regime but also constitutes an important technical and computational simplification of theapproach proposed in [45]. It has also the advantage of being applicable to a much wider class of algebraic BetheAnsatz solvable models as it solely relies on the universal structure of the form factors in these models. All themore than the number of models where these have been determined is constantly growing [14, 36, 53, 61, 73]. Themain result of this paper can be summarized as follows. We provide a method for constructing a new type of seriesrepresentation for the correlation functions of integrable models, that we call multidimensional Natte series. Thisrepresentation is THE one that is fit for an asymptotic analysis, as the first few terms of the asymptotic expansioncan be simply read-o ff without any e ff ort by looking at the terms of the series. Moreover, the computation of thehigher order asymptotics e ff ectively boils down to the case of a free fermionic model ( ie computation of subleadingasymptotics of the Fredholm determinant of an integrable integral operator) and thus bears the same combinatorialcomplexity. The main implication of our result for physics is that the asymptotics in the time-dependent case arenot only driven by excitations on the Fermi boundary (the latter coincides with the region of the spectrum thatcan be taken into account by using CFT / Luttinger liquid-based predictions), but also by excitations around thesaddle-point λ of the "plane-wave" combination xp ( λ ) − t ε ( λ ) of the dressed momentum p and dressed energy ε of the excitations. Also, we provide explicit expressions and identify the associated amplitudes with the infinitevolume limit of the properly normalized in the size of the system form factors of the field.We stress that although we have been able to set our method in a more rigorous framework then it was donein [45, 63], we still have to rely on a few conjectures. More precisely, we have been able to split the asymptoticanalysis part from the one of proving the convergence of certain series of multiple integrals representing thecorrelators. The part related to asymptotic analysis has been set into a rigorous framework. However, in orderto raise the results of this asymptotic analysis to the level of the two-point function of interest, we still need toassume the convergence of the series of multiple integrals we obtain.The main novelty of this method is that it provides a systematic way for carrying out the asymptotic analysis ofmultiple integrals or series thereof whose integrands contain some large-parameter dependent driving term beingdressed up by coupled functions of the integration variables. We provide a setting that allows one to interpret the"coupled" case as some deformation of the "uncoupled" one. This deformation is such that, provided one is able tocarry out the analysis in the "uncoupled" case (but with a su ffi ciently rich range of functions involved), one is ableto deform the "uncoupled" asymptotics back to the "coupled" case of interest. It is in this respect that the analysiscarried out in this article strongly relies on the results obtained in [62] (where the relevant "uncoupled" series ofmultiple integrals of interest has been analyzed) as well as on the fact that correlation functions of generalized freefermionic models (which correspond to the "uncoupled" case) are naturally representable in terms of Fredholmdeterminants [60].This paper is organized as follows. In section 2, we remind the definition and main properties of the model.We also introduce all the necessary notations allowing us to present the asymptotic behavior of the reduced densitymatrix. In section 3, we present our result and discuss the strategy of our method. Then, in section 4, we outlinethe main properties of the form factors in the model and write down the form factor series for the reduced densitymatrix. We explain how this series can be re-summed into the so-called multidimensional deformation of theNatte series. Once that such a representation is built, thanks to the very properties of the Natte series, it is possibleto literally read-o ff the first few terms of the asymptotic expansion. We gather all the auxiliary and technicalresults in several appendices. We discuss the large size-behavior of the form factors of the fields in appendixA. In appendix B, we derive finite-size Fredholm minor representations for the form factor based expansions ofcertain two-point functions in generalized free fermion models. In appendix C, we prove the existence of thethermodynamic ( ie infinite volume) limit for certain quantities of interest. We also provide various alternativeexpressions for this limit. In appendix D, we develop the theory of functional translation in spaces of holomorphicfunctions. The results established in this appendix constitute the main tools of our analysis. They allow for ane ff ective separation of variables in the intermediate steps so that one is able to carry out various re-summations of4he formulae by building on the results stemming from the generalized free fermion model studied in appendix B. The non-linear Schrödinger model corresponds to the Hamiltonian H NLS = L Z n ∂ y Φ † ( y ) ∂ y Φ ( y ) + c Φ † ( y ) Φ † ( y ) Φ ( y ) Φ ( y ) − h Φ † ( y ) Φ ( y ) o d y . (2.1)The model is defined on a circle of length L , so that the canonical Bose fields Φ , Φ † are subject to L -periodicboundary conditions. In the following, we will focus on the repulsive regime c > h >
0. The Hamiltonian H NLS commutes with the number of particles operator, and thus canbe diagonalized independently in every sector with a fixed number of particles N . In each of these sectors, themodel is equivalent to a N-body gas of bosons subject to δ -like repulsive interactions. The corresponding modelof interacting bosons was first proposed and studied by Girardeau [22] in the c = + ∞ case and then introducedand solved, through the coordinate Bethe Ansatz, by Lieb and Liniger [66] in the case of arbitrary c . It is alsopossible to build the eigenstates of the Hamiltonian by means of the algebraic Bethe Ansatz. This was first doneby Sklyanin [77] directly in the infinite volume. In the case of finite volume L , as observed by Izergin and Korepin[34], it is possible to put the continuous model on a lattice in such a way that the standard construction [20] of thealgebraic Bethe Ansatz holds. At the end of the computations, it is then possible to send the lattice spacing to zeroand recover the spectrum and eigenstates of the continuous model. The fact that this manipulation is indeed fullyrigorous has been shown by Dorlas [19].In the algebraic Bethe Ansatz approach, the Hamiltonian (2.1) appears as a member of a one-parameter com-muting family of operators λ
7→ T ( λ ). It is sometimes useful to consider a β -deformation of this family T β ( λ ),such that T β ( λ ) | β = = T ( λ ). The common eigenstates | ψ β ( { µ } ) i of T β ( λ ) in the N κ -particle sector are param-eterized by a set of real numbers (cid:8) µ ℓ a (cid:9) N κ a = which are the unique solutions to the β -deformed logarithmic Betheequations [4, 85] Lp (cid:0) µ ℓ a (cid:1) + N κ X b = θ (cid:0) µ ℓ a − µ ℓ b (cid:1) = π ℓ a − N κ + ! + i πβ with p ( λ ) = λ and θ ( λ ) = i ln (cid:18) ic + λ ic − λ (cid:19) . (2.2) p is called the bare momentum and θ the bare phase. The set of solutions corresponding to all choices of integers ℓ a ∈ Z such that ℓ < · · · < ℓ N κ yield the complete set of eigenstates in the N κ -particle sector [19].In each sector with a fixed number of particles N κ , the so-called ground state’s Bethe roots are given by thesolution to (2.2) corresponding to the choice of N κ consecutive integers ℓ a = a , with a = , . . . , N κ and β =
0. Thenumber N κ corresponding to the number of particles in the overall ground state of H NLS is imposed by the chemicalpotential h and scales with L . It will be denoted by N in the following. One shows that in the thermodynamic limit( N , L → + ∞ so that N / L → D ) the parameters { λ j } N associated to this ground state condensate on a symmetricinterval (cid:2) − q ; q (cid:3) called the Fermi zone.All other choices of sets of integers ℓ a lead to ( β -deformed) excited states. In principle, these excited statescan also be found in sectors with a di ff erent number N κ , N of particles. It is convenient to describe the excitedstates in the language of particle-hole excitations above the N κ -particle β -deformed ground state † . Namely, such † the β -deformed ground state corresponds to the choice ℓ a = a , with a = , . . . , N κ
5n excited state corresponds to a choice of integers ℓ j in (2.2) such that ℓ j = j for j ∈ [[ 1 ; N κ ]] \ h , . . . , h n and ℓ h a = p a for a = , . . . , n . (2.3)The integers p a and h a are such that p a < [[ 1 ; N κ ]] ≡ { , . . . , N κ } and h a ∈ [[ 1 ; N κ ]]. There is thus a one-to-onecorrespondence between integers ℓ j and the integers h a and p a describing particle-hole excitations.In this picture, the integers h a correspond to holes in the increasing sequence of integers defining the β -deformed ground state roots, whereas p a correspond to extra integers appearing in the equation and can be seenas defining some new position of "particles". Given a solution (cid:8) µ ℓ a (cid:9) N κ corresponding to a fixed choice of integers ℓ < · · · < ℓ N κ it is convenient to introduce their counting function: b ξ { ℓ a } ( ω ) ≡ b ξ { ℓ a } (cid:16) ω | (cid:8) µ ℓ a (cid:9) N κ (cid:17) = p ( ω )2 π + π L N κ X a = θ (cid:0) ω − µ ℓ a (cid:1) + N κ + L − i β L . (2.4)By construction, it is such that b ξ { ℓ a } (cid:0) µ ℓ a (cid:1) = ℓ a / L , for a = , . . . , N κ . Actually, b ξ { ℓ a } ( ω ) defines † a set of backgroundparameters { µ a } , a ∈ Z , as the unique ‡ solutions to b ξ { ℓ a } ( µ a ) = a / L . The latter allows one to define the rapidities µ p a , resp. µ h a , of the particles, resp. holes, entering in the description of (cid:8) µ ℓ a (cid:9) N κ . When the thermodynamic limit of the model is considered, it is possible to provide a slightly more precise de-scription of the solution to the Bethe equations for the ground state { λ a } Na = as well as for any particle-hole type β -deformed excited states (cid:8) µ ℓ a (cid:9) N κ a = above it with N κ − N being fixed and not depending on L or N . Introducing thecounting function for the ground state b ξ ( ω ) ≡ b ξ (cid:16) ω | { λ a } N (cid:17) = p ( ω )2 π + π L N κ X a = θ ( ω − λ a ) + N + L , ie b ξ ( λ a ) = aL , (2.5)it can be shown that, in the thermodynamic limit, it behaves as b ξ ( ω ) = ξ ( ω ) + O (cid:16) L − (cid:17) where ξ ( ω ) = p ( ω )2 π + D N / L → D . (2.6)There, the O (cid:16) L − (cid:17) is uniform and holomorphic in ω belonging to a strip of some fixed width around the real axis, p is the so-called dressed momentum, defined as the unique solution to the integral equation p ( λ ) − q Z − q θ ( λ − µ ) p ′ ( µ ) d µ π = p ( λ ) . (2.7)The parameter q corresponds to the right end of the Fermi interval (cid:2) − q ; q (cid:3) on which the ground state’s Betheroots condensate. It is fixed by the value of the chemical potential h by demanding that the dressed energy ε ( λ ),defined as the unique solution to the below integral equation, vanishes at ± q : ε ( λ ) − q Z − q K ( λ − µ ) ε ( µ ) d µ π = ε ( λ ) with ε ( λ ) = λ − h and ε ( ± q ) = . (2.8) † Note that di ff erent sets of roots (cid:8) µ ℓ a (cid:9) and { µ ℓ ′ a } lead to di ff erent sets of background parameters ‡ The uniqueness of solutions follows from the fact that the solution to (2.2) are such that µ ℓ a ( β ) = µ ℓ a (0) + i πβ/ L . This allows one toshow that b ξ { ℓ a } ( ω ) is strictly increasing on R + i πβ/ L and maps it onto R . Moreover, one can check that ℑ (cid:16)b ξ { ℓ a } (cid:17) , C \ ( R + i πβ/ L ).
6e also remind the relation p F = π D where p F = p ( q ) is the Fermi momentum.In the following, we will focus on the excited states in the N κ = N + β -deformed excited states, it is convenient to introduce the associated shiftfunction b F { ℓ a } ( ω ) ≡ b F (cid:16) ω | (cid:8) µ ℓ a (cid:9) N + (cid:17) = L hb ξ ( ω ) − b ξ { ℓ a } ( ω ) i = π N X a = θ ( ω − λ a ) − π N + X a = θ (cid:0) ω − µ ℓ a (cid:1) − + i β . (2.9)It can be shown that this counting function admits a thermodynamic limit F β that solves the linear integral equation F β λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n µ p a o(cid:8) µ h a (cid:9) − q Z − q K ( λ − µ ) F β µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n µ p a o(cid:8) µ h a (cid:9) d µ π = i β − − π θ ( λ − q ) − π n X a = h θ ( λ − µ p a ) − θ (cid:0) λ − µ h a (cid:1)i . There µ p a , resp. µ h a , are to be understood as the unique solutions to ξ (cid:0) µ p a (cid:1) = p a / L , resp. ξ (cid:0) µ h a (cid:1) = h a / L , where ξ is given by (2.6). Note that we have explicitly insisted on the auxiliary dependence of the thermodynamic limitof the shift function on the positions of the particles / holes. However, in the following, whenever the value of { µ p a } and (cid:8) µ h a (cid:9) will be dictated by the context, we will omit it. We also remind that the above shift functionmeasures the spacing between the ground state roots λ a and the background parameters µ a defined by b ξ { ℓ a } : µ a − λ a = F β ( λ a ) · (cid:2) L ξ ′ ( λ a ) (cid:3) − (cid:16) + O (cid:16) L − (cid:17)(cid:17) .The integral equation for the thermodynamic limit of the shift function F β can be solved in terms of the dressedphase φ ( λ, µ ) and dressed charge Z ( λ ) φ ( λ, µ ) − q Z − q K ( λ − τ ) φ ( τ, µ ) d τ π = π θ ( λ − µ ) and Z ( λ ) − q Z − q K ( λ − τ ) Z ( τ ) d τ π = . (2.10)Namely, F β ( λ ) ≡ F β λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { µ p a } (cid:8) µ h a (cid:9) ! = ( i β − / Z ( λ ) − φ ( λ, q ) − n X a = h φ ( λ, µ p a ) − φ (cid:0) λ, µ h a (cid:1)i (2.11)Here, we also remind two very nice relationships that exist between the dressed phase and dressed charge Z ( λ ) = + φ ( λ, − q ) − φ ( λ, q ) and Z − ( q ) = + φ ( − q , q ) − φ ( q , q ) . (2.12)The first one is easy to obtain and the second one has been obtained in [58, 79].The shift function allows one to compute many thermodynamic limits involving the parameters (cid:8) µ ℓ a (cid:9) . Forinstance, introducing the combination of bare momentum and energy u ( λ ) = p ( λ ) − t ε ( λ ) / x , one readily seesthat for a n particle / hole excited state (cid:8) µ ℓ a (cid:9) at β = N , L → + ∞ N + X a = u (cid:0) µ ℓ a (cid:1) − N X a = u ( λ a ) | β = = n X a = u ( µ p a ) − u ( µ h a ) . (2.13)Above and in the following, u stands for the combination of dressed momenta and energies u ( λ ) = p ( λ ) − t ε ( λ ) / x .It admits the integral representation u ( λ ) = u ( λ ) − q Z − q u ′ ( µ ) φ ( µ, λ ) d µ . (2.14)7he function u ′ admits a unique zero of first order on R . It is believed that this property is preserved for u . Clearly,in virtue of Rouché’s theorem, this holds true for c large enough. We will not purse the discussion of this propertyhere as it goes out of the scope of this paper and will use it as a working hypothesis. In other words, we assumethat given a fixed ratio t / x , there exists a unique λ such that u ′ ( λ ) = u ′′ ( λ ) < | u ′ ( λ ) | → + ∞ when ℜ ( λ ) → ±∞ that, for any value of c > u ′ has a finite number of real zeroes. The case when u ′ has multiple real zeroes of arbitrary order could be treatedwithin out method but would make the analysis heavier.As a concluding remark to this section, we would like to stress that all functions that have been introducedabove (the dressed momentum p , the dressed energy ε , the dressed charge Z and the dressed phase φ ) are holo-morphic in the strip U δ = n z ∈ C : (cid:12)(cid:12)(cid:12) ℑ ( z ) (cid:12)(cid:12)(cid:12) < δ o (2.15)around the real axis. The parameter δ satisfies c / > δ > ffi ciently small so that p is injectiveon U δ and that one has inf λ ∈ U δ (cid:2) ℜ ( Z ( λ )) (cid:3) >
0. We will tacitly assume such a choice in the following each timethe strip U δ will be used. The zero-temperature one-particle reduced density matrix in finite volume refers to the below ground state expec-tation value: ρ N ( x , t ) ≡ D ψ (cid:16) { λ a } N (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Φ ( x , t ) Φ † (0 , (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:16) { λ a } N (cid:17) E · (cid:13)(cid:13)(cid:13)(cid:13) ψ (cid:16) { λ a } N (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) − . (3.1)The parameters { λ a } N correspond to the set of Bethe roots parameterizing the ground state of (2.1). We recall thatthe fields evolve in space and time according to Φ ( x , t ) = e − ixP + it H NLS Φ (0 ,
0) e ixP − it H NLS , (3.2)where H NLS is the Hamiltonian of the model given in (2.1) and P is the total momentum operator. The action of P on the eigenstates of H NLS has been computed in [4].We denote by ρ ( x , t ) = lim N , L → + ∞ ρ N ( x , t ) the, presumably existing, thermodynamic limit of ρ N ( x , t ). We willnot develop further on the existence of this limit, and take this as a quite reasonable working hypothesis. In this article, we carry out several manipulations that lead us to propose a series representation for ρ ( x , t ) givinga straightforward access to its leading large-distance / long-time asymptotic behavior.The starting point of our analysis is the model in finite volume. We will first provide certain re-summationformulae for ρ N ( x , t ) starting from the form factor expansion of (3.1). The latter involves a summation over allthe excited states ( ie over all solutions to (2.2)-(2.3) at β = E ex and the one of the ground state by E gs we argue that all contributions issued from excited states such that E ex − E gs scales with L do not contributeto the thermodynamic limit of the form factor expansion of ρ N ( x , t ). In the light of these arguments, we are led toanalyze an e ff ective form factor series ρ N ;e ff ( x , t ) and a certain γ -deformation ρ N ;e ff ( x , t | γ ) thereof. Our conjectureis that ρ N ;e ff ( x , t | γ = = ρ N ;e ff ( x , t ) has the same thermodynamic limit as ρ N ( x , t ).8e study γ ρ N ;e ff ( x , t | γ ) by means of its Taylor coe ffi cients at γ = ρ ( m ) N ;e ff ( x , t ) ≡ ∂ m ∂γ m ρ N ;e ff ( x , t | γ ) (cid:12)(cid:12)(cid:12) γ = . (3.3)All rigorous, conjecture-free, results of this paper are relative to these Taylor coe ffi cients. We show that theseadmit a well defined thermodynamic limit ρ ( m )e ff ( x , t ). In addition, we provide two di ff erent representations for thislimit, each being a finite sum of multiple integrals. • The first representation is in the spirit of the ones obtained in [45, 63]. It corresponds to some truncation ofa multidimensional deformation of a Fredholm series for a Fredholm minor. • The second representation is structured in such a way that it allows one to read-o ff straightforwardly thefirst few terms of the asymptotic expansion of ρ ( m )e ff ( x , t ). The various terms appearing in this representationare organized in such a way that the identification of those that are negligible ( eg exponentially small) in the x → + ∞ limit is trivial.The above two results are derived rigorously without any approximation or additional conjecture. However,in order to push the analysis a little further and provide results that would have applications to physics, we needto rely on several conjectures. Namely, we assume that1. the series of multiple integrals that arises upon summing up the thermodynamic limits of the Taylor coe ffi -cients P + ∞ m = ρ ( m )e ff ( x , t ) / m ! is convergent;2. this sum moreover coincides with the thermodynamic limit of ρ N ;e ff ( x , t | γ =
1) and hence, due to our firstconjecture, with ρ ( x , t ).These conjectures allow us to claim that ρ ( x , t ) can be represented in terms of a series of multiple integrals. Thelatter series corresponds to a multidimensional deformation of the Natte series expansion for Fredholm minorsof integrable integral operators [62]. This multidimensional Natte series has all the virtues in respect to thecomputation of the long-time / large-distance asymptotic behavior of ρ ( x , t ); it is structured in such a way that onereadily reads-o ff from its very form, the sub-leading and the first few leading terms of the asymptoics.So as to conclude the description of our method, we would like to stress that the aforementioned conjecturesof convergence are supported by the fact that they can be proven to hold in the limiting case of a generalized freefermion model [62]. Unfortunately, the highly coupled nature of the integrands involved in our representationsdoes not allow one for any simple check of the convergence properties in the general + ∞ > c > / long-time asymptotic behavior of the one-particle reduced density matrix We have now introduced enough notations so as to be able to present the physically interesting part of our analysis.Let x > x / t is fixed. Let λ be the associated, presumably unique ( cf (2.14)), saddle-point of u ( λ ) = p ( λ ) − t ε ( λ ) / x . Assume in addition that λ , ± q and λ > − q . Then, under the validity of theaforementioned conjectures , the thermodynamic limit of the zero-temperature one-particle reduced density matrix9 ( x , t ) admits the asymptotic expansion ρ ( x , t ) = s − i π t ε ′′ ( λ ) − xp ′′ ( λ ) p ′ ( λ ) e ix [ u ( λ ) − u ( q ) ] (cid:12)(cid:12)(cid:12) F λ q (cid:12)(cid:12)(cid:12) [ i ( x + v F t )] (cid:2) F λ q ( − q ) (cid:3) [ − i ( x − v F t )] (cid:2) F λ q ( q ) (cid:3) (cid:16) ] q ; + ∞ [ ( λ ) + o (1) (cid:17) + e − ixp F (cid:12)(cid:12)(cid:12) F − qq (cid:12)(cid:12)(cid:12) [ i ( x + v F t )] (cid:2) F − qq ( − q ) − (cid:3) [ − i ( x − v F t )] (cid:2) F − qq ( q ) (cid:3) (1 + o (1)) + (cid:12)(cid:12)(cid:12) F ∅∅ (cid:12)(cid:12)(cid:12) [ i ( x + v F t )] (cid:2) F ∅∅ ( − q ) (cid:3) [ − i ( x − v F t )] (cid:2) F ∅∅ ( q ) + (cid:3) (1 + o (1)) + X ℓ + ,ℓ − ∈ Z η ( ℓ + + ℓ − ) ≥ ∗ C ℓ + ; ℓ − e ix ϕ ℓ + ; ℓ − x ∆ ℓ + ,ℓ − (1 + o (1)) (3.4)The critical exponents governing the algebraic decay in the distance of separation are expressed in terms of thethermodynamic limit F µ p µ h of the shift function (at β =
0) associated with an excited state of (2.1) having oneparticle at µ p and one hole at µ h , namely, F ∅∅ ( λ ) = − Z ( λ )2 − φ ( λ, q ) F − qq ( λ ) = − Z ( λ )2 − φ ( λ, − q ) F λ q ( λ ) = − Z ( λ )2 − φ ( λ, λ ) . (3.5)The type of algebraic decay in the explicit terms in (3.4) can be organized in two classes. There is a squareroot power-law decay ( t ε ′′ ( λ ) − xp ′′ ( λ )) − stemming from the saddle-point λ . All other sources of algebraicdecay appear in the so-called relativistic combinations x ± v F t and exhibit non-trivial critical exponents drivenby the shift function of the underlying type of excitation. We recall that ± v F corresponds to the velocity of theexcitations on the right / left Fermi boundary: v F = ε ′ ( q ) / p ′ ( q ).Each of the three explicit terms in these asymptotics has its amplitude ( (cid:12)(cid:12)(cid:12) F λ q (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) F ∅∅ (cid:12)(cid:12)(cid:12) or (cid:12)(cid:12)(cid:12) F − qq (cid:12)(cid:12)(cid:12) ) given by thethermodynamic limit of properly normalized in the length L moduli squared of form factors of the conjugated field Φ † . More precisely, • (cid:12)(cid:12)(cid:12) F λ q (cid:12)(cid:12)(cid:12) involves the form factor of Φ † taken between the N -particle ground state and an excited state abovethe N + λ and one hole at q . • (cid:12)(cid:12)(cid:12) F − qq (cid:12)(cid:12)(cid:12) corresponds to the case when one considers an excited state above the N + − q and one hole at q . • (cid:12)(cid:12)(cid:12) F ∅∅ (cid:12)(cid:12)(cid:12) corresponds to the case where the form factor average of Φ † is taken between the N and the N + ] q ; + ∞ [ stands for the characteristic function of the interval (cid:3) q ; + ∞ (cid:2) . It is there so as to indicate that, tothe leading order, the contribution stemming from the saddle-point only appears in the space-like regime λ > q .We stress however that hole-type excitations in a vicinity of the saddle-point also contribute in the time-like regimewhere λ ∈ (cid:3) − q ; q (cid:2) . This fact follows from the structure of the terms present in the sum over ℓ + , ℓ − .We would now like to discuss the sum over the integers ℓ + , ℓ − in (3.4). The latter represents the contributionsto the asymptotics associated to the so-called quicker harmonics. Ineed, every term in this sum oscillates with aphase ϕ ℓ + ; ℓ − = ℓ + u ( q ) + ℓ − u ( − q ) − ( ℓ + + ℓ − ) u ( λ ) . (3.6)10t is also caracterized by its own critical exponents ∆ ℓ + ; ℓ − = (cid:0) + ℓ + + ∆ + (cid:1) + (cid:0) ∆ − − ℓ − (cid:1) + | ℓ + + ℓ − | , (3.7)where, ∆ ± = − Z ( ± q )2 − ℓ − φ ( ± q , − q ) − ( ℓ + + φ ( ± q , q ) + ( ℓ + + ℓ − ) φ ( ± q , λ ) . (3.8)Our method of analysis only allows us to prove that the only harmonics present in the asymptotics are thoseoscillating with one of the frequencies ϕ ℓ + ,ℓ − and that they decay, to the leading order ( ie up to o (1) terms), withthe critical exponent ∆ ℓ + ,ℓ − . We are however unable to give an explicit prediction for the amplitudes C ℓ + ,ℓ − . Notethat the sum runs over all integers ℓ ± subject to the constraint η ( ℓ + + ℓ − ) ≥
0. The parameter η depends on theregime: η = λ > q ) and η = − | λ | < q ). Finally, the ∗ in thesums indicates that one should not sum up over those integers ℓ + , ℓ − giving rise to the frequencies that are presentin the first three lines of (3.4).Note that we have organized the large x (with x / t fixed) asymptotic expansion in respect to the various oscil-lating phases. Each phase appears with its own exponent driving the power-law decay in x . Our computationsallowed us to compute the leading ( ie up to o (1) corrections) behavior of each harmonic. Note that the o (1) termsstemming from one of the harmonics may be dominant even in respect to the leading terms coming from anotherharmonic. Remarks
The oscillating phases and amplitudes appearing in (3.4) are reminiscent of the type of excitations that give riseto their associated contribution. Each term in (3.4) can be associated with some macroscopic state of the model.For instance, the one occuring † in the first line of (3.4) corresponds to a macroscopic state characterized by oneparticle at λ and one hole at q . There are infinitely many microscopic realizations of such a macroscopic state.For instance, any excited state realized as one particle at λ , one hole at q , • n + particles µ ( r ) p a and holes µ ( r ) h a located at q in the thermodynamic limit: µ ( r ) p a , µ ( r ) h a −→ N , L → + ∞ q for a = , . . . , n + , • n − particles µ ( l ) p a and holes µ ( l ) h a located at − q in the thermodynamic limit: µ ( l ) p a , µ ( l ) h a −→ N , L → + ∞ − q for a = , . . . , n − .would give rise to the same (from the point of view of energy E = ε ( λ ) , momentum P = p ( λ ) − p ( q ),...)macroscopic state. In a joint collaboration with Kitanine, Maillet, Slavnov and Terras we have shown [43] thatindeed, in the zero-time case, the contribution of a given macroscopic state to the asymptotics is obtained bysumming up over all such zero-momentum excitations on each of the Fermi boundaries. Clearly, this picturepersists in the time-dependent case as well. The only di ff erence being that, in the time-dependent case, the numberof relevant macroscopic states contributing to the asymptotics is bigger (one has to include the contributions ofexcitations around the saddle-point in addition to the excitations on the Fermi boundary). Moreover, we would liketo draw the reader’s attention to the fact that it is precisely the sum over such zero momentum excitations on theFermi boundary that gives rise, through some intricate microscopic mechanism of summation, to the relativisticcombinations ( x + v F t ) α + (in what concerns the left Fermi boundary) and ( x − v F t ) α − (in what concerns the rightFermi boundary) arising in the asymptotics. This mechanism can be considered as yet another manifestation ofconformal field theory on the level of asymptotics. † The o (1) corrections being excluded etc ) of correlation functions, one is brought to the analysis of the con-tributions of "relevant" saddle-points. As one can expect from the saddle-point type analysis of one-dimensionalintegrals, the leading asymptotics are only depending on the local behavior around the saddle-point of the drivingterm. All other details of the integrand do not matter for fixing the exponent governing the algebraic decay. There-fore, it is quite reasonable to expect that models sharing the same types of saddle-points exhibit the same typeof critical behavior. The universality hypothesis [24] stating that models sharing the same symmetry class havethe same value for their critical exponents can be now re-interpreted as the fact that the symmetries of a modeluniquely determine the structure of the driving terms in the saddle-points that are relevant for the asymptotics.As a consequence, the leading power-law decay stemming from the local analysis around these saddle-points isalways characterized by the same critical exponents regardless of the fine, model dependent, function content ofthe integrals describing the correlation functions.We draw the reader’s attention to the fact that the terms appearing in the 2 nd and 3 rd lines of (3.4) correspondsolely to excitations on the Fermi boundaries and confirm the CFT / Luttinger liquid-based predictions for thelong-distance asymptotics ‡ at t = F ∅∅ ( q ) + = Z − ( q )2 , F ∅∅ ( − q ) = − Z − ( q )2 , F − qq ( q ) = Z − ( q )2 − Z ( q ) , F − qq ( − q ) − = − Z − ( q )2 − Z ( q ) . (3.9)However, we do stress that (3.4) clearly shows the need to go beyond the CFT / Luttinger liquid picture so asto provide the correct long-time / large-distance asymptotic behavior of the correlation functions in gapless one-dimensional quantum Hamiltonians. In particular, our results contain additional terms in respect to the predictionsobtained in [2]. Our result has a strong structural resemblance with the non-linear Luttinger liquid based predic-tions for the edge exponents [28] and amplitudes [12] arizing in the low momentum k and low energy ω behaviorof the spectral function ∗ . In this section, we will provide two new representations for the zero-temperature reduced density matrix (3.1)starting from its form factor issued expansion: ρ N ( x , t ) = X ℓ < ··· <ℓ N + ℓ a ∈ Z N + Q a = e ixu ( µ ℓ a ) N Q a = e ixu ( λ a ) (cid:12)(cid:12)(cid:12)(cid:12)D ψ (cid:16)(cid:8) µ ℓ a (cid:9) N + (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Φ † (0 , (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:16) { λ a } N (cid:17) E(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13) ψ (cid:16)(cid:8) µ ℓ a (cid:9) N + (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) · (cid:13)(cid:13)(cid:13)(cid:13) ψ (cid:16) { λ a } N (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) . (4.1)The above series runs through all the possible choices of integers ℓ a , a = , . . . , N + ℓ < · · · < ℓ N + .Below, we shall argue in favor of several reasonable approximations that allow us to reduce the form factorseries to another, e ff ective one, whose structure is simple enough so as to be able to continue the calculationsdirectly on it. ‡ Taking the t → t − e i x t . In the t → δ ( x ) function. The presence of this δ ( x ) function is expected from the form ofthe commutation relations between the fields. However, in the large- x limit of interest to us, it does not contribute. ∗ The latter corresponds to the space and time Fourier transform of h Φ ( x , t ) Φ † (0 , i ] 0; + ∞ [ ( t ) + h Φ † (0 , Φ ( x , t ) i ] −∞ ;0[ ( t ) .1 The e ff ective form factors It has been shown in [64] (slightly di ff erent determinant representations for these form factors have already ap-peared in [61, 73]) that the form factors of the operator Φ † taken between the N -particle ground state { λ a } N andany particle-hole type excited state (cid:8) µ ℓ a (cid:9) N + as described in (2.3) takes the form (cid:12)(cid:12)(cid:12)(cid:12)D ψ (cid:16)(cid:8) µ ℓ a (cid:9) N + (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Φ † (0 , (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:16) { λ a } N (cid:17) E(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13) ψ (cid:16)(cid:8) µ ℓ a (cid:9) N + (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13) ψ (cid:16) { λ a } N (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) = b G N ;1 { p a } n { h a } n ! h b F { ℓ a } , b ξ { ℓ a } , b ξ i · b D N { p a } n { h a } n ! h b F { ℓ a } , b ξ { ℓ a } , b ξ i . (4.2)This representation involves two functionals, the so-called smooth part of the form factor b G N ;1 and the so-calleddiscreet part b D N . These are functionals of the counting function b ξ for the ground state, of the counting function b ξ { ℓ a } for the excited state and of the associated shift function b F { ℓ a } .It has been show in [64], that, in the large L -limit and for any n particle-hole type excited state, with n boundedindependently of L , these functionals satisfy (cid:16) b G N ;1 b D N (cid:17) { p a } n { h a } n ! hb F { ℓ a } , b ξ { ℓ a } , b ξ i = b G N ;1 { p a } n { h a } n ! (cid:2) F , ξ, ξ F (cid:3) · b D N { p a } n { h a } n ! (cid:2) F , ξ, ξ F (cid:3) + O ln LL !! . (4.3)We stress that the functionals appearing on the rhs of the above equation act oni) the thermodynamic limit F ( λ ) of the shift function at β = { ℓ a } N + (2.11),ii) the thermodynamic limit ξ ( λ ) of the counting function (2.6),iii) the counting function associated with F : ξ F ( λ ) = ξ ( λ ) + F ( λ ) / L .We do stress that the shift function F depends implicitly on the rapidities of the particles { µ p a } n and holes (cid:8) µ h a (cid:9) n entering in the description of the excited state of interest, cf (2.11). We chose not to write this dependenceexplicitly in (4.3) as the auxiliary arguments of F are undercurrent by those of the functionals b D N and b G N , . Givenany holomorphic function ν ( λ ) in a neighborhood of R , the explicit expressions for b D N (cid:2) ν, ξ, ξ ν (cid:3) (and b G N (cid:2) ν, ξ, ξ ν (cid:3) )involves two sets of parameters { λ a } N and { µ ℓ a } N + which are defined as follows • µ k , k ∈ Z is the unique ∗ solution to ξ ( µ k ) = k / L , ie the second argument of the functionals; • λ k , k ∈ [[ 1 ; N ]] is the unique † solution to ξ ν ( λ k ) = k / L , ie the third argument of the functionals.We insist that here and in the following, the parameters µ k or λ p entering in the explicit expressions for thesefunctionals are always to be understood in this way. Also, we remind that the integers ℓ a are obtained from theintegers { p a } n and { h a } n as explained in (2.3). • The discreet part
