A thermal form factor series for the longitudinal two-pointfunction of the Heisenberg-Ising chain in the antiferromagnetic

massive regime

Constantin Babenko,† Frank Gohmann,† Karol K. Kozlowski∗ and Junji Suzuki‡

†Fakultat fur Mathematik und Naturwissenschaften,Bergische Universitat Wuppertal, 42097 Wuppertal, Germany

∗Univ Lyon, ENS de Lyon, Univ Claude Bernard,CNRS, Laboratoire de Physique, F-69342 Lyon, France

‡Department of Physics, Faculty of Science, Shizuoka University,Ohya 836, Suruga, Shizuoka, Japan

Dedicated to Professor Barry M. McCoy on the occasion of his 80th birthday

Abstract

We consider the longitudinal dynamical two-point function of the XXZ quan-tum spin chain in the antiferromagnetic massive regime. It has a series rep-resentation based on the form factors of the quantum transfer matrix of themodel. The nth summand of the series is a multiple integral accounting forall n-particle n-hole excitations of the quantum transfer matrix. In previousworks the expressions for the form factor amplitudes appearing under theintegrals were either again represented as multiple integrals or in terms ofFredholm determinants. Here we obtain a representation which reduces, in thezero-temperature limit, essentially to a product of two determinants of finitematrices whose entries are known special functions. This will facilitate thefurther analysis of the correlation function.

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1 Introduction

In this work we renew our attempts to obtain simple and manageable expressions forthe dynamical correlation functions of the XXZ chain. The XXZ chain is an anisotropicdeformation of the Heisenberg chain which is the fundamental model of 1d magnetism.The XXZ Hamiltonian for a chain of length L acts on the tensor product HL =

⊗Lj=1 Vj ,

Vj = C2, in which every factor is identified with a site in a 1d lattice. It is defined by

HL = JL∑j=1

σxj−1σ

xj + σyj−1σ

yj + ∆

(σzj−1σ

zj − 1

)− h

2

L∑j=1

σzj , (1)

where σα ∈ EndC2, α = x, y, z, are the Pauli matrices. The three real parameters of theHamiltonian are the anisotropy ∆, the exchange interaction J > 0, and the strength h > 0of an external magnetic field in the direction of the magnetic symmetry axis.

If the magnet is in contact with a heat bath of temperature T , it is in a ‘mixed state’with canonical density matrix

ρL(T ) =e−HL/T

tre−HL/T . (2)

The Heisenberg time evolution of a local operator xj ∈ EndVj is defined by

xj 7→ xj(t) = eiHLt xj e−iHLt . (3)

A typical quantity measured in experiments on quasi-1d magnets is the dynamical two-pointcorrelation function ⟨

xjyk(t)⟩T

= limL→∞

trρL(T )xjyk(t)

. (4)

of two local operators xj , yk.The analysis of finite-temperature dynamical correlation functions such as (4) is rather

involved. Even for integrable lattice models like the XXZ chain very little is known inthe general finite temperature case. Notable exceptions are so-called free-fermion modelsas, for instance, the XX chain (Hamiltonian (1) with ∆ = 0) [13, 25–27, 32, 34] or thetransverse field Ising model [28, 30, 31], where much of the long-time large-distanceasymptotics was worked out and numerically efficient representations of the correlationfunctions are available.

In the low-temperature limit, T → 0, when ρL(T ) becomes proportional to theprojector onto the groundstate subspace, the dynamical correlation functions of the XXZchain are better understood. They were mainly analysed by means of form factor seriesexpansions based on matrix elements of local operators between the ground state and theexcited states of the Hamiltonian. The ground state phase diagram is depicted in Figure 1.The analysis focused on the antiferromagnetic regions of the phase diagram. Form factorsfor the antiferromagnetic massive regime were first obtained within the q-vertex operatorapproach by Jimbo and Miwa [35]. In this approach the form factors are obtained in theform of multiple integrals which are interpreted in terms of even-numbered multiple-spinonexcitations and are called the 2n-spinon form factors. Since only the two-spinon formfactors are known explicitly in terms of special functions, most of the subsequent analysisfocused on the two-spinon contribution to the dynamical correlation functions. Moreover,

3

0

2

4

6

8

10

12

-2 -1 0 1 2 3 4 5

h / J

∆

ferromagnetic massive

antiferromagnetic massive

antiferromagnetic critical

Figure 1: The ground state phase diagram of the XXZ chain in the ∆-h/J plane. Theferromagnetic massive regime and the antiferromagnetic critical regime are separated bythe upper critical field hu = 4J(1 + ∆) (left red line). The lower critical field h` (right redline, defined in equation (54) below) determines the border between the antiferromagneticcritical and the antiferromagnetic massive regime.

the emphasis was on the calculation of the spatial and temporal Fourier transforms of thedynamical two-point functions, the so-called ‘dynamical structure factors,’ since these arethe functions that are measured more or less directly in neutron scattering experimentson certain quasi one-dimensional magnets. The two-spinon dynamical structure factor ofthe XXZ chain in the antiferromagnetic massive regime was studied in [7, 8, 12, 55]. Theimportant limiting case of the XXX chain (Hamiltonian (1) with ∆ = 1) was consideredin [38]. The four-spinon contribution for the XXX model was calculated in [9]. Two-and four spinon contributions together seem to give a rather quantitative description ofthe available experimental data [54]. The space-time asymptotics of the longitudinaldynamical two-point functions of the XXZ chain in the antiferromagnetic massive regimewas analysed in [17].

The q-vertex operator approach is designed to work directly in the thermodynamiclimit. Unfortunately, so far it has been successful only for massive integrable models andonly for the case of vanishing magnetic field. The two-spinon contribution to the dynamicalstructure factor of the XXZ chain in the antiferromagnetic massless regime at vanishingmagnetic field was obtained in [10] using expressions for the two-spinon form factors ofthe XYZ chain [53] and performing a massless limit. For the exploration of higher-spinoncontributions or of the antiferromagnetic massless phase at non-zero magnetic field theavailable results on dynamical correlation functions rely on Bethe Ansatz techniques.

An efficient Bethe Ansatz approach to the calculation of form factors of the finiteXXZ chain was developed in [44]. Such an approach has the advantage that is appliesequally well to all points in the ground state phase diagram. However, the calculation ofthe thermodynamic limit of the form factors in the general situation is rather sophisticated[16, 33, 39, 41, 43, 47, 60]. The summation of the form factor series imposed additional

4

difficulties that were overcome in [16, 40, 42, 49].In [44] the form factors pertaining to the finite XXZ chain of even length L were

obtained in a form proportional to determinants of matrices of size N ×N with N growingin L. These were analysed quite successfully by solving the Bethe Ansatz equations numer-ically [4] and resulted in predictions for the dynamical structure factors in finite magneticfields [11, 58] that could be compared with experiments. In the thermodynamic limit,L→∞, the N ×N determinants, generically, turn into Fredholm determinants of integraloperators [39, 41]. For a long time the Fredholm determinant corresponding to Baxter’sstaggered polarisation [1, 2], i.e. the Fredholm determinant arising in the calculation ofthe matrix element of σz between the two asymptotically degenerate groundstates of theXXZ chain in the antiferromagnetic massive regime of the XXZ chain, was the only knownexample that could be evaluated more explicitly in terms of known special functions [33].Only recently the well-known explicit result for the two-spinon form factors of the XXXchain [35] was reproduced from an algebraic Bethe Ansatz perspective [43].

Altogether the thermodynamic limit of the form factors generally results in rathertechnical expressions which can be seen as multiple residues related to the zeros of thecounting functions of the massive excitations in the respective ground state regimes [16,49].In spite of the technically complicated nature of the expressions for the form factors, theseries derived in [49] could be used for a detailed analysis of the space-time asymptoticsof the dynamical two-point functions of the XXZ chain in the antiferromagnetic masslessregime at finite magnetic field [50] and of the threshold singularities of their Fouriertransforms [48]. This analysis confirmed and refined, from a microscopic point of view,the non-linear Luttinger liquid phenomenology (for a review see [29]).

An alternative route to the zero temperature correlation functions of the XXZ chain isthrough the thermal form factor expansion [14, 18] developed initially for static correlationfunctions and later extended to the dynamical case in [22]. The thermal form factorexpansion is an expansion involving the form factors and eigenvalues of the quantumtransfer matrix. The latter is an auxiliary object originally introduced [46, 63, 64] in orderto study the thermodynamics of quantum spin chains. The quantum transfer matrix isgenerically non-Hermitian. Its eigenvalues are not necessarily all real. The eigenvalueof largest modulus, however, is always real and non-degenerate (which was rigorouslyestablished for temperature high enough in [21]). We call it the dominant eigenvalueand the corresponding eigenvector the dominant eigenvector. The dominant eigenvectordetermines the reduced density matrices of all finite sub-segments of the consideredquantum spin chain in the thermodynamic limit [23, 24]. Dynamical correlation functionscan be expressed as form factor series involving the matrix elements of the operators ofthe algebraic Bethe Ansatz between the dominant state and excited states of the quantumtransfer matrix [22].

The spectrum and the Bethe roots of the quantum transfer matrix of an integrablequantum spin chain are rather different from the spectrum and Bethe roots of the ordinarytransfer matrix associated with the Hamiltonian. Both depend parametrically on thetemperature and, in case of the XXZ chain, on the external magnetic field. The Betheroots form patterns in the complex plane which ‘evolve’ with increasing temperature.These patterns are currently not sufficiently well understood to allow for an analysis ofthe thermal form factor series of the dynamical two-point functions of the XXZ chain atarbitrary temperatures. For this reason we shall focus in this work on the case of the low-temperature limit of the XXZ chain in the antiferromagnetic massive regime, which is the

5

only parameter regime so far, where we have obtained a complete picture of all excitationsand explicit expressions for all eigenvalues [15]. Amazingly, the corresponding Bethe rootpatterns are much simpler than those belonging to the excited states of the Hamiltonian inthe same regime of the ground state phase diagram. They are characterized by the absenceof so-called strings. Accordingly, they also come with a different interpretation. Whilethe excitations of the Hamiltonian are interpreted in terms of spinons (see e.g. [16]), theexcitations of the quantum transfer matrix are parameterised by Bethe root patterns ofparticle-hole type. In the zero temperature limit every excitation is determined by twofinite sets of complex parameters, called particle- and hole roots, and by a ‘topologicalindex’ k ∈ Z/2Z. The particle- and hole roots are confined on two simple finite curves inthe upper and lower half plane, respectively, which are symmetric with respect to reflectionabout the real and imaginary axis (see Figure 2).

In [14] we found that the amplitudes in the form factor expansions factorize in threeparts which we called the universal part, the factorizing part and the determinant part. Inthe so-called Trotter limit, which is a necessary step of the formalism, the universal andfactorising parts were taking a very explicit form given in terms of exponents of one andtwo dimensional integrals. In the low-T limit, we showed that all these quantities can beevaluated in closed form in terms of products of q-gamma and q-Barnes functions. Inits turn, the determinant part was represented as a ratio of two Fredholm determinants inthe numerator over two Fredholm determinants in the denominator. While in the low-Tlimit the Fredholm determinants in the denominator turned out to be explicitly computablein terms of q-products the Fredholm determinants in the numerator had too complicatedkernels so as to go beyond the Fredholm determinant representation. This was not much ofa problem for the numerical evaluation of the correlation functions through its thermal formfactor expansions. However, this structural intricacy of the integrands in the form factorexpansion raised doubts about the usefulness of the representation for an analytic studyof other properties of the correlation functions, such as their long-distance and large-timeasymptotic behaviour.

By relying on a form of determinant factorisation proposed in [43], we obtain in thiswork a representation for the amplitudes arising in the thermal form factor expansion ofthe generating function of the dynamical longitudinal correlation functions. The latter isstructured in such a way that, in the low-T limit, the full amplitudes become explicitlycomputable in terms of products of q-gamma and q-Barnes functions and finite sizedeterminants built up of q-gamma and theta functions. These expressions for the formfactors are then used to express the dynamical two-point function 〈σz1σzm+1(t)〉T=0 as anovel series of multiple integrals with very explicit and simple integrands. More preciselywe shall derive the following

Theorem. The longitudinal dynamical two-point function of the XXZ chain in the antifer-romagnetic massive regime and in the zero-temperature limit can be expressed in terms ofa generating function G(0),⟨

σz1σzm+1(t)

⟩= 1

2∂2γαD

2mG

(0)(m+ 1, t, α)∣∣α=0

. (5)

Here γ parameterizes the anisotropy ∆ = ch(γ) and Dm denotes the difference oper-ator, formally defined by Dmfm = fm − fm−1. The generating function has a seriesrepresentation of the form

6

G(0)(m, t, α) =ϑ2

2(iγα/2)

ϑ22

+ϑ2

1(iγα/2)

ϑ22

(−1)m

+∑`∈Nk=0,1

(−1)km

(`!)2

∫C`h

d`u

(2πi)`

∫C`p

d`v

(2πi)`A(U,V|k) e−i

∑λ∈UV(mp(λ)+tε(λ|h)) , (6)

where ϑ1 and ϑ2 are Jacobi theta functions, where p and ε are the dressed momentum anddressed energy of the excitations, and where the form factor amplitudes are given by

A(U,V|k) =ϑ2

2(Σ0(U,V))

ϑ22

[ ∏λ,µ∈UV

Ψ(λ− µ)

]× det

`Ω(uj , vk|U,V) det

`Ω(vj , uk|U,V) . (7)

The sets U = uj`j=1, V = vj`j=1 gather all integration variables. The contours Chand Cp are intervals [−π/2, π/2] shifted down or up in the complex plane (cf. (224)).The notation

∑λ∈UV means that summands indexed by elements of U come with a plus

sign, while summands indexed by elements of V get a minus sign (for a formal definitionsee (52) below). The remaining functions Σ0, Ψ and Ω are explicitly defined in terms oftheta, q-gamma and q-Barnes functions. Their precise definition will be given in equations(183), (216) and (206) in the main body of the text. Σ0 and the functions Ω and Ω dependparametrically on the ‘twist parameter’ α.

This work is organised as follows. In Section 2 we recall the results of [22] on thethermal form-factor series representation of dynamical correlation functions. Section 3contains a review of the low-T analysis of the spectrum and Bethe root patterns of thequantum transfer matrix of the XXZ chain in the antiferromagnetic massive regime obtainedin [15]. We also provide appropriate forms of non-linear integral equations that will beneeded in the sequel. Section 4 contains the main technical part of this work, a low-Tanalysis of the amplitudes in the thermal form factor series. The new form of the amplitudesis then used in Section 5 in order to obtain our novel form factor series expansion of thelongitudinal two-point function. We also discuss the explicit evaluation of the remainingfinite determinants in terms of basic hypergeometric series. Section 6 is devoted to theisotropic limit, and Section 7 to some conclusions of this work. A few supplementarytechnical details are deferred to two appendices.

2 The thermal form factor series

In previous work [22] we have developed a thermal form factor approach to dynamicalcorrelation functions of Yang-Baxter integrable models. It is based on a ‘vertex-modelrepresentation’ of the canonical density matrix and the time evolution operator [57]. In thefollowing we shall apply this approach to the longitudinal two-point function of the XXZchain in the antiferromagnetic massive regime in the low-temperature limit.

2.1 The dynamical quantum transfer matrix

The basic object in this approach is a ‘dynamical quantum transfer matrix’ which can beassociated with any fundamental solution of the Yang-Baxter equation. For the XXZ chain

7

in the antiferromagnetic massive regime we use the parametrization

R(λ, µ) =

1 0 0 0

0 b(λ, µ) c(λ, µ) 0

0 c(λ, µ) b(λ, µ) 0

0 0 0 1

,b(λ, µ) = sin(µ−λ)

sin(µ−λ+iγ)

c(λ, µ) = sin(iγ)sin(µ−λ+iγ)

. (8)

of the R-matrix. Here γ > 0, and we set q = e−γ for later convenience.We define a staggered, twisted and inhomogeneous monodromy matrix acting on

‘vertical spaces’ with ‘site indices’ 1, . . . , 2N + 2, and on a ‘horizontal auxiliary space’indexed a,

Ta(λ|h) = ehσza/2T Rt12N+2,a(ξ2N+2, λ+ iγ/2)Ra,2N+1(λ+ iγ/2, ξ2N+1)

× · · · ×Rt12,a(ξ2, λ)Ra,1(λ+ iγ/2, ξ1) . (9)

The number N will be called the Trotter number. The superscript t1 denotes transpositionwith respect to the first space R is acting on, and ξ1, . . . , ξ2N+2 are 2N + 2 complex‘inhomogeneity parameters’. Following [22, 57] we fix these parameters to the values

ξ2k−1 = −ξ2k =

− itκN k = 1, . . . , N2

ε k = N2 + 1

it+1/TκN k = N

2 + 2, . . . , N + 1 ,

(10)

where1/κ = −2iJ sh(γ) . (11)

The corresponding transfer matrix

t(λ|h) = traTa(λ|h) (12)

is then what we call a dynamical quantum transfer matrix of the XXZ chain. As we havedemonstrated in [22] the dynamical two-point functions of the XXZ chain can be writtenas a series involving the eigenvalues and form factors of the dynamical quantum transfermatrix. This series will be the starting point of our actual considerations. In order to defineit we have to briefly review the algebraic Bethe Ansatz construction of the eigenvectorsand eigenvalues of t(λ|h).

