N Kitanine
email: kitanine@u-cergy.fr

K K Kozlowski
email: karol.kozlowski@ens-lyon.fr

J M Maillet

N A Slavnov
email: nslavnov@mi.ras.ru

V Terras

On the thermodynamic limit of form factors in the massless XXZ Heisenberg chain

We consider the problem of computing form factors of the massless XXZ Heisenberg spin-1/2 chain in a magnetic field in the (thermodynamic) limit where the size M of the chain becomes large. For that purpose, we take the particular example of the matrix element of the operator σ z between the ground state and an excited state with one particle and one hole located at the opposite ends of the Fermi interval (umklapp-type term). We exhibit its power-law decrease in terms of the size of the chain M , and compute the corresponding exponent and amplitude. As a consequence, we show that this form factor is directly related to the amplitude of the leading oscillating term in the long-distance asymptotic expansion of the correlation function σ z 1 σ z m+1 .

Introduction

The main purpose of this article is to open a way for the study of the form factors (or in general of the matrix elements of local operators) of the massless XXZ Heisenberg chain in a magnetic field in the limit where the size of the chain becomes large. For that purpose we will consider a particular matrix element of the local spin operator σ z between the ground state and an excited state with one particle and one hole located at the opposite ends of the Fermi interval (umklapp-type term). Although the present article is devoted to this special case, it will be clear that the method we propose can be applied to more general cases as well.

As already mentioned in our recent work [1], this particular form factor is of direct interest for the computation of the asymptotic behavior of the σ z two-point correlation function. Let us briefly recall how it appears in this context. In [1] we explained how to compute, from first principles, the long-distance asymptotic behavior of some two-point correlation functions: the longitudinal spin-spin correlation function in the massless phase of the XXZ spin-1/2 Heisenberg chain and also the density-density correlation function of the quantum one-dimensional Bose gas. In particular, for the spin chain, we obtained that the correlation function of the third components of spin behaves at large distance m as

σ z 1 σ z m+1 -σ z 1 2 = - 2Z 2 π 2 m 2 + 2|F σ | 2 • cos(2mp F ) m 2Z 2 + corrections. (1.1)
In this formula p F stands for the Fermi momentum and Z represents the value of the dressed charge on the Fermi boundary, the precise definition of these quantities being postponed to Section 2.2. We announced in [1] that the coefficient F σ in (1.1) is related to the thermodynamic limit of the properly normalized aforementioned special form factor of the σ z operator. We prove this statement in this article.

The fact that the coefficient F σ should be related to this form factor is not really surprising. It is well known that the leading asymptotic behavior of the correlation functions is defined by low-energy excitations in the spectrum of the Hamiltonian. From this assumption, the form of the asymptotics (1.1) was predicted by the Luttinger liquid approach [START_REF] Luther | [END_REF]3,4,5,6,7,8] and the conformal field theory [9,11,10,12] together with the analysis of finite size corrections [13,14,15,16,17,18,19,20,21]. There the oscillating term in the asymptotic formula corresponds to the process of a particle jumping from one Fermi boundary to the other. One can expect therefore that the numerical coefficient in this term should be somehow related to the corresponding form factor.

However, the precise relationship between the coefficient F σ and the form factor under consideration has been missing up to now. The first reason is that the methods mentioned above hardly give any prediction for the exact value of the coefficient F σ : an expression for this amplitude was already given in [30,31], but only for the case of zero magnetic field, and its physical meaning was not clear. The second reason is the absence of sufficient information about the thermodynamic limit of form factors in massless models. The matter is that, in distinction of massive models in infinite volume for which form factors were successfully calculated in series of works (see e.g. [22,23,[START_REF] Jimbo | Algebraic analysis of solvable lattice models[END_REF][START_REF] Izergin | [END_REF]), the thermodynamic limit of the form factors in the massless case have a non-trivial behavior with respect to the size of the system. This phenomenon was observed, apparently for the first time, in [28] for the model of one dimensional bosons. Recently similar results were obtained in [29] for the free fermion limit of the XXZ spin chain. This non-trivial behavior of the form factors with respect to the size of the system makes very difficult their analysis directly in the infinite volume contrary to the case of massive models.

One of the main goal of the present article is to develop a method to resolve this problem. We start with determinant representations for form factors for the f inite Heisenberg chain [36]. Then, acting in the spirit of works [28,[START_REF] Izergin | [END_REF], we proceed to the thermodynamic limit of the specific form factor under consideration. This method allows us to obtain its leading (power-law) behavior in terms of the size of the chain and to calculate the corresponding finite amplitude. Remarkably, the exponent governing the power-law decrease of the norm squared of this (umklapp-type) form factor in terms of the size M of the chain is equal to the critical exponent 2Z 2 for the corresponding oscillating term in (1.1).

The article is organized as follows. In Section 2, we recall some well known results concerning the XXZ chain, its algebraic Bethe ansatz solution and its thermodynamic limit. The main result of the article is discussed in Section 3: we define more precisely the form factor that we investigate and we write explicitly its relation with the amplitude appearing in (1.1); we also recall the exact expression of this amplitude as computed in [1]. The next sections are devoted to the proof of this result. In Section 4 we remind determinant representations for this form factor in the finite chain that are suited for our analysis. It enables us, in Section 5, to compute its thermodynamic limit. Several technical results concerning this limit are given in the appendices.

The XXZ chain: general results

The Hamiltonian of the integrable spin-1/2 XXZ chain, with anisotropy parameter ∆ and in an external longitudinal magnetic field h, is given by

H = H (0) -hS z , (2.1)
where

H (0) = M m=1 σ x m σ x m+1 + σ y m σ y m+1 + ∆(σ z m σ z m+1 -1) , (2.2)
S z = 1 2 M m=1 σ z m , [H (0) , S z ] = 0. (2.3)
In this expression, as well as in (1.1), σ x,y,z m stand for the local spin operators (in the spin-1/2 representation where they are represented by Pauli matrices) acting non-trivially on the m th site of the chain. The quantum space of states is H = ⊗ M m=1 H m where H m ∼ C 2 is the local quantum space at site m. We assume periodic boundary conditions and, for simplicity, the length of the chain M is chosen to be even. Since the simultaneous reversal of all spins is equivalent to a change of sign of the magnetic field, it is enough to consider the case h ≥ 0.

In the thermodynamic limit (M → ∞) the model exhibits different regimes depending on the values of the anisotropy ∆ and magnetic field h [START_REF] Baxter | Exactly solved models in statistical mechanics[END_REF]. We focus on the massless regime (|∆| < 1) and assume that the magnetic field is below its critical value h c [START_REF] Korepin | Quantum Inverse Scattering Method and Correlation Functions[END_REF]. All along this article we use the parameterization ∆ = cos ζ, 0 < ζ < π.

The finite XXZ chain in the algebraic Bethe Ansatz framework

The central object of the algebraic Bethe Ansatz method is the quantum monodromy matrix. In the case of the XXZ chain, it is a 2 × 2 matrix,

T (λ) = A(λ) B(λ) C(λ) D(λ) , (2.4)
with operator-valued entries A, B, C and D acting on the quantum space of states H of the chain. These operators depend on a complex parameter λ and satisfy a set of quadratic commutation relations driven by the six-vertex R-matrix.

The Hamiltonian H (0) of the XXZ chain is given by a trace identity [START_REF] Faddeev | Recent advances in field theory and statistical mechanics[END_REF] involving the transfer matrix

T (λ) = tr T (λ) = A(λ) + D(λ).
(2.5)

The transfer matrices commute for arbitrary values of the spectral parameter, [T (λ), T (µ)] = 0, and their common eigenstates coincide with those of the Hamiltonian. The solution of the quantum inverse scattering problem enables us to express local spin operators in terms of the entries of the monodromy matrix [36,37]:

σ α k = T k-1 (-iζ/2) • tr T (-iζ/2) σ α • T -k (-iζ/2). (2.6)
In the l.h.s. σ α k denote a local spin operator at site k, whereas σ α appearing in the r.h.s. should be understood as a 2 × 2 Pauli matrix multiplying the 2 × 2 monodromy matrix.

It turns out to be convenient to introduce a slightly more general object [26], the twisted transfer matrix

T κ (λ) = A(λ) + κD(λ), (2.7)
depending on an additional complex parameter κ. For a fixed value of κ, these twisted transfer matrices also commute with each others, [T κ (λ), T κ (µ)] = 0, and hence possess a common eigenbasis. The latter goes to the one of the Hamiltonian (2.2) in the κ → 1 limit.