The functional b D N represents the universal part of the form-factor: b D N { p a } n { h a } n ! (cid:2) F , ξ, ξ F (cid:3) = Q Nk = n [ π F ( λ k )] o N + Q a = π L ξ ′ (cid:0) µ ℓ a (cid:1) N Q a = π L ξ ′ F ( λ a ) N Y a = µ ℓ a − µ ℓ N + λ a − µ ℓ N + ! det N " µ ℓ a − λ b . (4.4) ∗ The uniqueness follows from the fact that the dressed momentum p ( λ ) is a biholomorphism on some su ffi ciently narrow strip U δ around the real axis and that p ( λ ) ∈ R ⇒ λ ∈ R . † The uniqueness follows from Rouché’s theorem when L is large enough. N , L behavior of (4.4) can be computed explicitly and is given in (A.2)-(A.4). However, it is the abovefinite product representation of b D N that is suited for carrying out resummations. • The smooth part
The functional b G N , represents the so-called smooth part of the form factor: b G N ; γ { p a }{ h a } ! (cid:2) F , ξ, ξ F (cid:3) = V N ;1 ( µ N + ) V N ; − ( µ N + )det N + (cid:2) Ξ ( µ ) (cid:2) ξ (cid:3)(cid:3) det N (cid:2) Ξ ( λ ) (cid:2) ξ F (cid:3)(cid:3) W n { µ p a } n { µ h a } n ! n Y a = Y ǫ = ± ( V N ; ǫ ( µ p a ) V N ; ǫ ( µ h a ) µ h a − µ N + + i ǫ c µ p a − µ N + + i ǫ c ) × W N { λ a } N { µ a } N ! det N h δ jk + γ b V jk [ F ] (cid:16) { λ a } N ; { µ ℓ a } N + (cid:17)i det N (cid:20) δ jk + γ b V jk [ F ] (cid:16) { λ a } N ; { µ ℓ a } N + (cid:17)(cid:21) . (4.5)Above, we have introduced several functions. For any set of generic parameters (cid:16) { z a } n ; { y a } n (cid:17) ∈ U n δ × U n δ W n { z a } n { y a } n ! = n Y a , b = ( z a − y b − ic ) ( y a − z b − ic )( y a − y b − ic ) ( z a − z b − ic ) and V N ; ǫ ( ω ) = N Y a = ω − λ b + i ǫ c ω − µ b + i ǫ c (4.6)Also we have set Ξ ( µ ) jk (cid:2) ξ (cid:3) = δ jk − K (cid:0) µ ℓ a − µ ℓ b (cid:1) π L ξ ′ (cid:0) µ ℓ b (cid:1) and Ξ ( λ ) jk (cid:2) ξ F (cid:3) = δ jk − K ( λ a − λ b )2 π L ξ ′ F ( λ b ) (4.7)Finally, for any set of generic parameters (cid:16) { z a } n ; { y a } n + (cid:17) ∈ U n δ × U n + δ the entries of the two determinants inthe numerator read b V k ℓ [ ν ] (cid:16) { z a } n ; { y a } n + (cid:17) = − i n + Q a = ( z k − y a ) n Q a , k ( z k − z a ) n Q a = ( z k − z a + ic ) n + Q a = ( z k − y a + ic ) K ( z k − z ℓ )e − i πν ( z k ) − b V k ℓ [ ν ] (cid:16) { z a } n ; { y a } n + (cid:17) = i n + Q a = ( z k − y a ) n Q a , k ( z k − z a ) n Q a = ( z k − z a − ic ) n + Q a = ( z k − y a − ic ) K ( z k − z ℓ )e i πν ( z k ) − z k = z j , j , k are only apparent, cf [45, 64]. ff ective form factors series It is believed † that when computing the T = K form factor expansion of a two-point function h G . S . | O O | G . S . i on the intermediate excited states (as in (4.1)), the contribution of those excited states whose energies di ff ermacroscopically from the ground state’s one ( ie by a quantity scaling as some positive power of L ) vanishes inthe L → + ∞ limit. This can, for instance, be attributed to an extremely quick oscillation of the phase factors andthe decay of form factors for states having large excitation momenta and energies. Therefore, we shall assumein the following that the only part of the form factor expansion in (4.1) that has a non-vanishing contribution to † The computations presented in appendices B.2 and B.3 can be seen as a proof of this statement in the case of a generalized freefermion model. ρ ( x , t ) of ρ N ( x , t ) corresponds to a summation over all those excited states which arerealized as some finite (in the sense that not scaling with L ) number n , n = , , . . . , of particle-hole excitationsabove the ( N + ie notscaling with L ) energy gap above the ground state in the N -particle sector.Even when dealing with excited states realized as a finite number n of particle-hole excitations above the( N + ff erent energy from the one of the N -particle ground state if the rapidities of the particles become very large ( ie scale with L ). This case corresponds,among others, to integers p a becoming very large and scaling with L . We will drop the contribution of such excitedstates in the following.Limiting the sum over all the excited states in the ( N + ff ectively neglects correcting terms in the lattice size L . It thus seemsvery reasonable to assume that, on the same ground, only the leading large- L asymptotic behavior of the formfactors will contribute to the thermodynamic limit of ρ N ( x , t ). It is clearly so when focusing on states with a lownumber n of particle / hole excitations. However, in principle, problems could arise when the number n becomes ofthe order of L . Our assumption lead to the following consequences: • we discard all summations over the excited states having a too large excitation energy. This means that weintroduce a "cut-o ff " in respect to the range of the integers entering in the description of the rapidities of theparticles. Namely, we assume that the integers p a are restricted to belong to the set ‡ B ext L ≡ n n ∈ Z : − w L < n < w L o \ [[ 1 ; N + w L ∼ L + . (4.9) • The oscillating exponent N + P a = u (cid:0) µ ℓ a (cid:1) − N P a = u ( λ a ) is replaced by its thermodynamic limit as given in (2.13). • We drop the contribution of the O (cid:16) L − · ln L (cid:17) terms in the large-size behavior of form factors given in (4.3).Note that, within our approximations, the localization of the Bethe roots (cid:8) µ ℓ a (cid:9) N + for an excited state whoseparticles’ (resp. holes’) rapidities are labeled by the integers { p a } na = (resp. { h a } na = ) does not depend on the specificchoice of the excited state one considers. Hence, we e ff ectively recover a description of the excitations that is inthe spirit of a free fermionic model.Our simplifying hypothesis suggest to raise the below conjecture Conjecture 4.1
The thermodynamic limit of the reduced density matrix ρ N ( x , t ) coincides with the thermody-namic limit of the e ff ective reduced density matrix ρ N ;e ff ( x , t ) : lim N , L → + ∞ ρ N ( x , t ) = lim N , L → + ∞ ρ N ;e ff ( x , t ) (4.10) where ρ N ;e ff ( x , t ) is given by the series ρ N ;e ff ( x , t ) = N + X n = X p < ··· < p n p a ∈B ext L X h < ··· < h n h a ∈B int L n Y a = e − ixu ( µ ha )e − ixu ( µ pa ) · (cid:16)b D N b G N ;1 (cid:17) { p a } n { h a } n ! " F ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { µ p a } (cid:8) µ h a (cid:9) ! ; ξ ; ξ F . (4.11) There B L = { n ∈ Z : − w L < n < w L } , B ext L = B L \ [[ 1 ; N + and B int L = [[ 1 ; N + . Also, the ∗ refers to therunning variable of F on which the two functionals act. ‡ Note that we could choose w L to scale as L + ǫ , where ǫ > ǫ = / cf appendix B.1 for a better discussion of the origin of such a property. ff ective form factor series (4.11) possesses several di ff erent features in respect to the form factor expansion-based series that would appear in a generalized free fermion model ( cf (B.20)). Namely, • the shift function F depends parametrically on the rapidities of the particles and holes entering in thedescription of each excited state one considers, cf (2.11). It is thus summation dependent . • Each summand is weighted by the factor b G N ;1 that takes into account the more complex structure of thescattering and of the scalar products in the interacting model. This introduces a strong coupling betweenthe summation variables { p a } n and { h a } n . Indeed, the explicit expression for b G N ;1 involves complicatedfunctions of the rapidities { µ p a } n and { µ h a } n , which, in their turn, depend on the aforementioned integers.A separation of variables that would allow one for a resummation of (4.11) is not possible for precisely thesetwo reasons. To overcome this problem, we proceed in several steps. First, we introduce a γ -deformation of thee ff ective form factor series such that ρ N ;e ff ( x , t | γ ) | γ = = ρ N ;e ff ( x , t ): ρ N ;e ff ( x , t | γ ) = N + X n = X p < ··· < p n p a ∈B ext L X h < ··· < h n h a ∈B int L n Y a = e − ixu ( µ ha )e − ixu ( µ pa ) (cid:16)b D N b G N ; γ (cid:17) { p a } n { h a } n ! " γ F ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { µ p a } n (cid:8) µ h a (cid:9) n ! ; ξ ; ξ γ F . (4.12)For any finite N and L , it is readily checked by using the explicit representations (4.4) for b D N and (4.5) for b G N ; γ that the γ -deformation ρ N ;e ff ( x , t | γ ) is holomorphic in γ belonging to an open neighborhood of the closed unitdisc † . Hence, its Taylor series around γ = γ =
1. We will then show in theorem C.1 that, givenany fixed m , the m th Taylor coe ffi cient of ρ N ;e ff ( x , t | γ ) at γ = ρ ( m ) N ;e ff ( x , t ) = ∂ m ∂γ m ρ N ;e ff ( x , t | γ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = , (4.13)can be re-summed into a representation where the existence of the thermodynamic limit ρ ( m )e ff ( x , t ) is readily seen.This fact is absolutely not-clear on the level of (4.13) as, due to (A.3)-(A.4), each individual summand vanishesas a complicated power-law in L that depends on the excited state considered. We will then show that one canrepresent the thermodynamic limit ρ ( m )e ff ( x , t ) in another way. This representation is given in terms of a finite sum ofmultiple integrals and corresponds to a truncation of the so-called multidimensional Natte series that we introducebelow. The latter description of ρ ( m )e ff ( x , t ) gives a straightforward access to its asymptotic expansion.The proof of the existence of the thermodynamic limit and the construction of the truncated multidimensionalNatte series for ρ ( m )e ff ( x , t ) constitute the rigorous and conjecture free part of our analysis. This is summarized intheorem 4.1.Working on the Taylor coe ffi cients ρ ( m ) N ;e ff ( x , t ) instead of the full function ρ N ;e ff ( x , t | γ ) taken at γ = ρ ( m ) N ;e ff ( x , t ) (andsubsequently on ρ ( m )e ff ( x , t ) once that the thermodynamic limit is taken). Indeed, by taking the m th γ -derivative at γ =
0, we always end up dealing with a finite number of sums. However, if we had carried out the forthcomingre-summation directly on the level of ρ e ff ( x , t ), we would have ended up with a series of multiple integrals insteadof a finite sum. The convergence of such a series constitutes a separate question that deserves, in its own right,another study. Nonetheless, in the present paper, in order to provide physically interesting results, we will takethis convergence as a reasonable conjecture in a subsequent part of the paper. † The apperent singularity of the determinants at e ± i π F ( λ k ) − = cf (4.8), are candelled by the pre-factors sin [ π F ( λ k )] present in b D N , cf (4.4). .3 An operator ordering Prior to carrying out the re-summation of the form factor expansion for ρ ( m ) N ;e ff ( x , t ), we need to discuss a wayof representing functional translations and generalizations thereof. These objects will allow us to separate thevariables in the sums occuring in (4.13), and carry out the various re-summations. A more precise analysis anddiscussion of these constructions is postponed to appendix D. In the following, we denote by O ( W ), the ring ofholomorphic functions in ℓ variables on W ⊂ C ℓ . Also, here and in the following f ∈ O ( W ), with W non-openmeans that f is a holomorphic function on some open neighborhood of W . Finally, for a set S on which thefunction f is defined we denote k f k S = sup s ∈ S | f ( s ) | .Throughout this paper we will deal with various examples ( b D N , b G ( β ) N , . . . ) of functionals F [ ν ] acting onholomorphic functions ν . The function ν will always be defined on some compact subset M of C whereas theexplicit expression for F [ ν ] will only involve the values taken by ν on a smaller compact ‡ K ⊂ Int ( M ). In fact,all the functionals that we will consider share the regularity property below: Definition 4.1
Let M, K be compacts in C such that K ⊂ Int ( M ) . Let W z be a compact in C ℓ z , ℓ z ∈ N ≡ { , , . . . } .An ℓ z -parameter family of functionals F [ · ] ( z ) depending on a set of auxiliary variables z ∈ W z is said to beregular (in respect to the pair ( M , K ) ) ifi) there exists constants C F > and C ′ > such that for any f , g ∈ O ( M ) k f k K + k g k K < C F ⇒ (cid:13)(cid:13)(cid:13) F (cid:2) f (cid:3) ( · ) − F (cid:2) g (cid:3) ( · ) (cid:13)(cid:13)(cid:13) W z < C ′ k f − g k K , (4.14) where the · indicates that the norm is computed in respect to the set of auxiliary variables z ∈ W z .ii) Given any open neighborhood W y of in C ℓ y , for some ℓ y ∈ N , if ν ( λ, y ) ∈ O (cid:16) M × W y (cid:17) is such that k ν k K × W y < C F , then the function ( y , z )
7→ F (cid:2) ν ( ∗ , y ) (cid:3) ( z ) is holomorphic on W y × W z . Here, the ∗ indicated the runningvariable λ of ν ( λ, y ) on which the functional F [ · ] ( z ) acts.The constant C F appearing above will be called constant of regularity of the functional. This regularity property is at the heart of the aforementioned representation for the functional translation andgeneralizations thereof that we briefly discuss below. However, prior to this discussion we need to define thediscretization of the boundary of a compact.
Definition 4.2
Let M be a compact with n holes (ie C \ M has n bounded connected components) and such that ∂ Mcan be realized as a disjoint union of n + smooth Jordan curves γ a : [ 0 ; 1 ] → ∂ M, ie ∂ M = F n + a = γ a ([ 0 ; 1 ]) .A discretization (of order s) of ∂ M will correspond to a collection of ( n +
1) ( s + points t j , a = γ a ( x j ) withj = , . . . , s + and a = , . . . , n + where x = ≤ x < · · · < x s ≤ = x s + is a partition of [ 0 ; 1 ] of mesh / s: (cid:12)(cid:12)(cid:12) x j + − x j (cid:12)(cid:12)(cid:12) ≤ / s. Suppose that one is given a compact M in C without holes whose boundary is a smooth Jordan curve γ : [ 0 ; 1 ] → ∂ M . Let K be a compact such that K ⊂ Int ( M ) and F a regular functional ( cf definition 4.1) in respect to ( M , K ),for simplicity, not depending on auxiliary parameters z . ‡ Here and in the following, Int ( M ) stands for the interior of the set M.
17t is shown in proposition D.1 that, then, for | γ | small enough one has the identity F " γ W n ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { y a } n { z a } n ! = lim s → + ∞ n Y a = e b g s ( y a ) − b g s ( z a ) · F (cid:2) γ f s (cid:3) | ς k = . (4.15)The function W n appearing above is defined in terms of an auxiliary function ψ ( λ, µ ) that is holomorphic on M × MW n λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { y a } n { z a } n ! = n X a = ψ ( λ, y a ) − ψ ( λ, z a ) whereas f s (cid:16) λ | { ς a } s (cid:17) ≡ f s ( λ ) = s X j = (cid:16) t j + − t j (cid:17) t j − λ · ς j i π . (4.16)Finally, b g s ( λ ) is a di ff erential operator in respect to ς a , with a = , . . . , s : b g s ( λ ) = s X j = ψ (cid:0) t j , λ (cid:1) ∂∂ς j . (4.17)The definitions of b g s and f s involve a set of s + t j of ∂ M .The limit in (4.15) is uniform in the parameters y a and z a belonging to M and in | γ | small enough. Actually, themagnitude of γ depends on the value of the constant of regularity C F . If the latter is large enough, one can evenset γ =
1. The limit in (4.15) also holds uniformly in respect to any finite order partial derivative of the auxiliaryparameters. In particular, n Y a = (cid:26) ∂ p a ∂ y p a a ∂ h a ∂ z h a a (cid:27) · ∂ m ∂γ m F " γ W n ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { µ p a } (cid:8) µ h a (cid:9) ! | γ = = lim s → + ∞ n Y a = (cid:26) ∂ p a ∂ y p a a ∂ h a ∂ z h a a (cid:27) n Y a = e b g s ( y a ) − b g s ( z a ) · ∂ m ∂γ m F (cid:2) γ f s (cid:3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ς k = γ = . (4.18) b b − q qK q C out K A C ( K A ) C in C out Figure 1: Example of discretized contours. In the lhs the compact M is located inside of its boundary C out whereasthe compact K corresponds to K q as defined in (4.27). In this case M has no holes. In the rhs the compact M isdelimited by the two Jordan curves C in and C out depicted in solid lines. The associated compact K (of definition4.1) corresponds to the loop C ( K A ) depicted by dotted lines. The compact M depicted in the rhs has one hole.This hole contains a compact K A inside.We refer to appendix D for a proof of the above statement. Here, we would like to describe in words howformula (4.15) works. By properly tuning the value of γ and invoking the regularity property of the functional F (cid:2) γ f s (cid:3) one gets that, for any s , { ς a } s
7→ F (cid:2) γ f s (cid:3) is holomorphic in a su ffi ciently large neighborhood of 0 ∈ C s .18his allows one to act with the translation operators Q nb = e b g s ( y b ) − b g s ( z b ) . Their action replaces each variable ς a occurring in f s by the combination P nb = (cid:2) ψ ( t a , y b ) − ψ ( t a , z b ) (cid:3) . Taking the limit s → + ∞ changes the sum over t a occurring in f s into a contour integral over C out , cf lhs of Fig. 1. Due to the presence of a pole at t = λ , thiscontour integral exactly reproduces the function W n that appears in the rhs of (4.15).Note that such a realization of the functional translation can also be build in the case of compacts M havingseveral holes as depicted in the rhs of Fig. 1. Also, there is no problem to consider regular functionals F [ · ] ( z )that depend on auxiliary sets of parameters z . In the course of our analysis, in addition to dealing with functional translations as defined above, we will alsohave to manipulate more involved expressions involving series of partial derivatives. Namely, assume that one isgiven a regular functional F (cid:2) f , g (cid:3) of two arguments f and g . Then, the expression : ∂ m γ F (cid:2) γ f s , b g s (cid:3) | γ = : is to beunderstood as the left substitution of the various ∂ ς a derivatives symbols stemming from b g s .More precisely, let e g s be the below holomorphic function of a , . . . , a s e g s ( λ ) = s X j = ψ (cid:0) t j , λ (cid:1) a j . (4.19)The regularity of the functional F ensures that the function { a p } 7→ ∂ m γ F (cid:2) γ f s , e g s (cid:3) is holomorphic in a , . . . , a s small enough. As a consequence, the below multi-dimensional series is convergent for a j small enough: ∂ m ∂γ m · F (cid:2) γ f s , e g s (cid:3) | γ = = X n j ≥ s Y j = a n j j n j ! ∂ n j ∂ a n j j ∂ m ∂γ m · F (cid:2) γ f s , e g s (cid:3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = a j = . (4.20)We stress that as f s (4.16) is a holomorphic function of ς , . . . , ς s , the functional of f s coe ffi cients of the aboveseries give rise to a family of holomorphic functions in the variables ς , . . . , ς s . This analyticity follows, again,from the regularity of the functional F (cid:2) f , g (cid:3) and the smallness of | γ | .The : · : ordering constitutes in substituting a j ֒ → ∂ ς j , j = , . . . , s in such a way that all di ff erential operatorsappear to the left. That is to say, : ∂ m ∂γ m · F (cid:2) γ f s , b g s (cid:3) | γ = : ≡ X n j ≥ s Y j = ∂ n j ∂ς n j j · s Y j = n j ! ∂ n j ∂ a n j j · ∂ m ∂γ m · F (cid:2) γ f s , e g s (cid:3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = a j = ς j = . (4.21)Although there where no convergence issues on the level of expansion (4.20), these can a priori arise on the levelof the rhs in (4.21). Clearly, convergence depends on the precise form of the functional F , and should thus bestudied on a case-by-case basis. However, in the case of interest to us, this will not be a problem due to the quitespecific class of functionals that we will deal with.At this point, two observations are in order. • (4.21) bears a strong resemblance with an s -dimensional Lagrange series. • The functional (of f s ) coe ffi cients appearing in the rhs of (4.21) are completely determined by the func-tional F (cid:2) γ f s , e g s (cid:3) whose expression only involves standard ( ie non-operator valued) functions. Should thisfunctional have two (or more) equivalent representations, then any one of them can be used as a startingpoint for computing the coe ffi cients in (4.20) and then carrying out the substitution (4.21).19ctually, for the class of functionals that we focus on, no convergence issues arise. Indeed, in all of the cases,the m th γ -derivative at γ = : · : ordered functionals of interest appears as a finite linear combinations (orintegrals thereof) of expressions of the type b E m = : ∂ m ∂γ m r Y a = e ǫ a b g s ( λ α a ) · e r Y b = e υ b b g s ( y b ) · F (cid:2) γ f s (cid:3) | γ = : where α a ∈ [[ 1 ; N ]] and ǫ a , υ b ∈ {± } . (4.22)Above y a are some auxiliary and generic parameters whereas λ α a are implicit functions of γ and ς , . . . , ς s . For L -large enough, λ α a is the unique solution to the equation ξ γ f s (cid:0) λ α a (cid:1) = α a / L .The prescription that we have agreed upon implies that one should first substitute b g s ֒ → e g s as defined in (4.19).Then, one computes the m th γ -derivative at γ = e g s . In the process, one has to di ff erentiate in respect to γ the functional F (cid:2) γ f s (cid:3) and the arguments of e g s (cid:0) λ α a (cid:1) .Using that λ α a | γ = = µ α a , one arrives to e E m ≡ ∂ m ∂γ m r Y b = e ǫ b e g s ( λ α b ) e r Y b = e υ b e g s ( y b ) · F (cid:2) γ f s (cid:3) | γ = = r Y b = e ǫ b e g s ( µ α b ) e r Y b = e υ b e g s ( y b ) m X n ,..., n s = s Y j = a n j j · c { n j } (cid:2) f s (cid:3) . (4.23)The sum is truncated at most at n j = m , j = , . . . , m due to taking the m th γ -derivative at γ =
0. It is readilyverified that the { n j }− dependent coe ffi cients c { n j } (cid:2) f s (cid:3) are regular functionals of f s with su ffi ciently large constantsof regularity. It remains to impose the operator substitution on the level of (4.23) a j ֒ → ∂ ς j with all di ff erentialoperators ∂ ς k , k = , . . . , s appearing to the left. It is clearly not a problem to impose such an operator orderon the level of the polynomial part of the above expression. Indeed, the regularity of the functionals c { n j } (cid:2) f s (cid:3) implies that these are holomorphic in ς , . . . , ς s belonging to an open neighborhood N of 0 ∈ C s . Hence, Q sk = ∂ m k ς k · c { n j } (cid:2) f s (cid:3) | ς k = is well-defined for any set of integers { m k } . In fact, in all the cases of interest for us, theneighborhood N is always large enough so as to make the Taylor series issued from the products of translationoperators Q ra = e ǫ a b g s ( µ α a ) Q e rb = e υ b b g s ( y b ) convergent. Their action can then be incorporated by a re-definition of f s leading to E m = m X n ,..., n s = s Y j = (cid:26) ∂ n j ∂ς n j j (cid:27) · c { n j } (cid:2) e f s (cid:3) | ς k = (4.24)with e f s ( λ ) = f s ( λ ) + s X b = ( t b + − t b )2 i π ( t b − λ ) r X k = ǫ k ψ (cid:0) t b , µ α k (cid:1) + e r X k = υ k ψ ( t b , y k ) . In this way, one obtains a (truncated to a finite number of terms) s-dimensional Lagrange series. The procedurefor dealing with such series and taking their s → + ∞ limits is described in proposition D.2. In the following, alloperator valued expressions ordered by : · : should be understood in this way. ffi cients In order to carry out the re-summation of the e ff ective form factor expansion with the help of functional translationsand generalizations thereof, we need to regularize the expression for the functional b G N ; γ with the help of anadditional parameter β . This regularization will allow us to represent it as a regular functional that, moreover, hasa form suitable for carrying out the intermediate calculations.20 he parameter β It is easy to see that (cid:16)b D N b G N ; γ (cid:17) { p a } n { h a } n ! h γ F ; ξ ; ξ γ F i = lim β → (b D N { p a } n { h a } n ! h γ F β ; ξ ; ξ γ F β i b G N ; γ { p a } n { h a } n ! h γ F β ; ξ ; ξ γ F β i ) (4.25)We now introduce a prescription for taking the β → b D N ,the functional b G N ; γ may exhibit singularities should it happen that γ − { e i πγ F β ( λ j ) − } = cf (4.5)-(4.8). For | γ | small enough, as it will always be the case for us, such potential zeroes correspond to the existence of solutions to F β ( λ j ) =
0. For β ∈ e U β with e U β = n z ∈ C : 10 ℜ ( β ) ≥ ℜ ( z ) ≥ ℜ ( β ) and (cid:12)(cid:12)(cid:12) ℑ ( z ) (cid:12)(cid:12)(cid:12) ≤ ℑ ( β ) o (4.26) ℜ ( β ) > ℑ ( β ) > F β ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { y a } n { z a } n ! = ω ∈ U δ , this uniformly in 0 ≤ n ≤ m and (cid:16) β, { y a } n , { z a } n (cid:17) ∈ e U β × U n δ × U n δ . It is clear that the optimal value of β preventing the existence of such solutions depends on the width δ of thestrip U δ and on the integer m .Hence, our strategy is as follows. We will always start our computations on a representation that is holomor-phic in the half-plane ℜ β ≥
0, as for instance (4.12)-(4.13). In the intermediate calculations whose purpose is toallow one to relate the initial representation to another one, we will assume that β ∈ e U β . This will allow us toavoid the problem of the aforementioned poles and represent b G N ; γ in terms of a regular functional that is moreoverfit for carrying out the intermediate calculations. Then, once that we obtain the final expression, we will checkthat this new representation is in fact holomorphic in the half-plane ℜ ( β ) ≥ e U β up to β =
0. As the same property holds for the initial representation, both will be equal at β = β -regularization and treating the β → ff ective form factor expansion-based representation for ρ ( m ) N ;e ff ( x , t ) (4.12) can be simplified with the use of the twoproperties below. The functional b G N ; γ Given A ∈ R + , we define the compact K A contained in U δ : K A = n z ∈ C : (cid:12)(cid:12)(cid:12) ℑ z (cid:12)(cid:12)(cid:12) ≤ δ , (cid:12)(cid:12)(cid:12) ℜ z (cid:12)(cid:12)(cid:12) ≤ A o , (4.27)and denote the open disk of radius r by D , r = { z ∈ C : | z | < r } .As follows from lemma A.2, given A > m ∈ N ∗ fixed, there exists • a complex number β with a su ffi ciently large real part and an imaginary part small enough • a positive number e γ > • a regular functional b G ( β ) γ ; A ≤ n ≤ m , (cid:16) γ, β, { µ p a } n , { µ h a } n (cid:17) ∈ D , e γ × e U β × K nA × K nA one has b G N ; γ { p a } n { h a } n ! h γ F β ; ξ ; ξ γ F β i = b G ( β ) γ ; A " H ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { µ p a } n { µ h a } n ! with H λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { µ p a } n { µ h a } n ! = n X a = λ − µ p a − λ − µ h a . (4.28)The ∗ in the argument of b G ( β ) γ ; A appearing above indicates the running variable of H on which this functional acts.The explicit expression for the functional b G ( β ) γ ; A is given in lemma A.1. The main advantage of such a representationis that all the dependence on the auxiliary parameters is now solely contained in the function H given in (4.28).The constant e γ is such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ F β ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { y a } n { z a } n !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) <