2.2 Eigenvectors and eigenvalues

The monodromy matrix (9) fulfills the two prerequisites for an algebraic Bethe Ansatz. Itsatisfies the Yang-Baxter algebra relations

Rab(λ, µ)Ta(λ|h)Tb(µ|h) = Tb(µ|h)Ta(λ|h)Rab(λ, µ) (13)

by construction, and it has a pseudo vacuum state

|0〉 =[(

10

)⊗(

01

)]⊗(N+1) (14)

8

on which its action is triangular. More precisely, setting

Ta(λ|h) =

(A(λ|h) B(λ|h)

C(λ|h) D(λ|h)

)a

, (15)

it is not difficult to see that

C(λ|h)|0〉 = 0 , A(λ|h)|0〉 = a(λ|h)|0〉 , D(λ|h)|0〉 = d(λ|h)|0〉 , (16)

where a(λ|h) and d(λ|h) are the pseudo vacuum expectation values

a(λ|h) = eh2T

sin(λ+ iγ

2 + ε)

sin(λ+ 3iγ

2 + ε)[ sin

(λ+ iγ

2 −itκN

)sin(λ+ 3iγ

2 −itκN

) sin(λ+ iγ

2 + it+1/TκN

)sin(λ+ 3iγ

2 + it+1/TκN

)]N2

,

d(λ|h) = e−h2T

sin(λ+ iγ

2 − ε)

sin(λ− iγ

2 − ε)[sin

(λ+ iγ

2 + itκN

)sin(λ− iγ

2 + itκN

) sin(λ+ iγ

2 −it+1/TκN

)sin(λ− iγ

2 −it+1/TκN

)]N2

. (17)

The eigenvalues and eigenstates of the quantum transfer matrix can be parameterizedby sets of Bethe roots. These are defined with the aid of an auxiliary function

a(λ∣∣λkMk=1, h

)=d(λ|h)

a(λ|h)

M∏k=1

sin(λ− λk − iγ)

sin(λ− λk + iγ)(18)

as the solutions of the ‘Bethe Ansatz equations’

a(λj∣∣λkMk=1, h

)= −1 , j = 1, . . . ,M . (19)

Note that the Bethe roots depend parametrically on t, T , γ and h. In order to simplify ournotation we number the solutions consecutively as λ(n)

j Mnj=1 and denote the corresponding

auxiliary functions byan(λ|h) = a

(λ∣∣λ(n)

k Mnk=1, h

). (20)

Left and right eigenvectors of the quantum transfer matrix can be constructed fromleft and right ‘off-shell Bethe vectors’ which, for a given subset ν = νjMj=1 ⊂ C, aredefined as⟨ν, h

∣∣ = 〈0|C(ν1|h) . . . C(νM |h) ,∣∣ν, h⟩ = B(νM |h) . . . B(ν1|h)|0〉 . (21)

If λ(n)j

Mnj=1 is a solution of the Bethe Ansatz equations (19), then

〈n, h| =⟨λ(n)

j Mnj=1, h

∣∣ , |n, h〉 =∣∣λ(n)

j Mnj=1, h

⟩(22)

are left and right eigenvectors of the quantum transfer matrix satisfying

〈n, h|t(λ|h) = 〈n, h|Λn(λ|h) , t(λ|h)|n, h〉 = Λn(λ|h)|n, h〉 , (23)

where

Λn(λ|h) =(1 + an(λ|h)

)a(λ|h)

Mn∏j=1

sin(λ− λ(n)

j + iγ)

sin(λ− λ(n)

j

) (24)

is the eigenvalue function.There is a unique so-called dominant eigenvalue Λ0, say, which is real and maximal in

the sense that Λ0(−iγ/2|h) > |Λn(−iγ/2|h)| for all n > 0. We know that M0 = N + 1.To further simplify the notation we drop the index ‘0’ for the dominant state, such thatΛ = Λ0, M = M0, |h〉 = |0, h〉, a(·|h) = a0(·|h), and λjMj=1 = λ(0)

j M0j=1.

9

2.3 The thermal form factor series

LetΣz(λ|h) = traσzaTa(λ|h) = 2T∂ht(λ|h) (25)

andρn(λ|h, h′) = Λn(λ|h′)/Λ(λ|h) . (26)

Here and hereafter we keep the magnetic fields h pertaining to the dominant state and h′

pertaining to the nth excited state as independent parameters. This is a common trick thatwill allow us to introduce a generating function of the longitudinal two-point function. Inthe end of our calculation in Section 5 we shall see that this generating function dependsonly on a ‘twist parameter’

α = (h− h′)/(2γT ) (27)

which is introduced here for later convenience. It becomes an independent variable in thezero-temperature limit.

Then, according to Theorem 1 of [22],⟨σz1σ

zm+1(t)

⟩T

= limN→∞

limε→0

∑n

〈h|Σz(−iγ/2|h)|n, h〉〈n, h|Σz(−iγ/2|h)|h〉Λn(−iγ/2|h)〈h|h〉Λ(−iγ/2|h)〈n, h|n, h〉

× ρn(−iγ/2|h, h)m(ρn(−iγ/2 + it/κN |h, h)

ρn(−iγ/2− it/κN |h, h)

)N2

. (28)

Due to pseudo spin conservation the sum over the excited states can be restricted to all nwith Mn = M . The limit N → ∞ is called the Trotter limit. It is only in this limit thatthe original quantum problem is recovered. The right hand side of (28) is a thermal formfactor series representation of the correlation function. Rather than working directly withthis series we shall introduce a generating function, which will simplify the subsequentcalculations to a certain extent.

We call the functions

Azzn (h) =〈h|Σz(−iγ/2|h)|n, h〉〈n, h|Σz(−iγ/2|h)|h〉Λn(−iγ/2|h)〈h|h〉Λ(−iγ/2|h)〈n, h|n, h〉

(29)

the form factor amplitudes of the longitudinal correlation function. The right hand side of(29) can be recast as

Azzn (h) = 2T 2∂2h′〈h|n, h′〉〈n, h′|h〉〈h|h〉〈n, h′|n, h′〉

[Λn(−iγ/2|h′)Λ(−iγ/2|h)

− 2 +Λ(−iγ/2|h)

Λn(−iγ/2|h′)

]h′=h

(30)

(see Appendix A for a derivation). This form of the amplitudes allows us to rewrite theform factor series (28) in terms of a generating function. We introduce the amplitudes ofits form factor series,

An(h, h′) =〈h|n, h′〉〈n, h′|h〉〈h|h〉〈n, h′|n, h′〉

, (31)

and the formal difference operator Dm acting as Dmfm = fm−fm−1 on sequences. Then⟨σz1σ

zm+1(t)

⟩T

= 2T 2∂2h′D

2mG(m+ 1, t, T, h, h′)

∣∣h′=h

, (32)

where

G(m, t, T, h, h′) = limN→∞

limε→0

∑n

An(h, h′)ρn(−iγ/2|h, h′)m

×(ρn(−iγ/2 + it/κN |h, h′)ρn(−iγ/2− it/κN |h, h′)

)N2

. (33)

10

2.4 Amplitudes at finite Trotter number

The two functions

e(x) = ctg (x)− ctg (x− iγ) , (34)

K(x) = ctg (x− iγ)− ctg (x+ iγ) = − e(x)− e(−x) (35)

will be used throughout these notes. We will call them the bare energy and the kernelfunction.

The main objective of this work is to obtain a manageable expression for the Trotterlimit, N →∞, of the amplitudes (31) appearing in the form factor series of the generatingfunction (33). To begin with, we will express the amplitudes for a given excited state|n, h〉 at finite Trotter number N in terms of the two sets of Bethe roots of the excited stateand the dominant state. For the XXZ chain this is possible due to the well-known scalarproduct formula of Nikita Slavnov [59] (for a recent constructive proof see [3]).

Lemma 1. Slavnov formula.Let 〈n, h| be a left eigenstate of the dynamical quantum transfer matrix (12) with Mn

Bethe rapidities and |ν, h〉 be an off-shell Bethe vector (21) with cardν = Mn. Thescalar product of these two states can be represented as

⟨n, h

∣∣ν, h⟩ =

[Mn∏j=1

d(λ

(n)j

∣∣h)Λn(νj |h)

]detMn

e(λ

(n)j −νk)

1+an(νk|h) −e(νk−λ

(n)j )

1+1/an(νk|h)

detMn

1

sin(λ(n)j −νk)

. (36)

Corollary 1. Let us now fix an excited state |n, h′〉 with Mn = M and denote the corre-sponding set of Bethe roots µjMj=1 = λ(n)

j Mnj=1 for brevity. Then

An(h, h′) =

[M∏j=1

ρn(λj |h, h′)ρn(µj |h, h′)

]detM

e(λj−µk)1+a(µk|h) −

e(µk−λj)1+1/a(µk|h)

detM

δjk +

K(λj−λk)a′(λk|h)

detM

1

sin(λj−µk)

×detM

e(µj−λk)

1+an(λk|h′) −e(λk−µj)

1+1/an(λk|h′)

detM

δjk +

K(µj−µk)a′n(µk|h′)

detM

1

sin(µj−λk)

. (37)

Proof. For the proof note that

〈h|ν, h′〉〈n, h′|ν, h〉〈h|ν, h〉〈n, h′|ν, h′〉

=〈h|ν, h〉〈n, h′|ν, h′〉〈h|ν, h〉〈n, h′|ν, h′〉

(38)

for any two off-shell Bethe states |ν, h′〉, |ν, h〉 with cardν = cardν = M .Then use (36) in (38) and send νk → µk, νk → λk. In order to obtain the limits in thedenominator write

detM

e(µj−νk)

1+an(νk|h) −e(νk−µj)

1+1/an(νk|h)

detM

1

sin(µj−νk)

=

detM

e(νk − µj)(1 + an(νk|h))− e(µj − νk)− e(νk − µj)

detM

1+an(νk|h)sin(νk−µj)

(39)

and use (35), the Bethe Ansatz equations (19) and l’Hospital’s rule, and similarly for theother determinant in the denominator.

11

3 Bethe root patterns and eigenvalues in the low-T limit

In [15] we considered the low-T spectrum of the quantum transfer matrix of the XXZchain in the antiferromagnetic massive regime. In this regime a characteristic feature of thesets of Bethe roots for fixed h > 0 and T → 0+ is the absence of strings, regular patternsin the complex plane which would appear at h = 0 or for the usual transfer matrix. Allexcitations of the quantum transfer matrix in the low-T limit can rather be interpreted asparticle-hole excitations.

In [22, 57] it was observed that the spectra and Bethe root patterns of the quantumtransfer matrix and of the dynamical quantum transfer matrix coincide in the Trotter limit,N → ∞. This follows from the fact that the driving terms in the associated non-linearintegral equations have the same Trotter limit which is independent of t. For this reason theresults of [15] can be used in the dynamical case as well. We shall recall them below. Theyare based on a thorough analysis of the solutions of the non-linear integral equations inthe entire complex plane. As not only their solutions but the non-linear integral equationsthemselves will be needed in the sequel, we include a short a posteriori derivation at theend of this section.

3.1 A reminder of the set of functions that determine the low-temperaturespectrum of correlation lengths and the universal part

The basic functions that eventually appeared in our previous low-T analysis of the correla-tion lengths and form factors of the XXZ chain in the antiferromagnetic massive regimewere q-gamma and q-Barnes functions. They may be expressed in terms of (infinite)q-multi factorials which, for |qj | < 1 and a ∈ C, are defined as

(a; q1, . . . , qp) =∞∏

n1,...,np=0

(1− aqn11 . . . q

npp ) . (40)

Based on this definition we introduce the q-gamma and q-Barnes functions Γq and Gq,

Γq(x) = (1− q)1−x (q; q)

(qx; q), (41a)

Gq(x) = (1− q)−12

(1−x)(2−x)(q; q)x−1 (qx; q, q)

(q; q, q). (41b)

They satisfy the normalisation conditions

Γq(1) = Gq(1) = 1 (42)

and the basic functional equations

[x]qΓq(x) = Γq(x+ 1) , Γq(x)Gq(x) = Gq(x+ 1) , (43)

where

[x]q =1− qx

1− q. (44)

is a familiar form of the q-number.

12

A closely related family of functions are the Jacobi theta functions ϑj(x) = ϑj(x|q),j = 1, . . . , 4, where

ϑ4(x|q) = (q2; q2)(e−2ixq; q2)(e2ix q; q2) (45)

and

ϑ1(x) = −iq14 eix ϑ4(x+ iγ/2) , ϑ2(x) = q

14 eix ϑ4(x+ iγ/2 + π/2) ,

ϑ3(x) = ϑ4(x+ π/2) . (46)

These functions are related to the q-gamma functions through the second functional relationof the latter, which can, for instance, be written as

ϑ4(x)

ϑ4=

Γ2q2

(12

)Γq2(

12 −

ixγ

)Γq2(

12 + ix

γ

) . (47)

Here we have employed a common convention [65] for theta constants,

ϑ′1 = ϑ′1(0) , ϑj = ϑj(0) , j = 2, 3, 4 . (48)

Using the above set of functions we can proceed with those functions that determine thephysical properties of the XXZ chain in the low-T limit. These are the dressed momentump, the dressed energy ε and the dressed phase ϕ. Dressed momentum and dressed energyare defined as

p(λ) =π

2+ λ− i ln

(ϑ4(λ+ iγ/2|q2)

ϑ4(λ− iγ/2|q2)

), (49)

ε(λ|h) =h

2− 2J sh(γ)ϑ3ϑ4

ϑ3(λ)

ϑ4(λ). (50)

The dressed phase is the function

ϕ(λ1, λ2) = i(π

2+ λ12

)+ ln

Γq4(1 + iλ12

2γ

)Γq4(

12 −

iλ122γ

)Γq4(1− iλ12

2γ

)Γq4(

12 + iλ12

2γ

) , (51)

where λ12 = λ1 − λ2, |Imλ2| < γ.In order to define the shift function, which fixes the Bethe root patterns in the low-T

limit, we first introduce a convention for sums and products that will be used throughoutthis work. Let X,Y ⊂ C be two discrete sets. We shall write∑

λ∈XYf(λ) =

∑λ∈X

f(λ)−∑λ∈Y

f(λ) ,∏

λ∈XYf(λ) =

∏λ∈X f(λ)∏λ∈Y f(λ)

. (52)

Then the shift function F is defined as

F (λ|X,Y) =1

2πi

∑µ∈YX

ϕ(λ, µ) . (53)

Note that the dressed energy depends parametrically on the magnetic field h. Thecondition ε(π2 |h`) = 0 determines the lower critical field

h` = 4J sh(γ)ϑ24 , (54)

i.e. the boundary between the antiferromagnetic massive and massless regimes (see Fig-ure 1). The curves Re ε(λ|h) = 0 are shown in Figure 2. These are the curves on whichthe Bethe roots ‘condense’ at low T .

13

-1.5 -1.0 -0.5 0.5 1.0 1.5Re x

-0.6

-0.4

-0.2

0.2

0.4

0.6

Im x

Figure 2: The curves Re ε(λ|h) = 0 for various values of the magnetic field. Here∆ = 1.7, h`/J = 0.76. The values of the magnetic field decrease proceeding from theinner to the outer curve: h/h` = 1.34, 1, 2/3, 1/3, 0. In the critical regime h` < h <hu = 4J(1 + ∆) the curves are closed. In this regime ε is not given by (50), but is ratherdefined as a solution of a linear integral equation. At the lower critical field h = h` theclosed curves develop two cusps, and a gap opens for 0 < h < h`, which is the parameterregime considered in this work.

Conjecture 1. Low-T Bethe root patterns at 0 < h < h` [15].

(i) All excitations of the quantum transfer matrix at low T and large enough Trotternumber N can be parameterized by an even number of complex parameters locatedinside the strip |Imλ| < γ/2 and by an index e ∈ Z/2Z. Referring to [15] we callthe parameters in the upper half plane particles, the parameters in the lower halfplane holes. We shall denote the set of particles by Y, the set of holes by X.

(ii) For excited states with Mn = M = N + 1 we have cardX = cardY = `.

(iii) Up to corrections of the order T∞ the particles and holes are determined by thelow-T higher-level Bethe Ansatz equations

ε(y|h) = 2πiT(`y + e/2 + F (y|X,Y)

), ∀ y ∈ Y , (55a)

ε(x|h) = 2πiT(mx + e/2 + F (x|X,Y)

), ∀ x ∈ X , (55b)

where `y,mx ∈ Z and where the `y and the mx are mutually distinct.

(iv) For solution Xn, Yn of (55) with index e = en we denote F (λ|Xn,Yn) = Fn(λ).The auxiliary function corresponding to this solution is

an(λ|h) = (−1)en e−ε(λ|h)/T+2πiFn(λ)(1 + O(T∞)

)(56)

uniformly for |Imλ| < γ/2, away from the points ±iγ/2.

14

In the limit T → 0+ the higher-level Bethe Ansatz equations (55) decouple, iπ`yTand iπmxT turn into independent continuous variables, and the particles and holes becomefree parameters on the curves

B± =λ ∈ C

∣∣Re ε(λ|h) = 0,−π/2 ≤ Reλ ≤ π/2, 0 < ±Imλ < γ. (57)

These curves are shown in Figure 2. As we can see, the massive regime is distinguishedfrom the massless regime by the opening of a ‘mass gap’ at the critical field h`.

3.2 Low-temperature limit of the eigenvalue ratios

The main result of our work [15] was an explicit formula for all correlation lengths, orrather all eigenvalue ratios, in the low-temperature regime. At low enough temperaturesall excitations are parameterized by solutions Xn, Yn of the higher-level Bethe Ansatzequations (55). Let k = en − e0. Then, for −γ < Imλ < 0, the corresponding eigenvalueratios in the Trotter limit and at finite magnetic field [15, 18] can be expressed as

ρn(λ|h, h′) = (−1)k exp

i∑

z∈XnYn

p(λ− z + iγ/2)

, (58)

this being valid up to multiplicative corrections of the form(1 + O(T∞)

). Note that the

value of the magnetic field h′ enters here only through the particle and hole parameters.From equation (58) we obtain the eigenvalue ratios entering the form factor series (33),

ρn(−iγ/2|h, h′) = (−1)k exp

i∑

z∈YnXn

p(z)

= (−1)k

∏z∈YnXn

ϑ1(z − iγ/2|q2)

ϑ4(z − iγ/2|q2), (59)

and

limN→∞ε→0

(ρn(−iγ/2 + it/κN |h, h′)ρn(−iγ/2− it/κN |h, h′)

)N2

= exp

(it/κ)∂λ ln ρn(λ|h, h′)λ=−iγ/2

= exp

2itJ sh(γ)

∑z∈XnYn

p′(z)

= exp

it∑

z∈YnXn

ε(z|h)

. (60)

Both formula are again valid up to multiplicative temperature corrections of the form(1 + O(T∞)

).