In the framework of the algebraic Bethe Ansatz, an arbitrary quantum state can be obtained from the states generated by a multiple action of B(λ) operators on the reference state | 0 with all spins up (respectively by a multiple action of C(λ) operators on the dual reference state 0 |). Consider the subspace H (M/2-N ) of the space of states H with a fixed number N of spins down. In this subspace, the eigenstates | ψ κ ({µ}) (respectively the dual eigenstates ψ κ ({µ}) |) of the twisted transfer matrix T κ (ν) can be constructed in the form

| ψ κ ({µ}) = N j=1 B(µ j )| 0 , ψ κ ({µ}) | = 0 | N j=1 C(µ j ) , (2.8)
where the parameters µ 1 , . . . , µ N satisfy the system of twisted Bethe equations

Y κ (µ j |{µ}) = 0, j = 1, . . . , N. (2.9)
Here, the function Y κ is defined as

Y κ (ν|{µ}) = a(ν) N k=1 sinh(µ k -ν -iζ) + κ d(ν) N k=1 sinh(µ k -ν + iζ), (2.10)
a(ν), d(ν) being the eigenvalues of the operators A(ν) and D(ν) on the reference state | 0 :

a(ν) = sinh M ν -iζ 2 , d(ν) = sinh M ν + iζ 2 .
(2.11)

The corresponding eigenvalue of T κ (ν) on | ψ κ ({µ}) (or on a dual eigenstate) is

τ κ (ν|{µ}) = a(ν) N k=1 sinh(µ k -ν -iζ) sinh(µ k -ν) + κ d(ν) N k=1 sinh(ν -µ k -iζ) sinh(ν -µ k ) . (2.12)
Note that all this construction is also valid at κ = 1, in which case we agree upon omitting the subscript κ in the corresponding quantities. Not all the solutions to the system (2.9) yield eigenstates of the operator T κ (ν). The detailed classification of the pertinent solutions is given in [START_REF] Tarasov | [END_REF] (see also [27]). For those solutions corresponding to the eigenstates (the so-called admissible solutions), the system of the twisted Bethe equations can be recast as

a(µ j ) d(µ j ) N k=1 sinh(µ k -µ j -iζ) sinh(µ k -µ j + iζ) = -κ, j = 1, . . . , N.
(2.13)

Thermodynamic limit

We outline here very briefly the thermodynamic limit of the model, more detailed analysis can be found in [START_REF] Yang | [END_REF][START_REF]Mathematical Physics in One Dimension[END_REF][START_REF] Korepin | Quantum Inverse Scattering Method and Correlation Functions[END_REF]. The ground state of the infinite chain with fixed magnetization σ z can be constructed as the M → ∞ limit of the finite chain ground state (2.8) at κ = 1. Hereby the limit of the ratio N/M is equal to an average density D whose value is related to the magnetization by σ z = 1 -2D.

In order to construct the ground state, one first takes the logarithm of the system (2.13) at κ = 1 and N and M fixed,

M p 0 (µ j ) - N k=1 ϑ(µ j -µ k ) = 2πn j , j = 1, . . . , N. (2.14)
Here -M/2 < n j ≤ M/2, with n j ∈ Z for N odd and n j ∈ Z + 1/2 for N even (recall that M is assumed even). The functions p 0 (λ) and ϑ(λ),

p 0 (λ) = i log sinh( iζ 2 + λ) sinh( iζ 2 -λ) , ϑ(λ) = i log sinh(iζ + λ) sinh(iζ -λ) , (2.15)
are respectively called bare momentum and bare phase (we choose the principal branch of the logarithm). Among all the eigenstates obtained through (2.14), we define the N -particle ground state to be the state with the lowest energy in the sector with N spins down. It is characterized by the (half-)integers n j = I j ≡ j -(N + 1)/2 [START_REF] Yang | [END_REF], and the corresponding roots λ j are real numbers belonging to some interval [-q, q] called the Fermi zone. At the next step the length of the chain M as well as the number of spins down N are sent to infinity at fixed limiting value of N/M . In this limit the Bethe roots λ j corresponding to the N -particle ground state condensate, λ j+1 -λ j = O(M -1 ) (see e.g. [START_REF]Mathematical Physics in One Dimension[END_REF]), and the system (2.14) turns into a linear integral equation for the spectral density ρ(λ) 1 .

A standard way to derive this integral equation is to introduce a ground state counting function ξ(λ). Namely, let λ( ξ) satisfy the functional equation

p 0 λ( ξ) - 1 M N k=1 ϑ λ( ξ) -λ k = 2π ξ - N + 1 2M , (2.16)
for a given value of ξ and with parameters λ k corresponding to the N -particle ground state. Evidently, ξ(λ) is the counting function, since λ( j M ) = λ j and ξ(λ j ) = j/M due to equation (2.14). The (finite-size) density of the N -particle ground state is defined by ρ(λ) = d ξ/dλ. From (2.16), it is equal to

2π ρ(λ) = p ′ 0 (λ) - 1 M N k=1 K(λ -λ k ), (2.17)
with

K(λ) = ϑ ′ (λ) = sin 2ζ sinh(λ + iζ) sinh(λ -iζ)
.

(2.18)

In the thermodynamic limit, the N -particle density ρ(λ) goes to the spectral density ρ(λ) of the infinite chain with support on a finite interval [-q, q]. In this limit, discrete sums of piecewise continuous functions can be replaced by integrals via the Euler-Maclaurin summation formula lim

N,M →∞ 1 M N k=1 f (λ k ) = lim N,M →∞ 1 M N k=1 f ξ-1 ( k M ) = q -q f (λ) ρ(λ) dλ. (2.19)
Thus, replacing the sum over k by an integral in (2.17), we obtain the integral equation for the density of the infinite chain,

ρ(λ) + 1 2π q -q K(λ -µ) ρ(µ) dµ = 1 2π p ′ 0 (λ).
(2.20)

The average density D = lim N,M →∞ N/M is then given by

D = lim N,M →∞ 1 M N k=1 1 = q -q ρ(λ) dλ, (2.21)
which defines the integration boundary q in (2.20).

Observe that the definition of the discrete density ρ(λ) implies that ρ(λ) can also be defined as the limiting value

ρ(λ j ) = lim N,M →∞ ξ(λ j+1 ) -ξ(λ j ) λ j+1 -λ j = lim N,M →∞ 1 M (λ j+1 -λ j ) . ( 2

.22)

The method described above allows one to construct the ground state of the infinite XXZ chain at fixed magnetization. For the XXZ chain in an external magnetic field, the magnetization of the ground state is not an independent variable but depends on the magnetic field h. Then the boundary of the Fermi zone q in the equation (2.20) is defined from the condition ε(q) = 0 (instead of from (2.21)), where ε(λ) is the dressed energy satisfying an integral equation similar to (2.20):

ε(λ) + 1 2π q -q K(λ -µ) ε(µ) dµ = h -2p ′ 0 (λ) sin ζ . (2.23)
The condition ε(q) = 0 provides the positiveness of the energy of any excited state. We see from (2.23) that the Fermi boundary q depends on the anisotropy parameter ∆ and on the magnetic field h, i.e. that q is a function of the parameters of the Hamiltonian (2.1). We remind that q remains finite for non-zero magnetic fields, while it tends to infinity when h → 0. Throughout the present article, q will be kept finite in all the computations. Let us finally recall the definitions of the dressed momentum and charge, which are two important functions characterizing the ground state. The dressed momentum is closely related to the density as

p(λ) = 2π λ 0 ρ(µ) dµ.
(2.24)

Obviously, the thermodynamic limit ξ(λ) of the counting function ξ(λ) can be expressed in terms of the dressed momentum according to ξ(λ) = [p(λ) + p(q)]/2π. The dressed charge satisfies the integral equation

Z(λ) + 1 2π q -q K(λ -µ) Z(µ) dµ = 1, (2.25)
and can be interpreted in the XXZ model as the intrinsic magnetic moment of the elementary excitations [START_REF] Korepin | Quantum Inverse Scattering Method and Correlation Functions[END_REF]. The quantities Z = Z(q) and p F = p(q) = πD (Fermi momentum) enter the asymptotic formula (1.1). The excitations above the ground state can be constructed by adjoining particles and holes to the Dirac sea [41,42,43,44,45,[START_REF] Yang | [END_REF]46]. In other words, an excited state above the ground state corresponds to the replacement of a finite number of the (half-)integers I j = j -(N + 1)/2 parameterizing the ground state by some different ones, not belonging to the original sequence. In general, this induces a shift of the Bethe parameters {µ} describing this excited state with respect to the ground state Bethe roots {λ}, and possible appearance of complex roots.

Statement of the result

We are now in position to be more precise about the result we prove in this article.

We study a particular form factor of the σ z operator, involving a very specific excited state, namely the one with one particle and one hole located on the opposite ends of the Fermi zone. For definiteness, we consider the case where the spectral parameters of the particle and hole are respectively equal to ±q. This means that the (half-)integers which characterize this state through (2.14), that we will denote I ′ j (j = 1, . . . , N ), coincide with the ground state ones (I ′ j = I j ) for j = 2, . . . , N , but differ at j = 1:

I ′ 1 = N + 1 -(N + 1)/2.
There exists another way of describing the same eigenstate: by shifting the ground state (half-)integers by 1, i.e. by setting I ′ j = j + 1 -(N + 1)/2. In the framework of this approach, we can consider the aforementioned excited state as a limit of a twisted ground state. Indeed, similarly to (2.14) different eigenstates of the twisted transfer matrix are characterized by sets of (half-)integers ñj in the logarithmic form of the twisted Bethe equations (2.13),

M p 0 (µ j ) - N k=1 ϑ(µ j -µ k ) = 2πñ j -iβ, j = 1, . . . , N, (3.1)
where κ = e β . In the case ñj = I j , we call the corresponding eigenstate the twisted ground state.

In that case and at β = 2πi, the equations (3.1) reduce to (2.14) with n j = I ′ j = j+1-(N +1)/2, and, thus, the twisted ground state becomes the excited state described above.

Our result is the following :

In the thermodynamic limit, the normalized form factor of the σ z operator between the ground state | ψ({λ}) and the excited state ψ({µ}) | defined above is related to the amplitude |F σ | 2 (which is a finite number) appearing in the asymptotic formula (1.1) for the longitudinal spin-spin correlation function as

lim N,M →∞ M 2π 2Z 2 | ψ({µ})|σ z |ψ({λ}) | 2 ψ({µ}) 2 • ψ({λ}) 2 = |F σ | 2 . (3.2)
In other words, it means that the modulus squared of the corresponding form factor behaves as

|F σ | 2 M 2π -2Z 2
for M large. It is remarkable that the exponent 2Z 2 governing the modulus squared of the form factor power-law decrease in terms of the size M of the chain is exactly equal to the exponent for the power-law behavior of the corresponding oscillating term in the correlation function (1.1), but there in terms of the distance m separating the two spin operators. This result will be proven throughout Sections 4 and 5. There, the above special form factor will be computed and its thermodynamic limit will be taken.