12 uniformly in (cid:16) γ, β, { y a } n , { z a } n (cid:17) ∈ D , e γ × e U β × K nA × K nA and 0 ≤ n ≤ m . (4.29)The functional b G ( β ) γ ; A is regular in respect to the to the pair (cid:0) M G A , C ( K A ) (cid:1) where C ( K A ) in a loop in U δ around K A as depicted in the rhs of Fig. 1 and M G A corresponds to the compact with one hole that is delimited by C in and C out . This hole contains K A . Finally, the parameters β ∈ C and e γ > C G A of b G ( β ) γ ; A satisfies to the estimates C G A π d (cid:0) ∂ M G A , C ( K A ) (cid:1)(cid:12)(cid:12)(cid:12) ∂ M G A (cid:12)(cid:12)(cid:12) + π d (cid:0) ∂ M G A , C ( K A ) (cid:1) > A , (4.30)where (cid:12)(cid:12)(cid:12) ∂ M G A (cid:12)(cid:12)(cid:12) stands for the length of the boundary ∂ M G A and d (cid:0) ∂ M G A , C ( K A ) (cid:1) > C ( K A ) to ∂ M G A .Similarly to the discussion carried out in section 4.3.2 and according to proposition D.1, one has that, uni-formly in n , p ∈ { , . . . , m } , and z j , y j , j = , . . . , m belonging to K A : ∂ p ∂γ p · b G ( β ) γ ; A H ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { z j } n { y j } n | γ = = lim r → + ∞ ( n Y j = e b g , r ( z j ) − b g , r ( y j ) · ∂ p ∂γ p b G ( β ) γ ; A [ ̟ r ] ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η a , p = γ = . (4.31)The compact M G A has one hole. Hence, as discussed in section D.3 one has to consider two sets of discretiza-tion points t , p , p = , . . . , r + C in and t , p , p = , . . . , r + C out . The function ̟ r appearing in (4.31) isa linear polynomial in the variables η a , p with a = , p = , . . . , r : ̟ r (cid:0) λ | { η a , p } (cid:1) = r X p = t , p + − t , p i π (cid:16) t , p − λ (cid:17) η , p + r X p = t , p + − t , p i π (cid:16) t , p − λ (cid:17) η , p . (4.32)Finally, b g , r ( λ ) is a di ff erential operator in respect to η a , p with a = , p = , . . . , r : b g , r ( λ ) = r X p = t , p − λ ∂∂η , p + r X p = t , p − λ ∂∂η , p . (4.33) The functional b D N One can draw a small loop C out around K q in U δ as depicted in the lhs of Fig. 1. Let M b D be the compact withoutholes whose boundary is delimited by C out . Then, given L large enough, the functional b D N , as defined by (4.4), is22 regular functional (in respect to the pair ( M b D , K q )) of γ F β with β ∈ e U β and | γ | ≤ e γ . The parameters β and e γ are as defined previously. This regularity is readily seen by writing down the integral representation: λ j = I ∂ K q ξ ′ γ F β ( ω ) ξ γ F β ( ω ) − j / L · d ω i π , j = , . . . , N (4.34)which holds provided that L is large enough (indeed then all λ j ’s are located in a very small vicinity of the interval (cid:2) − q ; q (cid:3) ). Therefore, according to the results developed in appendix D and outlined in section 4.3, one has that,uniformly in β ∈ e U β and 0 ≤ p , n ≤ m ∂ p ∂γ p (b D N { p a } n { h a } n ! " γ F β · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { µ p a } n (cid:8) µ h a (cid:9) n ! ; ξ ; ξ γ F β | γ = = lim s → + ∞ " n Y a = e b g , s ( µ pa ) − b g , s ( µ ha ) · ∂ p ∂γ p (b D N { p a } n { h a } n ! h γν s ; ξ ; ξ γν s i) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = ς k = . (4.35)The function λ ν s ( λ ) appearing above is holomorphic in some open neighborhood of K q in M b D and givenby ν s (cid:16) λ | { ς a } s (cid:17) ≡ ν s ( λ ) = ( i β − / Z ( λ ) − φ ( λ, q ) + s X j = (cid:16) t j + − t j (cid:17) t j − λ · ς j i π . (4.36)The parameters t j , j = , . . . , s correspond to a discretisation ( cf definition 4.2) of the loop C out around K q in U δ that has been depicted in the lhs of Fig. 1. ς j are some su ffi ciently small complex numbers and b g , s ( λ ) is adi ff erential operator in respect to ς a : b g , s ( λ ) = − s X j = φ (cid:0) t j , λ (cid:1) ∂∂ς j . (4.37)We remind that the parameters λ a appearing in the second line of (4.35) through the expression (4.4) for b D N ,are the unique † solutions to ξ γν s ( λ a ) = a / L . As such, the λ a ’s become holomorphic functions of { ς a } s when thesebelong to a su ffi ciently small neighborhood of the origin in C s . Representation for the Taylor coe ffi cients To implement the simplifications induced by the functional translations on the level of ρ ( m ) N ;e ff ( x , t ), we first observethat all of the rapidities µ p a and µ h a occurring in the course of summation in (4.12) belong to the interval [ − A L ; B L ]with L ξ ( − A L ) = − w L − / L ξ ( B L ) = w L + / A L > B L ). Hence, a fortiori, they belong to the compact K A L . We can thus represent the smooth part functional as b G ( β ) γ ;2 A L . We are interested solely in the m th γ -derivativeof (4.12) at γ =
0. As b D N (cid:0) { p a } n , { h a } n (cid:1) ∝ γ n − and b G ( β ) γ ;2 A L [ ̟ r ] has no singularities around γ =
0, all termsissuing from n particle / hole excitations with n ≥ m will not contribute to the value of the derivative. Hence, † Here, as previously, the uniqueness follows from Rouché’s theorem. By writing down an integral representation for ξ − γν s , one readilyconvinces oneself that, for γ small enough and given any fixed s , λ a is holomorphic in { ς a } s . It is also holomorphic in γ belonging to someopen neighborhood of γ =
23e can truncate the sum over n in (4.12) at n = m . Once that the sum is truncated, we represent the functional ∂ m γ · nb D N · b G ( β ) γ ;2 A L o | γ = with the help of identities (4.35) and (4.31). This leads to ρ ( m ) N ;e ff ( x , t ) = lim β → lim s → + ∞ lim r → + ∞ " m X n = X p < ··· < p n p a ∈B ext L X h < ··· < h n h a ∈B int L n Y a = b E − (cid:0) µ h a (cid:1)b E − ( µ p a ) · ∂ m ∂γ m (b D N { p a } n { h a } n ! h γν s ; ξ ; ξ γν s i b G ( β ) γ ;2 A L [ ̟ r ] ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = ς p = = η a , p . (4.38)We have set b E − ( λ ) = e − ixu ( λ ) − b g ( λ ) with b g ( λ ) ≡ b g , s ( λ ) + b g , r ( λ ) . (4.39)Above, in order to lighten the notation we have not written explicitly the dependence of ν s , ̟ r on the auxiliaryparameters ς p , η a , p nor the one of b E − ( λ ) on the discretization indices r and s . However, we have kept the hat so asto insist on the operator nature of b E − . We do insist that (4.38) has to be understood as it was discussed in section4.3.Starting from representation (4.38), ρ ( m ) N ;e ff ( x , t ) can be related with the m th γ -derivative of the form factor likerepresentation of the functional † X N h γν s , b E − i given in (B.20). Namely, for such an identification to hold, onehas to extend the upper bound in the summation over n from m up to N +
1. This does not alter the result as itcorresponds to adding up a finite amount of terms that are zero due to the presence of γ -derivatives. Then, oneshould use the identity ∂ m ∂γ m (b D N { p a } n { h a } n ! h γν s ; ξ ; ξ γν s i b G ( β ) γ ;2 A L [ ̟ r ] ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = ς p = = η a , p = : ∂ m ∂γ m ( Q N + a = b E − ( µ a ) Q Na = b E − ( λ a ) · Q Na = b E − ( λ a ) Q N + a = b E − ( µ a ) b D N { p a } n { h a } n ! h γν s ; ξ ; ξ γν s i b G ( β ) γ ;2 A L [ ̟ r ] ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = ς p = = η a , p : (4.40)Just as it is the case for the parameters λ j appearing in the expression for b D N , the ones appearing in the pre-factorsof the rhs in (4.40) are the unique solutions to ξ γν s ( λ s ) = s / L . (4.40) is an expression of the type (4.22), and todeal correctly with it one should implement a : · : prescription for the way the di ff erential operators ∂ ς a or ∂ η a , p should be substituted in the rhs of (4.40).With the help of identity (4.40), one is able to force the appearance of the product of function b E − whosepresence is necessary for identifying the sum over the particle-hole type labeling of integers in (4.38) with thefunctional ∂ m γ X N h γν s b E − i | γ = given in (B.20). This leads to the below representation: ρ ( m ) N ;e ff ( x , t ) = lim β → lim s → + ∞ lim r → + ∞ : ∂ m ∂γ m ( Q N + a = b E − ( µ a ) Q Na = b E − ( λ a ) X N h γν s , b E − i b G ( β ) γ ;2 A L [ ̟ r ] ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = ς p = = η a , p : . (4.41) It is shown in appendix C, theorem C.1 that ρ ( m ) N ;e ff ( x , t ) admits a well defined thermodynamic limit that we denote ρ ( m )e ff ( x , t ). This limit is given in terms of a multidimensional analogue of a (truncated) Fredholm series. This series † The latter is a functional of ν s and b g as discussed in subsection 4.3.2
24s close in spirit to the type of series that have appeared in [45, 63]. It is also shown in that appendix (propositionC.1) that it is allowed to exchange • the thermodynamic limit N , L → + ∞ , N / L → D with • the ∂ m γ di ff erentiation along with its associated operator substitution, • the computation of the translation generated by b g , r , • the computation of the s -dimensional Lagrange series associated with b g , s , • the computation of the r → + ∞ and s → + ∞ limits, • the analytic continuation in β from e U β up to β = ρ ( m )e ff ( x , t ) admits the representation ρ ( m )e ff ( x , t ) = lim w → + ∞ lim β → lim s → + ∞ lim r → + ∞ : ∂ m ∂γ m (b E − ( q ) · e − q R − q [ ixu ′ ( λ ) + b g ′ ( λ ) ] γν s ( λ )d λ X C ( w ) E h γν s , b E − i G ( β ) γ ;2 w [ ̟ r ] ) | γ = : . (4.42)This formula deserves a few comments. In the case of complex valued functions e E − , the functional X C ( w ) E (cid:2) γν s , e E − (cid:3) appearing in (4.42) corresponds to a Fredholm minor (B.34) of an integrable integral operator I + V acting on L (cid:0)(cid:2) − q ; q (cid:3)(cid:1) . The kernel V of this operator is given by (B.35). RR + i δ R − i δ b b bb b − q q λ − w w C ( w ) E C ( ∞ ) E Figure 2: The contour C ( w ) E consists of the solid line. The contour C ( ∞ ) E corresponds to the union of the solid anddotted lines. The localization of the saddle-point λ corresponds to the space-like regime. Both contours lie in U δ/ .The subscript C ( w ) E in X C ( w ) E (cid:2) γν s , e E − (cid:3) refers to an auxiliary compact contour entering in the definition of thekernel V . The parameter w delimiting the size of this contour plays the role of a regularization. The limit of anunbounded contour C ( ∞ ) E can only be taken after r and s are sent to infinity and the analytic continuation up to β = G ( β ) γ ;2 w . It can be thought of as the thermodynamiclimit of the functional b G ( β ) γ ;2 w . Its precise expression and properties are discussed in lemma A.1.We also would like to stress that the parameter β defining the region e U β from which one should carry outthe analytic continuation up to β = w as stated in lemma A.1. This dependence is chosen in such a25ay that the constant of regularity C G A for the functional G ( β ) γ ;2 w is large enough so as to make licit all the necessarymanipulations with the translation operators and generalizations thereof.We stress that formula (4.42) constitutes the most important result of appendix C. Indeed, it provides onewith a convenient representation for the thermodynamic limit ρ ( m )e ff ( x , t ). The latter constitutes the first step towardsextracting the large-distance x and long-time t asymptotic behavior of ρ ( m )e ff ( x , t ). The proof of such a representationfor the thermodynamic limit is however quite technical and lengthy. It can definitely be skipped on a first reading.Moreover, should one be solely interested in a "short path" to extracting the asymptotics, we stress that formula(4.42) can be readily obtained without the use of any complicated and long computations. It is enough to take thethermodynamic limit formally on the level of formula (4.41). Such a formal manipulation leads straightforwardlyto the representation (4.42). Theorem 4.1
The thermodynamic limit of the Taylor coe ffi cients ρ ( m )e ff ( x , t ) admits the below truncated multidi-mensional Natte series representation ρ ( m )e ff ( x , t ) = ∂ m ∂γ m ] q ; + ∞ [ ( λ ) √− π xu ′′ ( λ ) × e ix [ u ( λ ) − u ( q ) ] B h γ F λ q ; p i A h γ F λ q i ( x − tv F + i + ) (cid:2) γ F λ q ( q ) (cid:3) ( x + tv F ) (cid:2) γ F λ q ( − q ) (cid:3) G (0)1; γ λ q ! + e ix [ u ( − q ) − u ( q ) ] ( BA − ) h γ F − qq ; p i G (0)1; γ − qq ! ( x − tv F + i + ) (cid:2) γ F − qq ( q ) (cid:3) ( x + tv F ) (cid:2) γ F − qq ( − q ) − (cid:3) + ( BA + ) h γ F ∅∅ ; p i G (0)0; γ ∅∅ ! ( x − tv F + i + ) h γ F ∅∅ ( q ) + i ( x + tv F ) h γ F ∅∅ ( − q ) i + e − ixu ( q ) m X n = X K n X E n ( ~ k ) Z C ( w ) ǫ t H ( { ǫ t } ) n ; x ( { u ( z t ) } ; { z t } ) (cid:2) γ F z + z − (cid:3) B (cid:2) γ F z + z − ; p (cid:3) ( x − tv F + i + )[ γ F z + z − ( q ) ] ( x + tv F )[ γ F z + z − ( − q ) ] G (0) |{ z + }| ; γ { z + }{ z − } ∪ { q } ! d n z t (2 i π ) n | γ = . (4.43) There, we have introduced the notations (cid:8) z + (cid:9) = (cid:8) z t , t ∈ J { ~ k } : ǫ t = (cid:9) , (cid:8) z − (cid:9) = (cid:8) z t , t ∈ J { ~ k } : ǫ t = − (cid:9) , (cid:12)(cid:12)(cid:12)(cid:8) z + (cid:9)(cid:12)(cid:12)(cid:12) ≡ (cid:8) z t , t ∈ J { ~ k } : ǫ t = (cid:9) . (4.44) F ∅∅ , F λ q , F − qq have been defined in (3.5) and, in general, we agree uponF z + z − ( λ ) ≡ F λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { z + }{ z − } ∪ { q } ! = − Z ( λ )2 − X t ∈ J { k } ǫ t = φ ( λ, z t ) + X t ∈ J { k } ǫ t = − φ ( λ, z t ) . (4.45) The function G (0) n ; γ is related to the thermodynamic limit of the smooth part of the form factor. Its expression canbe found in (A.8) . The functionals B , A ± and A are given by B (cid:2) ν, p (cid:3) = (cid:2) κ [ ν ] ( − q ) (cid:3) ν ( − q ) (cid:2) κ [ ν ] ( q ) (cid:3) ν ( q ) G (1 + ν ( q )) G (1 − ν ( − q )) e i π ( ν ( q ) − ν ( − q ) ) (cid:2) qp ′ ( q ) (cid:3) ν ( q ) (cid:2) qp ′ ( − q ) (cid:3) ν ( − q ) (2 π ) ν ( q ) − ν ( − q ) e q R − q ν ′ ( λ ) ν ( µ ) − ν ′ ( µ ) ν ( λ ) λ − µ d λ d µ , (4.46) where G is the Barnes double Gamma function, A + (cid:2) ν, p (cid:3) = − q κ − [ ν ] ( q ) (cid:2) qp ′ ( q ) (cid:3) ν ( q ) + Γ + ν ( q ) − ν ( q ) ! − i πν ( q ) − , κ [ ν ] ( λ ) = exp ( − Z q − q ν ( λ ) − ν ( µ ) λ − µ d µ ) , (4.47)26 nd A − (cid:2) ν, p (cid:3) = − q κ [ ν ]( − q ) Γ − ν ( − q ) ν ( − q ) ! (cid:2) qp ′ ( − q ) (cid:3) ν ( − q ) − e − i πν ( − q ) − A [ ν ] = e − i π κ − [ ν ]( λ ) λ − q λ + q ! ν ( λ ) . (4.48) The second sum appearing in the last line of (4.43) runs through all the elements ~ k belonging to K n = ( ~ k = ( k , . . . , k n + ) : k n + ∈ N ∗ and k a ∈ N , a = , . . . , n such that n X a = ak a + k n + = n ) . (4.49) Once that an element of K n has been fixed, one defines the associated set of triplets J { ~ k } :J { ~ k } = (cid:26) ( t , t , t ) , t ∈ [[ 1 ; n + , t ∈ [[ 1 ; k t ]] , t ∈ [[ 1 ; t − n δ t , n + ]] (cid:27) . (4.50) The third sum runs through all the elements { ǫ t } t ∈ J { ~ k } belonging to the set E n ( ~ k ) = ( { ǫ t } t ∈ J { ~ k } : ǫ t ∈ {± , } ∀ t ∈ J { ~ k } with t X t = ǫ t = t = , . . . , n and k n + X p = ǫ n + , p , = ) . In other words, E n ( ~ k ) consists of n-uples of parameters ǫ t labeled by triplets t = ( t , t , t ) belonging to J { ~ k } .Each element of such an n-uple takes its values in {± , } . In addition, the components of this n-uple are subject tosummation constraints. These hold for any value of t or t and are di ff erent whether one deals with t = , . . . , nor with t = n + .The integral appearing in the n th summand occurring in the third line of (4.43) is n-fold. The contours ofintegration C ( w ) ǫ t depend on the choices of elements in E n ( ~ k ) and are realized as n-fold Cartesian products of one-dimensional compact curves that correspond to various deformations of the base curve C ( w ) E depicted in Fig. 2.In the w → + ∞ limit, these curves go to analogous deformations of the base curve C ( ∞ ) E . All these contours lie inU δ/ The integrand H ( { ǫ t } ) n ; x ( { u ( z t ) } ; { z t } ) [ ν ] is a regular functional of ν , that is simultaneously a function of u ( z t ) and z t with t running through the set J { ~ k } . This functional depends on the choice of an element { ǫ t } t ∈ J { ~ k } from E n ( ~ k ) and on x. It appears originally as a building block of the Natte series (cf appendix B.5 for more details). We stress that all summands involving the functional G ( β ) n ; γ are well defined at β =
0. The potential singularitiespresent in G ( β ) n ; γ are canceled by the zeroes of the pre-factors. Proof —
As a starting point for the proof, we need to introduce the below set of functions depending on the auxiliaryparameters a p , b , p and b , p . As it has been discussed in section 4.3.2, these functions will allow us to computethe (functional) coe ffi cients necessary for carrying out the operator substitution. We set e E − ( λ ) = e − ixu ( λ ) − e g ( λ ) with e g ( λ ) = e g , s ( λ ) + e g , r ( λ ) , (4.51)where e g , s ( λ ) = − s X p = φ (cid:0) t p , λ (cid:1) a p and e g , r ( λ ) = r X p = b , p t , p − λ + r X p = b , p t , p − λ . (4.52)27t is readily checked with the help of lemma A.1 and proposition B.3 that for γ small enough F given below is aregular functional of ν s , ̟ r and e g : F (cid:2) γν s , e g , ̟ r (cid:3) ( γ ) = e E − ( q ) · e − q R − q [ ixu ′ ( λ ) + e g ′ ( λ ) ] γν s ( λ )d λ X C ( w ) E (cid:2) γν s , e E − (cid:3) G ( β ) γ ;2 w [ ̟ r ] . (4.53)In particular F (cid:2) γν s , e g , ̟ r (cid:3) ( γ ) is holomorphic in γ , at least for γ small enough. In order to implement the operatorsubstitution, we have to compute the Taylor coe ffi cients of the series expansion of F (cid:2) γν s , e g , ̟ r (cid:3) ( γ ) into powersof b , p , b , p with p = , . . . , r and a p with p = , . . . , s . These Taylor coe ffi cients are solely determined by thefunctional F (cid:2) γν s , e g , ̟ r (cid:3) ( γ ) depending on the classical function e g (4.19). Therefore, one can use any equivalentrepresentation for F (cid:2) γν s , e g , ̟ r (cid:3) ( γ ) as a starting point for computing the various partial derivatives in respect to b j , p or a p . In other words, one can use any equivalent series representation † for the Fredholm minor X C ( w ) E (cid:2) γν s , e E − (cid:3) .Clearly, di ff erent series representations for the Fredholm minor will lead to di ff erent type of expressions for theTaylor coe ffi cients. However, in virtue of the uniqueness of the Taylor coe ffi cients, their values coincide. Asshown in [62], the Fredholm minor we’re interested in admits the so-called Natte series representation. The latterseries of multiple integrals is built in such a way that it gives a quasi-direct access to the asymptotic behavior of X C ( w ) E (cid:2) γν, e E − (cid:3) . It is thus clear that this is THE series representation that is fit for providing the large-distance / long-time asymptotic expansion of the two-point function. We will thus take this series representation as a startingpoint for our calculations.The first remarkable consequence of the use of the Natte series is that the exponential pre-factor in front of X C ( w ) E (cid:2) γν s , e E − (cid:3) in (4.53) exactly compensates the one appearing in the Natte series (B.48). Once that these pre-factors are simplified, one should take the m th γ -derivative of the remaining part of the Natte series representation(B.48) for X C ( w ) E (cid:2) γν s , e E − (cid:3) G ( β ) γ ;2 w [ ̟ r ]. One of the consequences of taking the m th - γ derivative is that the Natte seriesgiven in (B.48) becomes truncated at n = m due to the property ii ) of the functions H ( { ǫ t } ) n ; x ( cf appendix B.5): ∂ m ∂γ m F (cid:2) γν s , e g , ̟ r (cid:3) ( γ ) | γ = = e ix [ u ( λ ) − u ( q ) ]e e g ( λ ) − e g ( q ) ∂ m ∂γ m ( B (cid:2) γν s ; u + i + (cid:3) A (cid:2) γν s (cid:3) √− π u ′′ ( λ ) x · x γ ν s ( q ) + γ ν s ( − q ) G ( β ) γ ;2 w [ ̟ r ] ) | γ = × ] q ; + ∞ [ ( λ ) + e ix [ u ( − q ) − u ( q ) ]e e g ( − q ) − e g ( q ) ∂ m ∂γ m ( ( BA − ) (cid:2) γν s ; u + i + (cid:3) x (1 − γν s ( − q )) + γ ν s ( q ) G ( β ) γ ;2 w [ ̟ r ] ) | γ = + ∂ m ∂γ m ( ( BA + ) (cid:2) γν s ; u + i + (cid:3) x γ ν s ( − q ) + ( γν s ( q ) + G ( β ) γ ;2 w [ ̟ r ] ) | γ = + e − e g ( q ) − ixu ( q ) m X n = X K n X E n ( ~ k ) I C ( w ) ǫ t Y t ∈ J { k } n e ǫ t e g ( z t ) o · ∂ m ∂γ m ( H ( { ǫ t } ) n ; x ( { u ( z t ) } ; { z t } ) (cid:2) γν s (cid:3) B (cid:2) γν s ; u + i + (cid:3) x γ ν s ( q ) + γ ν s ( − q ) G ( β ) γ ;2 w [ ̟ r ] ) | γ = d n z t (2 i π ) n . (4.54)It follows from lemma A.1, representation (B.50) and the explicit formulae for the functionals B , A and A ± (4.46)-(4.48) that the functionals occurring in (4.54) are all regular ( cf definition 4.1). Moreover, as follows fromthe previous discussion relative to the procedure of taking the β → ℜ ( β ) > C G w of the functionals G ( β ) γ ;2 w is su ffi ciently large to beable to apply proposition D.1 and corollary D.1 (due to the estimates (4.30) for C G w , the constant γ occurring in(D.4) is greater then 1 for w large enough, which is the limit of interest) to this functional. Proposition D.1 andcorollary D.1 are also directly applicable to all functionals of γν s in as much as, at the end of the day, one sets γ = † One natural representation that can be used as a starting point for taking the derivatives is the Fredholm series-like representation for X C ( w ) E (cid:2) γν s , e E − (cid:3) . In fact, it is this series representation that has been used for the computations carried out in theroem C.1. a p ∂ ς p and b i , p ∂ η i , p on the level of (4.54) insuch a way that all the partial derivative operators appear to the left of all η i , p and ς p dependent functions. The firsttwo lines in (4.54) will give rise to translation operators. In the case of the ultimate line in (4.54), this operatorsubstitution will produce expressions of the type + ∞ X n p ≥ + ∞ X n a , p ≥ s Y p = n p ! ∂ n p ∂ς n p p r Y p = Y a = n a , p )! ∂ n a , p ∂η n a , p a , p I C ( w ) ǫ t s Y p = h Ω p ( { z t } ) i n p r Y p = Y a = h Ω ′ a , p ( { z t } ) i n a , p × ∂ m ∂γ m ( H ( { ǫ t } ) n ; x ( { u ( z t ) } ; { z t } ) (cid:2) γν s (cid:3) B (cid:2) γν s ; u + i + (cid:3) x γ ν s ( q ) + γ ν s ( − q ) G ( β ) γ ;2 w [ ̟ r ] ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = ς p = = η a , p d n z t (2 i π ) n . (4.55)Where Ω p and Ω ′ a , p take the form Ω p ( { z t } ) = φ (cid:16) t p , q (cid:17) − X t ∈ J { k } ǫ t φ (cid:16) t p , z t (cid:17) and Ω ′ a , p ( { z t } ) = − t a , p − q + X t ∈ J { k } ǫ t t a , p − z t . (4.56)One can compute the r , s → + ∞ limit of such series of integrals by applying corollary † D.1 and observingthat C ( w ) ǫ t is a Cartesian product of a finite number of compact one dimensional curves that are contained in U δ/ .In fact, the result of this corollary allows one to carry out the operator substitution in (4.54) directly under theintegration sign. In other words, one is allowed to replace e g , s ֒ → b g , s and e g , r ֒ → b g , r directly on the level of(4.54), this without pulling out the partial ς p or η a , p derivatives out of the integrals. Hence, one is brought tocomputing the action of translation operators. The latter can be estimated by applying proposition D.1. Again,there is no problem to apply this proposition either because we compute the m th γ -derivative at γ = γ can be as small as desired in the case of functionals of γν s ) or because the constant of regularity is large enoughfor G ( β ) γ ;2 w . As follows from this proposition, one can permute the partial γ -derivative symbols at γ = s and r translation operators. It then remains to take the r → + ∞ and the s → + ∞ limits. As ineach case the convergence is uniform, the limit can be taken directly under the finite sum, compact integrals andpartial γ -derivatives symbols.Then, in order to compute the e ff ect of the s → + ∞ limit we apply the identity (4.15) (also cf appendix D.3):lim s → + ∞ n Y a = e b g , s ( z a ) − b g , s ( y a ) · F (cid:2) γν s (cid:3) = F (cid:2) γ F β (cid:3) with F β ( λ ) ≡ F β λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { z a } n { y a } n ! , (4.57)valid for any regular functional F , | γ | small enough and z a , y a all lying in U δ . Here, we would like to remind that F β appearing above corresponds to the thermodynamic limit of the β -deformed shift function, cf (2.11). Similarly,lim r → + ∞ n Y a = e b g , r ( z a ) − b g , r ( y a ) · G ( β ) γ ;2 w [ ̟ r ] = G ( β ) γ ;2 w " H ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { z a } n { y a } n ! with H λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { z a } n { y a } n ! = n X a = λ − z a − λ − y a . (4.58)All this for (cid:16) { z a } n ; { y a } n (cid:17) ∈ K n w . Then, by applying lemma A.1 backwards, we get G ( β ) γ ;2 w " H ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { z a } n { y a } n ! = G ( β ) n ; γ { z a } n { y a } n ! . (4.59) † This corollary can be applied to G ( β ) γ ;2 w precisely because its constant of regularity is large enough. G ( β ) n ; γ has been defined in (A.8).Therefore, we obtain ρ ( m )e ff ( x , t ) = lim w → + ∞ lim β → ∂ m ∂γ m ( B (cid:2) γ e F λ q ; u + i + (cid:3) A (cid:2) γ e F λ q (cid:3) √− π u ′′ ( λ ) x e ix [ u ( λ ) − u ( q )] x (cid:2) γ e F λ q ( q ) (cid:3) + (cid:2) γ e F λ q ( − q ) (cid:3) G ( β )1; γ λ q ! + e ix [ u ( − q ) − u ( q )] x (cid:2) γ e F − qq ( q ) (cid:3) + (cid:2) − γ e F − qq ( − q ) (cid:3) · ( BA − ) (cid:2) γ e F − qq ; u + i + (cid:3) G ( β )1; γ − qq ! + ( BA + ) (cid:2) γ e F ∅∅ ; u + i + (cid:3) x (cid:2) + γ e F ∅∅ ( q ) (cid:3) + (cid:2) γ e F ∅∅ ( − q ) (cid:3) G ( β )0; γ ∅∅ ! + e − ixu ( q ) m X n = X K n X E n ( ~ k ) I C ( w ) ǫ t B (cid:2) γ e F z + z − ; u + i + (cid:3) x (cid:2) γ e F z + z − ( q ) (cid:3) + (cid:2) γ e F z + z − ( − q ) (cid:3) × H ( { ǫ t } ) n ; x (cid:0)(cid:8) u ( z t ) (cid:9) ; (cid:8) z t (cid:9) (cid:1)(cid:2) γ e F z + z − (cid:3) G ( β ) |{ z + }| ; γ { z + }{ z − } ∪ { q } ! d n z t (2 i π ) n ) | γ = . (4.60)Here e F z + z − ( λ ) = F z + z − ( λ ) + i β Z ( λ ) and F z + z − has been defined in (4.45).Once that the functional translations have been computed, one should carry out the analytic continuation ofthe expression in brackets from β ∈ e U β up to β = w to + ∞ . For this, we recall that the functions H ( { ǫ t } ) n ; x admit the below decomposition ( cf (B.50)): H ( { ǫ t } ) n ; x (cid:0)(cid:8) u ( z t ) (cid:9) ; (cid:8) z t (cid:9) (cid:1)(cid:2) γ e F z + z − (cid:3) = e H ( { ǫ t } ) n ; x (cid:16) { γ e F z + z − ( z t ) } , { u ( z t ) } , { z t } (cid:17) Y t ∈ J { k } (cid:16) κ [ e F z + z − ] ( z t ) (cid:17) − ǫ t Y z t ∈{ z + } (cid:18) e − i πγ e F z + z − ( z t ) − (cid:19) . (4.61)It follows from the way H ( { ǫ t } ) n ; x depends on the set of its ν -type arguments (4.61) and from the expression for thefunctional B [ ν, u ] (4.46) and G ( β ) n ; γ (A.8) that all of the expressions one deals with contain the combination G (cid:16) − γ e F z + z − ( − q ) (cid:17) G (cid:16) + γ e F z + z − ( q ) (cid:17) Y z t ∈{ z + } (cid:18) e − i πγ e F z + z − ( z t ) − (cid:19) det C q + ǫ h I + γ V (cid:2) γ e F z + z − (cid:3)i det C q + ǫ h I + γ V (cid:2) γ e F z + z − (cid:3)i In virtue of proposition A.1, the function appearing above is holomorphic in ( β, γ, { z + } , { z − } ) ∈ {ℜ ( β ) ≥ } ×D , × U |{ z + }| δ × K |{ z + }| q + ǫ .The function e H ( { ǫ t } ) n ; x (cid:16) { γ e F z + z − ( z t ) } , { u ( z t ) } , { z t } (cid:17) is analytic in ( γ, β ) ∈ D , e γ × e U β (here e γ is chosen so that (cid:12)(cid:12)(cid:12) γ e F z + z − ( z t ) (cid:12)(cid:12)(cid:12) < / z t , t ∈ J { ~ k } belonging to U δ ), and integrable in respect to the { z t } t ∈ J { ka } variables. The remaining part of G ( β ) n ; γ has also the same properties. As the integrals are compactly supportedit follows that the whole expression appearing inside of the "big" brackets in (4.60) is holomorphic in ( γ, β ) ∈D , e γ × e U β . As a consequence, the m th γ -derivative at γ = β =
0. To get the value of theanalytic continuation at this point it is in fact enough to set β = w → + ∞ . This operation will result in an extension of the integrationcontours from bounded ones C ( w ) ǫ t to ones going to infinity C ( ∞ ) ǫ t . Hence, one needs to check that the resultingintegrals will be convergent. Note that the function F z + z − ( z t ) are bounded whenever z t or any of the variablesbelonging to the set { z + } or { z − } goes to infinity. Also, the function G ( β ) n ;1 is bounded at infinity by a polyno-mial in z t of degree n , this uniformly in respect to γ -derivatives of order 0 , . . . , m . Therefore, as the functions e H ( { ǫ t } ) n ; x (cid:16) { e F z + z − ( z t ) } , { u ( z t ) } , { z t } (cid:17) go to zero exponentially fast in all directions where C ( ∞ ) ǫ t goes to ∞ , the integralsover C ( ∞ ) ǫ t are indeed convergent. 30 .7 Some more conjectures leading to the dominant asymptotics of ρ ( x , t ) Under the assumption that1. the Taylor series P + ∞ m = γ m ρ ( m )e ff ( x , t ) / m ! is convergent up to γ = ρ ( x , t ),3. the multidimensional Natte series given below is convergent.We get that ρ ( x , t ) is obtained from (4.43) by removing the m th γ -derivative symbol and setting γ =
1. It thenremains to identify the coe ffi cients in the first two lines with the properly normalized thermodynamic limit of formfactors of the field as given in (A.46), (A.47) and (A.48). One then obtains the below series of multiple integralrepresentation for the thermodynamic limit of the one-particle reduced density matrix: ρ ( x , t ) = s − i π t ε ′′ ( λ ) − xp ′′ ( λ ) × p ′ ( λ ) e ix [ u ( λ ) − u ( q )] (cid:12)(cid:12)(cid:12) F λ q (cid:12)(cid:12)(cid:12) [ − i ( x − tv F )] (cid:2) F λ q ( q ) (cid:3) [ i ( x + tv F )] (cid:2) F λ q ( − q ) (cid:3) ] q ; + ∞ [ ( λ ) + e − ixp F (cid:12)(cid:12)(cid:12) F − qq (cid:12)(cid:12)(cid:12) [ − i ( x − tv F )] (cid:2) F − qq ( q ) (cid:3) [ i ( x + tv F )] (cid:2) F − qq ( − q ) − (cid:3) + (cid:12)(cid:12)(cid:12) F ∅∅ (cid:12)(cid:12)(cid:12) [ − i ( x − tv F )] (cid:2) F ∅∅ ( q ) + [ i ( x + tv F )] (cid:2) F ∅∅ ( − q ) (cid:3) + e − ixu ( q ) + ∞ X n = X K n X E n ( ~ k ) I C ( w ) ǫ t G (0) |{ z + }| ;1 { z + }{ z − } ! H ( { ǫ t } ) n ; x ( { u ( z t ) } ; { z t } ) (cid:2) F z + z − (cid:3) B (cid:2) F z + z − ; p (cid:3) ( x − tv F + i + )[ F z + z − ( q ) ] ( x + tv F )[ F z + z − ( − q ) ] · d n z t (2 i π ) n . (4.62)It follows from the above representation and from conjecture B.1 that Corollary 4.1
The reduced density matrix admits the asymptotic expansion as given in subsection 3.2.Proof —
The proof is immediate as far as the multidimensional Natte series defining ρ ( x , t ) is convergent. Indeed,then, the fine structure of the functions H ( { ǫ t } ) n ; x given in (B.51) implies that all the contributions stemming fromintegrations are subdominant in respect to the first two lines in (4.62), this provided that (cid:12)(cid:12)(cid:12) F z + z − ( ± q ) (cid:12)(cid:12)(cid:12) < / { z ± } that belong to { λ , ± q } . This condidtion is not satisfied, especially if |{ z + }| becomes large. One should then invoke conjecture B.1 stating that, in fact, higher order oscillating terms in therepresentation (B.51) for H ( { ǫ t } ) n ; x are more dampen than it is apparent from the sum in (B.51). This is enoughto show that, indeed, the contributions to the oscillating tems at − p F , u ( λ ) − u ( q ) and 0 frequencies that arestemming from H ( { ǫ t } ) n ; x are subdominant in respect to the terms appearing in the first two lines of (4.62). There willof course arize terms oscillating at higher multiples of these frequencies exactly as it happens in (B.51). Thesehigher oscillating terms give rise to other critical exponents. For instance, one can convince oneself that, fromthis structure, one recovers the whole expected tower of critical exponents for the terms corresponding purely tooscillations at integer multiples of u ( q ) − u ( − q ) as predicted in [2] on the basis of CFT-based technique. Conclusion
In this article, we have continued developing a new method allowing one to build two types of series of multipleintegral representation for the correlation functions of integrable models starting from their form factor expansion.One of these series which we called the multidimensional Natte series yields a straightforward access to the31arge-distance / long-time asymptotic behavior of the two-point functions. In this way, we were able to extract thelong-time / large-distance asymptotic behavior of the reduced density matrix for the non-linear Schrödinger model.In order to provide applications to physically pertinent cases, the method we have developed has to recourseto a few conjectures. The first one is relative to the convergence of the series of multiple integrals representingthe correlators. This conjecture is supported by the free fermion case, where the convergence is rather quick,especially in the large-distance / long-time regime. The second conjecture concerns the possibility of using ane ff ective series instead of the one appearing in the form factors expansion of two-point functions. Both series havebeen assumed to have the same thermodynamic limit N , L → + ∞ . This conjecture is supported, on the physicalground, by the argument that sums over states whose energies scale as some power of the system-size ought togive a vanishing contribution to the sum over form factors once that the thermodynamic limit is taken. It wouldbe very interesting and important from the conceptual point of view to prove these two conjectures in the case ofmodels that are away from their free fermion points.However, we do insist that we have organized the analysis in such a way that all of the aforementionedconvergence issues are separated from the asymptotic analysis part. Therefore, all the part of this work relatedpurely to the asymptotic analysis is rigorous. Moreover, we do expect that the scheme of asymptotic analysiswe have developed can be applied in full rigor to many cases which are free of convergence issues. We do alsostress that, for the moment, the proofs of convergence of series of multiple integral representations for correlationfunctions of models away from their free-fermion point are, in general, an open problem. Apart from very specificrepresentations related to the spin-1 / rd Painlevé transcendent, a new type of specialfunction whose description and asymptotic behavior is interesting in its own right.