3.3 Non-linear integral equations

The low-T analysis can be based on a non-linear integral equation satisfied by the auxiliaryfunctions an(·|h). Let us briefly recall this equation. We start with definitions and usefulnotation. Let

θ(λ|γ) =

∫Γλ

dµ(

ctg (µ− iγ)− ctg (µ+ iγ)), (61)

where Γλ is a piecewise straight contour beginning at−π2 , running parallel to the imaginary

axis to −π2 + Imλ, then continuing parallel to the real axis from −π

2 + Imλ to λ. This

15

function has branch cuts along (−∞± iγ,−π ± iγ] ∪ [±iγ,+∞± iγ) and satisfies thequasi-periodicity, quasi-reflection and asymptotic conditions

θ(λ+ π|γ) = θ(λ|γ) +

2πi |Imλ| < γ

0 |Imλ| > γ,(62a)

θ(−λ|γ) = −θ(λ|γ) +

2πi |Imλ| < γ

0 |Imλ| > γ,(62b)

limReλ→±∞

θ(λ|γ) = ∓2γ . (62c)

It is constructed in such a way that

eθ(λ|γ) =sin(λ− iγ)

sin(λ+ iγ). (63)

For γ > 0 we also set

K0(λ|γ) =θ′(λ|γ)

2πi. (64)

Then K0(λ|γ) > 0 for λ ∈ R. Now define

ε(N,ε)0 (λ|h) = h+ T

θ(λ− ε

∣∣γ2

)− θ(λ+ ε

∣∣γ2

)(65)

+NT

2

θ(λ+ it

κN

∣∣γ2

)− θ(λ− it

κN

∣∣γ2

)+ θ(λ− it+1/T

κN

∣∣γ2

)− θ(λ+ it+1/T

κN

∣∣γ2

).

This function has the point-wise limit

limN→∞

limε→0

ε(N,ε)0 (λ|h) = h− θ′

(λ|γ2)/κ = h− 4πJ sh(γ)K0

(λ|γ2)

= ε0(λ|h) (66)

which is independent of t and T .For a shortcut in the derivation of the non-linear integral equation we shall now use our

insight from previous work [15] summarized above as an input. Recall that we have fixeda solution µjMj=1 = λ(n)

j Mnj=1 of the Bethe ansatz equations (19) which determines an

auxiliary function an(·|h′) and that we are keeping the magnetic field h pertaining to thedominant state and h′ pertaining to the nth excited state independent. Let us now assumethat for fixed small T and N large enough all Bethe roots µj are close to B±, such that|Imµj | < γ/2, j = 1, . . . ,M . Using (63) and (65) in (18) we see that

ln an(λ|h′) = −ε(N,ε)0 (λ|h′)

T+

N+1∑j=1

θ(λ− µj |γ)

− N

2

θ(λ+ iγ

2 −itκN

∣∣γ)+ θ(λ+ iγ

2 + it+1/TκN

∣∣γ)− θ(λ+ iγ2 + ε

∣∣γ) (67)

is a possible definition of the logarithm of an(·|h′). This definition guarantees thatln an(·|h′) is π-periodic and analytic for |Imλ| < γ/2 in the following sense. For agiven λ with |Imλ| < γ/2 and small T > 0 there is an N0 ∈ N such that ln an(·|h′) isanalytic in λ and

ln an(λ+ π|h′) = ln an(λ|h′) (68)

16

for all N > N0.We fix a simple closed and positively oriented contour C = Cl +C−+Cr +C+, where

Cl = [−π/2,−π/2− iγ/2− i0] , Cr = −Cl + π , (69)

C+ = [π/2,−π/2] , C− = −C+ − iγ/2− i0 . (70)

Here the regularization i0 means that the pole of an(λ|h′) at λ = − iγ2 + it

κN is inside C.Let

Zn = λ ∈ C|1 + an(λ|h′) = 0 ∩ IntC . (71)

With µjMj=1 we associate sets of ‘particles’ Yn and ‘holes’ Xn relative to C, setting

Yn = µjMj=1 ∩ ExtC , Xn = Zn \(µjMj=1 ∩ IntC

). (72)

Using the residue theorem in (67) then implies that

ln an(λ|h′) = −ε(N,ε)0 (λ|h′)

T+∑

z∈YnXn

θ(λ− z|γ) +

∫C

dµ

2πiθ(λ− µ|γ)

∂µan(µ|h′)1 + an(µ|h′)

(73)

for all λ ∈ C with |Imλ| < γ/2. Here we have used that, due to our assumptions,µj − iγ /∈ IntC.

Equation (73) turns into a non-linear integral equation for an(·|h′) upon applyingpartial integration to the integral. For this purpose a proper definition of the logarithmof 1 + an(·|h′) is required. Let us assume (56) to hold approximately for small T and Nlarge enough. Then |an(λ|h′)| = 1, −γ/2 < Imλ < 0 determines a curve Bn close toB− which intersects the line Reλ = −π

2 in a point −π2 − ix0 with −γ/2 < x0 < 0. Due

to the π-periodicity of an(·|h′) the curve Bn passes through π2 − ix0 as well. Define a

domain D↑ located between Bn and the real axis and a domain D↓ located between theline Imλ = −γ

2 and Bn. Then

|an(λ|h′)|

> 1 for λ ∈ D↑

< 1 for λ ∈ D↓.(74)

This allows us to define the logarithm of 1 + an(·|h′) on D↑ ∪ D↓ as an analytic andπ-periodic function,

Ln(1 + an)(λ|h′) =

ln an(λ|h′) + ln

(1 + 1/an(λ|h′)

)for λ ∈ D↑

ln(1 + an(λ|h′)

)for λ ∈ D↓,

(75)

where ln an(·|h′) is defined by (67) and the other logarithms on the right hand side aredefined by the principal branch. By construction Ln(1 + an)(·|h′) has jump discontinuitiesacross Bn. In particular, at ∓π

2 − ix0

Ln(1 + an)(∓π

2 − ix−0∣∣h′)− Ln(1 + an)

(∓π

2 − ix+0

∣∣h′) = 2πien (76)

for some en ∈ Z.Using the properties of the function Ln(1 + an)(·|h′) we may now calculate the index

of 1 + an(·|h′) along C,

17

∫C

dµ

2πi

∂µan(µ|h′)1 + an(µ|h′)

= Ln(1 + an)(−π

2 − ix−0∣∣h′)− Ln(1 + an)

(π2 − ix−0

∣∣h′)+ Ln(1 + an)

(π2 − ix+

0

∣∣h′)− Ln(1 + an)(−π

2 − ix+0

∣∣h′) = 0 . (77)

Here we have used the analyticity of Ln(1 + an)(·|h′) in D↑ und D↓ in the first equationand (76) in the second equation. Calculating the same integral by means of the residuetheorem we see that ∫

C

dµ

2πi

∂µan(µ|h′)1 + an(µ|h′)

= cardYn − cardXn , (78)

implying thatcardYn = cardXn = ` . (79)

Performing a similar calculation as in (77) and using also the quasi-periodicity (62) ofthe function θ(·|γ) we may now rewrite the integral on the right hand side of (73),∫

C

dµ

2πiθ(λ−µ|γ)

∂µan(µ|h′)1 + an(µ|h′)

= 2πien +

∫C

dµK0(λ−µ|γ) Ln(1 + an)(µ|h′) . (80)

Inserting this back into (73) we have converted the latter equation into a non-linear integralequation for the function an(·|h′),

ln an(λ|h′) = −ε(N,ε)0 (λ|h′)

T

+ 2πien +∑

z∈YnXn

θ(λ− z|γ) +

∫C

dµ K0(λ− µ|γ) Ln(1 + an)(µ|h′) . (81)

This equation determines an(·|h′) on C and then, by analytic continuation in the entirecomplex plane. Recall that, an(x|h′) = an(y|h′) = −1 for all x ∈ Xn, y ∈ Yn byconstruction.

Remark 1. The non-linear integral equation (81) is appropriate for taking the Trotter limitof the function an(·|h′). Formally we just have to replace the function ε(N,ε)

0 (·|h′) by itslimit ε0(·|h′) defined in (66).

Remark 2. Reversing the steps in the derivation of (77) and (81) we can go back to theBethe Ansatz equations (19). This means that every solution of the non-linear integralequation for which 1 + an(·|h′) has index zero corresponds to a solution of the BetheAnsatz equation and to an eigenstate of the dynamical quantum transfer matrix. In [15] weargued that, in the low-T limit, all eigenstates can be obtained this way.

We shall also need an ‘off-shell version’ of the auxiliary function an(·|h′) which isdefined by means of a non-linear integral equation similar to (81). In order to construct itwe first of all set

ε(N,ε)0 (λ|h) = h+ T

θ(λ+ iγ + ε

∣∣γ2

)− θ(λ+ iγ − ε

∣∣γ2

)+NT

2

θ(λ+ iγ − it

κN

∣∣γ2

)− θ(λ+ iγ + it

κN

∣∣γ2

)+ θ(λ+ iγ + it+1/T

κN

∣∣γ2

)− θ(λ+ iγ − it+1/T

κN

∣∣γ2

)(82)

18

Then, for every pair of sets U ⊂ IntC, V ⊂ ExtC with cardU = cardV = `, the functiona−(·|U,V, h′) is the solution of the non-linear integral equation

ln a−(λ|U,V, h′) =ε

(N,ε)0 (λ|h′)

T

− 2πie′n −∑

z∈VUθ(λ− z|γ)−

∫C

dµ K0(λ− µ|γ) Ln−(1 + a−)(µ|U,V, h′) . (83)

Here

Ln−(1 + a−)(λ|U,V, h′)

=

ln(1 + a−(λ|U,V, h′)

)for λ ∈ D−↑

ln a−(λ|U,V, h′))

+ ln(1 + 1/a−(λ|U,V, h′)

)for λ ∈ D−↓ ,

(84)

and D−↑ and D−↓ are separated by the line, where |a−(λ|U,V, h′))| = 1. The contour C is

the same as in (81) but properly regularized such as to include the point − iγ2 + it+1/T

κN forsufficiently large values of N . The integral equation (83) is constructed in such a way that

an(·|h′) = 1/a−(·|Xn,Yn, h′) (85)

if Xn and Yn are solutions of the subsidiary conditions a−(x|Xn,Yn, h′) = −1 for allx ∈ Xn, and a−(y|Xn,Yn, h′) = −1 for all y ∈ Yn.

Hence, it is natural to define an off-shell function

a+(·|U,V, h′) = 1/a−(·|U,V, h′) . (86)

Then

an(·|h′) = a+(·|Xn,Yn, h′) = a+n (·|h′) , (87a)

1/an(·|h′) = a−(·|Xn,Yn, h′) = a−n (·|h′) . (87b)

4 Low-temperature analysis of the amplitudes

In this section we derive explicit expressions for the amplitudes (31) in the low-temperaturelimit.

4.1 A factorization of determinants

Denoting the elements of Xn by xj , the elements of Yn by yj and the elements of Zn by zj(cf. (71), (72)) we may label them in such a way that

zj = µj , j = 1, . . . ,M − ` ,

yj = µM−`+j , j = 1, . . . , ` ,

xj = zM−`+j , j = 1, . . . , ` . (88)

Using this notation we shall separate the particle- hole contributions from the determi-nants on the right hand side of (37). We first define a number of auxiliary matrices through

19

their matrix elements. For these we shall use a convention in which upper indices refer torows and lower indices to columns. Let

Kjk =

K(λj − λk)a′(λk|h)

, j, k = 1, . . . ,M ,

V jk =

e(λj − µk)1 + a(µk|h)

− e(µk − λj)1 + 1/a(µk|h)

, j, k = 1, . . . ,M . (89)

Further define

V (i)j

k = V jk , j = 1, . . . ,M ; k = 1, . . . ,M − ` ,

V (p)j

k = V jM−`+k , j = 1, . . . ,M ; k = 1, . . . , ` ,

V (h)j

k =e(λj − xk)1 + a(xk|h)

− e(xk − λj)1 + 1/a(xk|h)

, j = 1, . . . ,M ; k = 1, . . . , ` (90)

andG = (IM +K)−1

(V (i), V (h)

), D = (0, I`)G

−1(IM +K)−1V (p) . (91)

Here and in the following In denotes the n × n unit matrix, and we use a block-matrixnotation. E.g., in

(V (i), V (h)

)we combine the M × (M − `) block V (i) and the M × `

block V (h) into an M ×M square matrix. Similarly, (0, I`) consists of an ` × (M − `)block of zeros, combined with the `× ` unit matrix into an `×M matrix. This conventionis particularly convenient for the proofs of the two following lemmata.

Lemma 2. A factorization of determinants.We have the following factorization of determinants,

detM

e(λj−µk)1+a(µk|h) −

e(µk−λj)1+1/a(µk|h)

detM

δjk +

K(λj−λk)a′(λk|h)

= detMGdet

`D . (92)

Proof. With the notations above we can write

detM

e(λj−µk)1+a(µk|h) −

e(µk−λj)1+1/a(µk|h)

detM

δjk +

K(λj−λk)a′(λk|h)

= detM

((IM +K)−1

)detM

((V (i), V (p))

). (93)

Now

detM

((V (i), V (p))

)= det

M+`

[V (i) V (p) V (h)

0 0 I`

]= det

M+`

[V (i) V (h) V (p)

0 −I` 0

]. (94)

Hence,

detM

((IM +K)−1

)detM

((V (i), V (p))

)= det

M+`

[G (IM +K)−1V (p)

(0,−I`) 0

]

= detMG det

M+`

[IM G−1(IM +K)−1V (p)

(0,−I`) 0

]

20

= detMG det

M+`

[IM G−1(IM +K)−1V (p)

0 (0, I`)G−1(IM +K)−1V (p)

](95)

which implies (92).

In order to perform a similar calculation with the second such ratio on the right handside of (37) we have to decompose the norm determinant in the denominator first. For thispurpose let

Ljk =K(µj − µk)a′n(µk|h′)

, j, k = 1, . . . ,M ,

L(ii)j

k =K(µj − µk)a′n(µk|h′)

, j, k = 1, . . . ,M − ` ,

L(ip)j

k =K(µj − yk)a′n(yk|h′)

, j = 1, . . . ,M − `; k = 1, . . . , ` ,

L(pi)j

k =K(yj − µk)a′n(µk|h′)

, j = 1, . . . , `; k = 1, . . . ,M − ` ,

L(pp)j

k =K(yj − yk)a′n(yk|h′)

, j, k = 1, . . . , ` ,

L(ih)j

k =K(µj − xk)a′n(xk|h′)

, j = 1, . . . ,M − `; k = 1, . . . , ` ,

L(hi)j

k =K(xj − µk)a′n(µk|h′)

, j = 1, . . . , `; k = 1, . . . ,M − ` ,

L(hh)j

k =K(xj − xk)a′n(xk|h′)

, j, k = 1, . . . , ` ,

L(ph)j

k =K(yj − xk)a′n(xk|h′)

, j, k = 1, . . . , ` ,

L(hp)j

k =K(xj − yk)a′n(yk|h′)

, j, k = 1, . . . , ` (96)

and

K∗ =

(L(ii) L(ih)

L(hi) L(hh)

),

C =

(L(hi) L(hh)

L(pi) L(ph)

), B =

(L(ih) −L(ip)

L(hh) −L(hp)

), D = I2` +

(−L(hh) L(hp)

−L(ph) L(pp)

),

J = D + C(IM +K∗)−1B . (97)

Finally define

W (i)j

k =e(µj − λk)

1 + an(λk|h′)− e(λk − µj)

1 + 1/an(λk|h′), j = 1, . . . ,M − `; k = 1, . . . ,M ,

21

W (p)j

k =e(yj − λk)

1 + an(λk|h′)− e(λk − yj)

1 + 1/an(λk|h′), j = 1, . . . , `; k = 1, . . . ,M ,

W (h)j

k =e(xj − λk)

1 + an(λk|h′)− e(λk − xj)

1 + 1/an(λk|h′), j = 1, . . . , `; k = 1, . . . ,M (98)

and

G∗ = (IM +K∗)−1

(W (i)

W (h)

), D∗ = W (p)G∗−1(IM +K∗)−1

(0

I`

). (99)

With this notation we can state our next lemma.

Lemma 3. Another factorization of determinants.The remaining determinants in (37) admit the factorization

detM

δjk +

K(µj − µk)a′n(µk|h′)

= det

MIM +K∗det

2`J , (100a)

detM

e(µj−λk)

1+an(λk|h′) −e(λk−µj)

1+1/an(λk|h′)

detMIM +K∗

= detMG∗ det

`D∗ . (100b)

Proof. The proof is similar to the proof of Lemma 2 and relies on elementary row- andcolumn manipulations of determinants.

In lemma 2 and 3 we have factorized the determinants in the original representation(37) of the form factor amplitudes for the generating function in a way that will allow us totake the Trotter limit and the zero temperature limit. Two of the determinants are of sizeM , which goes to infinity for N →∞. We shall call them the ‘large determinants’. Thethree remaining determinants are of size `, or 2` and will be called the ‘small determinants’.In the following subsections the five determinants will be considered one by one. We shallsee that the matrix elements of G, G∗, (IM +K)−1 and (IM +K∗)−1 can be expressedin terms of functions that solve linear integral equations and can be explicitly calculated inthe limit T → 0+.