In order to complete this statement, let us recall the explicit value of the amplitude |F σ | 2 obtained in [1]:

|F σ | 2 = 2G(2, Z) sin p F πZ 2 • 2π ρ(q) sinh(2q) -2Z 2 • e C 1 -C 0 • A(β) β=2πi . (3.3)
Here the notation G(a, x) represents the product of Barnes functions G(a, x) = G(a+x)G(a-x) (cf A.2). The constants C 0 and C 1 are defined in terms of the dressed charge Z:

C 0 = q -q Z(λ) Z(µ) sinh 2 (λ -µ -iζ) dλ dµ , (3.4)
C 1 = 1 2 q -q Z ′ (λ) Z(µ) -Z(λ) Z ′ (µ) tanh(λ -µ) dλ dµ + 2Z q -q Z -Z(λ) tanh(q -λ) dλ . (3.5)
Finally, the coefficient A(β) is given in terms of a ratio of Fredholm determinants:

A(β) = e β 2 [z(q-iζ)-z(-q-iζ)] det I + 1 2πi U (λ) -q (w, w ′ ) det I + 1 2π K 2 . (3.6)
In this expression, the integral operator I + K/2π occurring in the denominator acts on the interval [-q, q] with kernel (2.18), whereas the integral operator appearing in the numerator acts on a contour surrounding [-q, q] with kernel

U (λ) -q (w, w ′ ) = - e β z(w) [K κ (w -w ′ ) -K κ (-q -w ′ )] e β z(w+iζ) -e β+β z(w-iζ) , (3.7)
where

K κ (λ) = coth(λ + iζ) -κ coth(λ -iζ), κ = e β . (3.8)
The function z(w) appearing (3.6), (3.7), is the iπ-periodic Cauchy transform of the dressed charge Z,

z(w) = 1 2πi q -q coth(λ -w) Z(λ) dλ. (3.9)
Observe that it has a cut on the interval [-q, q] and satisfies the following properties:

z(w + iζ) -z(w -iζ) = 1 -Z(w) , (3.10
)

z+ (w) -z-(w) = Z(w) , w ∈ [-q, q] . (3.11)
The equation (3.10) follows from (2.25). In (3.11), z± stand for the limiting values of z when R is approached from the upper and lower half planes.

Remark 3.1. Equation (3.3) contains the limiting value of A(β) at β = 2πi. We did not set β = 2πi directly in (3.6)-(3.8), as it may cause problems at some particular values of ∆. For example, at ∆ = 0 (ζ = π 2 , Z(λ) = 1), the kernel (3.7) becomes ill-defined when β = 2πi. It should therefore be understood in the sense of the limit β → 2iπ, the latter being well defined.

Special form factor for the finite chain

The solution of the quantum inverse scattering problem (2.6) and determinant representations for scalar products [38,36] give a possibility to express, for the finite chain, arbitrary form factors of local spin operators as finite-size determinants [36]. We explain here how to obtain such a representation for the special form factor defined above. Then, following the ideas of [START_REF] Izergin | [END_REF]1], we transform the obtained determinant to a new one, more convenient for the calculation of the thermodynamic limit.

From now on, we will consider eigenstates of the twisted transfer matrix only for κ belonging to the unit circle, i.e. κ = e β , where β is pure imaginary. This restriction is not crucial, but convenient. The matter is that, due to the involution A † ( λ) = D(λ) ( † means Hermitian conjugation), the operator e -β/2 T κ (λ) becomes self-adjoint for |κ| = 1 and λ ∈ R. In such a case, the roots of the twisted Bethe equations (3.1) are real or contain complex conjugated pairs. In its turn, the involution B † ( λ) = -C(λ) guarantees the duality of eigenstates ψ κ ({µ})| = (-1) N |ψ κ ({µ}) † .

Special form factor and scalar product

Consider the following operator Q m , giving the number of spins down in the first m sites of the chain:

Q m = 1 2 m n=1 (1 -σ z n ) . (4.1)
It is easy to see that

e βQm = m n=1 1 + κ 2 + 1 -κ 2 • σ z n , κ = e β , (4.2)
and we have

2D m ∂ ∂β e βQm β=2πin = 1 -σ z m+1 , n ∈ Z, (4.3)
where the symbol D m stands for the lattice derivative: D m f (m) = f (m + 1) -f (m). Also, note that e βQm is a polynomial in κ which becomes the identity operator whenever β = 2πin, n ∈ Z.

The operator (4.2) admits, through the solution of the inverse problem (2.6), a simple representation in terms of the twisted and standard transfer matrices:

e βQm = T m κ -iζ 2 T -m -iζ 2 . (4.4)
We use this representation and consider the following matrix element of e βQm :

ψ κ ({µ})|e βQm |ψ({λ}) = ψ κ ({µ})|T m κ -iζ 2 T -m -iζ 2 |ψ({λ}) , (4.5)
where |ψ({λ}) is the ground state in the N -particle sector, and ψ κ ({µ})| is an eigenstate of the twisted transfer matrix T κ . Then

ψ κ ({µ})|e βQm |ψ({λ}) = τ κ ( -iζ 2 |{µ}) τ ( -iζ 2 |{λ}) m ψ κ ({µ})|ψ({λ}) . (4.6)
Setting ν = -iζ 2 in (2.12), we obtain

ψ κ ({µ})|e βQm |ψ({λ}) = e im P N j=1 [p 0 (µ j )-p 0 (λ j )] ψ κ ({µ})|ψ({λ}) . (4.7)
Let us now differentiate (4.7) with respect to β at β = 2πi. Hereby, at β = 2πi, the state ψ κ ({µ})| becomes an eigenstate of the standard transfer matrix. As it does not coincide with the ground state, it is orthogonal to the last one, and thus,

ψ({µ})|Q m |ψ({λ}) = e im P N j=1 [p 0 (µ j )-p 0 (λ j )] -1 ∂ ∂β ψ κ ({µ})|ψ({λ}) β=2πi . (4.8)
Using also (4.3) we find that the special form factor we want to compute can be obtained in terms of the normalized scalar product between the ground state | ψ({λ}) and the twisted ground state ψ κ ({µ}) | given by (3.1) with ñj = I j :

| ψ({µ})|σ z m+1 |ψ({λ}) | 2 ψ({µ}) 2 • ψ({λ}) 2 = -8 sin 2 P ex 2 • ∂ 2 ∂β 2 S (κ) N ({µ}, {λ}) β=2πi , (4.9)
with

S (κ) N ({µ}, {λ}) = ψ κ ({µ})|ψ({λ}) ψ κ ({µ}) • ψ({λ}) 2 , (4.10)
and where P ex = N j=1 [p 0 (µ j ) -p 0 (λ j )] is the total momentum of the excitation. Deriving (4.9) we have used that ψ κ ({µ})|ψ({λ}) is real at |κ| = 1.

In (4.10), the parameters µ j satisfy the system of twisted Bethe equations (3.1) with ñj = I j and, hence, are functions of β. In particular, they are taken to be

µ j (β) β=0 = λ j , j = 1, . . . , N. (4.11)
Then, in the l.h.s. of (4.9), the set {µ j (β = 2πi)} describes the excited state considered in Section 3 with a particle at q and a hole at -q.

Determinant representation

Thus, in order to study our special form factor, we can use the determinant representations for the scalar product between an eigenstate of the twisted transfer matrix with any arbitrary state of the form (2.8). Let us recall these representations.

Proposition 4.1. [38,36] Let µ 1 , . . . , µ N satisfy the system (2.9) and λ 1 , . . . , λ N be generic complex numbers. Then

0| N j=1 C(λ j )|ψ κ ({µ}) = ψ κ ({µ})| N j=1 B(λ j )|0 = N a=1 d(µ a ) N a>b sinh(µ a -µ b ) sinh(λ b -λ a )
• det N Ω κ ({µ}, {λ}|{µ}), (4.12)

where the N × N matrix Ω κ ({µ}, {λ}|{µ}) is defined as

(Ω κ ) jk ({µ}, {λ}|{µ}) = a(λ k ) t(µ j , λ k ) N a=1 sinh(µ a -λ k -iζ) -κ d(λ k ) t(λ k , µ j ) N a=1 sinh(µ a -λ k + iζ), (4.13) with t(µ, λ) = -i sin ζ sinh(µ -λ) sinh(µ -λ -iζ) . (4.14)
Note that, due to the Bethe equations (2.9), the entries of the matrix Ω κ ({µ} , {λ} | {µ}) are not singular at λ k = µ j . In particular, if λ j = µ j for all j = 1, . . . , N , then we obtain the square of the norm of the twisted eigenstate ψ κ ({µ})|ψ κ ({µ}) (recall that |κ| = 1), which reduces to

ψ κ ({µ})|ψ κ ({µ}) = (-1) N N j=1 2πiM ρκ (µ j ) a(µ j ) d(µ j ) N a,b=1 sinh(µ a -µ b -iζ) N a,b=1 a =b sinh(µ a -µ b ) × det N Θ (µ) jk , (4.15) with Θ (µ) jk = δ jk + K(µ j -µ k ) 2πM ρκ (µ k ) , 2πM ρκ (µ) = -i log ′ a(µ) d(µ) - N a=1 K(µ -µ a ). (4.16)
In the case of interest (scalar product of the untwisted ground state | ψ({λ}) with the κtwisted one ψ κ ({µ}) |), one has two different representations for the scalar product. On the one hand, it is possible to apply equation (4.12) when | ψ({λ}) in understood as an arbitrary state. On the other hand, one can consider | ψ({λ}) as an eigenstate (for κ = 1), interpret ψ κ ({µ}) | as an arbitrary state, and apply formula (4.12) with λ ↔ µ. Thus, one can obtain two different, but equivalent representations for the scalar product ψ κ ({µ}) |ψ({λ}) . We do not give here the explicit formulae (which can be easily derived from (4.12)-(4.14)) as these representations are not convenient for the calculation of thermodynamic limits. In order to obtain new formulae that are appropriate for such goal, one can extract the products of sinh -1 (µ k -λ j ) from the determinant of the matrix Ω κ (see [1,28,[START_REF] Izergin | [END_REF]). As a result, one obtains two new representations containing Fredholm determinants of integral operators of the form I + 1 2πi U (λ,µ) θ (w, w ′ ), with kernels