Acknowledgment
I acknowledge the support of the EU Marie-Curie Excellence Grant MEXT-CT-2006-042695. I would like tothank N.Kitanine, J.-M. Maillet, N. Slavnov and V. Terras for numerous stimulating discussions.
A Thermodynamic limit of the Form Factors of conjugated fields
A.1 Thermodynamic limit of form factors
It has been shown in [64] with the help of techniques introduced in [46, 78] that the normalized modulus squaredof the form factor of the conjugated field taken between the ground state { λ a } N and any finite n particle / hole type32xcited state { µ ℓ a } N + admits the below behavior in the thermodynamic limit N , L → + ∞ , N / L → D (cid:12)(cid:12)(cid:12)(cid:12)D ψ (cid:16)(cid:8) µ ℓ a (cid:9) N + (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Φ † (0 , (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:16) { λ } N (cid:17) E(cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13) ψ (cid:16)(cid:8) µ ℓ a (cid:9) N + (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13) ψ (cid:16) { λ } N (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) = D L (cid:2) F (cid:3) R N , n { µ p a } n ; { p a } n (cid:8) µ h a (cid:9) n ; { h a } n ! (cid:2) F (cid:3) G (0) n ;1 { µ p a } n (cid:8) µ h a (cid:9) n ! + O ln LL !! . (A.1)Above, F corresponds to the thermodynamic limit of the shift function associated to the excited state of interest(at β = F are undercurrent by the various functionals appearing above. We recallthat the parameters µ k appearing in the rhs of (A.1) are defined as the unique solutions to ξ ( µ a ) = a / L . The discreet part
The first two functionals appearing in (A.1) correspond to the leading in L behavior of the so-called singular part b D N (cid:2) b F { ℓ a } , b ξ { ℓ a } , b ξ (cid:3) of the form factor, namely b D N { p a } n { h a } n ! (cid:2) b F { ℓ a } , b ξ { ℓ a } , b ξ (cid:3) = D L (cid:2) F (cid:3) R N , n { µ p a } n ; { p a } n (cid:8) µ h a (cid:9) n ; { h a } n ! (cid:2) F (cid:3) + O ln LL !! . (A.2)Given any function ν ( λ ) holomorphic in some neighborhood of (cid:2) − q ; q (cid:3) , one has D L [ ν ] = (cid:2) κ [ ν ] ( − q ) (cid:3) ν ( − q ) (cid:2) κ [ ν ] ( q ) (cid:3) ν ( q ) + n Y a = λ N + − µ p a λ N + − µ h a ! q G (1 − ν ( − q )) G (2 + ν ( q )) /π (2 π ) ν ( q ) − ν ( − q ) · (cid:2) qL ξ ′ ( q ) (cid:3) ( ν ( q ) + + ν ( − q ) · e q R − q ν ′ ( λ ) ν ( µ ) − ν ′ ( µ ) ν ( λ ) λ − µ d λ d µ , (A.3)The parameter λ N + appearing above is defined as the unique solution to L ξ ν ( λ N + ) = N + κ [ ν ] ( λ ) is given by(4.47) and G stands for the Barnes double Gamma function. Finally, we agree upon R N , n { µ p a } n ; { p a } n (cid:8) µ h a (cid:9) n ; { h a } n ! [ ν ] = n Y a = ϕ (cid:0) µ h a , µ h a (cid:1) ϕ ( µ p a , µ p a )e ℵ [ ν ] ( µ pa ) ϕ ( µ p a , µ h a ) ϕ ( µ h a , µ p a )e ℵ [ ν ] ( µ ha ) n Q a < b ϕ ( µ p a , µ p b ) ϕ (cid:0) µ h a , µ h b (cid:1) n Q a , b ϕ ( µ p a , µ h b ) det n " h a − p b × n Y a = sin (cid:2) πν (cid:0) µ h a (cid:1)(cid:3) π ! · n Y a = Γ p a − N − + ν ( µ p a ) , p a , N + − h a − ν (cid:0) µ h a (cid:1) , h a + ν (cid:0) µ h a (cid:1) p a − N − , p a + ν ( µ p a ) , N + − h a , h a ! . (A.4)There ℵ [ ν ] ( ω ) = ν ( ω ) ln ϕ ( ω, q ) ϕ ( ω, − q ) ! + q Z − q ν ( λ ) − ν ( ω ) λ − ω d λ and ϕ ( λ, µ ) = π λ − µ p ( λ ) − p ( µ ) . (A.5)Above, we have used the standard hypergeometric-type representation for products of Γ -functions: Γ a , . . . , a n b , . . . , b n ! = n Y k = Γ ( a k ) Γ ( b k ) . (A.6)33 escription of G ( β ) n ;1 In order to give an explicit representation for G ( β ) n ; γ we need to introduce a few notations. First, let F β correspondto the thermodynamic limit of the β -deformed shift function associated to the choice of the rapidities { µ p a } for theparticles and { µ h a } for the holes. The auxiliary arguments of the shift function will be kept undercurrent. Also, let m ∈ N and U δ be the open strip (2.15) around R .Then there exists e γ > β ∈ C with ℜ ( β ) > ℑ ( β ) > i πγ F β ( ω ) − , n = , . . . , m and (cid:16) γ, ω, β, { µ p a } n , { µ h a } n (cid:17) ∈ D , e γ × U δ × e U β × U n δ × U n δ . (A.7)Let all parameters µ h a , a = , . . . , n belong to a compact K q + ǫ ⊃ (cid:2) − q ; q (cid:3) for some ǫ > C q + ǫ be a smallcounterclockwise loop around this compact K q + ǫ , then G ( β ) n ,γ admits the below representation G ( β ) n ; γ { µ p a } n (cid:8) µ h a (cid:9) n ! = e − i π P ǫ = ± C [ γ F β ] ( q + ǫ ic ) n Y a = Y ǫ = ± µ h a − q + ǫ ic µ p a − q + ǫ ic e i π C [ γ F β ]( µ ha + ǫ ic )e i π C [ γ F β ]( µ pa + ǫ ic ) e C [ γ F β ] × n Y a , b = (cid:0) µ p a − µ h b − ic (cid:1)(cid:0) µ h a − µ p b − ic (cid:1)(cid:0) µ p a − µ p b − ic (cid:1) (cid:0) µ h a − µ h b − ic (cid:1) · det C q + ǫ h I + γ V (cid:2) γ F β (cid:3)i det C q + ǫ h I + γ V (cid:2) γ F β (cid:3)i det [ I − K / π ] . (A.8)There C (cid:2) F β (cid:3) stands for the Cauchy transform on (cid:2) − q ; q (cid:3) and C (cid:2) F β (cid:3) is given by a double integral C (cid:2) F β (cid:3) ( λ ) = q Z − q d µ i π F β ( µ ) µ − λ and C (cid:2) F β (cid:3) = − q Z − q F β ( λ ) F β ( µ )( λ − µ − ic ) d λ d µ . (A.9)The integral kernels V and V read V [ ν ] (cid:0) ω, ω ′ (cid:1) = − π ω − q ω − q + ic n Y a = (cid:16) ω − µ p a (cid:17) (cid:0) ω − µ h a + ic (cid:1)(cid:0) ω − µ h a (cid:1) (cid:16) ω − µ p a + ic (cid:17) · e C [2 i πν ]( ω ) − C [2 i πν ]( ω + ic ) K ( ω − ω ′ )e − i πν ( ω ) − V [ ν ] (cid:0) ω, ω ′ (cid:1) = π ω − q ω − q − ic n Y a = (cid:16) ω − µ p a (cid:17) (cid:0) ω − µ h a − ic (cid:1)(cid:0) ω − µ h a (cid:1) (cid:16) ω − µ p a − ic (cid:17) · e C [2 i πν ]( ω ) − C [2 i πν ]( ω − ic ) K ( ω − ω ′ )e i πν ( ω ) − . (A.11)The representation (A.8) is valid for n = , . . . , m and (cid:16) γ, β, { µ p a } n , { µ h a } n (cid:17) ∈ D ,γ × e U β × U n δ × K nq + ǫ anddefines a holomorphic function of these parameters belonging to this set.It is also valid at γ =
1, provided that ℜ ( β ) > γ = D L [ F β ] R N , n [ F β ] G ( β ) n ;1 is holomorphic in ℜ β ≥
0, and can thus be analytically continued from e U β up to β =
0. It is in this sense that the formula (A.1) forthe leading asymptotics in the size L of the form factors of Φ † is to be understood. Proposition A.1 [64]Let m ∈ N , δ > small enough define the width of the strip U δ around R and (cid:16) { µ p a } n ; { µ h a } n (cid:17) ∈ U n δ × K nq + ǫ ,where ǫ > and the compact K q + ǫ is as defined by (4.27) .Let ν , h and τ be holomorphic function in the strip U δ around R and such that h ( U δ ) ⊂ (cid:8) z : ℜ ( z ) > (cid:9) and ℑ ( h ( z )) is bounded on U δ . Set ν β ( λ ) = ν ( λ ) + i β h ( λ ) .Then, there exists β ∈ C with ℜ ( β ) > large enough and ℑ ( β ) > small enough • e γ > and small enough • a small loop C q + ǫ ⊂ U δ around the compact K q + ǫ such that uniformly in β ∈ e U β the function λ e − i πγ ( ν + i β h )( λ ) − has no roots inside of C q + ǫ . In addition, thefunction (cid:0) { µ p a } n , (cid:8) µ h a (cid:9) n , γ, β (cid:1) G (1 − γτ ( − q )) G (2 + γτ ( q )) n Y a = (cid:16) e − i πγν β ( µ ha ) − (cid:17) · det C q + ǫ h I + γ V h γν β i (cid:0) { µ p a } n , (cid:8) µ h a (cid:9) n (cid:1)i (A.12) is a holomorphic function in U n δ × K nq + ǫ × D , e γ × e U β , this uniformly in ≤ n ≤ m.It admits a (unique) analytic continuation to U n δ × K nq + ǫ × D , × (cid:8) z ∈ C : ℜ ( z ) ≥ (cid:9) . In particular, it has awell defined β → limit. The β → limit of this analytic continuation is still holomorphic in (cid:0) { µ p a } n , (cid:8) µ h a (cid:9) n (cid:1) ∈ U n δ × K nq + ǫ . In (A.12) we have insisted explicitly on the dependence of the integral kernel V on the auxiliary parameters (cid:0) { µ p a } n , (cid:8) µ h a (cid:9) n (cid:1) , cf (A.10). The same proposition holds when the kernel V is replaced by V as it has been definedin (A.11). Alternative representation for G ( β ) n ; γ It so happens that the smooth part of the form factor’s asymptotics admits a representation as a functional actingon a unique function H . More precisely, Lemma A.1
Let m ∈ N and the strip U δ be fixed. Let A > be some constant defining the size of the compact K A (4.27) . Then, there exists A, δ , m dependent parameters • β ∈ C with ℜ ( β ) > large enough and ℑ ( β ) > small enough • e γ > small enoughsuch that uniformly in (cid:16) { y a } n , { z a } n (cid:17) ∈ K nA × K nA , | γ | ≤ e γ , β ∈ e U β and ≤ n ≤ m G ( β ) n ; γ { y a } n { z a } n ! = G ( β ) γ ; A " H · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { y a } n { z a } n ! with H λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { y a } n { z a } n ! = n X a = λ − y a − λ − z a . (A.13) The functional G ( β ) γ ; A acts on a bounded loop C ( K A ) ⊂ U δ around the compact K A . The functional G ( β ) γ ; A [ ̟ ] isa regular functional (cf definition 4.1) of ̟ in respect to the pair (cid:0) M G A , C ( K A ) (cid:1) where the compact M G A has itsboundaries given by C out and C in as depicted in the rhs of Fig. 1. For all ̟ ∈ O (cid:0) M G A (cid:1) such that k ̟ k C ( K A ) ≤ C G A ,where C G A is a constant of regularity of the functional G ( β ) γ ; A , one has G ( β ) γ ; A [ ̟ ] = e C [ γ G β ] Y ǫ = ± exp ( − I C ( K A ) d z i π ̟ ( z ) n i π C h γ G β i ( z + i ǫ c ) + ln ( z − q + i ǫ c ) o ) × exp ( − I C ( K A ) d y d z (2 i π ) ̟ ( y ) ̟ ( z ) ln ( z − y − ic ) ) det C A h I + γ V h γ G β , ̟ ii det C A h I + γ V h γ G β , ̟ ii det [ I − K / π ] exp n i π P ǫ = ± C (cid:2) γ G β (cid:3) ( q + i ǫ c ) o . (A.14)35 n the above formula, one should understand G β as a one parameter family of functionals of ̟ given byG β ( λ ) ≡ G β [ ̟ ] ( λ ) = ( i β − / Z ( λ ) − φ ( λ, q ) − I C ( K A ) d z i π ̟ ( z ) φ ( λ, z ) . (A.15) In the second line of (A.25) there appear Fredholm determinants of integral operators acting on a contour C A .The contour C A corresponds to a loop around C ( K A ) such that C A ⊂ U δ . The kernels read V [ ν, ̟ ] (cid:0) ω, ω ′ (cid:1) = − π ω − q ω − q + ic exp ( I C ( K A ) d z i π ̟ ( z ) ln (cid:18) ω − z ω − z + ic (cid:19) ) · e C [2 i πν ]( ω ) − C [2 i πν ]( ω + ic ) K ( ω − ω ′ )e − i πν ( ω ) − and V [ ν, ̟ ] (cid:0) ω, ω ′ (cid:1) = π ω − q ω − q − ic exp ( I C ( K A ) d z i π ̟ ( z ) ln (cid:18) ω − z ω − z − ic (cid:19) ) · e C [2 i πν ]( ω ) − C [2 i πν ]( ω − ic ) K ( ω − ω ′ )e i πν ( ω ) − The A, m and δ -dependent parameters β and e γ and the compacts C ( K A ) , M G A are such that the constant ofregularity C G A satisfies to the estimates given in (4.30) and is such that one has ∀ k ̟ k C ( K A ) < C G A (cid:13)(cid:13)(cid:13)e γ G β [ ̟ ] (cid:13)(cid:13)(cid:13) U δ < / k H k C ( K A ) < C G A uniformly in (cid:16) { y a } n , { z a } n (cid:17) ∈ K nA × K nA . Proof —
We first check that G ( β ) γ ; A is a regular functional. • G β [ ̟ ] is a regular functional as it is linear in ̟ and C ( K A ) is compact. • the estimates | e x − e y | ≤ e | x | + | y | | x − y | , majorations of integrals in terms of sup norm and derivation under theintegral sign theorems ensure that all of the exponential pre-factors in (A.25) are also regular functionals of ̟ .The associated constants of regularity can be taken as large as desired. It thus remains to focus on the Fredholmdeterminants. For this let us first assume that we are able to pick the contours C out / in delimiting the boundary ofthe compact M G A in such a way that there exists β ∈ C and e γ > i πγ G β [ ̟ ( ∗ , y ) ] ( λ ) − , ∀ ( λ, y , β, γ ) ∈ U δ × W y × e U β × D , e γ (A.18)this for any holomorphic function ̟ ( λ, y ) on M G A × W y , W y ⊂ C ℓ y , that satisfies k ̟ k C ( K A ) × W y < C G A .If this condition is satisfied, then the integral kernels γ V h γ G β , γ G β , ̟ i ( ω, ω ′ ) and γ V h γ G β , γ G β , ̟ i ( ω, ω ′ )are holomorphic in ω, ω ′ belonging to a small neighborhood of C A and y ∈ W y . The contour C A being compact,the two integral operators γ V h γ G β , γ G β , ̟ i and γ V h γ G β , γ G β , ̟ i are trace class operators that have an analyticdependence on y ∈ W y .Recall that if A , B are trace class operators ( k·k stands for the trace class norm) then | det [ I + A ] − det [ I + B ] | ≤ k A − B k e k A k + k B k + . (A.19)Also [76], if A ( y ), y ∈ W y ⊂ C ℓ y , is an analytic trace class operator then det (cid:2) I + A ( y ) (cid:3) is holomorphic on W y These two properties show that, indeed, in (A.25), the two Fredholm determinants of integral operators acting onthe contour C A are regular functionals of ̟ . 36ence, it remains to prove the existence of e γ and β such that condition (A.18) holds. Given ̟ ( λ, y ) ∈ O (cid:0) M G A × W y (cid:1) , the function ω e i πγ G β [ ̟ ]( ω ) − (cid:12)(cid:12)(cid:12)e γ G β (cid:2) ̟ ( ∗ , y ) (cid:3) ( λ ) (cid:12)(cid:12)(cid:12) < / ℑ (cid:16) G β (cid:2) ̟ ( ∗ , y ) (cid:3) ( λ ) (cid:17) > λ, y , β ) ∈ U δ × W y × e U β . (A.20)One has that, for β ∈ e U β ℑ (cid:16) G β [ ̟ ] ( λ ) (cid:17) > ℜ ( β ) inf λ ∈ U δ (cid:2) ℜ ( Z ( λ )) (cid:3) − (cid:0) ℑ ( β ) + / (cid:1) (cid:13)(cid:13)(cid:13) ℑ ( Z ) (cid:13)(cid:13)(cid:13) U δ −k φ k U δ −k ̟ k C ( K A ) × W y sup λ ∈ U δ I C ( K A ) | d z | π | φ ( λ, z ) | . Hence, ℑ (cid:16) G β [ ̟ ] ( λ ) (cid:17) > k ̟ k C ( K A ) × W y ≤ C G A with C G A = ( λ ∈ U δ I C ( K A ) | d z | π | φ ( λ, z ) | ) − · ( ℜ ( β ) inf λ ∈ U δ (cid:2) ℜ ( Z ( λ )) (cid:3) − (cid:0) ℑ ( β ) + / (cid:1) (cid:13)(cid:13)(cid:13) ℑ ( Z ) (cid:13)(cid:13)(cid:13) U δ − k φ k U δ ) . (A.21)Here ℜ ( β ) > C G A as defined above to be positive. Then, if k ̟ k C ( K A ) × W y ≤ C G A with C G A as given above, one hassup ω ∈ U δ y ∈ W y (cid:12)(cid:12)(cid:12) G β (cid:2) ̟ ( ∗ , y ) (cid:3) ( ω ) (cid:12)(cid:12)(cid:12) ≤ (cid:0) ℜ ( β ) + ℑ ( β ) + / (cid:1) k Z k U δ + k φ k U δ + k ̟ k C ( K A ) × W y sup λ ∈ U δ I C ( K A ) | d z | π | φ ( λ, z ) | < (cid:0) ℜ ( β ) + ℑ ( β ) + (cid:1) k Z k U δ . (A.22)Hence, if we take e γ − = (cid:0) ℜ ( β ) + ℑ ( β ) + (cid:1) k Z k U δ , the condition (cid:12)(cid:12)(cid:12) γ G β [ ̟ ] (cid:12)(cid:12)(cid:12) < / | γ | ≤ e γ and β ∈ e U β . It remains to tune ℜ ( β ) so that conditions C G A · π d (cid:0) ∂ M G A , C ( K A ) (cid:1)(cid:12)(cid:12)(cid:12) ∂ M G A (cid:12)(cid:12)(cid:12) + π d (cid:0) ∂ M G A , C ( K A ) (cid:1) > A and 2 m d( K A , C ( K A )) < C G A . (A.23)are satisfied.One can always choose the contours C out / in defining ∂ M G A in such a way that d (cid:0) ∂ M G A , C ( K A ) (cid:1) > c thisuniformly in A >
0. These contours can also be chosen such that there exists an A -independent constant c with (cid:12)(cid:12)(cid:12) ∂ M G A (cid:12)(cid:12)(cid:12) < c A . It is also clear that the contour C ( K A ) surrounding the compact K A can be chosen such that | C ( K A ) | < c A for some A -independent constant c and also d( K A , C ( K A )) > c ′ . It is then enough to take ℜ ( β ) > c β A with c β being properly tuned in terms of c , c ′ , c , c so that conditions (A.23) hold for any A su ffi ciently large.Note that the second condition in (A.23) guarantees that the function H as given in (A.13) satisfies k H k C ( K A ) < C G A uniformly in the parameters (cid:16) { µ p a } n , { µ h a } n (cid:17) ∈ K nA × K nA .Having proved that G ( β ) γ ; A is a regular functional with a regularity constant C G A > ffi ciently large, we canevaluate it on H . Then, it is readily seen that G β [ ̟ ] ( λ ) coincides with the shift function F β once upon taking ̟ = H as given in (A.13). All other integrals involving ̟ = H are computed by the residues at µ p a and µ h a . Allcalculations done, one recovers the representation (A.8) for the function G ( β ) n ; γ . We stress that the parameters e γ and β ensuring the regularity of the functional G ( β ) γ ; A are also such that G ( β ) n ; γ is well defined due to conditions (A.20).37 egular functional for b G N ; γ A very similar representation to the one given in the previous lemma exists for the functional b G N ; γ . Lemma A.2
Let m ∈ N and the strip U δ be fixed. Let A > be some constant defining the size of the compact K A (4.27) . Then, there exists A, m and δ -dependent constants • β ∈ C with ℜ ( β ) > large enough and ℑ ( β ) > small enough, • e γ > small enough,such that for L large enough and uniformly in (cid:16) { µ p a } n , { µ h a } n (cid:17) ∈ K nA × K nA , | γ | ≤ e γ and ≤ n ≤ m b G N ; γ { p a } n { h a } n ! (cid:2) γ F β , ξ, ξ γ F β (cid:3) = b G ( β ) γ ; A " H ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { µ p a } n (cid:8) µ h a (cid:9) n ! with H λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { µ p a } n (cid:8) µ h a (cid:9) n ! = n X a = λ − µ p a − λ − µ h a . (A.24) The functional b G ( β ) γ ; A acts on a bounded loop C ( K A ) ⊂ U δ around the compact K A . The functional b G ( β ) γ ; A [ ̟ ] isa regular functional (cf definition 4.1) of ̟ in respect to the pair (cid:0) M G A , C ( K A ) (cid:1) where the compact M G A has itsboundaries given by C out and C in as depicted in the rhs of Fig. 1. For all ̟ ∈ O (cid:0) M G A (cid:1) such that k ̟ k C ( K A ) ≤ C G A ,where C G A is a constant of regularity of the functional b G ( β ) γ ; A , b G ( β ) γ ; A [ ̟ ] = W N (cid:2) γ G β (cid:3) { λ a } N { µ a } N ! Y ǫ = ± exp ( − I C ( K A ) d z i π ̟ ( z ) n − ln (cid:16) V N ; ǫ (cid:2) γ G β (cid:3)(cid:17) ( z ) + ln ( z − µ N + + i ǫ c ) o ) × exp ( − I C ( K A ) d y d z (2 i π ) ̟ ( y ) ̟ ( z ) ln ( z − y − ic ) ) det C A h I + γ b V N h γ G β , γ G β , ̟ ii det C A (cid:20) I + γ b V N h γ G β , γ G β , ̟ i(cid:21) det N + (cid:2) Ξ ( µ ) (cid:2) ξ (cid:3)(cid:3) det N (cid:2) Ξ ( λ ) (cid:2) ξ γ G β (cid:3)(cid:3) Q ǫ = ± V − N ; ǫ h γ G β [ ̟ ] i ( µ N + ) . (A.25) In the above formula, one should understand G β as the one-parameters family of regular functionals of ̟ asdefined by (A.15) . We did not make the functional dependence of G β on ̟ explicit in (A.25) . The functionals W N and V N ; ǫ have been defined in (4.6) . We have added the (cid:2) γ G β (cid:3) symbol so as to make it clear that the parameters { λ a } N entering in their definition are functionals of γ G β through the relation λ a = ξ − γ G β ( a / L ) .In the second line of (A.25) there appear Fredholm determinants of integral operators acting on a contour C A . The contour C A corresponds to a loop around C ( K A ) such that C A ⊂ U δ . The kernels read b V N [ ν, ̟ ] (cid:0) ω, ω ′ (cid:1) = − π ω − µ N + ω − µ N + + ic exp ( I C ( K A ) d z i π ̟ ( z ) ln (cid:18) ω − z ω − z + ic (cid:19) ) · V N ;1 [ ν ] ( ω ) V N ;0 [ ν ] ( ω ) · K ( ω − ω ′ )e − i πν ( ω ) − and b V N [ ν, ̟ ] (cid:0) ω, ω ′ (cid:1) = π ω − µ N + ω − µ N + − ic exp ( I C ( K A ) d z i π ̟ ( z ) ln (cid:18) ω − z ω − z − ic (cid:19) ) · V N ; − [ ν ] ( ω ) V N ;0 [ ν ] ( ω ) · K ( ω − ω ′ )e i πν ( ω ) − The constant of regularity C G A satisfies to the estimates already given in (4.30) and is such that ∀ k ̟ k C ( K A ) < C G A (cid:13)(cid:13)(cid:13)e γ G β [ ̟ ] (cid:13)(cid:13)(cid:13) U δ < / k H k C ( K A ) < C G A uniformly in (cid:16) { µ p a } n , { µ h a } n (cid:17) ∈ K nA × K nA . roof — The proof is very similar to the one of lemma A.1. Hence, we only specify that for L -large enough, and as soonas condition (cid:12)(cid:12)(cid:12) γ G β [ ̟ ] ( λ ) (cid:12)(cid:12)(cid:12) < / λ ∈ U δ is satisfied, the parameters λ j are seen to be regular functionals of ̟ thanks to their integral representation λ j [ ̟ ] = I C q d z i π ξ ′ γ G β [ ̟ ] ( z ) ξ γ G β [ ̟ ] ( z ) − j / L . (A.28)All other details are left to the reader. A.2 Specific values of the functionals G ( β ) γ ; A and b G ( β ) γ ; A In this subsection, we estimate the value of the functional G ( β ) γ ; A [ ̟ ] for a specific type of function ̟ . This resultwill play a role later on. Lemma A.3
Let the function ν ( λ ) ≡ ν (cid:0) λ | { z k } n , { y k } n + (cid:1) be the unique solution to the linear integral equationdriven by the resolvent R of the Lieb kernel (ie ( I − K / π ) ( I + R / π ) = I): ν ( λ ) + γ q Z − q d µ π R ( λ, µ ) ν ( µ ) = ( i β − / Z ( λ ) + n X k = φ ( λ, z k ) − n + X k = φ ( λ, y k ) . (A.29) Let A > be large enough and such that (cid:16) { z k } n , { y k } n + (cid:17) ∈ K nA × K n + A . Let β ∈ C and e γ be the two numbersassociated to the constant A as stated in lemma A.1. Then defining ̟ ( λ ) = n + X a = λ − y a − λ − q − n X a = λ − z a − q Z − q γν ( τ )( λ − τ ) d τ , (A.30) the below identity holds G ( β ) γ ; A [ ̟ ] = − ic n Q a = n + Q b = ( y b − z a − ic ) ( z a − y b − ic ) n + Q a , b = ( y a − y b − ic ) n Q a , b = ( z a − z b − ic ) det n h δ k ℓ + γ b V k ℓ (cid:2) γν (cid:3)i det n (cid:20) δ k ℓ + γ b V k ℓ (cid:2) γν (cid:3)(cid:21) det [ I − K / π ] . (A.31) The non-trivial entries of the two determinants are given by (4.8) . The auxiliary variables (cid:0) { z k } n , { y k } n + (cid:1) on whichthese entries depend are undercurrent by the set of auxiliary variables on which depends ν .Proof — The function ν is bounded on the strip U δ . As a consequence, the associated function ̟ (A.30) is also boundedby an A independent constant. The estimates (4.30) for the constant of regularity C G A for the functional G ( β ) γ ; A ensurethat there exists A large enough such that k ̟ k C ( K A ) < C G A uniformly in ( { z k } n , { y k } n + ) ∈ K nA × K n + A . Thus, G ( β ) γ ; A [ ̟ ]is then well definied.A direct calculation leads toexp I C A d z i π ̟ ( z ) ln (cid:18) ω − z ω − z ± ic (cid:19) = ω − q ± ic ω − q n + Y a = ω − y a ω − y a ± ic n Y a = ω − z a ± ic ω − z a e C [ i πγν ] ( ω ± ic ) − C [ i πγν ] ( ω ) . (A.32)39y using the linear integral equation satisfied by ν and the representation (A.15) we get that G β [ ̟ ] ( λ ) = ν ( λ ) . (A.33)As a consequence, the kernel V and V simplify V h γ G β [ ̟ ] , ̟ i (cid:0) ω, ω ′ (cid:1) = − n + Y a = ω − y a ω − y a + ic n Y a = ω − z a + ic ω − z a K ( ω − ω ′ )2 π (cid:0) e − i πγν ( ω ) − (cid:1) (A.34)and V h γ G β [ ̟ ] , ̟ i (cid:0) ω, ω ′ (cid:1) = n + Y a = ω − y a ω − y a − ic n Y a = ω − z a − ic ω − z a K ( ω − ω ′ )2 π (cid:0) e i πγν ( ω ) − (cid:1) (A.35)The associated Fredholm determinants can now be reduced to finite-size determinants by computing the polesat ω = z a with a = , . . . , n (by definition of e γ and β , since | γ | ≤ e γ and β ∈ e U β , there are no poles of e i πγν ( ω ) − C A ).This leads to det C A h I + γ V (cid:2) γ G β [ ̟ ] , ̟ (cid:3)i = det n h δ k ℓ + γ b V k ℓ (cid:2) γν (cid:3) (cid:16) { z a } n , { y a } n + (cid:17)i (A.36)det C A h I + γ V (cid:2) γ G β [ ̟ ] , ̟ (cid:3)i = det n (cid:20) δ k ℓ + γ b V k ℓ (cid:2) γν (cid:3) (cid:16) { z a } n , { y a } n + (cid:17)(cid:21) . (A.37)The claim then follows once upon applying the identity − ic n Q a = n + Q b = ( y b − z a − ic ) ( z a − y b − ic ) n + Q a , b = ( y a − y b − ic ) n Q a , b = ( z a − z b − ic ) = exp ( − I C ( K A ) d y d z (2 i π ) ̟ ( y ) ̟ ( z ) ln ( z − y − ic ) ) × e C [ γ G β [ ̟ ] ]e − i π P ǫ = ± C [ γ G β [ ̟ ] ] ( q + i ǫ c ) Y ǫ = ± exp ( − I C ( K A ) d z i π ̟ ( z ) n i π C h γ G β [ ̟ ] i ( z + i ǫ c ) + ln ( z − q + i ǫ c ) o ) (A.38) Lemma A.4