4.2 The small determinant in the denominator

We start with the 2` × 2` determinant in (100a) whose structure is familiar to us fromprevious work [18, 22]. We first of all observe that

J = I2`

+

(−L(hh) + L(hv)(IM +K∗)−1L(vh) L(hp) − L(hv)(IM +K∗)−1L(vp)

−L(ph) + L(pv)(IM +K∗)−1L(vh) L(pp) − L(pv)(IM +K∗)−1L(vp)

), (101)

where

L(hv) =(L(hi), L(hh)

), L(pv) =

(L(pi), L(ph)

), (102a)

L(vh) =

(L(ih)

L(hh)

), L(vp) =

(L(ip)

L(hp)

). (102b)

22

In order to simplify (101) we define a resolvent kernel as the solution of the linearintegral equation

R∗(λ, µ) = K(λ− µ)−∫C

dν

2πi

K(λ− ν)R∗(ν, µ)

1 + an(ν|h′). (103)

Recall that R∗(λ, µ) then also satisfies the integral equation

R∗(λ, µ) = K(λ− µ)−∫C

dν

2πi

R∗(λ, ν)K(ν − µ)

1 + an(ν|h′). (104)

Let λ ∈ IntC. The contour C is constructed in such a way that µ ∈ IntC ∪ Yn impliesthat µ± iγ ∈ ExtC. Hence, R∗(·, µ) is holomorphic in IntC and on the boundary of thisdomain. Then the residue theorem applied to (103) implies that

R∗(λ, µ) = K(λ− µ)−M∑l=1

K(λ− zl)R∗(zl, µ)

a′n(zl|h′)(105)

for all µ ∈ Zn ∪ Yn and for all λ ∈ IntC. Equation (105) also defines the analyticcontinuation of R∗(·, µ) to the complex plane. Similarly, (104) implies that

R∗(λ, µ) = K(λ− µ)−M∑l=1

R∗(λ, zl)K(zl − µ)

a′n(zl|h′)(106)

for all λ ∈ Zn ∪ Yn, for all µ ∈ IntC, and also for µ ∈ C for which the right hand side isdefined by analytic continuation.

Upon defining a matrix R∗ with matrix elements

R∗jk =R∗(zj , zk)

a′n(zk|h′)(107)

equation (105) implies that

(IM +K∗)−1 = IM −R∗ . (108)

This can be used to derive the following

Lemma 4. Jacobian of higher level Bethe Ansatz equations [22].The small determinant det2`J has the representations

det2`J = det

2`

δjk − R∗(xj ,xk)a′n(xk|h′)

R∗(xj ,yk)a′n(yk|h′)

−R∗(yj ,xk)a′n(xk|h′) δjk +

R∗(yj ,yk)a′n(yk|h′)

(109)

and

det2`J =

[∏j=1

−1

a′n(xj |h′)a′n(yj |h′)

]det2`J , (110)

where

det2`J = det

2`

(∂uka

−(uj |U,V, h′) ∂vka−(uj |U,V, h′)

∂uka+(vj |U,V, h′) ∂vka

+(vj |U,V, h′)

)U=XnV=Yn

. (111)

23

Proof. Let us consider, for instance, the upper left element of J . First of all, (105) impliesthat

M∑l=1

L(hv)j

l

[(IM +K∗)−1

]lk

=

M∑l=1

K(xj − zl)a′n(zl|h′)

(δlk −R∗lk) =

R∗(xj , zk)

a′n(zk|h′)(112)

for j = 1, . . . , `, k = 1, . . . ,M . Using (106) we infer

M∑l,m=1

L(hv)j

l

[(IM +K∗)−1

]lmL(vh)m

k

=M∑m=1

R∗(xj , zm)

a′n(zm|h′)K(zm − xk)a′n(xk|h′)

=K(xj − xk)a′n(xk|h′)

− R∗(xj , xk)

a′n(xk|h′)(113)

which is equivalent to

δjk − L(hh)j

k +[L(hv)(IM +K∗)−1L(vh)

]jk

= δjk −R∗(xj , xk)

a′n(xk|h′). (114)

Thus, we have obtained the upper left element of J . The other elements are obtained in asimilar way.

Taking the derivative of (83) with respect to uk we obtain

∂uka−(λ|U,V, h′)

a−(λ|U,V, h′)= −∂uka

+(λ|U,V, h′)a+(λ|U,V, h′)

= −K(λ− uk)−∫C

dµ

2πiK(λ− µ)

∂uka−(µ|U,V, h′)

1 + a−(µ|U,V, h′). (115)

Using the first equation in the second equation and comparing with equation (103) weconclude that

R∗(λ, xk) = ±∂uka±(λ|U,V, h′)

a±(λ|U,V, h′)

∣∣∣∣U=XnV=Yn

. (116)

Thus, using the Bethe Ansatz equations,

∂uka+(uj |U,V, h′)a′n(xk|h′)

∣∣∣∣U=XnV=Yn

= −∂uka−(uj |U,V, h′)a′n(xk|h′)

∣∣∣∣U=XnV=Yn

= δjk −R∗(λj , xk)

a′n(xk|h′). (117)

The other entries of the matrix J can be treated in a similar way.

Note that Lemma 4 will allow us to convert our sum over excited states (33) into a sumover classes of excited states, each class being represented by a multiple integral.

4.3 The functionsG andG∗

In order to prepare for the calculation of the remaining determinants in (92) and in (100b)we introduce a pair of functions G, G∗ which, for every µ ∈ IntC, are defined as thesolutions of the linear integral equation

G(λ, µ) = e(µ− λ)−∫C

dν

2πi

K(λ− ν)G(ν, µ)

1 + a(ν|h), (118a)

G∗(λ, µ) = e(µ− λ)−∫C

dν

2πi

K(λ− ν)G∗(ν, µ)

1 + an(ν|h′). (118b)

24

Lemma 5. Matrices and functionsG andG∗.The matrix elements ofG andG∗ defined in equations (91) and (99) are naturally expressedin terms of the function G and G∗ defined in (118),

Gjk = −G(λj , zk) , G∗jk = −G∗(zj , λk) , (119)

j, k = 1, . . . ,M .

Proof. Recall that e(µ− λ) = ctg (µ− λ)− ctg (µ− λ− iγ). Thus, µ− iγ ∈ ExtC, ifµ ∈ IntC and G(·, µ) then has a single simple pole with residue −1 at µ. It follows that

G(λ, µ) =e(µ− λ)

1 + 1/a(µ|h)− e(λ− µ)

1 + a(µ|h)−

M∑l=1

K(λ− λl)G(λl, µ)

a′(λl|h). (120)

Here we have used the second equation (35). We compare (120) with equation (91), whichis equivalent to

Gjk +

M∑l=1

K(λj − λl)Glka′(λl|h)

=e(λj − zk)1 + a(zk|h)

− e(zk − λj)1 + 1/a(zk|h)

. (121)

Then the first equation (119) follows.Similarly,

G∗(λ, µ) =e(µ− λ)

1 + 1/an(µ|h′)− e(λ− µ)

1 + an(µ|h′)−

M∑l=1

K(λ− zl)G∗(zl, µ)

a′n(zl|h′), (122)

while, on the other hand, the definition of the matrix G∗ in (99) is equivalent to

G∗jk +

M∑l=1

K(zj − zl)G∗lka′n(zl|h′)

=e(zj − λk)

1 + an(λk|h′)− e(λk − zj)

1 + 1/an(λk|h′). (123)

Comparison of (122) and (123) shows that the second equation (119) holds as well.

For the actual calculation of the determinants of the matrices G, G∗ and of theirinverses that appear in the remaining small determinants, we need a closer characterizationof the functions G, G∗. Important tools in this context are the resolvent kernels R∗ definedin (103) and its counterpart associated with the dominant state, which may be defined asthe solution of the integral equation

R(λ, µ) = K(λ− µ)−∫C

dν

2πi

R(λ, ν)K(ν − µ)

1 + a(ν|h). (124)

Lemma 6. Representation of the functions G and G∗ by means of resolvent kernelsand implications.

(i) The functions G and G∗ introduced in (118) can be represented as

G(λ, µ) = e(µ− λ)−∫C

dν

2πi

R(λ, ν) e(µ− ν)

1 + a(ν|h), (125a)

G∗(λ, µ) = e(µ− λ)−∫C

dν

2πi

R∗(λ, ν) e(µ− ν)

1 + an(ν|h′). (125b)

25

(ii) For λ ∈ IntC the functions G(λ, ·) and G∗(λ, ·) are meromorphic inside C, wherethey both have a single simple pole with residue +1 at λ.

Proof. (i) is an immediate consequence of the definitions of the resolvent kernels. (ii)holds, since R(λ, ·) and R∗(λ, ·) are holomorphic on and inside C as can be seen from(103), (124).

Define two functions

R0(λ) =1

2π

+1

4πγ

ψq4(

1 + iλ2γ

)+ ψq4

(1− iλ

2γ

)− ψq4

(12 + iλ

2γ

)− ψq4

(12 −

iλ2γ

), (126)

where ψq(λ) = ∂λ ln Γq(λ) is the q-digamma function, and

gα(λ, µ) =ϑ′1ϑ2(µ− λ− α)

ϑ2(α)ϑ1(µ− λ), (127)

where α ∈ C is a parameter (recall the definitions of the q-gamma, q-Barnes and thetafunctions in Section 3.1).

Lemma 7. Low-T form of the integral equations forG andG∗.

(i) Let µ ∈ IntC. The functions G(·, µ), G∗(·, µ) satisfy the linear integral equations

G(λ, µ) = g0(λ, µ)−∑σ=±

σ

∫Cσ

dνR0(λ− ν)G(ν, µ)

1 + aσ(ν|h), (128a)

G∗(λ, µ) = g0(λ, µ)−∑σ=±

σ

∫Cσ

dνR0(λ− ν)G∗(ν, µ)

1 + aσn(ν|h′). (128b)

(ii) If µ ∈ IntC, then

G(λ, µ) = g0(λ, µ) + O(T∞) , G∗(λ, µ) = g0(λ, µ) + O(T∞) . (129)

(iii) The functions G(λ, ·), G∗(λ, ·) can be analytically continued to the upper half plane.In particular, for µ ∈ Yn we have the representation

G(λ, µ) = g0(λ, µ) +2πiR0(λ− µ)

1 + a(µ|h)−∑σ=±

σ

∫Cσ

dνR0(λ− ν)G(ν, µ)

1 + aσ(ν|h), (130)

implying the low-T asymptotic behaviour

G(λ, µ) = g0(λ, µ) +2πiR0(λ− µ)

1− a(µ|h)an(µ|h′)

+ O(T∞) . (131)

Proof. (i) The functions under the integral on the right hand side of (118a) are π-periodic.Hence, the contributions from Cl and Cr cancel each other, and we may replace C byC+ + C−. For T → 0+ the function a(λ|h) behaves as

a(λ|h) = e−ε(λ|h)T(1 + O(T∞)

), (132)

26

(see (56)) and Re ε(λ|h) < 0 on C+, Re ε(λ|h) > 0 on C−. This suggests to decompose∫C

dν

2πi

K(λ− ν)G(ν, µ)

1 + a(ν|h)=∑σ=±

∫Cσ

dν

2πi

K(λ− ν)G(ν, µ)

1 + a(ν|h)

=

∫C−

dν

2πiK(λ− ν)G(ν, µ) +

∑σ=±

σ

∫Cσ

dν

2πi

K(λ− ν)G(ν, µ)

1 + aσ(ν|h)(133)

and to rewrite the integral equation (118a) as

G(λ, µ) +

∫C−

dν

2πiK(λ− ν)G(ν, µ)

= e(µ− λ)−∑σ=±

σ

∫Cσ

dν

2πi

K(λ− ν)G(ν, µ)

1 + aσ(ν|h). (134)

This can be further transformed using Fourier series techniques. The functions K(λ)and e(µ− λ) have Fourier series representations

K(λ) =

∞∑l=−∞

Kl e2ilλ , e(µ− λ) =

∞∑l=−∞

el(µ) e2ilλ , (135)

where

Kl = 2i e−2γ|l| , el(µ) = 2i

0 l ≤ 0

e−2ilµ− e−2il(µ−iγ) l > 0. (136)

For the derivation recall that µ ∈ IntC. Inserting the Fourier series (135) into (134) we seethat G(λ, µ) has a Fourier series representation

G(λ, µ) =∞∑

l=−∞gl(µ) e2ilλ (137)

with Fourier coefficients that satisfy the equation

gl(µ) +Kl

2i

∫C−

dν

πe−2ilν G(ν, µ) = el(µ)−

∑σ=±

σ

∫Cσ

dν

2πi

Kl e−2ilν G(ν, µ)

1 + aσ(ν|h). (138)

Using that µ ∈ IntC we see that∫C−

dν

πe−2ilν G(ν, µ) = −2i e−2ilµ +gl(µ) . (139)

Inserting this back into (138), solving for gl(µ) and performing the back transformationwe arrive at (128a).

The integral equation (118b) for G∗(·, µ) is of the same form as the integral equation(118a) for G(·, µ), just a(·|h) is replaced by an(·|h′) whose low-T behaviour is displayedin (56). As above, one can use the π-periodicity of the functions G∗(·, µ), e and K to arguethat (128b) must hold.

(ii) The limits (129) are an obvious implication of equations (128).(iii) Using Lemma 6 and analytically continuing G(λ, ·) through C+ we see that (130)

must hold. The limit (131) follows from (130), using (56) and the fact that an(µ|h′) = −1,since µ ∈ Yn.

27

Following the same procedure as above we obtain the low-T form of the integralequation for R∗ which will be needed below.

Lemma 8. Low-T limit ofR∗.

R∗(λ, µ) = 2πiR0(λ− µ)−∑σ=±

σ

∫Cσ

dνR0(λ− ν)R∗(ν, µ)

1 + aσn(ν|h′), (140)

implying thatR∗(λ, µ) = 2πiR0(λ− µ) + O(T∞) . (141)

4.4 Determinant and inverse of g0

The determinants of the matrices with matrix elements G0jk = g0(λj , zk) and G∗0

jk =

g0(zj , λk) and the inverses of these matrices can be evaluated in explicit form. This will beimportant for our further reasoning. In order to write the resulting expressions compactlywe introduce the shorthand notation

Σ =M∑j=1

(λj − zj) (142)

and the function

φ(λ) =

M∏j=1

ϑ1(λ− λj)ϑ1(λ− zj)

. (143)

Lemma 9. Properties of the matricesG0 andG∗0.

(i) The elliptic Cauchy determinant.

detMg0(λj , zk) =

ϑ2(Σ)ϑ′M1ϑ2

∏1≤j<k≤M ϑ1(λj − λk)ϑ1(zk − zj)∏M

j,k=1 ϑ1(zj − λk). (144)

(ii) The inversion formulae.

G−10

j

k =gΣ(zj , λk)

(1/φ)′(zj)φ′(λk), G∗0

−1j

k =g−Σ(λj , zk)

φ′(λj)(1/φ)′(zk). (145)

(iii) The square of the determinant.

detMg0(λj , zk) det

Mg0(zj , λk) =

ϑ22(Σ)

ϑ22

M∏j=1

φ′(λj)(1/φ)′(zj) . (146)

Proof. (i) The determinant formula is a variant of a classical result due to Frobenius [19]. Itappeared in the literature on the XXZ chain in [45] and has many interesting generalizations[56]. For the sake of self-containedness of this work we present a proof in Appendix B.

(ii) is obtained by Cramer’s rule and (144). (iii) is a direct consequence of (144) andthe definition (143) of φ.

28

4.5 The large determinants

For λ, µ ∈ IntC and else by analytic continuation define a pair of kernel functions

U(λ, µ) =∑σ=±

σ

∫Cσ

dξ

1 + aσ(ξ|h)

∫C

dζ

2πiφ(ζ)gΣ(λ, ζ)R0(ζ − ξ)G(ξ, µ) , (147a)

U∗(λ, µ) =∑σ=±

σ

∫Cσ

dξ

1 + aσn(ξ|h′)

∫C

dζ φ(ζ)

2πig−Σ(λ, ζ)R0(ζ − ξ)G∗(ξ, µ) . (147b)

Lemma 10. Low-T factorization ofG andG∗.

Gjk = −M∑l=1

g0(λj , zl)

(δlk −

U(zl, zk)

(1/φ)′(zl)

), (148a)

G∗jk = −M∑l=1

g0(zj , λl)

(δjk −

U∗(λl, λk)

φ′(λl)

). (148b)

Proof. The proof is by straightforward calculation,

M∑l=1

g0(λj , zl)U(zl, zk)

(1/φ)′(zl)

=∑σ=±

σ

∫Cσ

dξ

1 + aσ(ξ|h)

M∑l=1

g0(λj , zl)

(1/φ)′(zl)

M∑m=1

gΣ(zl, λm)

φ′(λm)R0(λm − ξ)G(ξ, zk)

=∑σ=±

σ

∫Cσ

dξ

1 + aσ(ξ|h)R0(λj − ξ)G(ξ, zk) = g0(λj , zk)−G(λj , zk) . (149)

Here we have inserted the definition of U in the first equation, the inversion formula (145)in the second equation and Lemma 7 in the third equation. Equation (148a) then followswith Lemma 5. The proof of (148b) is similar.

Corollary 2. Low-T form of the large determinants.

detMG = det

M−g0(λj , zk) det

M

δjk −

U(zj , zk)

(1/φ)′(zj)

, (150a)

detMG∗ = det

M−g0(zj , λk) det

M

δjk −

U∗(λj , λk)

φ′(λj)

. (150b)

This form is now suitable for taking the Trotter limit and the limit T → 0+, since thefirst factors on the right hand side can be written as products (see (144)) and since thekernel functions U , U∗ vanish for T → 0+.

4.6 The small determinants

The kernels U(λ, y) and U∗(λ, y) are holomorphic functions of both of their arguments λ,µ for λ, µ ∈ IntC. For the first argument this is obvious from the definition (147), for the

29

second argument it follows from Lemma 6. Let us consider the corresponding resolventkernels defined as solutions of the linear integral equations

S(λ, µ) = U(λ, µ) +

∫C

dν φ(ν)

2πiU(λ, ν)S(ν, µ) , (151a)

S∗(λ, µ) = U∗(λ, µ) +

∫C

dν

2πiφ(ν)U∗(λ, ν)S∗(ν, µ) . (151b)

These resolvents can be used to derive representations of the matrices G−1 and G∗−1 thatare suitable for performing the low-T limit.