U (λ) θ (w, w ′ ) = - N a=1 sinh(w -µ a ) sinh(w -λ a ) • K κ (w -w ′ ) -K κ (θ -w ′ ) V -1 + (w) -κV -1 -(w) , (4.17)
U (µ) θ (w, w ′ ) = N a=1 sinh(w ′ -λ a ) sinh(w ′ -µ a ) • K κ (w -w ′ ) -K κ (w -θ) V -(w ′ ) -κV + (w ′ ) . (4.18)
Here θ is an arbitrary complex number, K κ is given by (3.8), and

V ± (w) ≡ V ± w | {λ} N {µ} N = N a=1 sinh(w -λ a ± iζ) sinh(w -µ a ± iζ) . (4.19)
These integral operators act on a contour surrounding the points {λ} (resp.{µ}) but no any other singularity of U (λ,µ) θ (w, w ′ ). More precisely, one has the following proposition (see Appendix A of [1] for the proof): Proposition 4.2. [1] Let λ 1 , . . . , λ N satisfy the system of untwisted Bethe equations and µ 1 , . . . , µ N satisfy the system of κ-twisted Bethe equations (2.9). Then

ψ({λ})|ψ κ ({µ}) = N a,b=1 sinh(µ a -λ b -iζ) sinh(λ a -µ b ) • N j=1 d(µ j ) d(λ j ) κ V + (λ j ) V -(λ j ) -1 × 1 -κ V -1 + (θ) -κV -1 -(θ) • det I + 1 2πi U (λ) θ (w, w ′ ) , (4.20)
and

ψ({λ})|ψ κ ({µ}) = N a,b=1 sinh(λ a -µ b -iζ) sinh(µ a -λ b ) • N j=1 d(µ j ) d(λ j ) V -(µ j ) V + (µ j ) -κ × 1 -κ V -(θ) -κV + (θ) • det I + 1 2πi U (µ) θ (w, w ′ ) . (4.21)
We would like to emphasize that equations (4.20) and (4.21) describe scalar products for the finite XXZ chain even though they contain Fredholm determinants of integral operators. In fact, it is not difficult to see that the integral operators U (λ,µ) θ are of finite rank and hence the Fredholm determinants can be reduced to usual determinants of N × N matrices [1].

Remark 4.1. We gave here two different representations for the same quantity, because we will use both of them. Namely, we will substitute (4.20) for one scalar product ψ κ ({µ})|ψ({λ}) in (4.10) and (4.21) for the other one, despite the evident fact that these scalar products are the same. This way seems to be slightly strange, but it leads directly to equation (3.3). Due to the equivalence of the representations (4.20) and (4.21) one can use, of course, only one of them, coming eventually to the same result. The last way, however, requires some additional identities, since the equivalence of the representations (4.20) and (4.21) is non-trivial.

It remains to apply all these results to our special form factor, which was expressed in terms of the normalized scalar product S (κ) N through (4.9). Substituting (4.20), (4.21), (4.15) in (4.10), and using the Bethe equations (2.13), we obtain a representation for S (κ) N as a product of three factors:

S (κ) N ({µ}, {λ}) = A (κ) N • D (κ) N • exp C (κ) 0,N , (4.22)
with

D (κ) N = det N 1 sinh(µ j -λ k ) 2 • N j=1 κV + (λ j ) V -(λ j ) -1 V -(µ j ) κV + (µ j ) -1 4π 2 M 2 ρ(λ j ) ρκ (µ j ) , (4.23)
C (κ) 0,N = N a,b=1 log sinh(λ a -µ b -iζ) sinh(µ b -λ a -iζ) sinh(λ a -λ b -iζ) sinh(µ a -µ b -iζ) , (4.24)
and

A (κ) N = (κ -1) 2 det I + 1 2πi U (λ) θ 1 (w, w ′ ) det I + 1 2πi U (µ) θ 2 (w, w ′ ) V -1 + (θ 1 ) -κV -1 -(θ 1 ) V -(θ 2 ) -κV + (θ 2 ) • det N Θ (λ) jk • det N Θ (µ) jk . (4.25)
Recall that the representations (4.20), (4.21) contain an arbitrary complex parameter θ. Here we have used two different parameters θ 1 and θ 2 for each of the scalar products. In the following, we shall set θ 1 = -q and θ 2 = q.

5 Thermodynamic limit of the special form factor

We compute now the large size behavior (N, M → ∞, N/M → D) of the special form factor represented in (4.9), (4.22). We need first to characterize how the parameters µ j of the κ-twisted ground state Bethe Ansatz equations behave in this limit. More precisely, how they are shifted with respect to the parameters λ j of the untwisted ground state. This will enable us to take the thermodynamic limit in the expression (4.22) for the normalized scalar product. Whereas it is quite straightforward to evaluate the limit of the factors A

Twisted ground state and shift functions

In order to study the κ-twisted ground state | ψ κ ({µ}) in the thermodynamic limit, it is convenient to introduce, similarly as for the untwisted case (see (2.16)), a twisted counting function ξκ (µ) satisfying ξκ (µ j ) = j/M . Its explicit expression is given by ξκ

(ω) = 1 2π p 0 (ω) - 1 2πM N k=1 ϑ(ω -µ k ) + N + 1 2M + iβ 2πM , (5.1)
with p 0 and ϑ given by (2.15). The corresponding discrete N -particle density ρκ (µ) = d ξκ dµ coincides with the function defined in (4.16). It also coincides, at β = 0, with the N -particle density of the ground state ρ(λ). It is easy to check that both functions ρκ (λ) (for κ = 1) and ρ(λ) go to the same function ρ(λ) (2.22) in the thermodynamic limit.

Using the counting functions for the ground state (2.16) and twisted ground state (5.1), one can define shift functions describing the displacement of the spectral parameters {µ} with respect to ground state ones {λ}. One can actually introduce two shift functions:

F (λ j ) = M ξ(µ j ) -ξ(λ j ) , Fκ (µ j ) = M ξκ (µ j ) -ξκ (λ j ) . (5.2)
Both have the same thermodynamic limit which we, from now on, denote by F (λ). One can easily verify that

F (λ j ) = lim N,M →∞ µ j -λ j λ j+1 -λ j . (5.
3)

The subtraction of the equation (2.14) with n j = I j (Bethe equation for the ground state) from (3.1) (Bethe equation for the twisted ground state, corresponding to the same number ñj = I j ) and a replacement of finite differences by derivatives and of sums by integrals as in (2.19), leads to the integral equation for the shift function F :

F (λ) + 1 2π q -q K(λ -µ)F (µ) dµ = β 2πi . (5.4)
Comparing this equation with the one for the dressed charge (2.25), one obtains

F (λ) = βZ(λ) 2πi .
(5.5)

Thence, for M large enough, the spectral parameters µ j are shifted with respect to the ground state ones λ j by a quantity ǫ j = µ j -λ j which can be computed for large M ǫ j = βZ(λ j ) 2πiM ρ(λ j )

+ O(M -2 ), j = 1, . . . , N.

(5.6)

Thermodynamic limit of

A (κ) N and C (κ) 0,N
We first compute the limits of C N . These limits are quite simple to calculate since it is enough to take into account only the leading order of the shift (5.6).

We have already seen that, in the thermodynamic limit N, M → ∞, discreet sums over the parameters λ j can be replaced by integrals via (2.19). It is also easy to see that lim

N,M →∞ N j=1 1 + 1 M f (λ j ) = exp    q -q f (λ) ρ(λ) dλ    , (5.7)
lim

N,M →∞ det N δ jk + 1 M V (λ j , λ k ) = det[I + V ρ] , (5.8)
provided f (λ) and V (λ, µ) are piecewise continuous on [-q, q]. In (5.8), det[I + V ρ] refers to the Fredholm determinant of the integral operator I + V ρ acting on the interval [-q, q] with kernel V (λ, µ)ρ(µ).

The thermodynamic limit of C (κ) 0,N is straightforward to obtain using (5.6):

lim

N,M →∞ C (κ) 0,N = lim N,M →∞ N j,k=1 log sinh(λ j -λ k -ǫ k -iζ) sinh(λ j -λ k + ǫ j -iζ) sinh(λ j -λ k -iζ) sinh(λ j -λ k + ǫ j -ǫ k -iζ) = -lim N,M →∞ N j,k=1 ǫ j ǫ k ∂ λ j ∂ λ k log sinh(λ j -λ k -iζ) = β 2 4π 2 C 0 , (5.9)
where C 0 is given by (3.4).