Let γ be small enough and L large enough such that ν ( L ) ( µ ) is the unique solution to the non-linearintegral equation ν ( L ) ( λ ) = ( i β − / Z ( λ ) − φ ( λ, q ) + N + X a = φ ( λ, µ a ) − n + X a = φ ( λ, y a ) − N X a = , i ,..., i n φ ( λ, e λ a ) (A.39) The parameters e λ a appearing above are functional of ν ( L ) through the relation ξ γν ( L ) ( e λ a ) = a / L, µ a are such that ξ ( µ a ) = a / L and the parameters y a ∈ U δ/ are arbitrary. Finally, L is assumed large enough so that all parameters (cid:16) { µ a } N + , { e λ a } N , { y a } n + (cid:17) of the N + + n-uple belong to K A L . Then, given β and e γ as in lemma A.2, one has he identity b G ( β ) γ ;2 A L " H ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { y a } n + ∪ { e λ a } N { e λ i a } n ∪ { µ a } N + ! = − ic n Q a = n + Q b = (cid:16) y b − e λ i a − ic (cid:17) (cid:16)e λ i a − y b − ic (cid:17) n + Q a , b = ( y a − y b − ic ) n Q a , b = (cid:16)e λ i a − e λ i b − ic (cid:17) det − N + h Ξ ( µ ) (cid:2) ξ (cid:3) i det − N h Ξ ( λ ) (cid:2) ξ γν ( L ) (cid:3)i × det n h δ k ℓ + γ b V k ℓ (cid:2) γν ( L ) (cid:3) (cid:16) { e λ i a } n , { y a } n + (cid:17) i det n h δ k ℓ + γ b V k ℓ (cid:2) γν ( L ) (cid:3) (cid:16) { e λ i a } n , { y a } n + (cid:17) i (A.40) Proof —
It has been shown in proposition D.3 that for | γ | small enough and L large enough the solution ν ( L ) to thenon-linear integral equation occurring in the rhs of (A.39) is unique and exists. Moreover this solution is boundedon U δ by an L -independent constant.As discussed in the proof of lemma A.1, the contour C (cid:0) K A L (cid:1) can always be taken such that, uniformly in L ,d( C (cid:0) K A L (cid:1) , K A L ) > c ′ > c ′ . Hence, the principal argument λ of H is uniformly away from thecompact K A L where the auxiliary arguments of H are located. As a consequence, it follows from the expressionfor H and the estimates for the spacing between the parameters µ a and e λ a µ a − e λ a = πγν ( L ) ( µ a ) / (cid:0) Lp ′ ( µ a ) (cid:1) + O (cid:16) L − (cid:17) , uniformly in a = , . . . , N (A.41)that k H k C ( K AL ) is bounded by an L -independent constant, this uniformly in L large enough. In particular, for L large enough, due to the estimates (4.30) for the constant C G AL of regularity for b G ( β ) γ ;2 A L , we get that k H k C ( K A ) < C G AL . One can thus acts with the functional b G ( β ) γ ;2 A L on H . A straightforward residue calculation shows that G β " H ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { y a } n + ∪ { e λ a } N { e λ i a } n ∪ { µ a } N + ! = ν ( L ) ( λ ) . (A.42)This means that all the λ a appearing in the expression (A.25) for the functional b G ( β ) γ ;2 A L [ H ] coincide with theparameters e λ a defined above. The claim of the lemma then follows from straightforward residue computationsand multiple cancelations. The Fredholm determinants reduce to finite rank determinants that can be computedby the residues at ω = e λ i a , a = , . . . , n . A.3 Leading asymptotic behavior of one particle / one hole form factors We now build on the formulae for the leading asymptotic behavior of form factors so as to provide, properlynormalized in the size of the model, expressions for the large- L limit of the form factors of the fields betweenthe N -particle ground state and N + λ of the function u ( λ ) given in (2.14). Such thermodynamic limits of properly normalized form factors appear as amplitudes inthe large-distance / long-time asymptotic expansion of the reduced density matrix. The explicit expressions that wewrite down will allow for such an identification. We do stress that all shift functions appearing below are taken at β =
0. The fact that (A.46)-(A.48) are well-defined in this limit follows from proposition A.1.In the following, let { λ } ≡ { λ a } N stand for the Bethe roots corresponding to the ground state in the N -particlesector. Let { µ ∅∅ } ≡ { µ ∅∅ } N + stand for the Bethe roots corresponding to the ground state in the ( N + F ∅∅ stands for the thermodynamic limit of the corresponding shift function cf (3.5), we41efine (cid:12)(cid:12)(cid:12) F ∅∅ (cid:12)(cid:12)(cid:12) = lim N , L → + ∞ (cid:18) L π (cid:19)(cid:2) F ∅∅ ( q ) + (cid:3) + (cid:2) F ∅∅ ( − q ) (cid:3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D ψ (cid:16)(cid:8) µ ∅∅ (cid:9)(cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Φ † (0 , (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:16)(cid:8) λ (cid:9)(cid:17) E(cid:13)(cid:13)(cid:13)(cid:13) ψ (cid:16)(cid:8) λ (cid:9)(cid:17)(cid:13)(cid:13)(cid:13)(cid:13) · (cid:13)(cid:13)(cid:13)(cid:13) ψ (cid:16)(cid:8) µ ∅∅ (cid:9)(cid:17)(cid:13)(cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A.43)Similarly, given the set { µ − qq } ≡ { µ − qq } N + corresponding to a particle-hole excitation such that p = h = N +
1, we denote by F − qq the thermodynamic limit of the corresponding shift function cf (3.5), and define (cid:12)(cid:12)(cid:12) F − qq (cid:12)(cid:12)(cid:12) = lim N , L → + ∞ (cid:18) L π (cid:19)(cid:2) F − qq ( q ) (cid:3) + (cid:2) F − qq ( − q ) − (cid:3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D ψ (cid:16)(cid:8) µ − qq (cid:9)(cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Φ † (0 , (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:16)(cid:8) λ (cid:9)(cid:17) E(cid:13)(cid:13)(cid:13)(cid:13) ψ (cid:16)(cid:8) λ (cid:9)(cid:17)(cid:13)(cid:13)(cid:13)(cid:13) · (cid:13)(cid:13)(cid:13)(cid:13) ψ (cid:16)(cid:8) µ − qq (cid:9)(cid:17)(cid:13)(cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A.44)Finally, given the set { µ λ q } ≡ { µ λ q } N + corresponding to a particle-hole excitation such that h = N + µ p a = λ we denote by F λ q the thermodynamic limit of the corresponding shift function cf (3.5), and define (cid:12)(cid:12)(cid:12) F λ q (cid:12)(cid:12)(cid:12) = lim N , L → + ∞ (cid:18) L π (cid:19)(cid:2) F λ q ( q ) (cid:3) + (cid:2) F λ q ( − q ) (cid:3) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D ψ (cid:16)(cid:8) µ λ q (cid:9)(cid:17) (cid:12)(cid:12)(cid:12)(cid:12) Φ † (0 , (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:16)(cid:8) λ (cid:9)(cid:17) E(cid:13)(cid:13)(cid:13)(cid:13) ψ (cid:16)(cid:8) λ (cid:9)(cid:17)(cid:13)(cid:13)(cid:13)(cid:13) · (cid:13)(cid:13)(cid:13)(cid:13) ψ (cid:16)(cid:8) µ λ q (cid:9)(cid:17)(cid:13)(cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A.45)By using (A.1) and expressions (A.3)-(A.5) we are lead to (cid:12)(cid:12)(cid:12) F λ q (cid:12)(cid:12)(cid:12) = e i π π p ′ ( λ ) A h F λ q i B h F λ q , p i G (0)1;1 λ q ! exp (cid:26) i π (cid:16)(cid:2) F λ q ( − q ) (cid:3) − (cid:2) F λ q ( q ) (cid:3) (cid:17)(cid:27) , (A.46) (cid:12)(cid:12)(cid:12) F − qq (cid:12)(cid:12)(cid:12) = A − h F − qq , p i B h F − qq , p i G (0)1;1 − qq ! exp (cid:26) i π (cid:16)(cid:2) F − qq ( − q ) − (cid:3) − (cid:2) F − qq ( q ) (cid:3) (cid:17)(cid:27) , (A.47)and finally (cid:12)(cid:12)(cid:12) F ∅∅ (cid:12)(cid:12)(cid:12) = A + h F ∅∅ , p i B h F ∅∅ , p i G (0)0;1 ∅∅ ! exp (cid:26) i π (cid:16)(cid:2) F ∅∅ ( − q ) (cid:3) − (cid:2) F ∅∅ ( q ) + (cid:3) (cid:17)(cid:27) . (A.48)The functionals B , A ± and A appearing above have been defined in (4.46), (4.47) and (4.48). B The generalized free-fermion summation formulae
In this appendix, we establish summation identities allowing one to recast the form factor expansion of an analogueof the field / conjugated-field two-point function that would appear in a generalized free fermion model in termsof a finite-size determinant. The representation we obtain constitutes the very cornerstone for deriving variousrepresentations for the correlation functions in the interacting case. In particular, it allows one for an analysisof their asymptotic behavior in the large-distance / long-time regime. We first establish re-summation formulaeallowing one to estimate discreet analogs of singular integrals. This will open the way for obtaining Fredholmdeterminant like representations out of the form factor based expansions.42 .1 Computation of singular sums Let ξ stand for the thermodynamic limit of the counting function (2.6) and E − be a non-vanishing and holomorphicfunction in some open neighborhood U δ ( cf (2.15)) of R such that ℜ (cid:16) ln E − − (cid:17) has, at most, polynomial growth, ie (cid:12)(cid:12)(cid:12)(cid:12) ℜ h ln E − − ( λ ) − iC λ k i(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12)(cid:12)(cid:12) ℜ (cid:16) i λ k − (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) + C , for some C , C , C ∈ R + and k > λ ∈ U δ . (B.1)We remind that the neighborhood U δ is always taken such that ξ is a biholomorphism on U δ In the following, we study the below singular sums over the set { µ a } : S ( L ) r h E − − i ( λ ) = X a ∈B L E − − ( µ a )2 π L ξ ′ ( µ a ) ( µ a − λ ) r with µ a being the unique solution to ξ ( µ a ) = a / L . (B.2)The summation runs through the set B L = { a ∈ Z : − w L ≤ a ≤ w L } where w L is some L -dependent sequence in N such that L = o ( w L ) and (cid:16) w L · L − (cid:17) k − = o ( L ). Proposition B.1
Let N q be a compact neighborhood of (cid:2) − q ; q (cid:3) lying in U δ , then under the above assumptionsand provided that L is large enough, one has, uniformly in λ ∈ N q S ( L )0 h E − − i ( λ ) = Z C bk ; L d µ π E − − ( µ ) + I ( L )0 h E − − i ( λ ) (B.3) S ( L )1 h E − − i ( λ ) = Z C bk ; L d µ π E − − ( µ ) µ − λ − i E − − ( λ )e i π L ξ ( λ ) − + I ( L )1 h E − − i ( λ ) (B.4) S ( L )2 h E − − i ( λ ) = ∂∂λ Z C bk ; L d µ π E − − ( µ ) µ − λ − i ∂ λ h E − − ( λ ) i e i π L ξ ( λ ) − + π E − − ( λ ) L ξ ′ ( λ )2 sin (cid:2) π L ξ ( λ ) (cid:3) + I ( L )2 h E − − i ( λ ) . (B.5) The integration goes along the curve C bk ; L depicted on Fig. 4. Also, given r ∈ N ,I ( L ) r h E − − i ( λ ) = Z C ↑ ; L d z π E − − ( z )( z − λ ) r − e − i π L ξ ( z ) + Z C ↓ ; L d z π E − − ( z )( z − λ ) r i π L ξ ( z ) − + Z C bd ; L d z π E − − ( z )( z − λ ) r . (B.6) The contours C ↑ / ↓ ; L are depicted in Fig. 3 whereas C bd ; L is depicted on Fig. 4.The functionals I ( L ) r h E − − i ( λ ) are such that I ( L ) r h E − − i ( λ ) = O (cid:16) ( L / w L ) k + r − (cid:17) , uniformly in λ ∈ N q .Proof — Let N q be a compact neighborhood of (cid:2) − q ; q (cid:3) in U δ . Then, for L large enough it is contained inside of thecontour C ↑ ; L ∪ C ↓ ; L as depicted in Fig. 3, and thus 43 b − A L B L C ↑ ; L C ↓ ; L b b − w L L − L w L L + L ξξ − R + i α R − i α Figure 3: Contour C ↑ ; L ∪ C ↓ ; L lying in U δ . b b b b − A L B L − A A C bk ; L N q C bd ; L C bd ; L b b b b − A L B L − A A C bk ; L N q C bd ; L C bd ; L Figure 4: Contours C bk ; L (solid lines) and C bd ; L (dashed lines) in the case of k odd ( lhs ) and k even ( rhs ) both inthe case C <
0. The dashed lines C bd ; L are pre-images of the segments [ ǫ υ w L + ǫ υ / ǫ υ w L + ǫ υ / + i ǫ ′ α ], with υ ∈ { l , r } and ǫ l = − ǫ r =
1. The sign of ǫ ′ depends on the left or right boundary, the parity of k and the signof C . S ( L )1 h E − − i ( λ ) = − iE − − ( λ )e i π L ξ ( λ ) − + Z C ↑ ; L ∪ C ↓ ; L E − − ( z )2 π ( z − λ ) · i π L ξ ( z ) − z = Z C ↓ ; L E − − ( z )2 π ( z − λ ) 1e i π L ξ ( z ) − z + Z C ↑ ; L E − − ( z )2 π ( z − λ ) ( e i π L ξ ( z ) e i π L ξ ( z ) − − ) d z − iE − − ( λ )e i π L ξ ( λ ) − = Z C bk ; L d µ π E − − ( µ ) µ − λ − i E − − ( λ )e i π L ξ ( λ ) − + I ( L )1 h E − − i ( λ ) . (B.7)In order to obtain the last line, we have deformed the contour C ↑ ; L into the contour C bk ; L ∪ C bd ; L as depicted inFig. 4. The intermediate points ± A entering in the definition of C bk ; L are chosen large (in order to include N q ) butfixed, in the sense that L independent.The representation for S ( L )2 h E − − i ( λ ) follows by di ff erentiation. The computations for S ( L )0 h E − − i ( λ ) are carriedout similarly with the sole di ff erence that there is no pole at z = λ .In now remains to prove the statement relative to the asymptotic behavior in L of the functionals I ( L ) r h E − − i .The main di ffi culty is that the function E − ( λ ) might have an exponential increase when λ belongs to the upper orlower half-plane. We establish the claimed estimates for the C ↑ ; L -part of the contour. This can be done similarlyfor C ↓ ; L , and we leave these details to the reader. 44e first perform the change of variables † z = ξ − ( s ) and set u L = w L / L + / (2 L ). The contour of integrationis then mapped to the contour depicted on the rhs of Fig. 3. We stress that the parameter α > C ↑ ; L ∪ C ↓ ; L lies in U δ . The aforementioned change of variables leads to Z C ↑ ; L d z π E − − ( z )( z − λ ) r · − e − i π L ξ ( z ) = − u L Z u L d s π E − − ξ ′ ◦ ξ − ( s + i α ) 1 (cid:2) ξ − ( s + i α ) − λ (cid:3) r n − e π L α e − i π sL o − + X ǫ = ± ǫ α Z i d s π E − − ξ ′ ◦ ξ − ( is + ǫ u L ) 1 (cid:2) ξ − ( is + ǫ u L ) − λ (cid:3) r n + e π sL o − . (B.8)We first establish a bound for the integral over the line [ − u L ; u L ]. It follows from the integral equation (2.7)satisfied by p that, p ( λ ) = λ ± π D − cD /λ + O (cid:16) λ − (cid:17) when ℜ ( λ ) → ±∞ . (B.9)Hence, uniformly in 0 ≤ τ ≤ α and for s ∈ R large, ξ − ( s + i τ ) = ψ s + i πτ + O( s − ) where ψ s = π s − π D (1 ± + cD π s ∈ R . (B.10)The condition (B.1) implies that there exists constants C > C ′ > (cid:12)(cid:12)(cid:12)(cid:12) ℜ h ln E − − ( λ ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12)(cid:12)(cid:12) ℜ (cid:16) i λ k (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) + C ′ , uniformly in λ ∈ U δ . (B.11)As a consequence, uniformly in 0 ≤ τ ≤ α , (cid:12)(cid:12)(cid:12)(cid:12) ℜ h ln E − − ◦ ξ − ( s + i τ ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12)(cid:12)(cid:12) ℑ h ( ψ s ) k + i π k τ ( ψ s ) k − + O (cid:0) s k − (cid:1)i(cid:12)(cid:12)(cid:12)(cid:12) + C ′ ≤ C τ k (2 π ) k | s | k − (cid:12)(cid:12)(cid:12)(cid:12) ℑ h + O (cid:0) s − (cid:1)i(cid:12)(cid:12)(cid:12)(cid:12) + C ′ . (B.12)There exists an s such that (cid:12)(cid:12)(cid:12) O( s − ) (cid:12)(cid:12)(cid:12) < | s | ≥ s , this uniformly in 0 ≤ τ ≤ α . Moreover, for such an s ,we define C ′′ = C ′ + max (cid:12)(cid:12)(cid:12)(cid:12) ℜ h ln E − − ◦ ξ − ( s + i τ ) i(cid:12)(cid:12)(cid:12)(cid:12) , (B.13)with the maximum being taken over | s | ≤ s and 0 ≤ τ ≤ α . Hence, for any s ∈ R and 0 ≤ τ ≤ α (cid:12)(cid:12)(cid:12)(cid:12) ℜ h ln E − − ◦ ξ − ( s + i τ ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ kC α (2 π ) k | s | k − + C ′′ . (B.14)Therefore, we obtain the estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − u L Z u L d s π E − − ξ ′ ◦ ξ − ( s + i α ) 1 (cid:2) ξ − ( s + i α ) − λ (cid:3) r n − e π L α e − i π sL o − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup z ∈ ξ − ( R + i α ) ( e C ′′ | z − λ | r ξ ′ ( z ) ) · w L + π L · e kC α (2 π ) k u k − L e π L α − = O (cid:0) L −∞ (cid:1) , (B.15) † we remind that ξ is a biholomorphism on U δ and that p ′ > R . (cid:0) w L · L − (cid:1) k − = o ( L ). It remains to estimate the integral over the lines [ 0 ; α ]: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ǫ = ± ǫ α Z i d s π E − − ξ ′ ◦ ξ − ( is + ǫ u L ) 1 (cid:0) ξ − ( is + ǫ u L ) − λ (cid:1) r n + e π sL o − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup s ∈ [ 0 ; α ] ǫ ∈{± } (cid:12)(cid:12)(cid:12) ξ ′ ◦ ξ − ( is + ǫ u L ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:2) ξ − ( is + ǫ u L ) − λ (cid:3)(cid:12)(cid:12)(cid:12) r × e C ′′ π α Z e Ck (2 π ) k ( u L ) k − τ + e π L τ d τ . (B.16)By making the change of variables y = L τ and then applying Lebesgue’s dominated convergence theorem, onecan convince oneself that the integral in the second line of (B.16) is a O (cid:0) L − (cid:1) .The last class of integrals to consider stems from integrations along C bd ; L . In order to carry the estimates, weneed to use the finer condition (B.1). Here, we only treat the case of k even and C <
0. All other cases are treatedvery similarly. An analogous reasoning to (B.12) leads, uniformly in 0 ≤ τ ≤ α to ℜ h ln E − − ◦ ξ − ( s ± i τ ) i = τ (cid:16) ∓ k (2 π ) k C s k − + O (cid:0) s k − (cid:1)(cid:17) for ∓ s > . (B.17)There exists s ′ such that for | s | ≥ s ′ one has (cid:12)(cid:12)(cid:12)(cid:12) O (cid:16) s k − (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k (2 π ) k (cid:12)(cid:12)(cid:12) C s k − (cid:12)(cid:12)(cid:12) /
2. As a consequence, for | s | ≥ s ′ and ∓ s > ℜ h ln E − − ◦ ξ − ( s ± i τ ) i ≤ − k (2 π ) k τ (cid:12)(cid:12)(cid:12) C s k − (cid:12)(cid:12)(cid:12) uniformly 0 ≤ τ ≤ α . (B.18)Therefore, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z C bd ; L d µ π E − − ( µ )( λ − µ ) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ǫ = ± ǫ α Z d τ i π (cid:16) E − − /ξ ′ (cid:17) ◦ ξ − ( ǫ u L − i ǫτ ) (cid:2) ξ − ( ǫ u L − i ǫτ ) − λ (cid:3) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ τ ∈ [ 0 ; α ] ǫ ∈{± } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ξ − ( ǫ u L − i ǫτ ) − λ i − r ξ ′ ◦ ξ − ( ǫ u L − i ǫτ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α Z d τ π e − k (2 π ) k u k − L | C | τ = O (cid:16) u − k − rL (cid:17) . (B.19) B.2 The generating function: form factor-like representation
From now on, we assume that the function E − takes the form E − − ( λ ) = e ixu ( λ ) + g ( λ ) where u ( λ ) is given by (2.14)and g is a bounded holomorphic function on the strip U δ around R (2.15). We also assume that ν ∈ O ( U δ ).We remind that the parameters { µ a } a ∈ Z (resp. { λ a } a ∈ Z ) are defined as the unique solutions to L ξ ( µ a ) = a , (resp. L ξ ν ( λ a ) = a ), where ξ is given by (2.6) and ξ ν ( λ ) = ξ ( λ ) + ν ( λ ) / L . We define the functional X N h ν, E − i as X N h ν, E − i = N + X n = X p < ··· < p n p k ∈B ext L X h < ··· < h n h k ∈B int L N Q a = E − ( λ a ) N + Q a = E − (cid:0) µ ℓ a (cid:1) b D N { p a } n { h a } n ! (cid:2) ν, ξ, ξ ν (cid:3) . (B.20)The functional b D N , n has been introduced in (4.4). The sums in (B.20) run through ordered n − uples of integers p < · · · < p n belonging to B ext L = B L \ [[ 1 ; N + n − uples of integers h < · · · < h n B in L = [[ 1 ; N + B L = { j ∈ Z : − w L ≤ j ≤ w L } and the sequence w L ∼ L . In particular,when L → + ∞ , w L grows much faster then N . The integers { p a } and { h a } define the sequence ℓ < · · · < ℓ N + asexplained in (2.3).The functional X N (cid:2) ν, E − (cid:3) admits two di ff erent representations. On the one hand, as written in (B.20), X N (cid:2) ν, E − (cid:3) is closely related to a form factor expansion of certain two-point functions in generalized free-fermion models.On the other hand, after some standard manipulations [60], one can also recast X N (cid:2) ν, E − (cid:3) in terms of a finite-sizedeterminant which goes to a Fredholm minor in the N , L → + ∞ limit.We derive this finite-size determinant representation for X N (cid:2) ν, E − (cid:3) below. Proposition B.2
Under the aforestated assumptions concerning the functions E − and ν , the functional X N (cid:2) ν, E − (cid:3) admits a finite-size determinant representationX N h ν, E − i = ( S ( L )0 h E − − i + ∂∂α ) | α = · det N " δ k ℓ + V ( L ) ( λ k , λ ℓ ) L ξ ′ ν ( λ ℓ ) + α P ( L ) ( λ k , λ ℓ ) L ξ ′ ν ( λ ℓ ) , (B.21) where V ( L ) ( λ, µ ) = πν ( λ )] sin (cid:2) πν ( µ ) (cid:3) i π ( λ − µ ) E − ( µ ) E − ( λ ) · n O ( L ) h ν, E − − i ( λ ) − O ( L ) h ν, E − − i ( µ ) o , (B.22) P ( L ) ( λ, µ ) = πν ( λ )] sin (cid:2) πν ( µ ) (cid:3) π E − ( λ ) E − ( µ ) · O ( L ) h ν, E − − i ( λ ) · O ( L ) h ν, E − − i ( µ ) . (B.23) Also, we have setO ( L ) h ν, E − − i ( λ ) = i Z C bk ; L d µ π E − − ( µ ) µ − λ + E − − ( λ )e − i πν ( λ ) − + iI ( L )1 h E − − i ( λ ) . (B.24) The contour of integration has been depicted on Fig. 5 and S ( L )0 (resp. I ( L ) r ) is given by (B.3) (resp. (B.6) ). b b b b b b − q − q q q − A L B L L ξ ( − A L ) = − w L − / L ξ ( B L ) = w L + / e C q C q C bk ; L Figure 5: Contour C bk ; L appearing in the definition of O ( L ) (cid:2) ν, E − − (cid:3) ( λ ), contour C q (solid line) and contour e C q (dashed line). The contour C bk ; L is such that, for (cid:12)(cid:12)(cid:12) ℜ λ (cid:12)(cid:12)(cid:12) ≥ q , it stays uniformly away from the real axis. Proof —
We first recast the sum over the integers { p a } and { h a } corresponding to particle-hole like excitations intothe equivalent sum over all possible choices of integers ℓ a : ℓ < · · · < ℓ N + with ℓ a ∈ B L = B int L ∪ B ext L , cf (2.3).As all the sums are finite, there is no problem in permuting the orders of summation. Therefore, X N (cid:2) ν, E − (cid:3) = X ℓ < ··· <ℓ N + ℓ a ∈B L N Q a = E − ( λ a ) N + Q a = E − (cid:0) µ ℓ a (cid:1) · b D N { p a } n { h a } n ! (cid:2) ν, ξ, ξ ν (cid:3) . (B.25)47he determinant entering in the definition of b D N can be represented as N Y a = µ ℓ a − µ ℓ N + λ a − µ ℓ N + · det N " µ ℓ a − λ b = det N + " (cid:0) − δ b , N + (cid:1) µ ℓ a − λ b + δ b , N + = + ∂∂α ! | α = det N " µ ℓ a − λ b − αµ ℓ N + − λ b . (B.26)There we have used that for any polynomial Q of degree 1, one has Q (1) = Q (0) + Q ′ (0).It follows from the above representation that the summand in (B.25) is a symmetric function of the N + µ ℓ a that is moreover vanishing whenever ℓ k = ℓ a , k , a . Therefore, we can replace thesummation over the fundamental simplex ℓ < · · · < ℓ N + in the ( N + th power Cartesian product B N + L by asummation over the whole space B N + L , provided that we divide the result by ( N + N + N × N determinant: X N h ν, E − i = N Y a = [ πν ( λ a )] b ξ ′ ν ( λ a ) · X n ∈B L E − − ( µ n )2 π L ξ ′ ( µ n ) + ∂∂α ! | α = det N h M jk + α e P jk ( µ n ) i , (B.27)with M k ℓ = δ k ,ℓ E − ( λ ℓ )2 π L S ( L )2 (cid:2) E − − (cid:3) ( λ ℓ ) + (cid:0) − δ k ,ℓ (cid:1) E − ( λ ℓ )2 π L ( λ k − λ ℓ ) (cid:26) S ( L )1 (cid:2) E − − (cid:3) ( λ k ) − S ( L )1 (cid:2) E − − (cid:3) ( λ ℓ ) (cid:27) , (B.28) S ( L ) r being given by (B.2), (B.3)-(B.5) and e P jk ( µ n ) being a µ n -dependent rank 1 matrix: e P jk ( µ n ) = − E − ( λ k ) (cid:16) µ n − λ j (cid:17) · S ( L )1 (cid:2) E − − (cid:3) ( λ k )2 π L . (B.29)Using the fact that e P jk ( µ n ) is a rank one matrix that contains all the dependence of the determinant on thesummation variable µ n , it is readily seen that X n ∈B L E − − ( µ n )2 π L ξ ′ ( µ n ) + ∂∂α ! | α = · det N h M jk + α e P jk ( µ n ) i = (cid:20) S ( L )0 (cid:2) E − − (cid:3) + ∂∂α (cid:21) | α = · det N h M jk + α P jk i (B.30)where P jk = − E − ( λ k )2 π L S ( L )1 (cid:2) E − − (cid:3) ( λ k ) · S ( L )1 (cid:2) E − − (cid:3) ( λ j ) . (B.31)Applying (B.4), (B.5) and then using that L ξ ( λ k ) = L ξ ν ( λ k ) − ν ( λ k ) = k − ν ( λ k ), we obtain that M k ℓ E − ( λ k ) E − ( λ ℓ ) = δ k ℓ ξ ′ ν ( λ ℓ )4 sin [ πν ( λ ℓ )] + E − ( λ ℓ ) E − ( λ k ) O ( L ) h ν, E − − i ( λ k ) − O ( L ) h ν, E − − i ( λ ℓ )2 i π L ( λ k − λ ℓ ) , (B.32)where O ( L ) h ν, E − − i is given by (B.24). Note that we have slightly deformed the form of the contours C bk ; L inrespect to Fig. 4. Very similarly, we find P jk E − ( λ j ) E − ( λ k ) = E − (cid:16) λ j (cid:17) E − ( λ k )2 π L O ( L ) h ν, E − − i ( λ k ) O ( L ) h ν, E − − i (cid:16) λ j (cid:17) = P ( L ) (cid:16) λ j , λ k (cid:17) L sin πν (cid:16) λ j (cid:17) sin πν ( λ k ) . (B.33)where P ( L ) (cid:16) λ j , λ k (cid:17) is given by (B.23). It then remains to factor out the pre-factors from the determinant.48 .3 Thermodynamic limit of X N h ν, E − − i Proposition B.3