Lemma 11. Low-T factorization of G−1 and G∗−1. The inverse matrices G−1 andG∗−1 admit the following representation in terms of the resolvent S,

G−1jk = − gΣ(zj , λk)

(1/φ)′(zj)φ′(λk)−∫C

dν φ(ν)

2πi

S(zj , ν)gΣ(ν, λk)

(1/φ)′(zj)φ′(λk), (152a)

G∗−1jk = − g−Σ(λj , zk)

φ′(λj)(1/φ)′(zk)−∫C

dν

2πiφ(ν)

S∗(λj , ν)g−Σ(ν, zk)

φ′(λj)(1/φ)′(zk). (152b)

Proof. Clearly S(λ, µ) is a holomorphic function of λ for λ ∈ IntC. Thus,

S(λ, µ) = U(λ, µ) +M∑l=1

U(λ, zl)S(zl, y)

(1/φ)′(zl), (153)

implying thatM∑l=1

(δjl −

U(zj , zl)

(1/φ)′(zj)

)(δlk +

S(zl, zk)

(1/φ)′(zl)

)= δjk . (154)

Combining the latter equation with (145) and (148a) we see that

G−1jk = −

M∑l=1

(δjl +

S(zj , zl)

(1/φ)′(zj)

)gΣ(zl, λk)

(1/φ)′(zl)φ′(λk), (155)

which is equivalent to (152a). The proof of (152b) is similar.

We fix a contour C′ ⊂ IntC in such a way that λ ⊂ IntC′. In preparation of the nextlemma we define for every λ ∈ ExtC′ a function

H(λ, xk) = g−Σ(λ, xk)−∫C′

dζ φ(ζ)

2πig−Σ(λ, ζ)R∗(ζ, xk)

(1/φ)′(xk)

a′n(xk|h′). (156)

Lemma 12. Low-T form of the matrices in the small determinants.

(i) The matrix in the first small determinant can be recast as

Djk = (1/φ)′(xj)D

jk =

∫C

dζ

2πiφ(ζ)gΣ(xj , ζ)G(ζ, yk)

+

∫C

dξ φ(ξ)

2πi

∫C′

dζ

2πiφ(ζ)S(xj , ξ)gΣ(ξ, ζ)G(ζ, yk) , (157)

j, k = 1, . . . , `.

30

(ii) The matrix in the second small determinant can be expressed as

D∗jk = (1/φ)′(xk)D∗jk = R∗(yj , xk)

(1/φ)′(xk)

a′n(xk|h′)

+

∫C

dξ

2πiφ(ξ)

(e(ξ − yj)

1 + 1/an(ξ|h′)− e(yj − ξ)

1 + an(ξ|h′)

)×(H(ξ, xk) +

∫C

dζ

2πiφ(ζ)S∗(ξ, ζ)H(ζ, xk)

), (158)

j, k = 1, . . . , `.

Proof. (i) It follows from Lemma 6 that G(λ, ·) can be analytically continued to ExtC.Using (118a) we conclude that

M∑l=1

(δjl +Kjl )G(λl, yk) =

e(yk − λj)1 + 1/a(yk|h)

− e(λj − yk)1 + a(yk|h)

= −V (p)j

k . (159)

Hence,

Djk =

[(0, I`)G

−1(IM +K)−1V (p)]jk

= −M∑m=1

G−1M−`+jm G(λm, yk) , (160)

j, k = 1, . . . , `. Inserting (152a) on the right hand side and using the residue theorem weend up with (157).

(ii) For the second matrix D∗ we start with the definition (99) and insert (108),

D∗jk =

[W (p)G∗−1(IM +K∗)−1

(0

I`

)]jk

=

M∑i,m=1

(e(yj − λi)

1 + an(λi|h′)− e(λi − yj)

1 + 1/an(λi|h′)

)G∗−1i

m

(δmM−`+k −

R∗(zm, xk)

a′n(xk|h′)

), (161)

j, k = 1, . . . , `. Then we substitute (152b) and convert the sums into integrals by means ofthe residue theorem. The only slightly subtle part of the calculation is the following:

M∑l=1

(e(yj − λl)

1 + an(λl|h′)− e(λl − yj)

1 + 1/an(λl|h′)

)g−Σ(λl, zm)

φ′(λl)

−∫C

dξ

2πiφ(ξ)

(e(yj − ξ)

1 + an(ξ|h′)− e(ξ − yj)

1 + 1/an(ξ|h′)

)g−Σ(ξ, zm)

= e(zm − yj) limξ→zm

1/φ(ξ)

1 + 1/an(ξ|h′)− e(yj − zm) lim

ξ→zm

1/φ(ξ)

1 + an(ξ|h′)

= K(yj − zm)(1/φ)′(zm)

a′n(zm|h′). (162)

Here the zeros of 1 + an(ξ|h′) are canceled by the poles of φ(ξ) and only the pole ofg−Σ(ξ, zm) at zm contributes to the sum over all residues. In the last equation we haveused (35). Taking into account (162) and the definition (156) of the function H , we easilysee, that (161) implies (158).

31

4.7 An intermediate summary

At this point we have factorized the determinants in a way that will allow us to take theTrotter limit and low-T limit. What remains to be done is to collect all the prefactors andto also rewrite them in a form that is appropriate for taking these limits. For this purposewe introduce the function

φs(λ) =

M∏k=1

sin(λ− λk)sin(λ− zk)

(163)

and two matrices U and U∗ with matrix elements

U jk =U(zj , zk)

(1/φ)′(zj), U∗jk =

U∗(λj , λk)

φ′(λj). (164)

Then the amplitudes (31) can be represented as

An(h, h′) = detM

(IM − U) detM

(IM − U∗)det`(D) det`(D

∗)

det2`(J)

× (−1)`[∏j=1

(1 + a(xj |h))(1 + a(yj |h))

][∏j=1

a′n(xj |h′)(1 + a(xj |h))(1/φ)′(xj)

]2

×[ M∏j=1

φs(zj + iγ)

φs(λj + iγ)

][ ∏µ∈YnXn

φs(µ− iγ)φs(µ+ iγ)

][ ∏µ,ν∈YnXn

1

sin(µ− ν + iγ)

]

× ϑ22(Σ)

ϑ22

[ M∏j=1

φ′(λj)(1/φ)′(zj)

][ M∏j=1

1 + an(λj |h′)a′(λj |h)

1 + a(zj |h)

a′n(zj |h′)

]. (165)

In order to see this one has to first realize that∏Mj=1

ρn(λj |h,h′)ρn(µj |h,h′)

detM

1

sin(λj−µk)

detM

1

sin(µj−λk)

=

M∏j=1

1 + an(λj |h′)a′(λj |h)

1 + a(µj |h)

a′n(zj |h′)

M∏k=1

sin(λj − µk + iγ) sin(µj − λk + iγ)

sin(λj − λk + iγ) sin(µj − µk + iγ), (166)

then use equations (37), (92), (100), (111), (150), (146), (157), (158), and the definitions(163), (164).

4.8 The Trotter limit

Next we shall rewrite the remaining sums and products over all Bethe roots in terms ofintegrals involving the auxiliary functions a and an. In fact, we will be always dealing withdifferences between excited state quantities and quantities pertaining to the dominant state.These will be taken care of by means of the function

z(λ) =1

2πi

(Ln(1 + an)(λ|h′)− Ln(1 + a)(λ|h)

), (167)

where the logarithms were defined in (75).

32

Lemma 13. Summation lemma.If f is holomorphic on and inside C, then

M∑j=1

(f(zj)− f(λj)

)= −k

(f(π2 − ix0)− f(−π

2 − ix0))−∫C

dµ f ′(µ) z(µ) , (168)

where x0 was introduced in Section 3.3 and k = en − e0.

Proof. Decompose the contour as in the calculation of the index in (77) and use partialintegration.

In order to have a compact notation in the following corollary and below we introducethe ‘indicator function’ 1 defined by

1condition =

1 if condition is satisfied0 else.

(169)

Corollary 3. Integral representations for Σ, φ and φs.

(i) The constant Σ defined in (142) has the integral representation

Σ = kπ +

∫C

dµ z(µ) . (170)

(ii) For the function φs defined in (163) we have for all λ ∈ C \ C

φs(λ) =

(1 + a(λ|h)

1 + an(λ|h′)

)1λ∈IntC(−1)k exp

−∫C

dµ ctg (λ− µ) z(µ)

, (171)

(iii) while for φ defined in (143) it holds for any λ with |Imλ| < γ/2 that

φ(λ) =

(1 + a(λ|h)

1 + an(λ|h′)

)1λ∈IntC(−1)k exp

−∫C

dµϑ′1(λ− µ)

ϑ1(λ− µ)z(µ)

. (172)

Proof. For (i) apply Lemma 13 to f(λ) = −λ. For the proof of (ii) consider any holo-morphic function ϕ that is π anti-periodic and has a single zero inside its periodicity stripat µ = 0. Then apply Lemma 13 to lnϕ(λ − µ) considered as a functions of µ withλ ∈ ExtC. This gives (171) for λ ∈ ExtC. The prefactor for λ ∈ IntC is obtained bymeans of analytic continuation. The proof of (iii) is similar. The restriction |Imλ| < γ/2guarantees that the zeros of ϑ1 at ±miγ, m ∈ N stay outside C.

Corollary 4. The double integrals.

M∏j=1

φs(zj + iγ)

φs(λj + iγ)= exp

∫C

dλ

∫C

dµ ctg ′(λ− µ+ iγ) z(λ) z(µ)

, (173)

M∏j=1

φ′(λj)(1/φ)′(zj)1 + an(λj |h′)

a′(λj |h)

1 + a(zj |h)

a′n(zj |h′)

= exp

−∫C′⊂C

dλ

∫C

dµ [lnϑ1]′′(λ− µ) z(λ) z(µ)

. (174)

Here C′ is a simple closed contour, tightly enclosed by C in such way that λ,Zn ⊂ IntC′.

33

Proof. Equation (173) follows from Lemma 13 and (171), since λj + iγ, zj + iγ ∈ ExtCfor j = 1, . . . ,M . For the proof of (174) notice that

φ′(λj)(1/φ)′(zj)1 + an(λj |h′)

a′(λj |h)

1 + a(zj |h)

a′n(zj |h′)

= limλ→λjz→zj

(1 + an(λ|h′))φ(λ)

1 + a(λ|h)

1 + a(z|h)

(1 + an(z|h′))φ(z)

= exp

∫C

dµ

(ϑ′1(zj − µ)

ϑ1(zj − µ)− ϑ′1(λj − µ)

ϑ1(λj − µ)

)z(µ)

, (175)

where we have used (172) in the last equation. Using once more (172) and the propertiesof the contour C′ we arrive at (174).

Corollary 5.[∏j=1

a′n(xj |h′)(1 + a(xj |h))(1/φ)′(xj)

]2

= exp

−2

∑λ∈Xn

∫C

dµϑ′1(λ− µ)

ϑ1(λ− µ)z(µ)

. (176)

Proof. Here the same reasoning as in the proof of the previous corollary applies.

Let us collect the single and double integrals in the exponents and denote their sums

∆1,C[z] = −2∑λ∈Xn

∫C

dµϑ′1(λ− µ)

ϑ1(λ− µ)z(µ)

−∑

λ∈YnXn

∑σ=±

∫C

dµ ctg (λ− µ+ σiγ) z(µ) , (177a)

∆2,C[z] =

∫C′⊂C

dλ

∫C

dµ(

ctg ′(λ− µ+ iγ)− [lnϑ1]′′(λ− µ))z(λ) z(µ) . (177b)

Lemma 14. Re-interpretation of the remaining large determinants as Fredholm de-terminants.

detM

(IM − U) = detC′

(id−U) , detM

(IM − U∗) = detC′

(id−U∗) , (178)

where the expressions of the right hand side of these equations are the Fredholm determi-nants of the integral operators U and U∗ acting on functions f holomorphic on IntC anddefined by

Uf(ξ) =

∫C′

dζ

2πiφ(ζ)f(ζ)U(ζ, ξ) , (179a)

U∗f(ξ) =

∫C′

dζ

2πiφ(ζ)f(ζ)U∗(ζ, ξ) , (179b)

where C′ is again a simple closed contour tightly enclosed by C.

Using this lemma and the above notation and corollaries as well as the definition (44)of the q-numbers in equation (165), we obtain our final result for the amplitudes at finiteTrotter number.

34

Proposition 1. Factorized amplitudes at finite (and infinite) Trotter number.

An(h, h′) = (−1)` e∆1,C[z]+∆2,C[z] det`(D) det`(D∗)

det2`(J)detC′

(id−U) detC′

(id−U∗)

× ϑ22(Σ)

ϑ22

[ ∏µ∈Xn∪Yn

(1− a(µ|h)

an(µ|h′)

)][ ∏µ,ν∈YnXn

1[12 −

i(µ−ν)2γ

]q4

]. (180)

Equation (180) still is a finite Trotter number representation of the amplitudes. Itsderivation is only based on certain assumptions on the location of the Bethe roots λ ofthe dominant state and µ of the excited states relative to the reference contour C, thatare described above. Conjecture 1, cited from our previous work [15], implies that theseassumptions are satisfied for all excited states if T is low enough and N is large enoughand that they continue to hold in the Trotter limit if T is low enough. This means thatin the Trotter limit the auxiliary functions a and an, which fully determine (180), are thesolutions of (81) with ε(N,ε) replaced by its limit ε0 (see (66)). Hence, the Trotter limit of(180) is obtained by replacing a and an by their limit functions.

4.9 The low-T limit

The calculation of the low-T limit of the form factor amplitudes is based on Conjecture 1and on the following corollary.

Corollary 6. The function z is π-periodic and on C+ ∪ C− has the low-T asymptotics

z(λ) = z0(λ)1λ∈C+ +1λ∈C+∪C− O(T∞), (181)

where

z0(λ) =k

2+h− h′

4πiT+ Fn(λ) (182)

with Fn as defined above (56).

In fact, by means of (181), most of the integrals that remain in the low-T limit reduce toconvolution-type integrals involving z0. In order to deal with those integrals we introducethe notation

Σ0 = Σ0(Xn,Yn) = −πk2− h− h′

4iT+

1

2

∑µ∈YnXn

µ , Υ(λ) =(e2iλ q4; q4)

(e2iλ q2; q4). (183)

Then2πi z0(λ) = −2iΣ0 +

∑µ∈YnXn

ln(Υ(λ− µ)

)− ln

(Υ(µ− λ)

). (184)

Lemma 15. Basic integration lemma.Let σ = sign

(Im (λ)

). For any λ /∈ C∫

C

dµ ctg (λ− µ) z(µ) = −σiΣ0 +∑

µ∈YnXn

ln(Υ(σ(λ− µ))

)+ O

(T∞). (185)

35

Proof. Equation (181) and (184) imply that∫C

dµ ctg (λ− µ) z(µ) = −2iΣ0

∫ π2

−π2

dµ

2πictg (µ− λ)

+∑

ν∈YnXn

∫ π2

−π2

dµ

2πi

ln(Υ(µ− ν)

)− ln

(Υ(ν − µ)

)ctg (µ− λ) + O

(T∞). (186)

The remaining integrals can be evaluated by means of the residue theorem. For this purposeone may use that ctg is a π-periodic function with its only simple poles at λ = 0 mod πand residue 1 at λ = 0, which behaves asymptotically as limImλ→±∞ ctg (λ) = ∓i. Withthis information the first integral on the right hand side may be calculated by shifting theintegration contour, for instance, to [−π/2, π/2] + i∞.

For the evaluation of the second integral note that Υ is a meromorphic, π-periodicfunction on C which is free of poles and zeros in H+ − iγ and decays exponentially fastfor Imλ→ +∞. Shifting the integration contour to [−π/2, π/2] + i∞ for the first termin the second integral on the right hand side and to [−π/2, π/2]− i∞ for the second termin the second integral on the right hand side we obtain a single residue contribution fromthe pole of the cotangent function, which is either in H+ or H−, and we have arrived at(185).

For σ = ± define

φ(σ)(λ) = eσiΣ0∏

µ∈XnYn

Γq4(

12 −

σi(λ−µ)2γ

)Γq4(

1 + σi(λ−µ)2γ

). (187)

Corollary 7. For any λ /∈ C and σ = sign(Im (λ)

)exp

−∫C

dµ ctg (λ− µ) z(µ)

=

eσiΣ0∏

µ∈XnYn

Γq4(

12 −

σi(λ−µ)2γ

)Γq4(

1− σi(λ−µ)2γ

) (1 + O(T∞)). (188)

If, in addition, |Imλ| < γ/2, then

exp

−∫C

dµϑ′1(λ− µ)

ϑ1(λ− µ)z(µ)

= φ(σ)(λ)

(1 + O(T∞)

). (189)

Proof. Equation (188) is a direct consequence of (185) and of the definition (41) of theq-gamma function.

For the proof of (189) we note that

ϑ′1(λ)

ϑ1(λ)= ctg (λ) +

∞∑k=1

(ctg (λ− kiγ) + ctg (λ+ kiγ)

)(190)

which follows from (45), (46). Assuming that |Imλ| < γ/2, multiplying by z(λ), integrat-ing over C and using (185), we obtain

36

∫C

dµϑ′1(λ− µ)

ϑ1(λ− µ)z(µ) = −σiΣ0 +

∑µ∈YnXn

ln(Υ(σ(λ− µ))

)+

∑µ∈YnXn

∞∑k=1

ln(Υ(λ− µ+ kiγ)

)+ ln

(Υ(µ− λ+ kiγ)

). (191)

By straightforward direct calculation

∞∏k=1

Υ(λ− µ+ kiγ) =1

(e2i(λ−µ) q4; q4). (192)

Finally, (191), (192) and (185) imply (189).

Corollary 8. If λ ∈ C+ ∪ C−, then

φ(λ) = (−1)kφ(+)(λ)1λ∈C+ +φ(−)(λ)1λ∈C−

(1 + O(T∞)

). (193)

Moreover,Σ = kπ + Σ0 + O(T∞) . (194)

Proof. Equation (193) follows directly from (172) and (189), taking into account that φ isanalytic across C+ and develops a jump singularity across B− in the limit T → 0+.

Equation (194) is obtained from (185) by taking the limit Imλ→ +∞ and taking intoaccount that in this limit ctg (µ− λ)→ i and ln

(Υ(λ− µ)

)→ 0.

Corollary 9.

e∆1,C[z] =

[ ∏λ∈Xn

φ(−)(λ)

]2[ ∏λ,µ∈XnYn

Γq4(

32 −

i(λ−µ)2γ

)Γq4(

1− i(λ−µ)2γ

)]2(1 + O(T∞)

). (195)

Proof. This follows easily from Corollary 7.