Let us now consider the different factors in (4.25). As the limit of ρ(λ) and ρκ (λ) coincides with the ground state density ρ(λ), both determinants det N Θ (λ,µ) jk go to the Fredholm determinant of the operator I + K/2π:

lim N,M →∞ det N Θ (λ,µ) jk = det [I + K/2π] .
(5.10)

A direct application of (5.7) leads to lim

N,M →∞ V ± (w) = e -β z(w±iζ) . (5.11) Similarly, lim N
,M →∞ N a=1 sinh(w -λ a ) sinh(w -µ a ) = e -β z(w) , w uniformly away from [-q, q]. (5.12)
Substituting (5.11) and (5.12) into (4.17) we have that, at

θ 1 = -q, lim N,M →∞ 1 2πi U (λ) -q (w, w ′ ) = 1 2πi U (λ) -q (w, w ′ ), (5.13)
where the kernel U

-q (w, w ′ ) is given by (3.7). It is also easy to see that, when θ 2 = q and β is purely imaginary, lim

N,M →∞ 1 2πi U (µ) q (w, w ′ ) = 1 2πi U (λ) -q † (-w, -w′ ), (5.14)
where † means hermitian conjugation. A simple algebra based on (3.10) leads to lim

N,M →∞ A (κ) N = sinh β 2 sinh βZ 2 2 A(β), (5.15)
where A(β) is given by (3.6).

Thermodynamic limit of the Cauchy determinant

As already mentioned, the thermodynamic limit of the factor D (κ) N (4.23) is the most complicated to obtain. Although the final result for this quantity can be expressed in terms of the limiting value of the shift function (5.5), the approximation (5.6) is not enough for the intermediate analysis. The reason is that D (κ) N contains the square of a Cauchy determinant which has poles at µ j = λ k . These singularities are compensated by the zeros of

N j=1 κV + (λ j ) V -(λ j ) -1 V -(µ j ) κV + (µ j ) -1 . (5.16)
However, this compensation takes place if and only if the parameters µ j exactly solve the set of twisted Bethe equations, which means that it is not enough to consider only the first order of the shift function as in (5.6). Therefore, to compute the limit of D

N , we should work with the exact formulae for the N -particle shift functions F and Fκ (5.2) defined in terms of the two counting functions ξ(ω) and ξκ (ω) (2.16), (5.1), their limiting values being only taken at the end of the computation.

In order to re-write (4.23) in terms of the shift functions F and Fκ , we also introduce the functions

φ(λ, µ) = sinh(λ -µ) ξ(λ) -ξ(µ) , φκ (λ, µ) = sinh(λ -µ) ξκ (λ) -ξκ (µ) , (5.17)
which fulfill φ-1 (λ, λ) = ξ′ (λ) = ρ(λ), φ-1 κ (λ, λ) = ξ′ κ (λ) = ρκ (λ). (5.18)
Note that φ and φκ have the same thermodynamic limit denoted by ϕ.

Proposition 5.1. The product D

N (4.23) admits the following factorization:

D (κ) N = H[ F ] • H[-Fκ ] • Φ[ φ] • Φ[ φκ ] • N j=1
φ(λ j , λ j ) φκ (µ j , µ j ) φ(λ j , µ j ) φκ (µ j , λ j ) .

(

5.19)

The functions H[ F ] and Φ[ φ] are defined as Proof -Using the definition (2.16), (5.1) of the counting functions, one can easily see that

H[ F ] = N j>k 1 + F (λ j ) -F (λ k ) j -k N j,k=1 j =k 1 + F (λ j ) j -k -1 N j=1 sin π F (λ j ) π F (λ j ) , (5.20)
Φ[ φ] = N j>k φ(λ j , λ k ) φ(µ k , µ j ) N j,k=1 j =k φ(µ j , λ k ) , (5.21
κV + (ω) V -(ω) = e 2πiM [ ξ(ω)-ξκ(ω)] , (5.22)
which leads to

N j=1 κV + (λ j ) V -(λ j ) -1 V -(µ j ) κV + (µ j ) -1 = N j=1 e 2πiM [ ξ(λ j )-ξκ(λj )] -1 e -2πiM [ ξ(µ j )-ξκ(µj )] -1 = N j=1 e 2πiM [ ξκ(µj )-ξκ(λj )] -1 e -2πiM [ ξ(µ j )-ξ(λ j )] -1 = N j=1
4 sin π Fκ (µ j ) sin π F (λ j ) .

(5.23)

Here we have used that ξ(λ j ) = ξκ (µ j ) = j/M , and the identity

N j=1 ξ(λ j ) -ξ(µ j ) + ξκ (µ j ) -ξκ (λ j ) = 0 (5.24)
which is a consequence of (2.16), (5.1).

The remaining part of the proof is quite trivial. It is based on the expression for the Cauchy determinant in terms of double products

det N 1 u j -v k = N j>k (u j -u k )(v k -v j ) N j,k=1 (u j -v k ) , (5.25)
where u j , v j are arbitrary complex numbers. Substituting the explicit expressions for all the factors into (5.19), and using again ξ(λ j ) = ξκ (µ j ) = j/M together with the definition of F and Fκ (5.2), one easily reproduces the original representation (4.23) for D

N . Since the functions φ(λ, µ) and φκ (λ, µ) are not singular at λ, µ ∈ [-q, q], the thermodynamic limits of the last three factors in the r.h.s. of (5.19) can easily be calculated similarly as in the previous subsection. Using (5.6) we find lim

N,M →∞ Φ[ φ] • Φ[ φκ ] • N j=1 φ(λ j , λ j ) φκ (µ j , µ j ) φ(λ j , µ j ) φκ (µ j , λ j ) = exp - β 2 4π 2 q -q Z(λ) Z(µ) ∂ λ ∂ µ log ϕ(λ,
[ F ] = H 1 [ F ] • H 2 [ F ], with H 1 [ F ] = exp - N j=1 ψ(j) F (λ j ) -F (λ N -j+1 ) • N j>k 1 + F (λ j ) -F (λ k ) j -k , (5.27)
H 2 [ F ] = exp N j=1 ψ(j) F (λ j ) -F (λ N -j+1 ) N j,k=1 j =k 1 + F (λ j ) j -k -1 N j=1 sin[π F (λ j )] π F (λ j ) , (5.28)
ψ(z) being the logarithmic derivative of the Γ-function (cf Appendix A.1).

Proposition 5.2. The thermodynamic limit of the factor H 1 [ F ] (5.27) is given by lim

N,M →∞ log H 1 [ F ] = - 1 4 q -q F (λ) -F (µ) ξ(λ) -ξ(µ) 2 ρ(λ) ρ(µ) dλ dµ.
(5.29)

Proof -Due to the smoothness of F and ξ, there is an ǫ, 0 < ǫ < 1, such that, for M large enough,

F (λ j ) -F (λ k ) j -k = 1 M F (λ j ) -F (λ k ) ξ(λ j ) -ξ(λ k ) < ǫ , uniformy in λ j , λ k .
(5.30)

The logarithm of the double product in the r.h.s. of (5.27) can therefore be expanded into its Taylor integral series of order 2 log

N j>k 1 + F (λ j ) -F (λ k ) j -k = 2 n=1 (-1) n+1 nM n N j>k F (λ j ) -F (λ k ) ξ(λ j ) -ξ(λ k ) n + 1 M 3 N j>k 1 0 dt (1 -t) 2 1 + t M • F (λ j )-F (λ k ) ξ(λ j )-ξ(λ k ) 3 F (λ j ) -F (λ k ) ξ(λ j ) -ξ(λ k ) 3 .
(5.31)

The expressions under the double sum over j and k being not singular at λ j , λ k ∈ [-q, q], the thermodynamic limit of the integral remaining terms in (5.31) vanish due to the M -3 prefactor. The contribution issued from the n = 1 term cancels with the pre-factor in (5.27) since

N j>k F (λ j ) -F (λ k ) j -k = N j,k=1 j =k F (λ j ) j -k = N j=1 ψ(j) F (λ j ) -F (λ N -j+1 ) .
(5.32)

Thus, the limit of log H 1 reduces to the one of the term n = 2 in the expansion (5.31), which ends the proof.

Proposition 5.3. The thermodynamic limit of the factor H 2 [ F ] (5.28) is given by

lim N,M →∞ H 2 [ F ] • N F 2 (-q)+F 2 (q) 2 = G 1 + F (-q) G 1 -F (q) • e F (-q)-F (q) 2 (1-log 2π) × exp    q -q F 2 (λ) -F 2 (q) ξ(λ) -ξ(q) - F 2 (λ) -F 2 (-q) ξ(λ) -ξ(-q) ρ(λ) dλ    , (5.33)
where G(z) is the Barnes function (A.7).

Proof -The product over k in (5.28) can be expressed in terms of Γ-functions as .34) This means that H 2 [ F ] can itself be factorized into two factors,

N j,k=1 j =k 1 + F (λ j ) j -k -1 N j=1 sin(π F (λ j )) π F (λ j ) = N j=1 Γ 2 (j) Γ j + F (λ j ) Γ j -F (λ N -j+1 ) . ( 5
H 2 [ F ] = H (1) 2 [ F ] • H (2) 2 [ F ], (5.35)
with

H (1) 2 [ F ] = N j=1 Γ(j) Γ(j + F (λ j )) e ψ(j) F (λ j ) , (5.36)
H (2)
2 [ F ] being obtained from (5.36) by the replacement F (λ j ) → -F (λ N -j+1 ).

We first consider H

2 [ F ]. Let F-= F ξ-1 (0) . Evidently, F-→ F (-q) in the thermodynamic limit. We use the fact that the shift function F (λ j ) is bounded uniformly for any value of N . Thus we can use the following bound | F (λ j )| < n for some n ∈ N and j = 1, . . . , N . Then we represent H

(1) 2 [ F ] as H (1) 2 [ F ] = N j=1 Γ(j)e ψ(j) F- Γ(j + F-) n-1 j=1 Γ(j + F-)e ( F (λ j )-F-)ψ(j) Γ j + F (λ j ) N j=n Γ(j + F-)e ( F (λ j )-F-)ψ(j) Γ j + F (λ j ) . (5.37)
Using (A.10), (B.4) we find

N j=1 Γ(j)e ψ(j) F- Γ(j + F-) = G(N + 1) G(1 + F-) G(N + 1 + F-) e F-(N ψ(N )+1-N ) , (5.38)
and thus, applying the asymptotic formulae (A.3), (A.9) we arrive at lim

N,M →∞ N F 2 - 2 N j=1 Γ(j)e ψ(j) F- Γ(j + F-) = G 1 + F (-q) • e F (-q) 2
(1-log 2π) .