The thermodynamic limit of X N h ν, E − i is well defined and can be expressed in terms of a Fred-holm determinant minor. Namely, X N h ν, E − i −→ N / L → D X C ( ∞ ) E h ν, E − i withX C ( ∞ ) E h ν, E − i = S C ( ∞ ) E h E − − i + q Z − q d λπ sin [ πν ( λ )] F + ( λ ) E − ( λ ) O C ( ∞ ) E (cid:2) ν, E − − (cid:3) ( λ ) · det [ I + V ] h ν, E − − i . (B.34) Here I + V is an integral operator on (cid:2) − q ; q (cid:3) acting on L (cid:0)(cid:2) − q ; q (cid:3)(cid:1) with a kernelV ( λ, µ ) = πν ( λ )] sin (cid:2) πν ( µ ) (cid:3) i π ( λ − µ ) E − ( λ ) E − ( µ ) · (cid:26) O C ( ∞ ) E h ν, E − − i ( λ ) − O C ( ∞ ) E h ν, E − − i ( µ ) (cid:27) (B.35) and the contour C ( w ) E dependent functionals O C ( w ) E h ν, E − − i ( λ ) and S C ( w ) E h E − − i are given byO C ( w ) E h ν, E − − i ( λ ) = i Z C ( w ) E d µ π E − − ( µ ) µ − λ + E − − ( λ )e − i πν ( λ ) − S C ( w ) E h E − − i = Z C ( w ) E d λ π E − − ( λ ) . (B.36) F + ( λ ) is the unique solution to the integral equation † sin [ πν ( λ )] F + ( λ ) + q Z − q V ( λ, µ ) sin (cid:2) πν ( µ ) (cid:3) F + ( µ ) d µ = sin [ πν ( λ )] E − ( λ ) O C ( ∞ ) E h ν, E − − i ( λ ) . (B.37) Also, C ( w ) E = C ( ∞ ) E ∩ n z ∈ C : (cid:12)(cid:12)(cid:12) ℜ ( z ) (cid:12)(cid:12)(cid:12) ≤ w o and C ( ∞ ) E have been depicted on Fig. 2. This representation can be seen as a generalization of the results obtained in [60]. Also, the contour C ( ∞ ) E canbe thought of as the L → + ∞ limit of the contour C bk ; L . Proof —
It is a direct consequence of the estimates obtained in appendix B.1 for I ( L ) r (cid:2) E − − (cid:3) together with the fact thatdet N h δ k ℓ + o (cid:16) L − (cid:17)i → (cid:0) L − (cid:1) that are uniform in the entries, that X N h ν, E − i −→ N / L → D S C ( ∞ ) E h E − − i + ∂∂α ! | α = · det [ I + V + α P ] (B.38)with I + V + α P acting on (cid:2) − q ; q (cid:3) and P ( λ, µ ) = π sin [ πν ( λ )] sin (cid:2) πν ( µ ) (cid:3) E − ( λ ) E − ( µ ) O C ( ∞ ) E h ν, E − − i ( λ ) O C ( ∞ ) E h ν, E − − i ( µ ) . (B.39)Note that there is no problem with the integration over an infinite contour C ( ∞ ) E in O C ( ∞ ) E h ν, E − − i ( µ ) and S C ( ∞ ) E h E − − i in as much as C ( ∞ ) E is built precisely in such a way to ensure the exponential decay of the integrand at infinity.Using that P is a one dimensional projector, we get thatdet [ I + V + α P ] = det [ I + V ] + α q Z − q ( I + V ) − ( λ, µ ) P ( µ, λ ) d λ d µ . (B.40)It then remains to take the α -derivative and use the definition of F + ( λ ). † By no means F + ought to be confused with the shift function .4 An algebraic representation for the Fredholm minor Proposition B.4
For L large enough, the finite N Fredholm minor X N h ν, E − i defined in (B.21) can be represented,through purely algebraic manipulations, as the below finite sum:X N h ν, E − i = N X n = i ( − n n ! X i ,..., i n i a ∈ [[ 1 ; N ]] I C q d n z (2 i π ) n Z C ( L ) d n + y (2 i π ) n + n + Y k = n f ( L ) ( y k , ν ( y k )) · E − − ( y k ) o n Y a = E − (cid:0) λ i a (cid:1) × n Y k = y n + − z k (cid:0) y n + − λ i k (cid:1) ( y k − z k ) · det n " z a − λ i b n Y k = (cid:2) πν (cid:0) λ i k (cid:1)(cid:3)(cid:0) z k − λ i k (cid:1) L ξ ′ ν (cid:0) λ i k (cid:1) . (B.41) Above, appear two contours, C q which stands for a small counterclockwise loop around (cid:2) − q ; q (cid:3) as depicted onFig. 5 and C ( L ) = C bk ; L ∪ C ↑ ; L ∪ C ↓ ; L ∪ C bd ; L ∪ f C q . Note that C bk; L is as it has been depicted on Figs. 3-4. As shownon Fig. 5, e C q stands for a small counterclockwise loop encircling C q . Finally, the function f ( L ) ( y , ν ) is supportedon C ( L ) and readsf ( L ) ( y , ν ) = C bk ; L ( y ) + − e − i π L ξ ( y ) C ↑ ; L ( y ) + i π L ξ ( y ) − C ↓ ; L ( y ) − − i πν − e C q ( y ) + C bd ; L ( y ) . (B.42) where A stands for the indicator function of A.Proof — The functional O ( L ) (cid:2) ν, E − − (cid:3) ( z ) as defined in (B.24) is holomorphic in some su ffi ciently small open neighbor-hood of (cid:2) − q ; q (cid:3) . Hence, there exists a small counterclockwise loop C q around (cid:2) − q ; q (cid:3) ( cf Fig. 5) such that thekernel V ( L ) ( λ, µ ) admits the integral representation V ( L ) ( λ, µ ) = πν ( λ )] sin (cid:2) πν ( µ ) (cid:3) E − ( λ ) E − ( µ ) I C q O ( L ) (cid:2) ν, E − − (cid:3) ( z )( z − λ ) ( z − µ ) d z (2 i π ) , for λ, µ ∈ { λ , . . . , λ N } . (B.43)In (B.43) we have used that λ , . . . , λ N are all inside of C q for L large enough. We first expand the N × N determinant appearing in the final expression for X N (cid:2) ν, E − (cid:3) into its discreet Fredholm series:det N " δ k ℓ + V ( L ) ( λ k , λ ℓ ) L ξ ′ ν ( λ ℓ ) + α P ( L ) ( λ k , λ ℓ ) L ξ ′ ν ( λ ℓ ) = N X n = X i ,..., i n i a ∈ [[ 1 ; N ]] det n h V ( L ) (cid:0) λ i a , λ i b (cid:1) + α P ( L ) (cid:0) λ i a , λ i b (cid:1)i n ! Q nk = (cid:2) L ξ ′ ν (cid:0) λ i k (cid:1)(cid:3) . (B.44)Next, observe thatdet n h V ( L ) (cid:0) λ i a , λ i b (cid:1) + α P ( L ) (cid:0) λ i a , λ i b (cid:1)i = I C q d n z (2 i π ) n n Y a = ( O ( L ) (cid:2) ν, E − − (cid:3) ( z a ) z a − λ i a ) × n Y a = n (cid:2) πν (cid:0) λ i a (cid:1)(cid:3) E − (cid:0) λ i a (cid:1)o × det n + (cid:0) z a − λ i b (cid:1) − α − iO ( L ) (cid:2) ν, E − − (cid:3) (cid:0) λ i b (cid:1) (B.45)It can be readily seen that for any z belonging to the interior of e C q O ( L ) h ν, E − − i ( z ) = Z C ( L ) i d y π f ( L ) ( y , ν ( y )) y − z E − − ( y ) with C ( L ) = C bk ; L ∪ C ↑ ; L ∪ C ↓ ; L ∪ e C q ∪ C bd ; L , (B.46)50nd f ( L ) is as given by (B.42). Then, using the multilinear structure of a determinant, one gets that ( S ( L )0 h E − − i + ∂∂α ) α = · det n + (cid:0) z a − λ i b (cid:1) − α − iO ( L ) h ν, E − − i (cid:0) λ i a (cid:1) = det n + (cid:0) z a − λ i b (cid:1) − − iO ( L ) h ν, E − − i (cid:0) λ i b (cid:1) S ( L )0 h E − − i = Z C ( L ) d y π E − − ( y ) f ( L ) ( y , ν ( y )) det n + (cid:0) z a − λ i b (cid:1) − (cid:0) y − λ i b (cid:1) − = Z C ( L ) d y π E − − ( y ) f ( L ) ( y , ν ( y )) n Y k = z k − y λ i k − y det n " z a − λ i b . (B.47)This leads to the claim, once upon inserting this representation into the discreet Fredholm series. B.5 The Natte series for a Fredholm minor
In this subsection, we recall the form of the Natte series representation for the Fredholm minor (B.34) involved inthe representation of form factor sums in generalized free fermionic models. We refer the reader to theorem 2.2and proposition 7.2 of reference [62] for further details relative to this Natte series expansion.Let E − = e − ixu ( λ ) − g ( λ ) be such that • u and g are holomorphic in the open neighborhood U δ/ of R ; • u has a unique saddle-point λ on the real axis which is of order 1, ie u ′′ ( λ ) < • the function ν is holomorphic in an open neighborhood N q ⊂ U δ/ of (cid:2) − q ; q (cid:3) .Also, let C ( w ) E = C ( ∞ ) E ∩ n z ∈ C : (cid:12)(cid:12)(cid:12) ℜ z (cid:12)(cid:12)(cid:12) < w o . The contours C ( ∞ ) E and C ( w ) E have been depicted in Fig. 2.For w > | λ | + q > x large enough, the Fredholm minor X C ( w ) E (cid:2) ν, E − (cid:3) defined in (B.34) admits the belowNatte series representation X C ( w ) E (cid:2) ν, E − (cid:3) = B (cid:2) ν, u + i + (cid:3) x ν ( q ) + ν ( − q ) e q R − q [ ixu ′ ( λ ) + g ′ ( λ ) ] ν ( λ )d λ ( A [ ν ] ] q ; + ∞ [ ( λ ) √− π xu ′′ ( λ ) e ixu ( λ ) + g ( λ ) + A + (cid:2) ν, u + i + (cid:3) x + ν ( q ) e ixu ( q ) + g ( q ) + A − [ ν, u ] x − ν ( − q ) e ixu ( − q ) + g ( − q ) + X n ≥ X K n X E n ( ~ k ) Z C ( w ) ǫ t H ( { ǫ t } ) n ; x ( { u ( z t ) } ; { z t } ) [ ν ] Y t ∈ J { k } e ǫ t g ( z t ) d n z t (2 i π ) n ) . (B.48)The + i + regularization of u only matters in the time-like regime (where | λ | < q ). The functionals B , A ± and A are given respectively by (4.46) (4.47) and (4.48). The notations and the structure of the sums appearing in thesecond line of (B.48) are exactly as explained in theorem 4.1.The Natte series is convergent for x large enough in as much as, for n large enough, X K n X E n ( ~ k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H ( { ǫ t } ) n ; x [ ν ] Y t ∈ J { k } e ǫ t g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L (cid:0) C ( ∞ ) ǫ t (cid:1) ≤ c (cid:18) c x (cid:19) nc . (B.49)There c and c are some n -independent constants. They only depend on the values taken by u , and g in somesmall neighborhood of the base curve C ( ∞ ) E and by ν on a small neighborhood of (cid:2) − q ; q (cid:3) , whereas c =
34 min (cid:18) / , − τ = ± (cid:12)(cid:12)(cid:12) ℜ (cid:2) ν ( τ q ) (cid:3)(cid:12)(cid:12)(cid:12) − Υ ǫ (cid:19) where Υ ǫ = (cid:26) (cid:12)(cid:12)(cid:12) ℜ (cid:2) ν ( z ) − ν ( τ q ) (cid:3)(cid:12)(cid:12)(cid:12) : | z − τ q | ≤ ǫ , τ = ± (cid:27) . ǫ > ffi ciently small but arbitrary otherwise. We stress that, should these norms change, then so wouldchange the constants c , c and c but the overall structure of the estimates in x would remain.The Natte series expansion (B.48) has a well defined w → + ∞ limit: all the concerned integrals are convergentas the functions H ( { ǫ t } ) n ; x approach zero exponentially fast in respect to any variable that runs to ∞ along C ( ∞ ) ǫ t .Moreover, this limit does not alter in any way the estimates (B.49) ensuring the convergence of the Natte series(the constants c - c are w -independent).We now list several properties of the functions H ( { ǫ t } ) n ; x :i) H ( { ǫ t } ) n ; x ( { u ( z t ) } ; { z t } ) [ ν ] is a function of { u ( z t ) } and { z t } . It is also a regular functional of ν .ii) H ( { ǫ t } ) n ; x ( { u ( z t ) } ; { z t } ) (cid:2) γν (cid:3) = O ( γ n ) and the O holds in the (cid:0) L ∩ L ∞ (cid:1)(cid:0) C ( ∞ ) ǫ t (cid:1) sense.iii) H ( { ǫ t } ) n ; x can be represented as: H ( { ǫ t } ) n ; x ( { u ( z t ) } ; { z t } ) [ ν ] = e H ( { ǫ t } ) n ; x ( { ν ( z t ) } ; { u ( z t ) } ; { z t } ) Y t ∈ J { k } ( κ [ ν ] ( z t )) − ǫ t × Y t ∈ J { k } ǫ t = (cid:16) e − i πν ( z t ) − (cid:17) . (B.50)with e H ( { ǫ t } ) n ; x is a holomorphic function for (cid:12)(cid:12)(cid:12) ℜ ( ν ) (cid:12)(cid:12)(cid:12) ≤ / κ [ ν ] ( λ ) is given by (4.47).iv) One has H ( { ǫ t } )1; x = O (cid:0) x −∞ (cid:1) and for n ≥ H ( { ǫ t } ) n ; x = O (cid:0) x −∞ (cid:1) + [ n / X b = b X p = n / − b X m = b − [ n ] e ix [ u ( λ ) − u ( − q ) ] x ν ( − q ) η b · e ix [ u ( q ) − u ( − q ) ] x [ ν ( q ) + ν ( − q ) ] m − η p · X τ ∈{± } e τ x n − b · h H ( { ǫ t } ) n ; x i m , p , b ,τ . (B.51)The O (cid:0) x −∞ (cid:1) appearing above holds in the (cid:0) L ∩ L ∞ (cid:1)(cid:0) C ( ∞ ) E (cid:1) sense. In order to lighten the formula, we have droppedthe argument-dependent part. However, we do stress that the O (cid:0) x −∞ (cid:1) as well as (cid:2) H ( { ǫ t } ) n ; x (cid:3) m , p , b ,τ depend on the sameset of variables as H ( ǫ t ) n ; x . Also, we agree upon η = λ > q , η = − | λ | < q and we made use of theshorthand notation e + = e ixu ( q ) x − ν ( q ) , e − = e ixu ( − q ) x ν ( − q ) and e = (cid:0) + η (cid:1) e ixu ( λ ) . (B.52)Finally, the functions (cid:2) H ( { ǫ t } ) n ; x (cid:3) m , p , b ,τ are only supported on a small vicinity of the points ± q and λ . In such acase, the contour of integration reduces to an integration for each variable z t to a small circle ∂ D v τ around v τ ( v ± = ± q , v = λ ). Their dependence on x is as follows. If a variable z t is integrated in a vicinity of v τ , Thefunction (cid:2) H ( { ǫ t } ) n ; x (cid:3) m , p , b ,τ contains a fractional power of x ± [2 ν ( z t ) − ν ( v τ )] , multiplied by a function of z t which has anasymptotic expansion into inverse powers of x . This asymptotic expansion holds on ∂ D v τ . The coe ffi cients inthis asymptotic expansions contain poles at z t = v τ . By computing the integrals associated to the terms in thisasymptotic expansion through the poles at z t = v τ one obtains that function coe ffi cients associated to x − r termsproduce, in fine , a contribution that is a (ln x / x ) r . Finally, the structure of these poles is such that, upon computingall the partial derivatives and for any holomorphic function h in the vicinity of the points ± q , λ , one should makethe replacement: X t ∈ J { ~ k } ǫ t h ( z t ) ֒ → η b ( h ( λ ) − h ( − q )) + (cid:0) m − η p (cid:1) ( h ( q ) − h ( − q )) + (cid:0) δ τ ;1 + δ τ ; − + (cid:0) + η (cid:1) δ τ ;0 / (cid:1) h ( v τ ) . (B.53)There is one last property which we conjecture to be true for the detailed representation (B.51) of H ( { ǫ t } ) n ; x butthat has not been proven so far. Namely, 52 onjecture B.1 For a given n the sum in (B.51) only contains those combinations of the integers m , p , b and τ that satisfy to the constraint (cid:0) m − η p + δ τ, (cid:1) + b + (cid:0) m + η ( b − p ) − δ τ, − (cid:1) ≤ n . (B.54) C Multidimensional Fredholm series for lim N → + ∞ ρ ( m ) N ;e ff ( x , t ) We begin this appendix by deriving the so-called discreet multidimensional Fredholm series representation for ρ ( m ) N ;e ff ( x , t ). We will prove in theorem C.1 that this representation has a well defined thermodynamic limit thatwe denote ρ ( m )e ff ( x , t ). This analysis will allow us to provide (proposition C.1) yet another representation for thethermodynamic limit ρ ( m )e ff ( x , t ). This alternative representation for ρ ( m )e ff ( x , t ) is used in subsection 4.6 so as toconstruct the multidimensional Natte series for ρ ( m )e ff ( x , t ). Theorem C.1 ρ ( m ) N ;e ff ( x , t ) admits a well defined thermodynamic limit ρ ( m )e ff ( x , t ) that is given by a multidimensionalFredholm series ρ ( m )e ff ( x , t ) = m X n = c ( − n n ! ∂ m ∂γ m q Z − q d n λ (2 i π ) n I C q d n z (2 i π ) n Z C d n + y (2 i π ) n + e ix U ( { λ a } n ; { y a } n + | γ ) Q n + k = f ( y k , γν ( y k )) Q nk = ( z k − λ k ) ( y k − z k ) ( y n + − λ k ) det n " ( y n + − z k ) z a − λ b × n Q a = n + Q b = ( y b − λ a − ic ) ( λ a − y b − ic ) n + Q a , b = ( y a − y b − ic ) n Q a , b = ( λ a − λ b − ic ) n Y k = n (cid:2) πγν ( λ k ) (cid:3)o det n h δ k ℓ + γ b V k ℓ (cid:2) γν (cid:3)i det n h δ k ℓ + γ b V k ℓ (cid:2) γν (cid:3) i det (cid:2) I + γ R / π (cid:3) det [ I − K / π ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = . (C.1) The function f appearing above is supported on the contour C = C ( ∞ ) E ∪ e C q . The contour C q is a small looparound (cid:2) − q ; q (cid:3) whereas e C q is a small loop around C q . Both C q and e C q lie below the curve C ( ∞ ) E as depicted onFig. 6. All of the aforementioned contours lie inside of the strip U δ (2.15) . The function f is supported on C andreads f ( y , ν ( y )) = C ( ∞ ) E ( y ) − − i πν ( y ) − e C q ( y ) with C = C ( ∞ ) E ∪ e C q . (C.2) There A stands for the indicator function of the set A. The function ν appearing in the n th -summand of (C.1) corresponds to the unique solution of the linear integral equation driven by the resolvent R of the Lieb kernel ( ie [ I − K / π ] [ I + R / π ] = I ) : ν ( λ ) + γ q Z − q d µ π R ( λ, µ ) ν ( µ ) = − Z ( λ ) / + n X a = φ ( λ, λ a ) − n + X a = φ ( λ, y a ) . (C.3) Hence, ν depends on the integration variables λ a (with a = , . . . , n) and y a (with a = , . . . , n + ), ie ν ( λ ) ≡ ν (cid:0) λ | { λ a } n ; { y a } n + (cid:1) . We kept this dependence implicit in (C.1) so as to shorten the formulae. The entries of thefinite-size determinants are as defined in (4.8) . They depend on the same set of auxiliary variables as ν . Finally,we agree upon U (cid:16) { λ a } n ; { y a } n + | γ (cid:17) = n + X a = u ( y a ) − n X a = u ( λ a ) + (1 − γ ) q Z − q u ′ ( λ ) ν ( λ ) d λ . (C.4)53 R + i δ R − i δ b bb b − q q − w w C q e C q C ( w ) E C ( ∞ ) E Figure 6: The contour C ( w ) E consists of the solid line. The contour C ( ∞ ) E corresponds to the union of the solid anddotted lines. The loop C q is depicted in solid lines whereas the loop e C q is depicted in dotted lines. Proof —
In order to implement the substitution of the operators ∂ ς p and ∂ η j , p ( cf section 4.3) in the representation (4.41)we introduce, exactly as it was done in the proof of theorem 4.1, the functions e E − ( λ ) (4.51) (whose definitioninvolves the functions e g = e g , s + e g , r cf (4.52)) as well as ν s (4.36) and ̟ r (4.32).We then consider the discreet Fredholm series representation for ∂ m γ X N h γν s , e E − i | γ = obtained in propositionB.4. This will allow us to compute the relevant Taylor coe ffi cients ( cf subsection 4.3 equation (4.20) and (4.21))arising in the representation (4.41) for ρ ( m ) N ;e ff ( x , t ). One has that ∂ m ∂γ m N + Y a = e E − ( µ a ) · N Y a = e E − − ( λ a ) · X N h γν s , e E − i b G ( β ) γ ;2 A L [ ̟ r ] | γ = = m X n = i ( − n n ! X i ,..., i n i a ∈ [[ 1 ; N ]] I C q d n z (2 i π ) n Z C ( L ) d n + y (2 i π ) n + e L ( m ) Γ ( L ) h F i ,... i n b G ( β ) γ ;2 A L i . (C.5)The contours C ( L ) , C q have been defined in proposition B.4. We stress that the summation over n in (C.5) couldhave been stopped at n = m since, prior to taking the γ -derivative at γ =
0, the n th term of the series (B.41) is asmooth function of γ that behaves as O ( γ n ). We have set F i ,..., i n (cid:2) γν s (cid:3) = n Y k = (cid:2) πγν s (cid:0) λ i k (cid:1)(cid:3) L ξ ′ γν s (cid:0) λ i k (cid:1) (cid:0) z k − λ i k (cid:1) Q n + k = f ( L ) ( y k , γν s ( y k )) n Q k = (cid:0) y n + − λ i k (cid:1) ( y k − z k ) × det n " y n + − z j z j − λ i k e ix U ( L ) ( { λ k } ; { µ k } ; { y k }| γ ) , (C.6)the function f ( L ) ( y , ν ( y )) is given in (B.42) and we have set U ( L ) ( { λ k } ; { µ k } ; { y k } | γ ) = n + X k = u ( y k ) − n X k = u (cid:0) µ i k (cid:1) − u ( µ N + ) + N X k = , i ,..., i n u ( λ k ) − u ( µ k ) . (C.7)54ast but not least, e L ( m ) Γ ( L ) h F i ,... i n G ( β ) γ ;2 A L i = m X n ,..., n s = s Y p = a n p p n p ! ∂ m ∂γ m ( Q n + k = e e g , s ( y k ) e e g , s ( µ N + ) Q nk = e e g , s ( µ ik ) s Y p = n Γ ( L ) (cid:2) γν s (cid:3) (cid:0) t p (cid:1)o n p × N + Y j = e − e g , r ( µ j ) N Y j = , i ,..., i n e e g , r ( λ j ) n + Y j = e e g , r ( y j ) · F i ,..., i n (cid:2) γν s (cid:3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = ς a = b G ( β ) γ ;2 A L [ ̟ r ] ) | γ = . (C.8)The functional Γ ( L ) is evaluated at the discretization points ( cf definition 4.2, subsection 4.4 and subsection4.2) t p , p = , . . . , s for the contour C out encircling the compact K q . C out has been depicted in the lhs of Fig. 1.The functional Γ ( L ) reads Γ ( L ) [ ν ] ( µ ) = N X j = j , i ,..., i n φ ( µ, µ j ) − φ ( µ, λ j ) with µ j and λ j defined by ξ ( µ j ) = j / L and ξ ν ( λ j ) = j / L . (C.9)We do stress that the variables y k , with k = , . . . , n +
1, and µ p or λ p with p = , . . . , N + K A L , where A L is such that L ξ ( − A L ) = − w L − /
2. As aconsequence, the singularities at λ = t i , p of the functions e g , r ( λ ) (C.8) are always disjoint from the variables y k , µ p or λ i b . Indeed, t , p and t , p with p = , . . . , r stand for discretization points of the contour C out / in appearing in the rhs of Fig. 1, cf subsection 4.4. These two contours are such that d (cid:0) C out / in , K A L (cid:1) > L .According to the prescription that has been adopted in section 4.3, one has to compute the m th γ -derivative ofrepresentation (C.8) prior to implementing the operator substitution. For this, consider any smooth function w ( γ )such that w ( γ ) = O ( γ n ) at γ =
0. By applying the Faa-dí-Bruno formula, we get that1 m ! ∂ m ∂γ m ( w ( γ ) N Y a = , i ,..., i n e e g , r ( λ a ) b G ( β ) γ ;2 A L [ ̟ r ] ) | γ = = X { ℓ a }′ w ( ℓ N + ) (0) ℓ ! ℓ N + ! N Y p = , i ,..., i n ℓ p ! ∂ ℓ p ∂γ ℓ p n e e g , r ( λ p ) o | γ = × ∂ ℓ ∂γ ℓ n b G ( β ) γ ;2 A L [ ̟ r ] o | γ = = X { ℓ a }′ w ( ℓ N + ) (0) ℓ ! ℓ N + ! X { k a , j } ′ N Y p = , i ,..., i n ( ∂ | k p | e e g , r ( τ p ) ∂τ | k p | p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ p = µ p × ℓ p Y j = (cid:18) λ ( j ) p j ! (cid:19) k p , j ) × ∂ ℓ ∂γ ℓ n b G ( β ) γ ;2 A L [ ̟ r ] o | γ = . (C.10)There the ′ in front of the sums indicates that these are constrained. The first sums runs through all choices of N + ℓ p ≥ ℓ i p = , for p = , . . . , n , ℓ N + ≥ n and N + X p = ℓ p = m . (C.11)The second sum runs through all the possible choices of sequences of integers k p , j with p = , . . . , N and j = , . . . , ℓ p such that ℓ p X j = jk p , j = ℓ p . (C.12)Finally, we agree upon (cid:12)(cid:12)(cid:12) k p (cid:12)(cid:12)(cid:12) = ℓ p X j = k p , j and have set λ ( j ) p = ∂ j γ p h λ p (cid:16) γ p (cid:17)i | γ p = with λ p defined by ξ γ p ν s (cid:16) λ p (cid:17) = p / L . (C.13)55n (C.13), we have explicitly insisted on the fact that λ p is a function of the parameter γ p . By substituting therepresentation (C.10) on the level of (C.5)-(C.8), one can implement the operator substitution a k ֒ → ∂ ς k and b j , k ֒ → ∂ η j , k on the level of (C.8).The functionals F i ,..., i n and b G ( β ) γ ;2 A L are regular in the sense of definition 4.1. Moreover, as L and hence 2 A L arelarge enough, and β defining e U β is chosen in such a specific way † that the constant of regularity C G AL of b G ( β ) γ ;2 A L satisfies (4.30), one gets that { ς a } s