Lemma 16.

e∆2,C[z] =

[ ∏λ,µ∈XnYn

Gq4(

1− i(λ−µ)2γ

)Gq4

(2− i(λ−µ)

2γ

)Gq4

(32 −

i(λ−µ)2γ

)Gq4

(52 −

i(λ−µ)2γ

)](1 + O(T∞)). (196)

Proof. We first of all evaluate the integral

I2(τ) =

∫C′⊂C

dλ

∫C

dµ ctg ′(λ− µ+ iτ) z(λ) z(µ) , (197)

where τ ∈ C is such that λ − µ + iτ 6= 0 for µ ∈ C, λ ∈ C′. The regularization C′ ⊂ C

may be lifted if Im τ 6= 0 which we shall assume in the following. Since ctg ′ = −1/ sin2

is an even function, I2(τ) is even in τ , whence I2(τ) = I2(στ), where σ = sign(Im τ).Recall that z is π-periodic. Hence, a partial integration gives

I2(τ) = −∫C′⊂C

dλ z′(λ)

∫C

dµ ctg (λ− µ+ iστ) z(µ) . (198)

37

Now we use Lemma 15 for the µ-integral and Corollary 6 and equation (184) for z(λ).Then

I2(τ) =∑

µ,ν∈XnYn

∫ π2

−π2

dλ

2πiln(Υ(λ− µ+ iστ)

)∂λ

ln(Υ(λ− ν)

)− ln

(Υ(ν − λ)

)+ O(T∞)

=∑

µ,ν∈XnYn

∞∑n=1

ln(Υ(ν − µ+ iστ + (2n− 1)iγ)

)− ln

(Υ(ν − µ+ iστ + 2niγ)

)+ O(T∞) . (199)

In the second equation we have shifted the integration contour to [−π/2, π/2] + i∞ andhave used the residue theorem and the analytic properties of ln Υ described in the proof ofLemma 15.

Equation (199) implies that

eI2(τ) =

[ ∏λ,µ∈XnYn

A(λ− µ+ iστ)

](1 + O(T∞)

), (200)

where

A(ξ) = (e2iξ q4; q4)(e2iξ q6; q4, q4)2

(e2iξ q4; q4, q4)2. (201)

Note that

A(ξ + iγ) = (e2iξ q6; q4)(e2iξ q8; q4, q4)2

(e2iξ q6; q4, q4)2=

(e2iξ q6; q4)(e2iξ q4; q4, q4)2

(e2iξ q4; q4)2(e2iξ q6; q4, q4)2(202)

and

A(ξ)∞∏k=1

A(ξ + ikγ)A(ξ − ikγ) =1

(e2iξ q4; q4)(203)

by a straightforward calculation. Using (190) and equations (200), (202) and (203) itfollows that

e∆2,C[z] = eI2(γ) e−I2(0)−∑∞k=1(I2(kγ)+I2(−kγ))

=

[ ∏λ,µ∈XnYn

(e2i(λ−µ) q6; q4)(e2i(λ−µ) q4; q4, q4)2

(e2i(λ−µ) q4; q4)(e2i(λ−µ) q6; q4, q4)2

](1 + O(T∞)

), (204)

Replacing the q-factorials by q-gamma and q-Barnes functions by means of (41) and usingthe functional equation (43) we establish the claim.

It remains to consider the low-T limit of the determinants. All our efforts so far weredriven by the desire to be left with Fredholm determinants that trivialize in the low-T limit.

Lemma 17. Low-T limit of the Fredholm determinants.

detC′

(id−U) = 1 + O(T∞) , detC′

(id−U∗) = 1 + O(T∞) . (205)

Proof. This holds, since the kernels of the operators U and U∗ vanish exponentially fast inthe low-T limit as can be seen from the definitions in (147).

38

In order to express the low-T limit of the finite determinants in a compact way weintroduce two functions Ωn(λ, µ) = Ω(λ, µ|Xn,Yn) and Ωn(λ, µ) = Ω(λ, µ|Xn,Yn)setting

Ω(λ, µ|Xn,Yn) = gΣ0(λ, µ)−∫ π/2

−π/2

dζ

2πi

φ(−)(µ)

φ(+)(ζ)gΣ0(λ, ζ) e(ζ − µ) , (206a)

Ω(λ, µ|Xn,Yn) = g−Σ0(λ, µ)−∫ π/2

−π/2

dζ

2πi

φ(−)(ζ)

φ(+)(µ)g−Σ0(λ, ζ) e(µ− ζ) . (206b)

Then we can formulate the following lemma.

Lemma 18. Low-T limit of the matrices in the small determinants.

Dlm =

(−1)k Ωn(xl, ym)

φ(+)(ym)(1− e2πi z0(ym)

) ( 1 + O(T∞)), (207a)

D∗lm =(−1)k+1 Ωn(yl, xm)

φ(+)(yl)(1− e2πi z0(xm)

) ( 1 + O(T∞)). (207b)

Proof. S(xj , ζ) = O(T∞) by (147) and (151). Inserting this together with (131), (193)and (194) into (157) we obtain the following low-T limit for the elements of the ‘first smalldeterminant’,

Dlm =

∫C

dζ

2πiφ(ζ)gΣ(xl, ζ)G(ζ, ym) + O(T∞) =

(−1)k∑σ=±

∫Cσ

dζ

2πiφ(σ)(ζ)gΣ0(xl, ζ)

[g0(ζ, ym) +

2πiR0(ζ − ym)

1− e−2πi z0(ym)

]+ O(T∞) . (208)

The integral can be simplified exploiting the properties of the functions in the integrandunder shifts by iγ,

gα(λ, µ) = − e2iα gα(λ+ iγ, µ) = − e−2iα gα(λ, µ+ iγ) , (209a)

2πiR0(λ) = −2πiR0(λ+ iγ)− e(λ+ iγ) , (209b)

φ(−)(λ) = e−2iΣ0 φ(+)(λ+ iγ) . (209c)

Inserting these relations into the integral over C− we obtain∫C−

dζ

2πiφ(−)(ζ)gΣ0(xl, ζ)

[g0(ζ, ym) +

2πiR0(ζ − ym)

1− e−2πi z0(ym)

]

=

∫C−+iγ

dζ

2πiφ(+)(ζ)gΣ0(xl, ζ)

[g0(ζ, ym) +

2πiR0(ζ − ym) + e(ζ − ym)

1− e−2πi z0(ym)

]

=gΣ0(xl, ym)

φ(+)(ym)

[1− 1

1− e−2πi z0(ym)

]−∫C+

dζ

2πiφ(+)(ζ)gΣ0(xl, ζ)

[g0(ζ, ym) +

2πiR0(ζ − ym) + e(ζ − ym)

1− e−2πi z0(ym)

]. (210)

39

Substituting this back into (208) and using that

e2πi z0(x) =φ(−)(x)

φ(+)(x)(211)

we arrive at (207a).As for the entries of the ‘second small determinant’ we first of all evaluate the low-T

limit of the function H introduced in (156). We start with

(1/φ)′(xm)

a′n(xm|h′)= lim

λ→xm

1

φ(λ)(1 + an(λ|h′)=

(−1)k exp∫

Cdµ

ϑ′1(xm−µ)ϑ1(xm−µ) z(µ)

1− a(xm|κ)

an(xm|κ′)

=(−1)k(1 + O(T∞))

φ(−)(xm)(1− e−2πi z0(xm))=

(−1)k(1 + O(T∞))

φ(−)(xm)− φ(+)(xm). (212)

Here we have employed (172) and the Bethe Ansatz equations in the second equation,(189) in the third equation and (211) in the last equation. Using the equation (212) as wellas (141), (193), and (194) we obtain

H(λ, xm) = H0(λ, xm) + O(T∞) , (213)

where

H0(λ, xm) = g−Σ0(λ, xm)

−[∫

C′+

dζ φ(+)(ζ) +

∫C′−

dζ φ(−)(ζ)

]g−Σ0(λ, ζ)R0(ζ − xm)

φ(−)(xm)− φ(+)(xm). (214)

Since S∗(ξ, ζ) = O(T∞) by (147) and (151), this allows us to conclude that

D∗lm = (−1)k

2πiR0(yl − xm)

φ(−)(xm)− φ(+)(xm)+

∫C+

dξ

2πiφ(+)(ξ)e(ξ − yl)H0(ξ, xm)

−∫C−

dξ

2πiφ(−)(ξ)e(yl − ξ)H0(ξ, xm)

+ O(T∞)

= (−1)k

2πiR0(yl − xm)

φ(−)(xm)− φ(+)(xm)− H0(yl, xm)

φ(+)(yl)

+ O(T∞) . (215)

Next we use (209) and e(−λ) = e(λ+ iγ) and apply similar manipulations as in the aboveproof of (207a) to equation (214). After some calculation we arrive at (207b).

4.10 The amplitudes in the low-T limit

If we now use Corollaries 8, 9 and Lemmata 16-18 in Proposition 1 together with the basicfunctional equations (43), some cancellations occur, and we obtain our final result for theamplitudes in the low-T limit. It assumes a more compact shape if we introduce a function

Ψ(λ) = Γq4(

12 −

iλ2γ

)Γq4(1− iλ

2γ

)G2q4

(1− iλ

2γ

)G2q4

(12 −

iλ2γ

) . (216)

We summarize it in the following theorem.

40

Theorem 1. Amplitudes at low T .The amplitudes in the form factor expansion of the generating function of the longitudinalcorrelations function have the low-T asymptotic behaviour

An(h, h′) =ϑ2

2(Σ0)

ϑ22

[ ∏λ,µ∈XnYn

Ψ(λ− µ)

]det`

Ωn(xj , yk)

det`

Ωn(yj , xk)

det2`J

×(1 + O(T∞)

). (217)

This is the main result of this work. It has many implications. We would like toemphasize that (217) is explicit and is defined entirely in terms of functions of the q-gammafamily. In particular, as compared to previous expressions for form-factor amplitudes, thereare neither multiple integrals nor Fredholm determinants involved. We like to think of (217)as of a ‘factorized form’ of the amplitudes. The function Ω remotely resembles the functionω (cf. equation (50) of [5], where the function is denoted Ψ rather than ω) appearing in thetheory of factorized static correlation functions [36]. The functions Ω, Ω are still definedas single integrals over explicit functions. In fact, the integrals can be evaluated in terms ofq-hypergeometric functions. We further comment on this and give explicit examples below.The general case will be considered in a separate subsequent work. We finally note thattaking the isotropic limit toward the Heisenberg chain is straightforward. This limit will beconsidered in an extra section below.

5 The form factor series

5.1 The series for the generating function

The amplitudes (217) together with the eigenvalue ratios at finite Trotter number are theinput for the form factor series (33) of the generating function. A key feature of theamplitudes is the appearance of the factor det2`J in the denominator. It is this factor thatallows us to convert the series over all excitations into a series over multiple integrals, eachaccounting for a whole class of particle-hole excitations. We have described this in somedetail in [18, 22]. The main idea is that

J(U,V, k) =

(∂uka

−(uj |U,V, h′) ∂vka−(uj |U,V, h′)

∂uka+(vj |U,V, h′) ∂vka

+(vj |U,V, h′)

)(218)

is the Jacobi matrix ∂(U,V)/∂(u,v) of a transformation C2` 7→ C2`, (u,v) 7→ (U,V),where

Uj(u,v) = 1 + a−(uj |U,V, h′) , j = 1, . . . , ` , (219a)

Vk(u,v) = 1 + a+(vk|U,V, h′) , k = 1, . . . , ` . (219b)

This transformation maps solutions of the higher level Bethe Ansatz equations to theorigin in C2`. For this reason the terms in the thermal form factor series of the generatingfunctions may be interpreted as multiple residues. Under certain assumptions which arediscussed in detail in Appendix D of [22] the sum over all excitations then becomes a sumover multiple integrals. For the derivation of the final series it is important to start withmultiple integral representations of the summands at finite Trotter number and perform the

41

Trotter limit on the integrals. This is feasible due to the fact, that the Trotter number entersour expressions for the amplitudes and eigenvalue ratios only parametrically through thefunctions a±.

For the following consideration let us make the Trotter number dependence of thevarious functions explicit by providing an extra index N . We shall denote, in particular,

AN (U,V|k) = An(h, h′) det2`J∣∣∣Xn→UYn→V

(220)

the ‘off-shell’ finite Trotter number form factor density. In a similar way ρN (λ|U,V|k) arethe off-shell eigenvalue ratios at finite Trotter number. For the limits N →∞, ε→ 0 weintroduce the notation

limN→∞

limε→0

AN (U,V|k) = A(U,V|k) . (221)

Recall that B± are the curves in H± on which the particle and hole roots condense inthe low-T limit, i.e. for T very small and N large enough. Let us assume that these curvesare oriented in the direction of growing real part. We define two simple positively orientedcurves Γ(B±) which go around B± and enclose all particles and holes, respectively, if Tis small enough and N is large enough. Then

G(m, t, T, h, h′) = limN→∞

limε→0

AN (∅, ∅|0) + AN (∅, ∅|1)(−1)m

+∑`∈Nk=0,1

(−1)km

(`!)2

∫Γ(B+)`

d`u

(2πi)`

∫Γ(B−)`

d`v

(2πi)`ρN(− iγ

2 + itκN

∣∣U,V∣∣k)N/2ρN(− iγ

2 −itκN

∣∣U,V∣∣k)N/2AN (U,V|k) ρN

(− iγ

2

∣∣U,V∣∣k)m∏`j=1

(1 + a−(uj |U,V, h′)

)(1 + a+(vj |U,V, h′)

) . (222)

The reason why we had to come back to finite Trotter number here is that we neededto control the singularities of the function ρN

(− iγ

2 −itκN

∣∣U,V∣∣k)−N/2. As a functionof uj it has an N/2-fold pole at − iγ

2 −itκN which is compensated by a similar pole

in a−(uj |U,V, h′). As a function of vk is has an N/2-fold pole at iγ2 −

itκN which is

compensated by a similar pole in a+(vk|U,V, h′). If we would have taken the Trotter limitfirst, the poles stemming from the amplitude ratios would have developed into essentialsingularities, and we would not have been able to use the residue theorem in order totransform the sum into a sum over integrals.

In (222) the Trotter limit and the limit ε→ 0 can be taken, resulting in

G(m, t, T, h, h′) = A(∅, ∅|0) + A(∅, ∅|1)(−1)m

+∑`∈Nk=0,1

(−1)km

(`!)2

∫Γ(B+)`

d`u

(2πi)`

∫Γ(B−)`

d`v

(2πi)`A(U,V|k)

ρ(− iγ

2

∣∣U,V∣∣k)m exp

itκ∂λ ln ρ

(λ|U,V|k)

∣∣λ=−iγ/2

∏`j=1

(1 + a−(uj |U,V, h′)

)(1 + a+(vj |U,V, h′)

) . (223)

Finally we can also take the limit T → 0+. For the amplitudes we can then use (217) andfor the eigenvalue ratios the results of section 3.2. The integrals further simplify, since

42

the functions 1/(1 + a±) behave like Fermi functions for particles and holes. Using (56)and the fact that the dressed energy ε(λ|h) is negative for λ between B− and B+ andpositive for λ below B− or above B+ we see that 1/

(1 + a±(λ|U,V, h′)

)= 1 + O(T∞)

if λ ∈ B± + i0, whereas these functions are of order O(T∞) if λ ∈ B± − i0. Thus, theintegrals over Γ(B±) can be replaced by integrals over B±+i0 in the low-T limit. Becauseof the π-periodicity of the integrands, these contours can then be deformed into a ‘particlecontour’ Cp which lies above iγ

2 and a ‘hole contour’ Ch located above − iγ2 both of which,

to a certain extend, can be chosen to our convenience. For the sake of definiteness we fixthem to

Cp = [−π2 ,

π2 ] + iγ(1+0+)

2 , Ch = [−π2 ,

π2 ]− iγ(1+0−)

2 (224)

here. Hence, in the zero-temperature limit, all remaining dependence on the magneticfields h, h′ is in α = h−h′

2γT which has to be seen as an independent variable. For this reasonwe shall use the notation

G(0)(m, t, α) = limT→0+

G(m, t, T, h, h′) (225)

for the generating function in the zero-temperature limit. As a consequence of the abovediscussion the latter can be characterized by the following theorem.

Theorem 2. Generating function at zero temperature.The generating function of the longitudinal dynamical two-point function in the zero-temperature limit has the series representation

G(0)(m, t, α) =ϑ2

2(iγα/2)

ϑ22

+ϑ2

1(iγα/2)

ϑ22

(−1)m

+∑`∈Nk=0,1

(−1)km

(`!)2

∫C`h

d`u

(2πi)`

∫C`p

d`v

(2πi)`A(U,V|k) e−i

∑λ∈UV(mp(λ)+tε(λ|h)) , (226)

where

A(U,V|k) =ϑ2

2(Σ0(U,V))

ϑ22

[ ∏λ,µ∈UV

Ψ(λ− µ)

]× det

`Ω(uj , vk|U,V) det

`Ω(vj , uk|U,V) . (227)

Remark. Note that

det`Ω(uj , vk|U,V)

∣∣α=0

= det`Ω(vj , uk|U,V)

∣∣α=0

= 0 (228)

which follows from Lemma 19 below. Using (228) we conclude with (226) that

G(0)(m, t, 0) = 1 , (229)

in accordance with our expectation.

Lemma 19. Left null vectors of Ω and Ω at α = 0.Fix two sets U ⊂ D− =

z ∈ C

∣∣|Re z| ≤ π2 ,−

γ2 ≤ Im z ≤ 0

, V ⊂ D+ = D− + iγ

2 suchthat cardU = cardV = `. Let

f(λ) =∏

µ∈VUϑ1(λ− µ|q2) (230)

43

and, for all u ∈ U, v ∈ V,

tu =1

(1/f)′(u), tv =

1

f ′(v). (231)

Then ∑w∈U

twΩ(w, v|U,V)∣∣α=0

= 0 ,∑w∈V

twΩ(w, u|U,V)∣∣α=0

= 0 , (232)

for all u ∈ U, v ∈ V.