(5.39)

The limit of the second finite product in (5.37) evidently is equal to 1 due to F (λ j ) -F-= O(M -1 ) for j = 1, . . . , n -1. Finally, in order to compute the limit of the last product in (5.37) we can use Taylor integral representation for the logarithms of Γ-functions (recall that

| F (λ j )| < n ≤ j) N j=n Γ(j + F-)e ( F (λ j )-F-)ψ(j) Γ j + F (λ j ) = exp    1 2 N j=n ψ ′ (j) F 2 --F 2 (λ j ) - 1 2 N j=n 1 0 dt(1 -t) 2 ψ ′′ F (λ j )t + j) F 3 (λ j ) -ψ ′′ F-t + j F 3 -    . (5.40)
The limit of the first sum in (5.40) is obtained by applying Lemma B.2 proved in Appendix B. Indeed, as the counting function is invertible and satisfies ξ(λ j ) = j/M , we write F (λ j ) = F ξ-1 ( j M ) . Then, using (B.9), we find lim

N,M →∞ N j=n ψ ′ (j) F 2 --F 2 (λ j ) = - D 0 F 2 ξ -1 (x) -F 2 ξ -1 (0) x dx.
(5.41)

Recall that D = lim N,M →∞ N/M is the average density, so that, after the change of variables ξ -1 (x) = λ, we obtain lim

N,M →∞ N j=1 ψ ′ (j) F 2 --F 2 (λ j ) = - q -q F 2 (λ) -F 2 (-q) ξ(λ) -ξ(-q) ρ(λ) dλ. (5.42)
Finally, the same transformation on the function F allows us to apply, uniformly in t, Lemma B.3 to the second sum in (5.40), hence proving that its thermodynamic limit is zero.

Substituting these results into (5.37), we obtain that lim

N,M →∞ H (1) 2 [ F ] • N F 2 (-q) 2 = G 1 + F (-q) • exp F (-q) 2 (1 -log 2π) - 1 2 q -q F 2 (λ) -F 2 (-q) ξ(λ) -ξ(-q) ρ(λ) dλ . (5.43)
The thermodynamic limit of H

F ] can be deduced from the previous calculation due to the fact that F (λ N -j+1 ) = F ξ-1 ( N +1 M -j M ) . The limit of H[-Fκ ] is obtained in the same spirit as above. Since the limits of F , ρ and ξ coincide with those of their κ-deformed analogs, the result is given by (5.29) and (5.33) provided that one makes the replacement F → -F . This enables us to obtain the thermodynamic limit of D

N : Proposition 5.4. The product D (κ) N admits the following thermodynamic limit:

lim N,M →∞ D (κ) N M -β 2 Z 2 2π 2 = G 2 1, βZ 2πi [ρ(q) sinh 2q] β 2 Z 2 2π 2 e -β 2 4π 2 C 1 , (5.44)
where C 1 is given by (3.5). We remind that G(a, z) = G(a -z)G(a + z) and Z = Z(q).

Proof -Using the expression (5.5) of F and the symmetry properties of Z, ρ and ξ,

Z(λ) = Z(-λ), ρ(λ) = ρ(-λ), and ξ(λ) = D -ξ(-λ),
we obtain from Propositions 5.2, 5.3 that lim

N,M →∞ D (κ) N N -β 2 Z 2 2π 2 = G 2 1, βZ 2πi exp β 2 8π 2 q -q Z(λ) -Z(µ) ξ(λ) -ξ(µ) 2 ρ(λ) ρ(µ) dλ dµ × exp - β 2 4π 2 q -q Z(λ)Z(µ) ∂ λ ∂ µ log ϕ(λ, µ) dλ dµ - β 2 2π 2 q -q Z 2 (λ) -Z 2 ξ(λ) -ξ(q) ρ(λ) dλ .
(5.45)

After some simple algebra the combination of integrals in (5.45) can be reduced to the constant C 1 (3.5), and the result takes the announced form.

Results for the normalized scalar product and special form factor

Putting together the individual limits (5.44), (5.9) and (5.15) yields lim

N,M →∞ S (κ) N • M -β 2 Z 2 2π 2 = 2 sinh β 2 βZ G 2, βZ 2πi 2 ρ(q) sinh 2q β 2 Z 2 2π 2 e -β 2 4π 2 (C 1 -C 0 ) A(β).
(5.46)

The limits being uniform in β and its derivatives, we take the second β derivative at β = 2πi and obtain lim

N,M →∞ -∂ 2 β S (κ) N β=2πi • M 2Z 2 = 1 2 G(2, Z) πZ 2 e C 1 -C 0 A(β)
ρ(q) sinh 2q

2Z 2 β=2πi
.

(5.47)

We still need to compute the excitation momentum P ex in (4.9) to finish our proof. We have lim N,M →∞ P ex = lim N,M →∞ N j=1 p 0 (µ j ) -p 0 (λ j ) = q -q p ′ 0 (λ) Z(λ) dλ.

(5.48)

The integral equations (2.20), (2.25) for the density and the dressed charge then lead to q -q p ′ 0 (λ) Z(λ) dλ = 2π q -q ρ(λ) dλ = 2p F , (5.49) due to (2.24). Substituting (5.49) and (5.47) into (4.9) and comparing the result with the amplitude (3.3), we finally arrive at (3.2).

Conclusion

In the present article, we have initiated the development of a method to study form factors in the massless regime of the XXZ Heisenberg spin chain. In particular we have given explicitly the thermodynamic limit of a special form factor in the XXZ chain. This form factor corresponds to the matrix element of the operator σ z between the ground state and an excited state containing one particle and one hole on the different ends of the Fermi zone. It is clear however that our method can be applied to other form factors as well, where excited states contain arbitrary number of particles and holes in arbitrary positions. This method can also be applied to other integrable models, for example to the system of one-dimensional bosons. One of the unsolved problems is to prove that at zero magnetic field our result coincides with the amplitude predicted in [30,31]. The limit h → 0 corresponds to the limit q → ∞ in (5.46). It is easy to see that the constants C 0 , C 1 , A, ρ(q) sinh 2q, being finite for finite q, become divergent at q → ∞. We were able to prove that the total combination giving F σ remains finite for h = 0, but we did not obtain a simple expression for its value in this limit. We succeeded nevertheless to compute explicitly this quantity in the vicinity of the free fermion point (ζ = π 2 ) up to the second order in ǫ = ζ π -1 2 . Our computation confirms the result [30,31] up to that order. n(2a -1) 2 ψ (n-1) (N +a)-ψ (n-1) (a+1) -n(n -1) 2 ψ (n-2) (N +a)-ψ (n-2) (a+1) .

(B.7)

B.2 More complicated sums

We now consider sums involving in addition some regular function(s).

  (see Subsection 5.2), we will see in Subsection 5.3 that the factor D (κ) N (4.23) requires more attention.

)

  and the function H[-Fκ ] (respectively Φ[ φκ ]) can be obtained from (5.20) (resp. (5.21)) by the replacement F (λ) → -Fκ (µ) (resp. φ → φκ ).

B. 1 - 1 k=0ψ 1 k=0ψ 1 k=0e

 1111 Finite sums involving ψ-functionLemma B.1. For n ≥ 0, a = -1, -2, . . . , and an arbitrary complex α, one has the following identityN k=1 ψ (n) (k+a)e αk = 1 e α -1 • ψ (n) (N +a)e α(N +1) -ψ (n) (a)e α -(-1) n n! N -1 k=1 e α(k+1) (k + a) n+1 . (B.1)Proof -Denote the l.h.s. of (B.1) as f (α),f (α) = N k=1 ψ (n) (k + a) e αk . (B.2)Shifting k by k + 1 and using (A.1), we obtainf (α) = e α N (n) (k + a + 1) e αk = e α N -(n) (k + a) e αk + (-1) n n! Nα(k+1) (k + a) n+1 = e α f (α) + e α ψ (n) (a) -e α(N +1) ψ (n) (N + a) + (-1) n n! N -1 k=0 e α(k+1) (k + a) n+1 . (B.3)Then (B.1) follows immediately.Taking derivatives of (B.1) with respect to α at α = 0 one obtains formulae for the sums of the type ψ (n) (k + a)k p . Let us give explicitly several of them.One has for p = 0, n = 0,N k=1 ψ(k + a) = (N + a)ψ(N + a) -aψ(a) -N,(B.4)and for p = 0, n ≥ 1,N k=1 ψ (n) (k +a) = (N +a)ψ (n) (N +a)-aψ (n) (a+1)+n ψ (n-1) (N +a)-ψ (n-1) (a+1) . (B.5)One also has for p = 1, n = 1,N k=1 kψ ′ (k + a) = (N + a)(N + 1 -a) 2 ψ ′ (N + a) + a(a -1) 2 ψ (n) (a + 1) -2a -1 2 ψ(N + a) -ψ(a + 1) + N -12, (B.6) and for p = 1, n ≥ 2, N k=1 kψ (n) (k + a) = (N + a)(N + 1 -a) 2 ψ (n) (N + a) + a(a -1) 2 ψ (n) (a + 1)

Lemma B. 2 . 8 )

 28 Let f ∈ C 1 ([0, a]) for a > 0. Let us consider the sum S In the limit N, M → ∞, N/M → D with D ∈ [0, a[, it tends to S

Although there are evidences that the ground state can be described in that way, strictly speaking, to our knowledge, this statement has been proved only in the case ∆ ≤ 0[START_REF] Yang | [END_REF][START_REF]Mathematical Physics in One Dimension[END_REF].