7→ F i ,..., i n (cid:2) γν s ( ∗ | { ς a } ) (cid:3) and { η , p } r ∪ { η , p } r b G ( β ) γ ;2 A L h ̟ r (cid:0) ∗ | { η a , p } (cid:1)i (C.14)are holomorphic in respect to { ς a } s ∈ N s , { η , p } r ∪{ η , p } r ∈ N r , where N is an r and s independent neighborhoodof 0 ∈ C . As the constant of regularity C G AL is large enough and | γ | can be taken small enough, the size of theneighborhood N is large enough in order to ensure the convergence of the series of di ff erential operators issuingfrom the exponentials e e g , s and e e g , r , once upon the operator substitution is carried out. In virtue of corollary D.1,and similarly to the summations (4.55)-(4.56), the action of the translation operators can be computed directlyunder the integral sign in (C.5) (the integration contours being Cartesian products of one dimensional compactcurves) and prior to taking the partial τ p or γ -derivatives in (C.10). There are also the di ff erential operators arisingfrom the substitutions a p ֒ → ∂ ς p in (C.8) for those parameters a p that are written down explicitly. The resulting ∂ ς p -derivatives should appear outside of the integrals that are written down in (C.5). However, the integrand ofthese compactly supported integrals is a continuous function of the integration variables that is holomorphic inrespect to { ς p } s , this uniformly in respect to the integration variables. As a consequence, one can exchange thederivation and integration symbols in this case as well.Note that the constraints (C.13) on the k p , j ’s ensure that in (C.10) there is at most m − n integers k p , j that di ff erfrom zero. As a consequence, there will be at most m translation operators in respect to the η i , k variables to takeinto account once that the operator substitution is made. More precisely, the substitution b j , k ֒ → ∂ η j , k shifts theparameters η j , k in ̟ r (cid:0) λ, { η j , k } (cid:1) (4.32) to the below value η j , k = n + X p = t j , k − y p − n X p = t j , k − µ i p − t j , k − µ N + + N X p = ℓ p , t j , k − τ p − t j , k − µ p , (C.15)where the ulimate sum in (C.15) only involves m terms at most. Under the substitution a p ֒ → ∂ ς a , the exponentialsin (C.8) produce a translation of the function ν s ֒ → e ν s , where e ν s ( λ ; { ς a } ) = ν s ( λ ; { ς a } ) + s X j = t j + − t j i π (cid:0) t j − λ (cid:1) (cid:26) φ (cid:0) t j , µ N + (cid:1) − φ (cid:0) t j , y n + (cid:1) + n X a = φ (cid:0) t j , µ i a (cid:1) − φ (cid:0) t j , y a (cid:1)(cid:27) . (C.16)After carrying out all these manipulations, we are led to the representation ρ ( m ) N ;e ff ( x , t ) = lim β → lim s → + ∞ lim r → + ∞ m X n = i ( − n n ! X i ,..., i n i a ∈ [[ 1 ; N ]] I C q d n z (2 i π ) n Z C ( L ) d n + y (2 i π ) n + L ( m ) Γ ( L ) h F i ,... i n b G ( β ) γ ;2 A L i . (C.17) † in particular it depends on L, cf lemma A.2. However, | γβ | · L − is still very small cf lemma A.2. L ( m ) Γ ( L ) is a truncated Lagrange series: L ( m ) Γ ( L ) h F i ,... i n G ( β ) γ ;2 A L i = m X n ,..., n s = s Y p = n p ! ∂ n p ∂ς n p p X { ℓ a }′ m ! ℓ ! ℓ N + ! ∂ ℓ N + ∂γ ℓ N + s Y p = n Γ ( L ) (cid:2) γ e ν s (cid:3) (cid:0) t p (cid:1)o n p F i ,..., i n (cid:2) γ e ν s (cid:3) | γ = × X { k a , j } ′ N Y a = , i ,..., i n ℓ a Y j = λ ( j ) a j ! k a , j · ∂ ℓ ∂γ ℓ · N Y a = , i ,..., i n ∂ | k a | ∂τ | k a | a × n b G ( β ) γ ;2 A L (cid:2) ̟ r (cid:0) ∗ | (cid:8) η i , k (cid:9)(cid:1)(cid:3)o (cid:12)(cid:12)(cid:12)(cid:12) γ = τ a = µ a . (C.18)We now take the r → + ∞ limit of (C.17)-(C.18). The very construction of ̟ r ( λ | { η j , k } ) along with the choiceof parameters η i , k given by (C.15) associated with the fact that G ( β ) γ ;2 A L is a regular functional with a su ffi cientlylarge regularity constant, leads to ( cf proof of proposition D.1)lim r → + ∞ b G ( β ) γ ;2 A L [ ̟ r ] = b G ( β ) γ ;2 A L " H ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { y a } n + ∪ { τ a } a : ℓ a , (cid:8) µ i a (cid:9) n ∪ { µ N + } ∪ { µ a } a : ℓ a , ! , (C.19)this uniformly in y a , µ a λ a and τ a belonging to K A L . The function H has been defined in (A.13)This uniform convergence also holds in respect to any finite order partial derivative in these parameters. Theuniformness of this limit in respect to the integration parameters occurring in (C.17) allows one to take it directlyunder the integral sign over a compact domain.As a consequence, we get thatlim r → + ∞ X { k a , j } ′ N Y a = , i ,..., i n ℓ a Y j = λ ( j ) a j ! k a , j ∂ ℓ ∂γ ℓ N Y a = , i ,..., i n ∂ | k a | ∂τ | k a | a × n b G ( β ) γ ;2 A L h ̟ r (cid:16) ∗ | { η j , k } (cid:17)io (cid:12)(cid:12)(cid:12)(cid:12) γ = τ a = µ a = N Y a = , i ,..., i n ℓ a ! ∂ ℓ a ∂γ ℓ a a ( b G ( β ) γ ;2 A L " H ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { y a } n + ∪ { λ a ( γ a ) } N { µ a } N + ∪ (cid:8) λ i a (cid:0) γ i a (cid:1)(cid:9) n ! | γ a = . (C.20)To get the rhs of this equality we have, in addition to exchanging the limits and derivatives, applied the Faa-dí-Bruno formula backwards. The constant of regularity of b G ( β ) γ ;2 A L being large enough, the action of b G ( β ) γ ;2 A L on H aswritten in the second line of (C.20) is indeed well defined.After collecting the various γ a derivatives into a single one, we arrive to the representationlim r → + ∞ L ( m ) Γ ( L ) h F i ,... i n b G ( β ) γ ;2 A L i = m X n ,..., n s = s Y a = ( n a ! ∂ n a ∂ς n a a ) ∂ m ∂γ m ( s Y a = n Γ ( L ) (cid:2) γ e ν s (cid:3) ( t a ) o n a · J i ,..., i n (cid:2) γ e ν s (cid:3) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = ς a = , (C.21)where we have set J i ,..., i n (cid:2) γ e ν s (cid:3) = F i ,..., i n (cid:2) γ e ν s (cid:3) b G ( β ) γ ;2 A L " H ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { y a } n + ∪ { λ a } N { µ a } N + ∪ (cid:8) λ i a (cid:9) n ! . (C.22)Since, no confusion is possible on the level of (C.21)-(C.22), the γ -dependence of the parameters λ p , p = , . . . , N is kept implicit again. We also remind that these are functions of e ν s .The truncated s -dimensional Lagrange series (C.21) together with its s → + ∞ limit has been studied inappendix D.5.2. It follows from the latter analysis that the s → + ∞ limit is uniform in respect to the parameters( { y k } n + , { z k } n ) on which J i ,..., i n depends. Therefore, this limit can be taken under the integrals signs. Similarly,57ne can exchange the limit with the m th γ -derivative symbol. It follows from the results gathered in appendixD.5.2 thatlim s → + ∞ lim r → + ∞ L ( m ) Γ ( L ) h F i ,... i n G ( β ) γ ;2 A L i = ∂ m ∂γ m ( J i ,..., i n h γν ( L ) i · det − C q " I − γ δ Γ ( L ) (cid:2) ρ (cid:3) δρ ( ζ ) ( µ ) ρ = γν ( L ) ) | γ = . (C.23)The answer is expressed with the help of ν ( L ) , the unique solution (for γ -small enough) to the non-linear integralequation driven by the functional Γ ( L ) : ν ( L ) ( λ ) = ( i β − / Z ( λ ) − φ ( λ, q ) + N + X a = φ ( λ, µ a ) − n + X a = φ ( λ, y a ) − N X a = , i ,..., i n φ ( λ, λ a ) with λ a = ξ − γν ( L ) ( a / L ) µ a = ξ − ( a / L ) . (C.24)Also, in (C.23), appears the Fredholm determinant of the linear integral operator acting on a small loop C q around (cid:2) − q ; q (cid:3) whose kernel is given in terms of the functional derivative δ Γ ( L ) (cid:2) ρ (cid:3) ( µ ) /δρ ( ζ ). The definition of thefunctional derivative is given in (D.11).Lemma A.4 allows one to reexpress the functional b G ( β ) γ ;2 A L appearing (C.22) in the case where the parameters λ a and µ a are defined exactly as in (C.24) in terms of the unique solution ν ( L ) .This leads to the below representation: ρ ( m ) N ;e ff ( x , t ) = lim β → m X n = c ( − n n ! ∂ m ∂γ m X i ,..., i n i a ∈ [[ 1 ; N ]] I C q d n z (2 i π ) n Z C ( L ) d n + y (2 i π ) n + n Y k = h πγν ( L ) (cid:0) λ i k (cid:1)i L ξ ′ γν ( L ) (cid:0) λ i k (cid:1) (cid:0) z k − λ i k (cid:1) det n " z a − λ i b × n Y k = y n + − z k ( y k − z k ) (cid:0) y n + − λ i k (cid:1) ! e ix U ( L ) ( { λ a } N ; { µ a } N + ; { y a } n + | γ )det C q " I − γ δ Γ ( L ) (cid:2) ρ (cid:3) δρ ( ζ ) ( µ ) ρ = γν ( L ) n Q a = n + Q b = (cid:0) y b − λ i a − ic (cid:1) (cid:0) λ i a − y b − ic (cid:1) n + Q a , b = ( y a − y b − ic ) n Q a , b = (cid:0) λ i a − λ i b − ic (cid:1) × Q n + k = f ( L ) (cid:16) y k , γν ( L ) ( y k ) (cid:17) det N + (cid:2) Ξ ( µ ) (cid:2) ξ (cid:3) (cid:3) det N (cid:2) Ξ ( λ ) [ ξ γν ( L ) ] (cid:3) · (cid:18) det n h δ k ℓ + γ b V k ℓ (cid:2) γν ( L ) (cid:3)i det n h δ k ℓ + γ b V k ℓ (cid:2) γν ( L ) (cid:3)i(cid:19) (cid:16) { λ i a } n , { y a } n + (cid:17) (C.25)Above we have written down the dependence of both determinants on { λ i a } and { y a } as a common argument.There is no problem to carry out the analytic continuation in (C.30) from β ∈ e U β up to β =
0: the potentialsingularities that could appear in the determinants are canceled by the prefactor Q nk = sin h πγν ( L ) (cid:0) λ i k (cid:1)i . From nowon, we can thus set β = β = L → + ∞ behavior of ν ( L ) It was shown in appendix D.5.2, equation (D.46), that ν ( L ) admits a large L asymptotic expansion ν ( L ) ( λ ) = ν (cid:0) λ | (cid:8) λ i a (cid:9) n ; { y a } n + (cid:1) + O (cid:0) L − (cid:1) . There the O is holomorphic and uniform in some open neighborhood of the real axisand the function ν ( λ ) = ν (cid:0) λ | (cid:8) λ i a (cid:9) n , { y a } n + (cid:1) stands for the unique solution to the linear integral equation (C.3)(here we have already set β = ν ( L ) , we are thus able to replace everywhere ν ( L ) by ν , up to O (cid:16) L − (cid:17) corrections. 58uilding on the large L asymptotics of Γ ( L ) and ν ( L ) it is shown in subsection D.5.2, (D.44)-(D.45), thatdet C q " I − γ δ Γ ( L ) (cid:2) ρ (cid:3) δρ ( ζ ) ( µ ) ρ = γν ( L ) = det[ − q ; q ] (cid:20) I + γ R π (cid:21) (cid:16) + O (cid:16) L − (cid:17)(cid:17) , (C.26)with a O that has the same uniformness properties as stated before. Above, we did not insist that the Fredholmdeterminant det (cid:2) I + γ R / π (cid:3) corresponds to an action on (cid:2) − q ; q (cid:3) . L → + ∞ limit of U ( L ) The thermodynamic limit of U ( L ) ( { λ a } ; { µ a } ; { y a } | γ ) is readily computed by using that ξ ( µ a ) − ξ γν ( L ) ( λ a ) = = − γν ( L ) ( µ a ) L + p ′ ( µ a )2 π ( µ a − λ a ) + O (cid:16) L − (cid:17) . (C.27)The remainder O (cid:16) L − (cid:17) is uniform in a ∈ [[ 1 ; N ]] and holomorphic in respect to the variables y a and z a belongingto U δ/ . By using the Euler-MacLaurin formula, the linear integral equation (C.3) satisfied by ν and the integralrepresentation (2.13) for u one gets that U ( L ) (cid:16) { λ a } N ; { µ a } N + ; { y a } n + | γ (cid:17) = U (cid:16)(cid:8) µ i a (cid:9) n ; { y a } n + | γ (cid:17) + O (cid:16) L − (cid:17) . (C.28)with a O that, again, is uniform and holomorphic in respect to µ i a or y a belonging to U δ/ . It is also holomorphicin ℜ ( β ) ≥ λ a and µ a on (cid:2) − q ; q (cid:3) , it is likewise easy to check thatdet N + (cid:2) Ξ ( µ ) (cid:2) ξ (cid:3) (cid:3) det N (cid:2) Ξ ( λ ) [ ξ γν ( L ) ] (cid:3) = det [ I − K / π ] · (cid:16) + O (cid:16) L − (cid:17)(cid:17) . (C.29) L → + ∞ limit of the remaining terms It is also readily seen due to the densification of the parameters λ a on (cid:2) − q ; q (cid:3) that the sums over the discreetsets λ i a can be replaced by integrals over (cid:2) − q ; q (cid:3) up to O (cid:0) L − (cid:1) corrections. Finally, it remains to estimate thecontributions of the functions f ( L ) . If one focuses on the contributions of the integrals over y a , a = , . . . , n + C ↑ / ↓ ; L and C bd ; L , then one readily convinces oneself that one deals with the type of integralsstudied in the proof of proposition B.1. Namely, these are precisely the integrals appearing when deriving theestimates for the functionals I ( L ) k given in (B.6). Clearly, each of these integrals can be estimated successively.By repeating word for word the proof given in proposition B.1, one has that each of these integrals produces aO ( L / w L ) = o (1) contribution. Hence, this part of the contour C ( L ) does not contribute to the thermodynamiclimit.As a consequence one obtains the following representation for ρ ( m ) N ;e ff ( x , t ): ρ ( m ) N ;e ff ( x , t ) = m X n = c ( − n n ! ∂ m ∂γ m q Z − q d n λ I C q d n z (2 i π ) n Z C d n + y (2 i π ) n + e ix U ( { λ a } n ; { y a } n + | γ ) Q n + k = f ( y k , γν ( y k )) n Q k = ( z k − λ k ) ( y k − z k ) ( y n + − λ k ) det n " y n + − z b z a − λ b × n Q a = n + Q b = ( y b − λ a − ic ) ( λ a − y b − ic ) n + Q a , b = ( y a − y b − ic ) n Q a , b = ( λ a − λ b − ic ) n Y k = n (cid:2) πγν ( λ k ) (cid:3)o det n h δ k ℓ + γ b V k ℓ (cid:2) γν (cid:3)i det n (cid:20) δ k ℓ + γ b V k ℓ (cid:2) γν (cid:3)(cid:21) det (cid:2) I + γ R / π (cid:3) det [ I − K / π ] × (1 + o (1)) | γ = . (C.30)59n order to obtain the representation (C.1) for the thermodynamic limit it remains to drop the o (1) corrections.We now provide an alternative representation for the thermodynamic limit ρ ( m )ext ( x , t ). Proposition C.1
The function ρ ( m )ext ( x , t ) admits the representation ρ ( m )e ff ( x , t ) = lim w → + ∞ lim β → lim s → + ∞ lim r → + ∞ : ∂ m ∂γ m (b E − ( q ) e − q R − q [ ixu ′ ( λ ) + b g ′ ( λ ) ] γν s ( λ )d λ X C ( w ) E h γν s , b E − − i G ( β ) γ ;2 w [ ̟ r ] ) | γ = : . (C.31) The contour C ( w ) E = C ( ∞ ) E ∩ n z ∈ C : (cid:12)(cid:12)(cid:12) ℜ ( z ) (cid:12)(cid:12)(cid:12) ≤ w o corresponds to a compact approximation of C ( ∞ ) E as depictedin Fig. 2. It is such that lim w → + ∞ C ( w ) E = C ( ∞ ) E . The functional X C ( w ) E has been defined in (B.34) .The functional G ( β ) γ ;2 w [ ̟ r ] appearing in (C.31) acts on the loop C ( K w ) and has been defined in lemma A.1. The compact approximation C ( w ) E of the contour C ( ∞ ) E appearing in (C.31) is there to ensure the well-definitenessof the translation operators. Indeed, in the setting discussed in subsection 4.14 and appendix D, the translationoperators are, a priori , only defined for functionals that involve the values of their argument on some compactsubset of C . As a consequence, a priori , the w → + ∞ limit and r → + ∞ limit do not commute.Also, the β → w → + ∞ limits do not commute. These limit should be understood as follows.Given w fixed and large enough, one considers the regular functional G ( β ) γ ;2 w as introduced in lemma A.1. The valueof w defines an associated β ∈ C and e γ > G ( β ) γ ;2 w is a regular functional for β ∈ e U β and | γ | ≤ e γ with aregularity constant large enough (in particular satisfying (4.30)). These ℜ ( β ) and e γ are such that ℜ ( β ) → + ∞ and e γ → w → + ∞ . Proof —
Let e E − − be as given in (4.51)-(4.52), ν s as in (4.36) and ̟ r (4.32). In order to implement the operatorsubstitution, we first expand the functional X C ( w ) E (cid:2) ν, e E − (cid:3) appearing in the rhs of (C.31) into a series very similarto the one occurring in the proof of proposition B.4. The sole exception is that, this time, the sums over λ i a ’sare directly replaced by integrals over (cid:2) − q ; q (cid:3) of the corresponding variables. Also, the function f ( L ) (resp. itsassociated contour C ( L ) ) should be replaced by f (resp. C ( w ) = C ( w ) E ∪ e C q ). At the end of the day, one deals withthe multi-dimensional Lagrange series belowlim w → + ∞ lim s → + ∞ lim r → + ∞ m X n = i ( − n n ! q Z − q d n λ (2 i π ) n I C q d n z (2 i π ) n Z C ( w ) d n + y (2 i π ) n + e L ( m ) Γ h F G ( β ) γ ;2 w i . (C.32)The functional F appearing above reads F (cid:2) γν s (cid:3) = Q n + k = f ( y k , γν s ( y k )) n Q k = ( z k − λ k ) ( y k − z k ) n Y k = y n + − z k y n + − λ k ! · det n " (cid:2) πγν s ( λ k ) (cid:3) z a − λ k e − ix R q − q u ′ ( λ ) γν s ( λ )d λ e − ixu ( q ) n Q a = e − ixu ( λ a ) n + Q a = e − ixu ( y a ) . And we have set e L ( m ) Γ h F G ( β ) γ ;2 w i = m X n ,..., n s = s Y p = ( (cid:2) a p (cid:3) n p n p ! ) ∂ m ∂γ m ( Q n + k = e e g , s ( y k ) e e g , s ( q ) Q nk = e e g , s ( λ k ) s Y p = n Γ (cid:2) γν s (cid:3) (cid:0) t p (cid:1)o n p × Q n + k = e e g , r ( y k ) e e g , r ( q ) Q nk = e e g , r ( λ k ) e − q R − q e g ′ , r ( λ ) γν s ( λ )d λ · F (cid:2) γν s (cid:3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = ς a = G ( β ) γ ;2 w [ ̟ r ] ) | γ = . (C.33)60he functional Γ is evaluated at the discretization points t p , p = , . . . , s for the contour C out appearing in the lhs of Fig. 1.One can implement the operator substitution on the level of (C.33) as it was done in the proof of theoremC.1. The well-foundedness of these manipulations (in particular the justification of the exchange of various limits,partial derivatives and integrals over compact contours) is justified along very similar lines. Once upon taking the r → + ∞ limit we end-up with the below multidimensional Lagrange series L ( m ) Γ h F G ( β ) γ ;2 w i = m X n ,..., n s = s Y p = n p ! ∂ n p ∂ς n p p ∂ m ∂γ m ( s Y p = n Γ (cid:2) γ e ν s (cid:3) (cid:0) t p (cid:1)o n p · F (cid:2) γ e ν s (cid:3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = ς a = G ( β ) γ ;2 w [ ̟ ] ) | γ = . (C.34)There, we agree upon e ν s ( λ ; { ς a } ) = ν s ( λ ; { ς a } ) + s X b = t b + − t b i π ( t b − λ ) ( φ ( t b , q ) + n X a = φ ( t b , λ a ) − n + X a = φ ( t b , y a ) ) . (C.35)Also, the function ̟ is to be considered as a functional of e ν s ̟ (cid:2)e ν s (cid:3) ( λ ) = n + X k = λ − y k − λ − q − n X k = λ − λ k − q Z − q γ e ν s ( τ )( τ − λ ) d τ . (C.36)The multidimensional Lagrange series (C.34) has been studied in appendix D.5.1. Its s → + ∞ limit is uniformin respect to the auxiliary parameters { λ a } n , { z a } n and { y a } n + . Hence, just as in the proof of theorem D.1, one isallowed to exchange the s → + ∞ limit with the integration over the compact contours. One can then apply theresults of appendix D.5.1 leading tolim s → + ∞ L ( m ) Γ h F G ( β ) γ ;2 w i = ∂ m ∂γ m ( F (cid:2) γν (cid:3) G ( β ) γ ;2 w [ ̟ [ ν ]]det[ − q ; q ] (cid:2) I + γ R / π (cid:3) ) | γ = . (C.37)The function ν appearing above is the unique solution to the linear integral equation ν ( λ ) + γ q Z − q d µ π R ( λ, µ ) ν ( µ ) = ( i β − / Z ( λ ) + n X a = φ ( λ, λ a ) − n + X a = φ ( λ, y a ) . (C.38)One can build on this result so as to simplify the obtained expression. The expression for the functional function G ( β ) γ ;2 w (cid:2) ̟ [ ν ] (cid:3) is simplified with the help of lemma A.3.By using the linear integral equation satisfied by ν together with the representation of u in terms of φ and u (2.14), we get that the oscillating factor present in F (cid:2) γν (cid:3) coincides with the one appearing in theorem C.1: n + X a = u ( y a ) − n X a = u ( λ a ) − u ( q ) − γ q Z − q u ′ ( λ ) ν ( λ ) d λ = U (cid:16) { λ a } n , { y a } n + | γ (cid:17) − i β p F . (C.39)61e are thus led to the below representation for the rhs of (C.31)lim w → + ∞ lim β → e x β p F m X n = c ( − n n ! ∂ m ∂γ m q Z − q d n λ I C q d n z (2 i π ) n Z C ( w ) d n + y (2 i π ) n + e ix U ( { λ a } n ; { y a } n + | γ ) Q n + k = f ( y k , γν ( y k )) n Q k = ( z k − λ k ) ( y k − z k ) ( y n + − λ k ) det n " y n + − z b z a − λ b × n Q a = n + Q b = ( y b − λ a − ic ) ( λ a − y b − ic ) n + Q a , b = ( y a − y b − ic ) n Q a , b = ( λ a − λ b − ic ) n Y k = h (cid:2) πγν ( λ k ) (cid:3)i det n h δ k ℓ + γ b V k ℓ (cid:2) γν (cid:3)i det n (cid:20) δ k ℓ + γ b V k ℓ (cid:2) γν (cid:3)(cid:21) det (cid:2) I + γ R / π (cid:3) det [ I − K / π ] . (C.40)Tha auxiliary arguments of the entries b V k ℓ [ ν ] and b V k ℓ [ ν ] are undercurrent by those of ν .One can carry out the analytic continuation from β ∈ e U β up to β = Q nk = sin (cid:2) πγν ( λ k ) (cid:3) .There is no problem to take the w → + ∞ limit of the above integrals. Indeed C ( ∞ ) E is chosen in such a waythat e ixu ( y a ) , a = , . . . , n + y a when y a → ∞ along C ( ∞ ) E . As the rest of theintegrand is a O (cid:0) y na (cid:1) , a = , . . . , n + C ( ∞ ) E are convergent. Once upon taking the β → w → + ∞ limits, we recover the representation given in (C.1). D Functional Translation operator
In this appendix, we build a convenient for our purposes representation of a functional translation. Our repre-sentation applies to su ffi ciently regular classes of functionals acting on holomorphic functions. Our constructionutilizes multidimensional Lagrange series (see eg. [1]). D.1 Lagrange series
Theorem D.1 [1]Let D , r = { z ∈ C : | z | < r } . Assume that • ϕ j (cid:0) { ς a } s (cid:1) , j = , . . . , s and f (cid:0) { ς a } s (cid:1) are holomorphic functions of { ς a } s belonging to the Cartesian product D s , r ; • there exists a series of radii r j < r such that for (cid:12)(cid:12)(cid:12) ς j (cid:12)(cid:12)(cid:12) = r j , j = , . . . , s, one has (cid:12)(cid:12)(cid:12) ϕ j ( { ς a } ) (cid:12)(cid:12)(cid:12) < r j .Then, the multidimensional Lagrange series L s = X n ,..., n s ∈ N s Y r = ( n r ! ∂ n r ∂ς n r r ) · s Y r = ϕ n r r ( { ς a } ) · f ( { ς a } ) (cid:12)(cid:12)(cid:12) ς p = is convergent and its sum is given by L s = f ( { z a } )det s " δ jk − ∂∂ς k ϕ j ( { ς a } ) |{ ς a } = { z a } . (D.1) Above, ( z , . . . , z s ) stands for the unique solution to the system z j = ϕ j ( { z a } ) such that (cid:12)(cid:12)(cid:12) z j (cid:12)(cid:12)(cid:12) < r for all j. Theuniqueness and existence of this solution is part of the conclusion of this theorem. .2 Some preliminary definitions Throughout this appendix, M and K will always stand for two compacts of C such that K ⊂ Int ( M ), M has n holes ( ie C ⊂ M has n bounded connected components) and ∂ M can be realized as disjoint union of n + † γ a : [ 0 ; 1 ] → ∂ M = ∐ n + a = γ a ([ 0 ; 1 ]).Let h be a holomorphic function on M and set f s (cid:0) λ | { ς a , p } (cid:1) = s X p = n + X a = ( t a , p + − t a , p )2 i π ( t a , p − λ ) ς a , p + s X p = n + X a = ( t a , p + − t a , p )2 i π ( t a , p − λ ) h ( t a , p ) , (D.2)The points t a , p correspond to the discretization points for ∂ M associated with the Jordan curves γ a , as given indefinition 4.2. It follows readily that the function λ f s ( λ | { ς a } ) is holomorphic in λ ∈ K . Moreover, given anyholomorphic function ν ( λ, y ) ∈ O (cid:16) M × W y (cid:17) where W y is a compact in C ℓ y , ℓ y ∈ N , one has that f s (cid:0) λ | { ν ( t a , p , y ) } (cid:1) −→ s → + ∞ Z ∂ M ν ( ζ, y ) + h ( ζ )2 i π ( ζ − λ ) d ζ = ν ( λ, y ) + h ( λ ) uniformly in λ ∈ K and y ∈ W y . (D.3)This convergence holds since ( ζ, λ, y ) ν ( ζ, y ) + h ( ζ ) / ( ζ − λ ) is uniformly continuous on ∂ M × K × W y .We recall that, given a holomorphic function h on M (and hence also on some open neighborhood of M ), and S a subset of M , we denote k h k S = sup s ∈ S | h ( s ) | . D.3 Pure translations
We are now in position to establish a representation for translation operators for functionals acting on holomorphicfunctions.