Proof. Using (46) and (47) we can recast the function f in the form

f(λ) = (−1)k e−2iΣ0−γα∏

w∈UVΓq4(

1− i(λ−w)2γ

)Γq4(

i(λ−w)2γ

). (233)

Then the definitions (187) of the functions φ(±) together with the first functional equation(43) of the q-gamma function imply that

f(λ)

φ(+)(λ)= (−1)k eiΣ0+γα

∏w∈VU

Γq4(

12 −

i(λ−w)2γ

)Γq4(− i(λ−w)

2γ

) , (234a)

f(λ+ iγ)

φ(+)(λ)= (−1)k eiΣ0+γα

∏w∈VU

Γq4(

1 + i(λ−w)2γ

)Γq4(

12 + i(λ−w)

2γ

) , (234b)

φ(−)(λ)

f(λ)= (−1)k eiΣ0+γα

∏w∈VU

Γq4(

i(λ−w)2γ

)Γq4(

12 + i(λ−w)

2γ

) , (234c)

φ(−)(λ)

f(λ− iγ)= (−1)k eiΣ0+γα

∏w∈VU

Γq4(

12 −

i(λ−w)2γ

)Γq4(

1− i(λ−w)2γ

) . (234d)

From these representations we can read off the analytic properties of the functions on theleft hand side which will become important below.

Further note that

f(λ+ π) g±Σ0(λ+ π, µ) = f(λ) g±Σ0(λ, µ) , (235a)

f(λ+ 2iγ) gΣ0(λ+ 2iγ, µ) = e2γα f(λ) gΣ0(λ, µ) , (235b)

g−Σ0(λ− 2iγ, µ)

f(λ− 2iγ)= e2γα g−Σ0(λ, µ)

f(λ). (235c)

Denote by L ⊂ C the rectangle with corners −π2 −

iγ2 , π2 −

iγ2 , π2 + 3iγ

2 , −π2 + 3iγ

2 andby ∂L its positively oriented boundary. Then, for µ ∈ D+,∫

∂L

dζ

2πif(ζ)gΣ0(ζ, µ) =

∑w∈U

twgΣ0(w, µ)− f(µ) + e−2iΣ0 f(µ+ iγ)

=(1− e2γα

) ∫ π2

−π2

dζ

2πif(ζ − iγ

2

)gΣ0

(ζ − iγ

2 , µ). (236)

44

Here we have used the residue theorem in the first equation and the quasi periodicity (235a),(235b) in the second equation. Using (236) and the fact that f(v) = 0 for all v ∈ V, wesee that∑

w∈UtwΩ(w, v) = − e−2iΣ0 f(v + iγ)− I(v)

+(1− e2γα

) ∫ π2

−π2

dζ

2πif(ζ − iγ

2

)Ω(ζ − iγ

2 , µ), (237)

where

I(v) = φ(−)(v)

∫ π2

−π2

dζ

2πi

f(ζ)

φ(+)(ζ)− e−2iΣ0 f(ζ + iγ)

φ(+)(ζ)

e(ζ − v) . (238)

The latter integral can be calculated by means of the residue theorem, since f(ζ)/φ(+)(ζ) isholomorphic and bounded in H+, whereas f(ζ+ iγ)/φ(+)(ζ) is holomorphic and boundedin H−. For this reason we can shift the integration contour to [−π/2, π/2] + i∞ for thecalculation of the first contribution to the integral and to [−π/2, π/2]− i∞ for the secondcontribution. Altogether we obtain

I(v) =φ(−)(v)f(v)

φ(+)(v)− φ(−)(v)f(v + iγ)

φ(+)(v + iγ)= − e−2iΣ0 f(v + iγ) . (239)

In the second equation we have used equation (209) and the fact that f(v) = 0 for allv ∈ V. Substituting this back into (237) we see that

∑w∈U

twΩ(w, v) =(1− e2γα

) ∫ π2

−π2

dζ

2πif(ζ − iγ

2

)Ω(ζ − iγ

2 , µ)

(240)

for all v ∈ V which implies the first equation (232).For the proof of the second equation we start by integrating the function g−Σ0(·, µ)/f(·)

with µ ∈ D− over the boundary of the rectangle ∂L with corners −π2 + iγ

2 , −π2 −

3iγ2 ,

π2 −

3iγ2 , π2 + iγ

2 and use similar arguments as above.

5.2 The series for the two-point function

It follows from (32) and (27) that⟨σz1σ

zm+1(t)

⟩= 1

2∂2γαD

2mG

(0)(m+ 1, t, α)∣∣α=0

. (241)

Using Lemma 19 we thus obtain the following form factor series for the two-point function.

Corollary 10. In the zero-temperature limit the longitudinal dynamical two-point functionhas the series representation

⟨σz1σ

zm+1(t)

⟩= (−1)m

ϑ′21ϑ2

2

+∑`∈Nk=0,1

(−1)km

(`!)2

∫C`h

d`u

(2π)`

∫C`p

d`v

(2π)`Azz(U,V|k) e−i

∑λ∈UV(mp(λ)+tε(λ|h)) , (242)

45

where

Azz(U,V|k) = (−1)(`−1)4 sin2

(kπ

2+∑

λ∈UV

p(λ)

2

)ϑ2

2(Σ0)

ϑ22

[ ∏λ,µ∈UV

Ψ(λ− µ)

]×(∂γα det

`Ω(uj , vk|U,V)

)(∂γα det

`Ω(vj , uk|U,V)

)∣∣∣α=0

. (243)

In the latter expression we take the α-derivatives explicitly and use the symmetry of theintegrand to simplify the series. As a result we obtain a formula in which the first columnsof the determinants of Ω and Ω are modified. Defining

Σ00 = Σ0

∣∣α=0

, gmodΣ0

0(λ, µ) = gΣ0

0(λ, µ)

ϑ′2(µ− λ− Σ00)

ϑ2(µ− λ− Σ00)

(244)

and, for j = 1, . . . , `,

Ωmod(uj , vk|U,V) =

−2igΣ0

0(uj , vk) + gmod

Σ00

(uj , vk)

−∫ π/2−π/2

dζ2πi

φ(−)(vk)

φ(+)(ζ)gmod

Σ00

(uj , ζ) e(ζ − vk) for k = 1,

Ω(uj , vk|U,V)α=0 for k = 2, . . . , `,

(245a)

Ωmod

(vj , uk|U,V) =

2ig−Σ0

0(vj , uk) + gmod

−Σ00(vj , uk)

−∫ π/2−π/2

dζ2πi

φ(−)(uk)

φ(+)(ζ)gmod−Σ0

0(vj , ζ) e(uk − ζ) for k = 1,

Ω(uj , vk|U,V)α=0 for k = 2, . . . , `

(245b)

we can formulate the following theorem.

Theorem 3. Longitudinal two-point functions at zero temperature.The series

⟨σz1σ

zm+1(t)

⟩= (−1)m

ϑ′21ϑ2

2

−∑`∈Nk=0,1

(−1)km+`

((`− 1)!)2

∫C`h

d`u

(2π)`

∫C`p

d`v

(2π)`e−i

∑λ∈UV(mp(λ)+tε(λ|h))

× sin2

(kπ

2+∑

λ∈UV

p(λ)

2

)ϑ2

2(Σ00)

ϑ22

[ ∏λ,µ∈UV

Ψ(λ− µ)

]

× det`Ωmod(uj , vk|U,V) det

`Ωmod

(vj , uk|U,V) (246)

represents the longitudinal dynamical two-point functions of the XXZ chain in the antifer-romagnetic massive regime at zero temperature.

5.3 On the explicit evaluation of det` Ω and det` Ω

In this section we shall present explicit expressions for det` Ω and det` Ω for ` = 1, 2.The matrix elements Ω(xj , yk|x, y) and Ω(yk, xj |x, y) with cardx =

cardy = ` are evaluated from (206). Reflecting the equi-distant patterns of poles

46

in the integrands, they assume very regular forms when written in terms of the basichypergeometric series [20],

rΦs

(a1, a2, · · · , arb1, b2, · · · , bs

; q, z

)=

∞∑k=0

(a1, · · · , ar; q)k(b1, · · · , bs, q; q)k

((−1)kq

k(k−1)2

)s+1−rzk. (247)

Here and hereafter we adopt the notation

(a; q)m =m−1∏k=0

(1−aqk) , (a1, a2, · · · , ak; q)m = (a1; q)m(a2; q)m · · · (ak; q)m (248)

for q-Pochhammer symbols.The resulting Ω(xj , yk|x, y) is given by a linear combination of 2`Φ2`−1(∗; q4, ζ),

where arguments ∗ depend on yj, xj and α. The argument ζ also depends on α.The determinants, det` Ω and det` Ω, are already proven to be null if α = 0 in

Section 5.1. One can check that the row vectors (and column vectors) of the matrixΩ(xj , yk|x, y)`j,k=1 are not simply proportional to each other. This implies theexistence of nontrivial identities among the `-products of 2`Φ2`−1 at α = 0. Even for` = 2, interesting identities, such as non-terminating q-Saalschutz formulae, naturallyappear. This might pose an interesting problem in the theory of basic hypergeometricseries.

We, however, need to go beyond α = 0, as the derivatives of the determinants det` Ωand det` Ω at α = 0 are of our concern. There are many equivalent expressions forΩ(xj , yk|x, y). This arbitrariness does not matter much for ` = 1, whereas one needsto take suitable linear combinations and to select appropriate forms, in order to proceedwith ` ≥ 2 and α 6= 0. Their derivation thus becomes slightly involved, and we decidedto present the details in a separate publication which will also include explicit numericalresults. Here we shall summarize the formulae for ` = 1, 2 and their relation to previousresults.

We will sometimes use exponentiated variables,

Pj = e2iyj , Hk = e2ixk .

For any function f(y1, · · · , x1 · · · , α) we write

f(y1, · · · , x1 · · · , α) = f(−y1, · · · ,−x1 · · · ,−α) . (249)

The same notation is applied to a function with exponentiated variables,

F (P1, · · · , H1, · · · , eαγ) = F

(1

P1, · · · , 1

H1, · · · , e−αγ

). (250)

By f(y1, y2, · · · ) we mean that f is symmetric w.r.t. y1 ↔ y2.We then introduce

µ(y, x, α) =ϑ′1(0|q)ϑ2(0|q)

2ϑ4(0|q2)ϑ1(∑

k(yk − xk) + iαγ|q2), (251)

where y and x mean that the function is symmetric with respect to permutations ofy and x.

47

We first state the result for ` = 1. Set further

o1(y1, x1) =1(

q2H1P1, q4 P1

H1; q4)∞

(252)

and

Φ`=11 (w1, w2, α) = 2Φ1

(q−2, Z1

Z2

q2Z1Z2

; q4, q4e−2αγ

), (253)

where Zj = e2iwj .

Lemma 20. Let ` = 1. For arbitrary α the function Ω(x1, y1|x, y) is explicitly givenby

Ω(x1, y1|x, y) = µ(y1, x1, α)ϑ1(iαγ|q2)

ϑ1(y1 − x1|q2)

(Φ`=11 (y1, x1, α)

− e−2iΣ0o1(y1, x1)

o1(y1, x1)Φ`=1

1 (y1, x1, α)

). (254)

This is clearly zero when α = 0. Thanks to the q-Gauss identity we have a simpleexpression for the first term in the small-α expansion,

Ω(x1, y1|x, y) = iαγ(1− Z−1/2(−1)k)

sin(w)

ϑ′1(0|q)ϑ1(w|q)

(q4, q2Z; q4)∞(q2, q4Z; q4)∞

+ O(α2)

=: iαγ ω(x1, y1) + O(α2) , (255)

where w = y1 − x1 and Z = e2iw. The expression for Ω(y1, x1|x, y) is simplyrelated,

Ω(y1, x1|x, y) = −Ω(x1, y1|x, y) . (256)

We shall comment on the consistency with the result from the vertex operator approach,which implies the form factor expansion in the spinon basis. For simplicity we considerthe formula in the static case,

〈σz1σzm+1〉 = (−1)m(q2; q2)4

(−q2; q2)4

+∑`∈Nk=0,1

(−1)km

(2`)!

∫ π2

−π2

d2`u

(2π)2`eim

∑2`j=1 p(uj) Fzz(ui2`i=1|k) , (257)

where Fzz(ui2`i=1|k) denotes the contribution from 2` spinons. For arbitrary `, thereexists a multiple-integral formula with complicated integration contours for this quantity.When ` = 1, it reduces to a simple formula due to Lashkevich [17, 53],

Fzz(u1, u2|k) =sin2 1

2

(p(u1) + p(u2) + kπ

)sin2(u12)ϑ2

3

(u12+πk

2 |q)

cos 12(u12 + iγ + kπ) cos 1

2(u12 − iγ + kπ)

× 32q(q2; q2)2∏σ=±

(q4; q4, q4)2

(q2; q4, q4)2

(q4e2iσu12 ; q4, q4)2

(q2e2iσu12 ; q4, q4)2

(q4e2iσu12 ; q4)

(q2e2iσu12 ; q4), (258)

48

where u12 = u1 − u2.One can easily show that this is consistent with the result obtained here if the following

(Conjecture 1 in [18]) is valid,

Fzz(u1, u2|k) = Azz(u1 − iγ, u2|k) + Azz(u2 − iγ, u1|k), (259)

for 0 < Imuj < γ, j = 1, 2 (recall that Azz was defined in (243)).This was verified numerically in [18], as there existed only an expression involving

Fredholm determinants for Azz . With the present result we have now an expression forAzz , involving ω in (255) instead, whose analytic properties are fully under control,

Azz(u1 − iγ, u2|k) = −4 sin2 12

(p(u2)− p(u1 − iγ) + kπ

)× ω(u1 − iγ, u2)2ϑ

22(−πk

2 + u21+iγ2 |q)

ϑ22(0|q)

Ψ2(0)

Ψ(u12 − iγ)Ψ(u21 + iγ). (260)

Thus, now is the right occasion to prove the validity of (259). We utilize the anti-periodicityof the dressed momentum, p(u− iγ) = −p(u), the explicit forms of Ψ and ω, and

Γ2q4(1

2)

Γq4(12 −

ix2γ )Γq4(1

2 + ix2γ )

=ϑ4(x|q2)

ϑ4(0|q2),

1

Γq4(1− ix2γ )Γq4(1 + ix

2γ )=

ϑ1(x|q2)

sin(x)ϑ′1(0|q2),

Gq4(1− ix2γ )Gq4(1 + ix

2γ )

Gq4(12 −

ix2γ )Gq4(1

2 + ix2γ )

= (1− q4)34 (q4; q4)

∏σ=±

(q4e2iσx; q4, q4)

(q2e2iσx; q4, q4). (261)

Then, after simple manipulations, one arrives at

Azz(u1 − iγ, u2|k) = 32q(q2; q2)2 sin2 12

(p(u1) + p(u2) + kπ

)× sin(u21) tg

(u21+iγ+kπ2

)ϑ2

3

(u12+πk

2 |q)

×∏σ=±

(q4; q4, q4)2

(q2; q4, q4)2

(q4e2iσu12 ; q4, q4)2

(q2e2iσu12 ; q4, q4)2

(q4e2iσu12 ; q4)

(q2e2iσu12 ; q4). (262)

By interchanging u1 ↔ u2 in the above and by summing up, one immediately verifies(259). We thus obtain,

Corollary 11. The 1-ph excitation brings the same contribution to the form factor expan-sion of the longitudinal correlation function as the 2-spinon excitation.

In order to present the result for the 2-ph excitations, we need a few more objects. Theanalogue of (252) is defined by

o2(y1, y2, x) =

(q2 P2

P1, q4 P1

P2; q4)∞(

q2H1P1, q2H2

P1, q4 P1

H1, q4 P1

H2; q4)∞. (263)

This time we need two kinds of basic hypergeometric series,

Φ1(y1, y2, x, α) = 4Φ3

(q−2, P1

H1, P1H2, q2 P1

P2

q2 P1H1, q2 P1

H2, P1P2

; q4, q4 e−2αγ

), (264)

49

Φ2(y1, y2, x, α) = 4Φ3

(q6, q4 P2

H1, q4 P2

H2, q2 P2

P1

q6 P2H1, q6 P2

H2, q8 P2

P1

; q4, q4 e−2αγ

). (265)

The first one is the ` = 2 counterpart of Φ`=11 . In addition, we set

r2(y1, y2, x) =q2(1− q2)2 P2

P1

(1− P2P1

)(1− q4 P2P1

)

∏k=1,2

(1− P2Hk

)

(1− q2 P2Hk

). (266)

We then introduce linear combinations of these objects,

Φ[1](y1, y2, x, α) =o2(y1, y2, x)o2(y1, y2, x)

Φ1(y1, y2, x, α)− e2iΣ0Φ1(y1, y2, x, α) ,

Φ[2](y1, y2, x, α) = e−2αγr2(y1, y2, x)o2(y2, y1, x)o2(y2, y1, x)

Φ2(y1, y2, x, α)

− e2iΣ0e2αγ r2(y1, y2, x)Φ2(y1, y2, x, α) . (267)

Note the similarly between the above Φ[1] and the content of the bracket in equation (254)for Ω. Finally, we introduce

δ0([y], [x], α) =ϑ1(y2 − x2 + iαγ|q2)

ϑ1(y2 − x2|q2)

ϑ1(y1 − x1 + iαγ|q2)

ϑ1(y1 − x1|q2)

− ϑ1(y2 − x1 + iαγ|q2)

ϑ1(y2 − x1|q2)

ϑ1(y1 − x2 + iαγ|q2)

ϑ1(y1 − x2|q2). (268)

By [y] we mean that δ0 is anti-symmetric in y1 and y2. Similarly for [x].We are now in position to write down the result of det2 Ω(xj , yk|x, y).