Acknowledgements J. M. M., N. S. and V. T. are supported by CNRS. N. K., K. K. K., J. M. M. and V. T. are supported by the ANR program GIMP ANR-05-BLAN-0029-01. N. K. and V. T. are supported by the ANR program MIB-05 JC05-52749. We also acknowledge the support from the GDRI-471 of CNRS "French-Russian network in Theoretical and Mathematical Physics". N. S. is also supported by the Program of RAS Mathematical Methods of the Nonlinear Dynamics, RFBR-08-01-00501a, Scientific Schools 795.2008.1. K. K. K. is supported by the French ministry of research. N. K and N. S. would like to thank the Theoretical Physics group of the Laboratory of Physics at ENS Lyon for hospitality, which makes this collaboration possible.

A ψ-function and Barnes function

We recall in this appendix the definitions of the ψ-function and the Barnes function, as well as several standard formulae that are useful for our study.

A.1 The ψ-function and its derivatives

The ψ-function is defined as the logarithmic derivative ψ(z) = d dz log Γ(z) of the Γ-function. The multiplication property of the Γ-function implies that,

It follows from (A.1) that

When z → ∞ with -π < arg(z) < π, one has

For n ≥ 1, the n th -derivative of the ψ-function admits the following integral representation:

The latter implies in particular that, for z > 0,

It is also easy to see that

A.2 The Barnes function

The Barnes function G(z) is the unique solution of

The Barnes function has the following asymptotic behavior when z → ∞, -π < arg(z) < π:

Due to (A.7), one has

B Sums with logarithmic derivatives of Γ-function

In this appendix, we explain how to compute the thermodynamic limit of some sums involving the ψ-function and its derivatives.

It is easy to see that the first sum vanishes in the limit considered. Indeed, setting f = sup [0,a] |f |, we get using (B.6), (A.6) and (A.3),

Thus, the limit of S

(N,M ) 1

[f ] reduces to one of the second term in (B.10), which is quite straightforward due to Euler-Maclaurin summation formula lim

and Lemma B.2 is proved.

Then, for m > 1, the sum

Proof -We have The last sum is readily computed by (B.7). It is easy to see that it is a O(log N ) at most.


O Risser-Maroix
email: orissermaroix@gmail.com

A Marzouki

H Djeghim

C Kurtz

N Loménie

Learning an adaptation function to assess image visual similarities

Keywords: Visual Similarity, Features Adaptation, Image Retrieval, Analogies

. In this new article, we expand on this previous work by using and comparing new pre-trained feature extractors on other datasets.

Motivation

Analogies are constantly employed by humans to find connections / similarities between images, when learning concepts or for creative purposes. Our perception being extremely complex to model, it remains difficult to imitate by machines. However, capturing image similarities is a cornerstone in various computer vision tasks, such as retrieval, classification, spotting, etc. This task is challenging since an image has many more interpretations than its tex-tual description and the sought similarity may depend on a hidden intention. State-of-the-art approaches often rely on the comparison of images described as vectors of features learned via convolutional neural networks (CNNs) with objectives close to classification task. Since such features are generally learned for image categorization, they are biased by the semantics of the decision process of the classifiers. In this article, we focus on the task of learning image visual similarities and we involve this in an image retrieval context. Inspired by previous works on neural features adaptation to psycho-cognitive representations, we propose a way to learn a perceptual similarity function between images that share visual connections but that are semantically unrelated. To our knowledge, we are among the first to propose a methodology to face this issue of understanding the underlying psycho-visual process of matching images of different natures.

Background

In human perception, the notion of similarity between concepts or even images has been studied for a while and remains extremely difficult to define [START_REF] Tversky | Studies of similarity[END_REF][START_REF] Rogowitz | Perceptual image similarity experiments[END_REF][START_REF] Tirilly | On the consistency and features of image similarity[END_REF][START_REF] Liu | The influence of task requirements on priming in object decision and matching[END_REF][START_REF] Mur | Human object-similarity judgments reflect and transcend the primate-IT object representation[END_REF]. During the last decade, scientists from cognitive and computer sciences started to analyze the differences between human and machine perception and to investigate on how modern neural architectures could help to capture human judgments of similarity ; such a similarity can be either guided by general concepts [START_REF] Peterson | Evaluating (and improving) the correspondence between deep neural networks and human representations[END_REF][START_REF] Zheng | Revealing interpretable object representations from human behavior[END_REF][START_REF] Hebart | Revealing the multidimensional mental representations of natural objects underlying human similarity judgements[END_REF] or performed by strictly visual correspondences [START_REF] Kubilius | Deep neural networks as a computational model for human shape sensitivity[END_REF][START_REF] Pramod | Do computational models differ systematically from human object perception ?[END_REF][START_REF] Sadovnik | Finding your lookalike : Measuring face similarity rather than face identity[END_REF][START_REF] Zhang | The unreasonable effectiveness of deep features as a perceptual metric[END_REF][START_REF] Scott | Can machine learning account for human visual object shape similarity judgments ?[END_REF]. As already stated by [START_REF] Rosenfeld | Totally looks like -how humans compare, compared to machines[END_REF], strategies based on visual stimuli yield good results with very simple images (with well segmented background) [START_REF] Pramod | Do computational models differ systematically from human object perception ?[END_REF] or when focusing on a narrow class [START_REF] Peterson | Evaluating (and improving) the correspondence between deep neural networks and human representations[END_REF][START_REF] Sadovnik | Finding your lookalike : Measuring face similarity rather than face identity[END_REF][START_REF] Hamilton | Conditional image retrieval[END_REF] (such as only faces, animals, arts). The Totally Looks Like image dataset (denoted as TLL) is an interesting example of this research area [START_REF] Rosenfeld | Totally looks like -how humans compare, compared to machines[END_REF]. TLL is composed of 6k+ image pairs resulting from human propositions, where similarity between images is not based on semantic categorization (as this is the case in most classical image retrieval datasets) but only based on visual clues derived from the image contents. The images belong to different domains such as photography, cartoons, sketches, logos, etc. making the task even harder but more represen-tative of human ability to make connections between semantically unrelated objects (some illustrative samples are provided in Fig. 1). From this dataset, one can note that the process of human visual similarity mixes multiple different levels of analysis ranging from color, texture, to shape, layout, etc. in which context, cultural aspect and possibly humor and irony can play an important role.

Ones could argue that similarity is a too subjective and an ill-posed problem. But given a dataset of image pairs such as TLL, authors from [START_REF] Rosenfeld | Totally looks like -how humans compare, compared to machines[END_REF] conducted human experiments and found that when other image candidates were proposed to form pairs, humans remained consistent in their choices and selected invariably the right target image to make the pair. Authors from [START_REF] Liu | The influence of task requirements on priming in object decision and matching[END_REF] found that results in different visual studies are highly influenced by the task requirements ; in our case the task is defined by the visual similarity without taking into account the semantic similarity. In addition, [START_REF] Tirilly | On the consistency and features of image similarity[END_REF] showed that, for the task of grading the similarity between images, the quantitative analysis of the similarity scores reported by subjects reached a consensus.

Authors from [START_REF] Rosenfeld | Totally looks like -how humans compare, compared to machines[END_REF] tried to reproduce the human similarity judgments from a given image pairs dataset with different neural architectures and, as mentioned earlier, they found that the learned convolutional features were not adapted for this task. Later [START_REF] Rosenfeld | High-level perceptual similarity is enabled by learning diverse tasks[END_REF] improved the similarity scores by using even more descriptors crafted for different uses (color only, shape only, etc.), and by using the right descriptor for each pair as oracle. None of [START_REF] Rosenfeld | Totally looks like -how humans compare, compared to machines[END_REF][START_REF] Rosenfeld | High-level perceptual similarity is enabled by learning diverse tasks[END_REF] worked on features adaptation, neither learned a model on a part of the dataset, making us the first to propose a baseline for this task. As comparative work, starting from the observation that traditional metrics (L2, PSNR, etc.) disagree with human judgments, authors from [START_REF] Zhang | The unreasonable effectiveness of deep features as a perceptual metric[END_REF] already learned a low-level perceptual measure of similarity from image patches affected by distortions. The poor results obtained by [START_REF] Rosenfeld | High-level perceptual similarity is enabled by learning diverse tasks[END_REF] on TLL by using the low level perceptual learned similarity from [START_REF] Zhang | The unreasonable effectiveness of deep features as a perceptual metric[END_REF], as well as with higher level semantic features confirm the previous study of [START_REF] Mur | Human object-similarity judgments reflect and transcend the primate-IT object representation[END_REF] stating that human visual similarity was not based in the visual cortex but may be the result of processing done in the primate inferior temporal (IT) cortex. In our case, we hypothesize that even if neural networks are learned to be robust to cases where images are semantically similar but visually dissimilar, they still carry useful (and reusable) information about texture, shapes, etc. When dealing with relatively small datasets such as TLL, based on a strategy originally introduced in [START_REF] Mur | Human object-similarity judgments reflect and transcend the primate-IT object representation[END_REF], we propose to use different layers of pre-trained networks with the opposite objective of categorization as a rough approximation of the visual cortex and learn only an adaptation function corresponding to the approximation of the the primate IT cortex through the metric learning framework. Similarity learning is closely related to distance metric learning where the goal is to learn a distance function over objects that measures how similar two objects are. In our case, we look for a model able to bring closer each training image pair, while moving away all other images that FIGURE 1 -Examples of image pairs from the "Totally Looks Like" dataset. Similarity can be based on the color, shape, texture, layout, facial similarity, etc. would form a less good pair than the ground-truth one. For each image, we only have a single solution which makes our problem close to one-shot learning where metric learning has been considered [START_REF] Koch | Siamese neural networks for one-shot image recognition[END_REF] by using Siamese networks [START_REF] Chopra | Learning a similarity metric discriminatively, with application to face verification[END_REF]. Another wildely used architecture in deep metric learning is the triplet network [START_REF] Hoffer | Deep metric learning using triplet network[END_REF] where for each positive pair, a negative image is also provided to learn simultaneously to gather the positive images while increasing the distance to the negative image. In a similar fashion, [START_REF] Hebart | Revealing the multidimensional mental representations of natural objects underlying human similarity judgements[END_REF] was able to provide powerful representations capturing human behavior by asking participants to select the one-odd-out from a triplet of images and thus learning a model to mimic it. It has been shown that those architectures suffer from sampling issues that could be partially solved with complex mining strategies [START_REF] Schroff | FaceNet : A unified embedding for face recognition and clustering[END_REF][START_REF] Chao-Yuan Wu | Sampling matters in deep embedding learning[END_REF]. For the sake of simplicity, we rely here on the hypothesis that each image in a pair has a stronger connection than with any other image in the dataset. We then learn a function able to bring closer the right image to the query than any other image. By doing so, we aim to bypass the sampling issue and use a generalization of the triplet loss similar to the N-Loss [START_REF] Sohn | Improved deep metric learning with multi-class n-pair loss objective[END_REF]. Inspired from previous works in cognitive sciences, we propose the first baseline for the task of visual similarity between images without taking into account semantic similarity. This research work was recently accepted for publication at the ICIP 2021 international conference. In this new article, we expand on this previous work by using and comparing new pre-trained feature extractors on other datasets.