Proposition D.1
Let F [ · ] ( z ) , z ∈ W z ⊂ C ℓ z be a regular functional in respect to the pair ( M , K ) and let thefunctions f s , ν and h as well as the compacts M and K be defined as above. Then, for any (cid:16) m , k , . . . , k ℓ y (cid:17) ∈ N ℓ y + lim s → + ∞ ℓ y Y j = ∂ k j ∂ y k j j · s Y p = n + Y a = e ν ( t a , p , y ) ∂ ς a , p · ∂ m ∂γ m F h γ f s (cid:0) ∗ | { ς a , p } (cid:1)i ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ς a , p = = ℓ y Y j = ∂ k j ∂ y k j j ∂ m ∂γ m F (cid:2) γν ( ∗ , y ) + γ h ( ∗ ) (cid:3) ( z ) . Above, the · inside of the argument of indicates the running variable on which the functional F [ · ] ( y ) acts. Thisconvergence holds uniformly in ( γ, y , z ) belonging to compact subsets of D ,γ × Int (cid:0) W y (cid:1) × Int( W z ) , where γ = C F k ν k M × W y + k h k M π d( ∂ M , K ) | ∂ M | + π d( ∂ M , K ) , (D.4) | ∂ M | stands for the length of ∂ M, d( ∂ M , K ) for the distance of K to ∂ M and C F > is the constant of regularityof F . Finally, D ,γ = { z ∈ C : | z | ≤ γ } .Proof — We first consider the case m = k = · · · = k ℓ =
0. We assume that s is taken large enough so that s X p = n + X a = (cid:12)(cid:12)(cid:12) t a , p − t a , p + (cid:12)(cid:12)(cid:12) ≤ | ∂ M | . (D.5) † we remind that γ a satisfies γ a (0) = γ a (1) and γ a | [ 0;1[ is injective. | γ | < γ and (cid:12)(cid:12)(cid:12) ς a , p (cid:12)(cid:12)(cid:12) ≤ k ν k M × W y , one has (cid:12)(cid:12)(cid:12)(cid:12) γ f s (cid:16) λ | { ς a , p } (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | γ | (cid:18) sup a , p (cid:12)(cid:12)(cid:12) ς a , p (cid:12)(cid:12)(cid:12) + k h k M (cid:19) × s X p = n + X a = (cid:12)(cid:12)(cid:12) t a , p + − t a , p (cid:12)(cid:12)(cid:12) π (cid:12)(cid:12)(cid:12) λ − t a , p (cid:12)(cid:12)(cid:12) ≤ | γ | sup a , p (cid:12)(cid:12)(cid:12) ς a , p (cid:12)(cid:12)(cid:12) + k h k M ! | ∂ M | π d( ∂ M , K ) < C F , (D.6)Hence, (cid:16) γ, { ς a , p } , z (cid:17)
7→ F h γ f s (cid:0) ∗ | { ς a , p } (cid:1)i ( z ) is holomorphic in (cid:16) γ, { ς a , p } , z (cid:17) ∈ D , γ × D s ( n + , k ν k M × W z , (D.7)this for any s large enough. As a consequence, the below multi-dimensional Taylor series is convergent uniformlyin ( γ, y , z ) ∈ D , γ × W y × W z and s Y p = n + Y a = e ν ( t a , p , y ) ∂ ς a , p F h γ f s (cid:16) ∗ | { ς a , p } (cid:17)i ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ς a , p = ≡ + ∞ X n a , p ≥ s Y p = n + Y a = ( (cid:2) ν (cid:0) t a , p , y (cid:1)(cid:3) n a , p ( n a , p )! ∂ n a , p ∂ς n a , p a , p ) · F h γ f s (cid:16) ∗ | { ς a , p } (cid:17)i ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ς a , p = = F h γ f s (cid:0) ∗ | { ν (cid:0) t a , p , y (cid:1) } (cid:1)i ( z ) . (D.8)Moreover, for any y ∈ W y and γ ∈ D , γ , one has the bound (cid:13)(cid:13)(cid:13) γ f s (cid:0) ∗ | { ν (cid:0) t a , p , y (cid:1) } (cid:1)(cid:13)(cid:13)(cid:13) K + | γ | (cid:0) k ν ( · , y ) k K + k h k K (cid:1) < C F .As a consequence, by (4.14) (cid:13)(cid:13)(cid:13) F (cid:2) γ f s (cid:0) ∗ | { ν (cid:0) t a , p , y (cid:1) } (cid:1)(cid:3) ( z ) − F (cid:2) γν ( ∗ , y ) + γ h ( ∗ ) (cid:3) ( z ) (cid:13)(cid:13)(cid:13) D , γ × W y × W z ≤ γ C ′ (cid:13)(cid:13)(cid:13) f s (cid:0) λ | { ν ( t a , p , y ) } (cid:1) − ν ( λ, y ) − h ( λ ) (cid:13)(cid:13)(cid:13) K × W y −→ s → + ∞ , due to (D.3). The norm in the first line is computed in respect to ( γ, y , z ) ∈ D , γ × W y × W z . The one in thesecond line in respect to ( λ, y ) ∈ K × W y . We insisted explicitly on the variable-dependence of the functions so asto make this fact clear.It remains to show that the convergence also holds uniformly on all compacts of D , γ × Int ( W y ) × Int ( W z )when considering partial derivatives in respect to γ, y , . . . , y ℓ y of finite total order.One can exchange any such partial derivatives with the Taylor series in (D.8) in as much as its partial sumsdefine a sequence of holomorphic functions that is uniformly convergent on D , γ × W y × W z . The same argumentscan be applied to the sequence of holomorphic functions F (cid:2) γ f s (cid:0) ∗ | { ν (cid:0) t a , p , y (cid:1) } (cid:1)(cid:3) ( z ) . Corollary D.1
Assume that the conditions and notations of proposition D.1 hold. Let C ( ℓ y ) = C × · · · × C ℓ y and e C ( ℓ z ) = e C ×· · ·× e C ℓ z be Cartesian products of compact curves in C such that C ( ℓ y ) ⊂ Int ( W y ) and e C ( ℓ z ) ⊂ Int ( W z ) .Then one has lim s → + ∞ + ∞ X n a , p = s Y p = n + Y a = n a , p )! ∂ n a , p ∂ς n a , p a , p · Z C ( ℓ y ) d ℓ y y Z e C ( ℓ z ) d ℓ z z ℓ y Y j = ∂ k j ∂ y k j j · s Y p = n + Y a = h ν (cid:16) t a , p , y (cid:17)i n a , p ∂ m ∂γ m F (cid:2) γ f s (cid:0) ∗ | { ς a , p } (cid:1)(cid:3) ( y , z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ς a , p = = Z C ( ℓ y ) d ℓ y y Z e C ( ℓ z ) d ℓ z z ℓ y Y j = ∂ k j ∂ y k j j ∂ m ∂γ m F (cid:2) γν ( ∗ , y ) + γ h ( ∗ ) (cid:3) ( y , z ) . (D.9) this uniformly in γ belonging to compact subsets of D ,γ . F depends on a third set of variables belonging to a compact, the results hold as well in respect tothis third set uniformly on the compact. Proof —
Proposition D.1 allows one to conclude, in virtue of the uniform convergence of the sequences, that for γ belonging to compact subsets of D ,γ one has the equalitylim s → + ∞ + ∞ X n a , p = Z C ( ℓ y ) d ℓ y y Z e C ( ℓ z ) d ℓ z z s Y p = n + Y a = n a , p ! ∂ n a , p ∂ς n a , p a , p · ℓ y Y j = ∂ k j ∂ y k j j · s Y p = n + Y a = h ν (cid:16) t a , p , y (cid:17)i n a , p ∂ m ∂γ m F (cid:2) γ f s (cid:0) ∗ | { ς a , p } (cid:1)(cid:3) ( y , z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ς a , p = = Z C ( ℓ y ) d ℓ y y Z e C ( ℓ z ) d ℓ z z ℓ y Y j = ∂ k j ∂ y k j j ∂ m ∂γ m F (cid:2) γν ( ∗ , y ) + γ h ( ∗ ) (cid:3) ( y , z ) . (D.10)The integrals occurring in the first line of (D.10) are over compact curves and the integrand is smooth in respectto the integration variables ( y , z ) and the auxiliary parameters ς a , p . As a consequence, the partial ς a , p -derivativescan be pulled outside of the integration symbols. D.4 Weighted translation
One can generalize the notion of functional translation with the help of multi-dimensional Lagrange series andconsider more complex objects. For this purpose, we need to introduce some more definitions. Also, from nowon we only focus on the case of a compact M without holes.Let Γ [ · ] ( µ ) be a one parameter family of functionals such that: • There exists a constant C Γ > ν ( λ, y ) is holomorphic in ( λ, y ) ∈ M × W y , with W y ⊂ C ℓ y and k ν k K × W y < C Γ then ( λ, y ) Γ (cid:2) ν ( ∗ , y ) (cid:3) ( λ ) is holomorphic in M × W y . • There exists a contour C in Int ( K ) such that for k ρ k K + k τ k K ≤ C Γ one has Γ (cid:2) ρ (cid:3) ( µ ) − Γ [ τ ] ( µ ) = Z C ( ρ − τ ) ( ζ ) δ Γ [ ν ] δν ( ζ ) ( µ ) | ν = τ d ζ + o (cid:0) k ρ − τ k K (cid:1) . (D.11) δ Γ [ ν ] ( µ ) /δν ( ζ ) will be called the functional derivative of Γ . This functional derivative is such that, for any τ holomorphic on M with k τ k K < C Γ , there exists an open neighborhood V ( C ) of the contour C appearingin (D.11) such that( µ, ζ ) δ Γ [ ν ] δν ( ζ ) ( µ ) | ν = τ is holomorphic in ( µ, ζ ) ∈ M × V ( C ) . (D.12) • There exists a constant C ′ Γ > k τ k K + k ν k K ≤ C Γ one has k Γ [ ν ] ( µ ) k M ≤ C ′ Γ k ν k K and k Γ [ τ ] ( µ ) − Γ [ ν ] ( µ ) k M ≤ C ′ Γ k ν − τ k K . (D.13)The properties of the functional Γ [ · ] ( λ ) ensure the solvability of an associated integral equation65 emma D.1 Let the compacts M , K and the one parameter family of functional Γ [ · ] ( λ ) be as defined above. Leth ∈ O ( M ) and r , γ be such that γ ( r + k h k M ) + | ∂ M | π d( ∂ M , K ) ! ≤ r min (1 , C Γ )2 (cid:16) r + C ′ Γ (cid:17) , and 2 C ′ Γ γ + | ∂ M | π d( ∂ M , K ) ! < min (1 , C Γ )2 , (D.14) Then for | γ | ≤ γ , there exists a unique solution ρ to the equation ρ ( λ ) = Γ (cid:2) γρ ( ∗ ) + γ h ( ∗ ) (cid:3) ( λ ) . This solution isholomorphic in ( λ, γ ) ∈ M × D ,γ and such that k ρ k M < r.Proof — Suppose that ρ and ρ ′ are two solutions. Then for | γ | < γ one has, by construction of γ , that | γ | k ρ + h k K + | γ | k ρ ′ + h k K < C Γ . As a consequence, (cid:13)(cid:13)(cid:13) ρ − ρ ′ (cid:13)(cid:13)(cid:13) M = (cid:13)(cid:13)(cid:13) Γ (cid:2) γ ( ρ + h ) (cid:3) − Γ (cid:2) γ (cid:0) ρ ′ + h (cid:1)(cid:3)(cid:13)(cid:13)(cid:13) M ≤ C ′ Γ γ (cid:13)(cid:13)(cid:13) ρ − ρ ′ (cid:13)(cid:13)(cid:13) M < (cid:13)(cid:13)(cid:13) ρ − ρ ′ (cid:13)(cid:13)(cid:13) M . (D.15)Therefore, ρ = ρ ′ on M , this uniformly in | γ | ≤ γ .In order to prove the existence, one considers the sequence of holomorphic functions on M : h = h and, for n ≥ h n ( λ ) = h ( λ ) + Γ (cid:2) γ h n − ( · ) (cid:3) ( λ ). It is readily seen by straightforward induction that, for all n ∈ N and | γ | ≤ γ , γ k h n k K ≤ C Γ / k h n + − h n k M ≤ k h n − h n − k M / . (D.16)Hence h n is a Cauchy sequence in the space of holomorphic functions on Int ( M ) × D γ . It is thus convergent tosome holomorphic function e h on Int ( M ) × D γ . Since e h ( λ ) = h ( λ ) + Γ (cid:2) γ e h (cid:3) ( λ ), it can be analytically continuedto a holomorphic function on M × D γ . Then, the function ρ = e h − h solves ρ ( λ ) = Γ (cid:2) γ ( ρ + h ) (cid:3) ( λ ). It alsofollows that then k ρ k M < r Proposition D.2
Let f s be as in (D.2) and assume that the functional Γ (cid:2) ρ (cid:3) ( µ ) satisfies to the assumptions givenabove. Let F [ · ]( z ) , with z ∈ W z ⊂ C ℓ z , be a regular functional in the sense of definition 4.1. Set L Γ ( γ, z ) = lim s → + ∞ : s Y r = e Γ [ γ f s ( ∗|{ ς p } )] ( t r ) ∂ ς r F h γ f s (cid:16) ∗ | { ς p } (cid:17)i ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ς p = : , (D.17) where : · : indicates that the expression is ordered in such a way that all the partial derivatives appear to the left(cf. subsection 4.3) and t r are the discretization points of ∂ M.Then, there exists γ > such that L Γ ( γ, z ) defines a holomorphic function of ( γ, z ) ∈ D ,γ × Int ( W z ) .The convergence of the rhs of (D.17) to L Γ ( γ, z ) is uniform on compact subsets of D ,γ × Int ( W z ) , and this inrespect to any partial γ or z -derivative of finite order. L Γ ( γ, z ) is given by L Γ ( γ, z ) = F (cid:2) γρ (cid:3) ( z )det C (cid:20) I − γ δ Γ [ ν ] δν ( µ ) ( λ ) (cid:21) | ν = γρ with ρ being the unique solution to ρ ( λ ) = h ( λ ) +Γ (cid:2) γρ (cid:3) ( λ ) . (D.18) In the denominator appears the Fredholm determinant of the linear integral operator acting on the contour C with an integral kernel δ Γ [ ν ] ( λ ) /δν ( µ ) | ν = γρ . The contour C is defined in (D.11) .Proof —
66e have, by definition, L Γ ( γ, z ) = lim s → + ∞ L s ( γ, z ) with L s ( γ, z ) = X n ,..., n s ∈ N s Y r = ( n r ! ∂ n r ∂ς n r r ) · s Y r = Γ n r h γ f s (cid:16) ∗ | { ς p } (cid:17)i ( t r ) · F h γ f s (cid:16) ∗ | { ς p } (cid:17)i ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ς a = . (D.19)The above series representation for L s ( γ, z ) corresponds to a particular case of a multidimensional Lagrangeseries.We start by checking the convergence conditions. Let C denote a common constant of regularity for thefunctionals F and Γ , ie for any ν ( λ, y ) ∈ O (cid:16) M × W y (cid:17) , with W y ⊂ C ℓ y such that k ν k M × W y ≤ C one has F (cid:2) ν ( ∗ , y ) (cid:3) ( z ) ∈ O (cid:16) W y × W z (cid:17) and Γ (cid:2) ν ( ∗ , y ) (cid:3) ( λ ) ∈ O (cid:16) M × W y (cid:17) . (D.20)Then let r > γ > C Γ being replaced with C . Let s be large enough so that P sa = | t a − t a + | ≤ | ∂ M | and | γ | ≤ γ . It then follows from (D.6) that k γ f s ( · | { ς a } ) k K < C for | ς a | < r . It is alsoeasy to see that for (cid:12)(cid:12)(cid:12) ς p (cid:12)(cid:12)(cid:12) ≤ r and for any t p ∈ ∂ M , one has (cid:12)(cid:12)(cid:12) Γ (cid:2) γ f s (cid:0) ∗ | { ς p } (cid:1)(cid:3) ( t k ) (cid:12)(cid:12)(cid:12) ≤ r /
2. Therefore, • Γ (cid:2) γ f s (cid:0) ∗ | { ς p } (cid:1)(cid:3) ( t k ), k = , . . . , s and F (cid:2) γ f s (cid:0) ∗ | { ς p } (cid:1)(cid:3) ( z ) are holomorphic functions of { ς a } in D s , r ; • for | ς k | = r / k ∈ [[ 1 ; s ]] one has (cid:12)(cid:12)(cid:12) Γ (cid:2) γ f s (cid:0) ∗ | { ς p } (cid:1)(cid:3) ( t k ) (cid:12)(cid:12)(cid:12) ≤ r / < r / L s ( γ, z ) = F h γ f s (cid:16) ∗ | { τ p } (cid:17)i ( z )det s " δ jk − ∂∂ς k Γ (cid:2) γ f s (cid:0) ∗ | { ς p } (cid:1)(cid:3)(cid:0) t j (cid:1) | ς p = τ p (D.21)where ( τ , . . . , τ s ) is the unique solution to the system τ j = Γ (cid:2) γ f s (cid:0) ∗ | { τ p } (cid:1)(cid:3) ( t j ) with (cid:12)(cid:12)(cid:12) τ j (cid:12)(cid:12)(cid:12) < r for all j .It is easy to see that, in fact, L s ( γ, z ) is a uniform limit of holomorphic functions of ( γ, z ) ∈ D , γ × W z .Therefore, L s ( γ, z ) is holomorphic on all compact subsets of D , γ × W z . Moreover, there one can permute anypartial γ or z -derivatives with the summations in (D.19). It is also clear for the previously obtained bounds that, L s ( γ, z ) is well defined for any s large enough and this independently of the choice of the points t k .We now show that its s → + ∞ limit exists and then we will compute it. It is readily inferred from the integralrepresentation τ j = Z | ζ a | = r ζ js Q p = n ζ p − Γ (cid:2) γ f s ( ∗ | { ζ a } ) (cid:3) ( t p ) o d s ζ (D.22)that τ j ≡ τ j ( γ ), j = , . . . , s , solving the system τ j = Γ (cid:2) γ f s (cid:0) ∗ | { τ p } (cid:1)(cid:3) ( t j ), is a holomorphic function of γ for | γ | ≤ γ . Hence, the function ρ s ( λ ; γ ) = Γ (cid:2) γ f s (cid:0) ∗ | { τ p } (cid:1)(cid:3) ( λ ) is holomorphic in ( λ, γ ) ∈ M × D ,γ . Also, byconstruction, ρ s ( t j ; γ ) = τ j ( γ ) and k ρ k M ×D ,γ < r . (D.23)Now let ρ be the unique solution to ρ ( λ ) = Γ (cid:2) γρ + γ h (cid:3) ( λ ) with k ρ k M ≤ r , as follows from lemma D.1.Then, keeping the γ dependence implicit, we consider ρ ( λ ) − ρ s ( λ ) = Γ (cid:2) γ ( ρ + h ) (cid:3) ( λ ) − Γ (cid:2) γ f s (cid:0) ∗ | { ρ ( t p ) } (cid:1)(cid:3) ( λ ) +Γ (cid:2) γ f s (cid:0) ∗ | { ρ ( t p ) } (cid:1)(cid:3) ( λ ) − Γ (cid:2) γ f s (cid:0) ∗ | { ρ s ( t p ) } (cid:1)(cid:3) ( λ ) | {z } ψ s ( λ ) . (D.24)67s k ρ s k M ≤ r , it follows that k ψ s k M ≤ C ′ Γ γ (cid:13)(cid:13)(cid:13) f s (cid:0) ∗ | { ρ ( t p ) } (cid:1) − f s (cid:0) ∗ | { ρ s ( t p ) } (cid:1)(cid:13)(cid:13)(cid:13) K ≤ C ′ Γ γ | ∂ M | π d ( ∂ M , K ) k ρ − ρ s k M < k ρ − ρ s k M . (D.25)Hence, k ρ − ρ s − ψ s k M ≥ k ρ − ρ s k M /
2. On the other hand, it follows from (D.24) that k ρ − ρ s − ψ s k M = (cid:13)(cid:13)(cid:13)(cid:13) Γ (cid:2) γρ + γ h (cid:3) − Γ h γ f s (cid:0) ∗ | { ρ ( t p ) } (cid:3)i(cid:13)(cid:13)(cid:13)(cid:13) M ≤ γ C ′ Γ (cid:13)(cid:13)(cid:13) ρ + h − f s (cid:0) ∗ | { ρ ( t p ) } (cid:1)(cid:13)(cid:13)(cid:13) K −→ s → + ∞ ρ s converges uniformly to ρ on M . Hence, in virtue of the regularity of F , F (cid:2) γ f s (cid:0) ∗ | { ρ s ( t p ) } (cid:1)(cid:3) ( z ) −→ s → + ∞ F (cid:2) γρ (cid:3) ( z ) uniformly in ( γ, z ) ∈ D ,γ × W z . (D.27)It remains to compute the limit of the determinant. It follows from the functional derivative property (D.11)that ∂∂ς k Γ (cid:2) γ f s (cid:0) ∗ | { ς p } (cid:1)(cid:3) ( t j ) |{ ς p } = { τ p } = γ ( t k + − t k ) Z C d µ i π t k − µ δ Γ δν ( µ ) h ν + γ f s (cid:0) ∗ | { τ p } (cid:1)i ( t j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν = . (D.28)By expanding the determinant appearing in (D.21) into its discreet Fredholm series we getdet s (cid:20) δ jk − ∂∂ς k Γ (cid:2) γ f s (cid:0) ∗ | { ς p } (cid:1)(cid:3) ( t j ) (cid:21) | { ς p } = { τ p } = s X p = ( − γ ) p p ! Z C d p µ det p h A s (cid:0) µ q , µ ℓ (cid:1)i (D.29)with A s (cid:16) µ q , µ ℓ (cid:17) = s X k = t k + − t k i π ( t k − µ ℓ ) δδν ( µ q ) Γ (cid:2) ν + f s (cid:0) ∗ | { ρ s ( t p ) } (cid:1)(cid:3) ( t j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν = −→ s → + ∞ Z ∂ M d ζ i π ζ − µ ℓ δ Γ (cid:2) ν + γρ (cid:3) δν ( µ q ) ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν = = δ Γ (cid:2) ν + γρ (cid:3) δν ( µ q ) ( µ ℓ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν = (D.30)The above convergence is uniform in ( µ q , µ ℓ ) ∈ C × C . Therefore, by elementary estimates, we obtain thatthe determinant of interest does indeed converge to the Fredholm determinant given in (D.18), this uniformly in | γ | ≤ γ .Therefore, we obtain that L s ( γ, z ) is a sequence of holomorphic functions on D ,γ × Int ( W z ). that convergesuniformly. As a consequence, L Γ ( γ, z ) is holomorphic on every compact subset of D ,γ × Int ( W z ) and one canpermute any partial- γ or z derivative of finite order with the s → + ∞ limit on these compacts. D.5 Examples
We now treat two examples that are of direct interest for the resummation of the form factor series. In the belowexamples, φ ( λ, µ ) refers to the dressed phase (2.10). We remind that it is holomorphic on U δ × U δ . In the following,the compacts K and M ⊂ U δ are such that (cid:2) − q ; q (cid:3) is contained in their interior. We will also consider functions h that are holomorphic on M . 68 .5.1 Γ (cid:2) ρ (cid:3) ( µ ) as a linear functional of ρ Let Γ (cid:2) ρ (cid:3) ( λ ) = R q − q ∂ λ φ ( µ, λ ) ρ ( µ ) d µ . Then, given a regular functional F [ · ]( z ), z ∈ W z ⊂ C ℓ z , there exists γ > γ, z ) ∈ D ,γ × W z L Γ ( γ, z ) = F (cid:2) ρ (cid:3) ( z )det[ − q ; q ] (cid:2) I − γ∂ λ φ (cid:3) with ρ ( λ ) − γ q Z − q ∂ λ φ ( µ, λ ) ρ ( µ ) d µ = h ( λ ) . (D.31)The limit defining L Γ ( γ, z ) as in (D.18) is uniform in respect to such parameters z ∈ W z and | γ | ≤ γ . Proof —
In order to apply proposition D.2, one should check the assumptions on the functional Γ . It is readily seenthat, independently of the norm of ν and ρ k Γ [ ν ] k M ≤ q k ∂ λ φ k M × M k ν k K and (cid:13)(cid:13)(cid:13) Γ [ ν ] − Γ (cid:2) ρ (cid:3)(cid:13)(cid:13)(cid:13) M ≤ q k ∂ λ φ k M × M k ν − ρ k K and δ Γ [ τ ] δτ ( ζ ) ( µ ) = (cid:16) ∂ µ φ (cid:17) ( ζ, µ ) . The validity of the holomorphicity conditions is readily checked by standard derivation under the integral theo-rems. One is thus in position to apply proposition D.2 and the claim follows.
D.5.2 Non-linear functional Γ ( L ) (cid:2) ρ (cid:3) ( µ )We now treat the case of the non-linear functional below Γ ( L ) (cid:2) ρ (cid:3) ( µ ) = X j ∈ J φ (cid:0) µ, µ j (cid:1) − φ (cid:0) µ, λ j (cid:1) = X j ∈ J I C q φ ( µ, ω ) ( ξ ′ ( ω ) ξ ( ω ) − j / L − ξ ′ ρ ( ω ) ξ ρ ( ω ) − j / L ) d ω i π . (D.32)There J = [[ 1 ; N ]] \ { i , . . . , i n } , ξ is given by (2.6), ξ F = ξ + F / L and 0 ≤ j / L ≤ D with N / L → D . Finally, C q isa small counterclockwise Jordan curve around (cid:2) − q ; q (cid:3) such that Int ( K ) ⊃ C q . Note that µ a , resp. λ a , appearingin (D.32) stand for the unique solutions to ξ ( µ a ) = a / L , resp. to ξ ρ ( λ a ) = a / L . Proposition D.3
Let F [ · ]( z ) , z ∈ W z ⊂ C ℓ z be a regular functional and assume that N , L are large (and such thatN / L → D). Then, there exists γ > such that for ( γ, z ) ∈ D ,γ × W z and L large enough L Γ ( L ) ( γ, z ) = F (cid:2) γρ ( L ) (cid:3) ( z )det C q (cid:20) I − γ δ Γ ( L ) [ ν ] δν ( ζ ) ( µ ) (cid:21) ν = γρ ( L ) , (D.33) where ρ ( L ) is the unique solution to ρ ( L ) ( λ ) = h ( λ ) + Γ ( L ) (cid:2) γρ ( L ) (cid:3) ( λ ) . This solution is such that ρ ( L ) ( λ ) = ρ ( λ ) + O (cid:0) L − (cid:1) , (D.34) where ρ solves the linear integral equation (cid:18) I + γ R π (cid:19) · ρ = h and the O (cid:0) L − (cid:1) is a holomorphic function of γ and λ .Moreover, this estimate and holds uniformly in | γ | ≤ γ and λ ∈ U δ . Finally, δ Γ ( L ) δν ( ζ ) [ ν ] ( µ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν = γτ = (cid:0) ∂ ζ φ (cid:1) ( µ, ζ )2 i π L X j ∈ J ξ γτ ( ζ ) − j / L . (D.35) Above I + R / π stands for the resolvent of the Lieb kernel acting on (cid:2) − q ; q (cid:3) . And one has det C q " I − γ δ Γ ( L ) [ ν ] δν ( ζ ) ( µ ) ν = γρ ( L ) = det[ − q ; q ] (cid:2) I + γ R / π (cid:3) · + O L !! . (D.36)69 roof — In order to apply proposition D.2, we ought to check that Γ ( L ) satisfies to all the necessary conditions. For thiswe observe that ǫ = inf | ξ ( ω ) − λ | > ω ∈ C q , λ ∈ [ 0 ; D ] . (D.37)Then, we choose a constant C > L large enough so that C < ǫ L /
2. It then follows that the functions ξ ν ( ω ) − j / L , for j = , . . . , N , have no zeroes on C q and some immediate neighborhood thereof provided that ν ( z , y ) ∈ O (cid:16) M × W y (cid:17) with W y ⊂ C ℓ y and k ν k M × W y < C . It then follows by the derivation under the integral signtheorems that Γ ( L ) (cid:2) ν ( ∗ , y ) (cid:3) ( µ ) is holomorphic in ( µ, y ) ∈ M × W y .In order to establish bounds on Γ ( L ) (cid:2) ν ( ∗ , y ) (cid:3) ( µ ) for ν holomorphic and such that k ν k M × W y < C , it is convenientto represent ξ ′ ( ω ) ξ ( ω ) − j / L − ξ ′ ν ( ω ) ξ ν ( ω ) − j / L = − ν ′ ( ω ) L ( ξ ( ω ) − j / L ) + ν ( ω ) Z d tL ξ ′ ν ( ω )( ξ t ( ω ) − j / L ) . (D.38)As C q ⊂ Int ( K ), there exists a constant c > ν holomorphic on K , one has k ν ′ k C q ≤ c k ν k K . Also,inf | ξ t ( ω ) − s | > ǫ/ ω ∈ C q , s ∈ [ 0 ; D ] and | t | < ǫ L / . (D.39)Hence, for any ν ∈ O (cid:16) M × W y (cid:17) such that k ν k K × W y < C < ǫ L / (cid:12)(cid:12)(cid:12) Γ ( L ) (cid:2) ν ( ∗ , y ) (cid:3) ( µ ) (cid:12)(cid:12)(cid:12) ≤ k φ k M × M | J | π (cid:12)(cid:12)(cid:12) C q (cid:12)(cid:12)(cid:12) ( c L ǫ k ν k K × W y + (cid:13)(cid:13)(cid:13) ξ ′ ν (cid:13)(cid:13)(cid:13) C q × W y · k ν k K × W y L ( ǫ/ ) ≤ k φ k M × M N (cid:12)(cid:12)(cid:12) C q (cid:12)(cid:12)(cid:12) π L ǫ c ( + ǫ (cid:2) k ξ k K + ǫ/ (cid:3)) k ν k K × W y . (D.40)This provides an estimate for the constant C ′ Γ ( L ) entering in the bounds for (cid:13)(cid:13)(cid:13) Γ ( L ) [ ν ] (cid:13)(cid:13)(cid:13) M × W y . Next one has Γ ( L ) (cid:2) ρ (cid:3) ( µ ) − Γ ( L ) [ τ ] ( µ ) = X j ∈ J I C q φ ( µ, ω ) ( ( τ ′ − ρ ′ ) ( ω ) ξ τ ( ω ) − j / L + ( ρ − τ ) ( ω ) ξ ′ τ ( ω )( ξ τ ( ω ) − j / L ) ) d ω i π L + R ( L ) ( µ ) . (D.41)Where, R ( L ) ( µ ) = X j ∈ J I C q φ ( µ, ω ) ( ρ − τ ) ( ω ) ( ρ − τ ) ′ ( ω ) L ( ξ τ ( ω ) − j / L ) + ρ ( ω ) Z τ ( ω ) d tL ( t − ρ ( ω )) ξ ′ ρ ( ω )( ξ t ( ω ) − j / L ) d ω i π = O (cid:16) k ρ − τ k K (cid:17) , (D.42)this uniformly in µ ∈ M and L large enough. Therefore, δ Γ ( L ) δν ( ζ ) (cid:2) γν (cid:3) ( µ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν = γρ = γ i π L (cid:16) ∂ ζ φ (cid:17) ( µ, ζ ) X j ∈ J ξ γρ ( ζ ) − j / L . (D.43)It follows that there exists a su ffi ciently small open neighborhood V (cid:16) C q (cid:17) of C q such that the functional derivativeis holomorphic in ( µ, ζ ) ∈ M × V (cid:16) C q (cid:17) . Moreover, we get that there exists an L -independent constant e C such that (cid:13)(cid:13)(cid:13) Γ ( L ) (cid:2) ρ (cid:3) − Γ ( L ) [ τ ] (cid:13)(cid:13)(cid:13) M ≤ e C k ρ − τ k K
70e are now in position to apply proposition D.2. It follows that L Γ ( L ) can be expressed in terms of the uniquesolution ρ ( L ) to ρ ( L ) ( µ ) = h ( µ ) + Γ ( L ) (cid:2) γρ ( L ) (cid:3) ( µ ) with (cid:13)(cid:13)(cid:13) ρ ( L ) (cid:13)(cid:13)(cid:13) M < r uniformly in | γ | ≤ γ .This means that, δ Γ ( L ) (cid:2) γν (cid:3) δν ( ζ ) ( µ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν = ρ ( L ) −→ L → + ∞ γ (cid:16) ∂ ζ φ (cid:17) ( µ, ζ ) q Z − q d u i π ξ ′ ( u ) ξ ( ζ ) − ξ ( u ) uniformly in ( µ, ζ ) ∈ C q . (D.44)In this limit, the contour integral C q in the Fredholm determinant can be computed and since the Fredholm deter-minant of a trace class operator is continuous in respect to the trace class norm (which is bounded by the sup normin the case of integral operators acting on compact contours)det C q h I − δ Γ ( L ) (cid:2) γν (cid:3) ( µ ) /δν ( ζ ) | ν = ρ ( L ) i −→ L → + ∞ det[ − q ; q ] (cid:2) I − γ∂ λ φ ( µ, λ ) (cid:3) = det[ − q ; q ] (cid:2) I + γ R / π (cid:3) . (D.45)Where R is the resolvent of the Lieb kernel.We now characterize the leading behavior of the solution ρ ( L ) when N , L → + ∞ . By repeating the type ofmanipulations carried our previously, and using that ρ ( L ) is bounded on K uniformly in L , we get that the non-linear integral equation for ρ ( L ) takes the form ρ ( L ) ( µ ) = h ( µ ) + γ L X j ∈ J I C q d ω i π ρ ( L ) ( ω ) ( ∂ ω φ ) ( µ, ω ) ξ ( ω ) − j / L + O L ! . There the O is uniform in µ ∈ U δ . The Riemann sum can be estimated by using the Euler-McLaurin formula andthe uniform boundedness of ρ ( L ) on K . After carrying out the resulting contour integral over C q we obtain ρ ( L ) ( µ ) = h ( µ ) − γ q Z − q d s π R ( µ, s ) ρ ( L ) ( s ) + O (cid:16) L − (cid:17) . (D.46)The O appearing in (D.46) is holomorphic in µ ∈ U δ . Indeed, ρ ( L ) just as all the other terms in (D.46) areholomorphic on U δ . This proves that ρ ( L ) admits an asymptotic expansion in L such that ρ ( L ) ( ω ) = ρ ( ω ) + O (cid:16) L − (cid:17) ,where ρ is the solution to the integral equation ( I + γ R / π ) · ρ = h . As ρ ( L ) and ρ are both holomorphic in U δ , sois the remainder. Moreover, one can convince oneself that this O is uniform in µ ∈ U δ . Therefore, the regularityof the functional F [ · ] ( z ) (4.14) leads to F (cid:2) γρ ( L ) (cid:3) ( z ) = F (cid:2) γρ (cid:3) ( z ) + O (cid:16) L − (cid:17) . References [1] I.A. Aizenberg and A.P. Yuzhakov,
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