Lemma 21. Let ` = 2. Then, for arbitrary α, det2 Ω(xj , yk|x, y) is given by

det2 Ω(xj , yk|x, y) = δ0([y], [x], α)µ(y, x, α)2 e−4iΣ0

× det

∣∣∣∣∣Φ[1](y1, y2, x, α) Φ[2](y1, y2, x, α)

Φ[2](y2, y1, x, α) Φ[1](y2, y1, x, α)

∣∣∣∣∣ . (269)

We remark that this is one of the possible equivalent forms and that it has severaladvantages over others. First, the anti-symmetry in x and y, that originated fromthe definition, is reflected only in δ0 and the other parts are symmetric in x and y,respectively. Second, it is clear from (268) that δ0([y], [x], 0) = 0 and thus the determinantis explicitly shown to be zero if α = 0. Third, the first term in the expansion w.r.t. α canbe immediately found: except in δ0, one only has to set α = 0,

det2 Ω(xj , yk|x, y) = iαγ ln′[ϑ1(y2 − x2|q2)ϑ1(y1 − x1|q2)

ϑ1(y2 − x1|q2)ϑ1(y1 − x2|q2)

]µ(y, x, 0)2

× H1H2

P1P2det

∣∣∣∣∣Φ[1](y1, y2, x, 0) Φ[2](y1, y2, x, 0)

Φ[2](y2, y1, x, 0) Φ[1](y2, y1, x, 0)

∣∣∣∣∣+ O(α2) , (270)

where ln′ stands for the logarithmic derivative.

50

We find that det2 Ω(yk, xj |x, y) is also proportional to δ0 and thus we can safelyset α = 0 in the rest. In place of (267) we set

ΦT[1](x1, x2, y) =

o(x1, x2, y)o(x1, x2, y)

Φ1(x1, x2, y, 0)

+ (−1)k−1 ei∑k(yk−xk) Φ1(x1, x2, y, 0) ,

ΦT[2](x1, x2, y) = r2(x1, x2, y)

o(x2, x1, y)o(x2, x1, y)

Φ2(x1, x2, y, 0)

+ (−1)k−1ei∑k(yk−xk)r2(x1, x2, y)Φ2(x1, x2, y, 0) . (271)

Then the small-α expansion of det2 Ω(yk, xj |x, y) reads

det2 Ω(yj , xi|x, y) = iαγ ln′[ϑ1(y2 − x2|q2)ϑ1(y1 − x1|q2)

ϑ1(y2 − x1|q2)ϑ1(y1 − x2|q2)

]µ(y, x, 0)2

× H1H2

P1P2det

∣∣∣∣∣ΦT[1](x1, x2, y) ΦT

[2](x1, x2, y)ΦT

[2](x2, x1, y) ΦT[1](x2, x1, y)

∣∣∣∣∣+ O(α2) . (272)

In a subsequent paper, we will apply the above formulae to the numerical investigationof the dynamical correlation functions and exemplify their efficiency. The details of theirderivation will also be explained there.

Finally, we comment on our previous Azzold [18] involving the Fredholm determinants.

It might be interesting to compare it numerically against the present one (denoted by Azznew

for comparison), as the actual computation of Azzold required a discretized approximation.

We consider ` = 2, fix x1 = 1 − iγ2 , y1 = 1

2 + iγ2 and evaluate the relative difference,∣∣1 − Azznew

Azzold

∣∣, with x2 = u0 − iγ2 and y2 = v0 + iγ

2 for various u0 and v0. The Fredholmdeterminants are approximated by determinants of 50× 50 matrices, see Fig. 3.

Figure 3:∣∣1− Azznew

Azzold

∣∣ for q = 0.5, k = 0, ` = 2.

The maximum value of the relative difference is ∼ 3× 10−8. We thus conclude thatthey agree with rather nice precision and that the discretization scheme proposed in [6] isnumerically efficient in our case.

51

6 The isotropic limit

The isotropic point ∆ = 1, h = 0 of the ground state phase diagram of the XXZ chainis located at the boundary between the antiferromagnetic massive and massless regimes(see Figure 1). It can be accessed from the antiferromagnetic massive regime, e.g. by firstsending h→ 0 and then γ → 0. The first limit is trivial at zero temperature, since in thiscase the correlation functions are independent of the magnetic field. The isotropic limitrequires a rescaling of the integration variables u, v → γu, γv. Sending γ → 0 meanssending q → 1. In this limit the functions of the q-gamma family become functions ofthe ordinary gamma family, e.g. limq→1 Γq(u) = Γ(u) and limq→1Gq(u) = G(u). Forthose functions involving Jacobi theta functions the limit can be easily calculated afteremploying a special modular transformation,

ϑ1(λ|q) = −i

√π

γe−λ

2

γ ϑ1

(πiλγ

∣∣q′) , ϑ2(λ|q) =

√π

γe−λ

2

γ ϑ4

(πiλγ

∣∣q′) ,ϑ3(λ|q) =

√π

γe−λ

2

γ ϑ3

(πiλγ

∣∣q′) , ϑ4(λ|q) =

√π

γe−λ

2

γ ϑ2

(πiλγ

∣∣q′) , (273)

whereq′ = e

−π2

γ . (274)

We indicate the isotropic limit by putting a hat over the respective function, f(u) =limγ→0 f(γu). Then the momentum and dressed energy go to

p(u) =π

2− i ln

(ch(π2 (u+ i

2))

ch(π2 (u− i

2))) , ε(u|0) = − 2πJ

ch(πu). (275)

The function Ψ becomes

Ψ(u) = Γ(

12 −

iu2

)Γ(1− iu

2

)G2(1− iu

2

)G2(

12 −

iu2

) . (276)

Severe simplifications of the series in the isotropic limit arise from the theta-functionfactors. First of all, as is well known, the staggered polarization vanishes in this limit,ϑ′1/ϑ2 → 0. Moreover,

ϑ2(Σ0)

ϑ2

∣∣∣∣α=0

→

1 for k = 0

0 for k = 1,(277)

implying that the whole ‘staggered part’ of the series (246) vanishes for γ → 0, since theremaining factors for k = 1 have a finite limit. For this reason is enough to give an explicitdescription of these remaining factors only for k = 0. If we denote

Ω(u, v|U,V) = limγ→0

γ Ω(γu, γv|γU, γV)∣∣k=0, α=0

, (278a)

Ω(u, v|U,V) = limγ→0

γ Ω(γu, γv|γU, γV)∣∣k=0, α=0

, (278b)

then

Ω(u, v|U,V) = g(u, v)−∫ ∞−∞

dt

2πi

φ(−)(v)

φ(+)(t)g(u, t)e(t− v) , (279a)

52

Ω(v, u|U,V) = g(v, u)−∫ ∞−∞

dt

2πi

φ(−)(t)

φ(+)(u)g(v, t)e(u− t) , (279b)

where

g(u, v) =π

sh(π(v − u)

) , e(u) =1

u− 1

u− i, (280a)

φ(σ)(u) =∏

w∈UVΓ(

12 −

σi(u−w)2

)Γ(

1 + σi(u−w)2

), σ = ± . (280b)

For j = 1, . . . , ` we further define

Ωmod(uj , vk|U,V) =

∫∞−∞

dtπφ(−)(vk)

φ(+)(t)

g(uj ,t)vk−t for k = 1,

Ω(uj , vk|U,V) for k = 2, . . . , `,(281a)

Ωmod

(vj , uk|U,V) =

∫∞−∞

dtπ

φ(−)(t)

φ(+)(uk)

g(vj ,t)uk−t for k = 1,

Ω(vj , uk|U,V) for k = 2, . . . , `.(281b)

Inserting all of the above into equation (246) we have arrived at the following theorem.

Theorem 4. Two-point functions in the isotropic limit.The dynamical two-point correlation functions of the isotropic Heisenberg chain in thezero-temperature limit have the form factor series representation

⟨σz1σ

zm+1(t)

⟩=∑`∈N

(−1)`−1

((`− 1)!)2

∫C`h

d`u

(2π)`

∫C`p

d`v

(2π)`e−i

∑λ∈UV(mp(λ)+tε(λ|h))

× sin2

∑λ∈UV

p(λ)

2

[ ∏λ,µ∈UV

Ψ(λ− µ)

]

× det`Ωmod(uj , vk|U,V) det

`Ωmod

(vj , uk|U,V) , (282)

where Cp = R + i(1+0+)2 , Ch = R− i(1+0−)

2 .

7 Conclusions

We have obtained novel form factor series representations for the longitudinal two-pointcorrelation functions of the XXZ chain in the antiferromagnetic massive regime and of theXXX chain at vanishing magnetic field. These series take simple forms at T = 0. They areseries of multiple integrals of increasing even multiplicity, 2, 4, 6, . . . . In previous worksthe integrands in the general terms were of rather intricate forms. They consisted either ofsums of multiple contour integrals with complicated contours [35], of sums over multipleresidues [16], or, in the best case [18], of functions involving Fredholm determinants intheir definition. In the present work the integrands in the 2`-fold integrals are basicallyproducts of two order-` determinants with entries that can be represented as simple integralsover special functions from the q-gamma family.

53

Expressions for form factors not involving multiple integrals or Fredholm determinantswere known for a long time for massive integrable quantum field theories [61], where theyhad been obtained as solutions of functional equations [62]. They also appear in the contextof the Fermionic basis approach to the form factors of the Sine-Gordon model [37]. Thus,taken the close relationship between the XXZ and Sine-Gordon models, the existence of aform factor series for the correlation functions of the XXZ chain in the antiferromagneticmassive regime with form factors represented as finite determinants might have beenanticipated. We wish to stress, however, that we have used an eigenbasis of the quantumtransfer matrix rather than the Hamiltonian eigenbasis. For this reason our form factorseries comes with a different interpretation of the particle content of the model, whichis in terms of particles and holes rather than in terms of spinons. The latter are usuallyinterpreted as a lattice manifestation of the quantum solitons in the Sine-Gordon model.This may mean that our representation is still structurally different from everything wewould expect to obtain by analogy with the Sine-Gordon model.

We recently learned about a thesis [52] in which a representation of the form factoramplitudes involving only finite determinants was obtained in an eigenbasis of the Hamil-tonian of the XXX chain. It will be interesting to see, if that representation can be broughtto a more explicit form and how it then compares with our result.

Due to the relative simplicity of our novel expressions, answers to a number of long-standing questions appear now within reach and new questions can be asked.

(i) We believe that the novel expressions are more efficient for a numerical computationof the correlation functions.

(ii) We hope that we will be able to prove the convergence of the series employingmethods that were recently developed in the context of integrable massive quantumfield theories [51].

(iii) We hope that we can now attack the problem of calculating the long-time, large-distance asymptotics of two-point correlations in the XXX chain, which requires toestimate all terms in the series as they all contribute.

(iv) We think that the new series representation may make it possible to prove our olderconjecture [18] about the connection of the form factor amplitudes in the spinon andparticle-hole bases.

(v) As the elements of the matrices Ω and Ω can be entirely expressed in terms of basichypergeometric functions the question arises, if there are even more simple andexplicit expressions for the corresponding determinants in the general case. Theexploration of this question may take us deep into the theory of basic hypergeometricseries and may potentially yield new identities between basic hypergeometric seriesor new derivations of some of the known identities.

In future work we would also like to gain a better understanding of the generic finitetemperature case and of the high-T asymptotics. We would further like to explore thepossibility if similar series representations exist for the two-point functions in the masslessregime. This will require a better understanding of the spectrum and general Bethe rootpatterns of the excited states of the quantum transfer matrix of the XXZ chain.

54

Acknowledgments. We would like to thank Jesko Sirker for providing his DMRG datafor comparison and Ole Warnaar for helpful discussions about basic hypergeometric seriesidentities. CB and FG acknowledge financial support by the DFG in the framework of theresearch unit FOR 2316. The work of KKK is supported by the CNRS and by the ‘Projetinternational de cooperation scientifique No. PICS07877’: Fonctions de correlationsdynamiques dans la chaıne XXZ a temperature finie, Allemagne, 2018-2020. JS is gratefulfor support by a JSPS Grant-in-Aid for Scientific Research (C) No. 18K03452 and by aJSPS Grant-in-Aid for Scientific Research (B) No. 18H01141.

Note added. While the preparation of this manuscript had been delayed for unfore-seeable reasons, we found and proved a generalization to arbitrary ` of the results ofSection 5.3 on the explicit evaluation of the remaining finite determinants in terms of basichypergeometric series. In order to not further increase the length of the present manuscript,and also for its independent significance, we shall publish this additional result separately.

55

Appendix A The generating function for the longitudinal two-point function

Here we provide a derivation of equation (30) of the main text. Consider two eigenstates|l, h〉 and |n, h〉 of the dynamical quantum transfer matrix t(λ|h) which are non-degeneratefor l 6= n. Then

∂h′/2T 〈l, h|t(λ|h′)|n, h′〉∣∣h′=h

= 〈l, h|Σz(λ|h)|n, h〉+ Λl(λ|h)〈l, h|∂h/2T |n, h〉

= δl,n〈l, h|l, h〉∂h/2TΛn(λ|h) + Λn(λ|h)〈l, h|∂h/2T |n, h〉 , (A.1)

implying that

〈l, h|Σz(λ|h)|n, h〉 =

δl,n〈l, h|l, h〉∂h/2TΛl(λ|h) +(Λn(λ|h)− Λl(λ|h)

)〈l, h|∂h/2T |n, h〉 . (A.2)

Similarly,

〈n, h|Σz(λ|h)|l, h〉 =

δl,n〈l, h|l, h〉∂h/2TΛl(λ|h) +(Λn(λ|h)− Λl(λ|h)

)(∂h/2T 〈n, h|

)|l, h〉 . (A.3)

Setting l = 0, multiplying (A.2) and (A.3) and supplying the correct normalization weobtain

Azzn = δn,0

(∂h/2TΛ(−iγ/2|h)

Λ(−iγ/2|h)

)2

+〈h|(∂h/2T |n, h〉

)(∂h/2T 〈n, h|

)|h〉

〈h|h〉〈n, h|n, h〉

(Λn(−iγ/2|h)

Λ(−iγ/2|h)− 2 +

Λ(−iγ/2|h)

Λn(−iγ/2|h)

), (A.4)

Now let

%n(h′) =〈h|n, h′〉〈n, h′|h〉〈h|h〉〈n, h′|n, h′〉

, ςn(h′) =Λn(−iγ/2|h′)Λ(−iγ/2|h)

− 2 +Λ(−iγ/2|h)

Λn(−iγ/2|h′). (A.5)

It follows that

%0(h) = 1 , ς0(h) = ς ′0(h) = 0 , 4T 2ς ′′0 (h) = 2

(∂h/2TΛ(−iγ/2|h)

Λ(−iγ/2|h)

)2

, (A.6)

whereas for n 6= 0

%n(h) = %′n(h) = 0 , 4T 2%′′n(h) =2〈h|

(∂h/2T |n, h〉

)(∂h/2T 〈n, h|

)|h〉

〈h|h〉〈n, h|n, h〉, (A.7)

and ςn(h) 6= 0. Thus,2T 2∂2

h′%n(h′)ςn(h′)∣∣h′=h

= Azzn (h) (A.8)

as a consequence of (A.4), which is equivalent to (30).

56

Appendix B Determinant of g0

In this section we shall derive a product representation of the determinant

∆M (x,y) = detMg0(xj , yk) , (B.1)

where x = (x1, . . . , xM ), y = (y1, . . . , yM ). For later convenience we also introduce thenotation

xl = (x1, . . . , xl−1, xl+1, . . . , xM ) (B.2)

and similarly for ym.Due to the multi-linearity of the determinant, the function ∆M (x,y) inherits the pole

structure and quasi-periodicity from g0. In order to understand the latter recall that

ϑ1(x+ π) = −ϑ1(x) , ϑ1(x+ iγ) = −q−1 e−2ix ϑ1(x) , (B.3a)

ϑ2(x+ π) = −ϑ2(x) , ϑ2(x+ iγ) = q−1 e−2ix ϑ2(x) . (B.3b)

It follows that

g0(x+ π, y) = g0(x, y) , g0(x+ iγ, y) = −g0(x, y) . (B.4)

Thus,

∆M (x + πel,y) = ∆m(x,y) , ∆M (x + iγel,y) = −∆m(x,y) , (B.5)

where el is the lth canonical unit vector.We are looking for a generalization of the Cauchy-det formula. For this reason we

define

χM (x,y) =

∏1≤j<k≤M ϑ1(xj − xk)ϑ1(yk − yj)∏M

j,k=1 ϑ1(yj − xk). (B.6)

Then, by (B.3a),

χM (x + πel,y) = −χM (x,y) , (B.7a)

χM (x + iγel,y) = −q e2i∑Mj=1(xj−yj) χM (x,y) . (B.7b)

We note that ∆M is an elliptic function (with periods π, 2iγ) in every xl. As a functionof xl it therefore has as many poles as zeros per cell. The same is definitively not true forχM . χM cannot be elliptic as it has less zeros than poles per cell. Also

∆1(x, y) = g0(x, y) =ϑ′1ϑ2(y − x)

ϑ2ϑ1(y − x), χ1(x, y) =

1

ϑ1(y − x). (B.8)

This suggests to replace χM (x,y) by

ΦM (x,y) = ϑ2

(∑Mj=1(xj − yj)

)χM (x,y) , (B.9)

which for all l = 1, . . . ,M has the periodicity properties

ΦM (x + πel,y) = ΦM (x,y) , ΦM (x + iγel,y) = −ΦM (x,y) (B.10)

57

following from (B.3b) and (B.7).We infer that the function

FM (x,y) =∆M (x,y)

ΦM (x,y)(B.11)

is double periodic with periods π, iγ and meromorphic in every xl, l = 1, . . . ,M , hence anelliptic function of xl with periods π, iγ. FM as a function of xl has at most a single simplepole congruent to

∑Mj=1 yj −

∑Mj=1,j 6=l xj − π/2 per cell. But such an elliptic function

must be a constant, since the sum of the residua of an elliptic function at its poles in anycell is zero. It follows that FM (x,y) is independent of x. Repeating all the arguments fory instead of x, we see that FM (x,y) is independent of y as well, so it is a mere constant.

This constant can be calculated, e.g., by comparing the residua of ∆M and ΦM atxl = yk. This way we see that FM = (ϑ′1)M/ϑ2. Thus,

∆M (x,y) =ϑ2(Σ)

ϑ2

∏1≤j<k≤M

(ϑ1(xj − xk)/ϑ′1

)(ϑ1(yk − yj)/ϑ′1

)∏Mj,k=1

(ϑ1(yj − xk)/ϑ′1

) , (B.12)

where we have introduced the shorthand notation Σ =∑M

j=1(xj − yj).

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