Learning image visual similarities

We propose to learn an adaptation function able to compute visual similarity from image pairs and to involve it in an image retrieval task (Fig. 2). Inspired by [START_REF] Mur | Human object-similarity judgments reflect and transcend the primate-IT object representation[END_REF], the features extraction part (Fig. 2 (a)) could be interpreted as the visual cortex while the learned adaptation matrix and the measure deciding on the visual similarity between two images (Fig. 2 (b)) can be viewed as the primate IT cortex.

Image retrieval task

We assume that we have an image dataset, structured as pairs of visually similar images (considered as the groundtruth (GT)). We consider each pair as the juxtaposition of a left image and and a right image, leading to two sets of images. As in [START_REF] Rosenfeld | Totally looks like -how humans compare, compared to machines[END_REF], we formulate this problem as an image retrieval task. For each query in the left set of images, we rank all the images in the set of right, according to a given similarity measure φ(•, •), and reciprocally. From this ranking, we want that for each query, the best returned candidate is the one expected by the GT pair. We know from [START_REF] Tversky | Features of similarity[END_REF] that asymmetry in human judgment of similarity is important. For example, in Fig. 1, most humans will think that the zucchini looks like the penguin rather than the penguin looks like the zucchini. In our case, as we do not have the direction information of which image from left or right is looking to the other one, we cannot use an asymmetric function such as the Tversky Ratio Model [START_REF] Tversky | Features of similarity[END_REF] and φ(•, •) is here a simple cosine similarity. However, we will embed this asymmetry in the evaluation function.

Image representation

As pre-processing, we extract visual features for each image (right, left) from pre-trained networks. Different layers were used for some architectures based on Residual Networks while only the last layer was adopted for more complex ones due to the difficulty to localize the middle layers ( FaceNet and transfomers). For more details, please refer to Table 1 in the annex.

We reduce the feature maps each layer to a simple vector with the same dimension as the number of channels, by averaging features maps on the spatial dimensions followed by L2 normalization. By doing so, each image is represented as the concatenation of features extracted from different layers to capture information, from low to high levels. To avoid overfitting, we reduce the number of dimension of the original vectors (up to 15k+) with a PCA. Left and right vectors are respectively named d lef t , d right and form the image representations. This pre-processing step (Fig. 2 (a)) is done once before training and can be interpreted as a rough approximation of visual features extracted by the human visual cortex. Indeed, the object recognition and visual similarity assessment is done in the Ventral Stream which is composed of the Visual Cortex and the Primate IT Cortex [START_REF] Lindsay | Convolutional neural networks as a model of the visual system : past, present, and future[END_REF]. The Visual Cortex is decomposed into sub-regions (V1-to-V4) ; V4 is particularly known to respond to orientation, color, disparity and simple shapes and it is directly connected to the Primate IT Cortex. Since the different features V4 responds to are located in different levels of a CNN (color and orientation are lower levels and extracted from early layers while shapes are higher layer levels) we propose to approximate V4 region by taking as input the different layers of a pre-trained CNN (aiming to extract color, orientation, shape, etc.) and directly pass them to our adaptation module. It then becomes obvious that this learned adaptation module directly wired to the V4 visual features simulated by the frozen pre-trained CNN is our model of the Primate IT cortex we propose to learn in a contrastive setting.

Adaptation function learning

We look for an adaptation function able to bring closer the left embeddings (d lef t ) to the right ones (d right ) of the corresponding pairs closer than any other data. We model our adaptation module as the multiplication between a (learnable) weight matrix W and input features d lef t , d right followed by a ReLU activation. We will refer to the adaptation of the d lef t , d right features as a lef t , a right (Fig. 2 (b)). Previous works used contrastive or triplet loss [START_REF] Chopra | Learning a similarity metric discriminatively, with application to face verification[END_REF][START_REF] Hoffer | Deep metric learning using triplet network[END_REF] when learning from pairs or triplets. To avoid sampling issue, as the whole pre-processed embeddings are light in memory, we build a similarity matrix by measuring similarity of each left image to each right image with the φ(•, •) similarity function. The corresponding GT is the identity matrix. We can thus learn to classify by using a softmax activation, followed by a Cross-Entropy loss function L. We learn to find the right image from left queries and left images from right queries simultaneously, by averaging the directed L left to right and L right to left losses as in Eq. 1.

L left to right = CrossEntropy(softmax(φ(a lef t , a right ) • σ), 1)

L right to left = CrossEntropy(softmax(φ(a right , a lef t ) • σ), 1) L = (L left to right + L right to left )/2
(1) We used a parameter σ in Eq. 1, often referred as a temperature parameter in the literature. This specific parameter σ has already been used with success in previous works such as in [START_REF] Wu | Improving generalization via scalable neighborhood component analysis[END_REF]. In our case, by using a large σ, we are able to peak the softmax distribution to near binary values. This can be viewed as a form of regularization. Let us consider the case where the pair is successfully matched, without σ the loss would still continue to try to bring the adapted embeddings even closer while they already are the best matches. But, when a large σ is used, the correct best match will have a value close to 1 not penalizing anymore the loss for right classification and thus not trying to continue to learn on adapted embeddings which are already optimal. We show the positive influence on learning and generalization by using a large σ in Table . 1.

TTL Dataset

We considered in our experimental study the Totally Looks Like (TLL) image dataset introduced earlier. It is composed of 6016 image pairs, perceptually similar but semantically unrelated (some illustrative samples are provided in Fig. 1). The images come from very different domains such as photography, cartoons, paintings, sketches, logos, etc. The dataset can be split into two sub-datasets : one of 1817 pairs containing only well centered faces (noted TLL f aces ) and one of 4199 pairs of images captured from the wild (noted TLL obj ). Since [START_REF] Sadovnik | Finding your lookalike : Measuring face similarity rather than face identity[END_REF] already focused on the particular case of facial similarity with data richer than pairs on a bigger dataset, we will focus on the more general case of the 4199 remaining pairs. To subtract the TLL f aces subset from the whole TLL, we labeled as face-pairs only pairs where faces were detected in both right and left images with a Haar Cascade classifier. In this context, the remaining set (TLL obj ) still contains pairs where faces are compared to other animals, objects, paintings, etc. due to strange facial expressions or other features.

Protocol

For each experiment, the TLL obj dataset of 4199 pairs was divided into 75-25 train-test sets. We did not use a validation set due to the non significativness of the scores obtained on too small validation or testing bases, furthermore, if we decrease the training set size, the exhaustivity of the different ways that human compare images is too partial. We used a PCA with 256 dimensions to compress the precomputed embeddings into d lef t , d right . Experimentally, we found that 256 was able to remove the noise reducing both, the input space and overfitting of our adaptation. Thus we will use those compressed vectors both as a second baseline and as inputs of our method. We used then an adap-FIGURE 4 -Results before (B) and after (A) adaptation. The first column corresponds to the query, the second to the GT and the other images correspond to the returned candidates.

tation vector size of 1024, thus the matrix W is sized from 256 to 1024. As we do not have a validation set, we experimentally set the number of epochs to 150 and we selected the optimal number of epochs by bootstrapping for each model. This provides a good compromise between low overfitting and the best possible model if we had an oracle to select it. Due to the small number of examples in the testing base and the stochasticity induced by the random split and matrix weights initialization, we run our experiments 20 times with different splits and matrix initializations.

Evaluation and results

As discussed previously, human similarity judgment has been found to be asymmetric [START_REF] Tversky | Features of similarity[END_REF]. Since we do not have the asymmetry indications in the TLL dataset, we propose to evaluate the results in an optimistic way : if the pairing is found in one direction or another, the pairing is considered as successful. We refer to this optimistic recall as Asymmetric Recall (aR). Results are reported on Fig. 3 and Tab. 1. We computed the Asymmetric Recall on 20 random tests splits and display the mean scores surrounded by two standard deviations. From these results, we observe the strong influence of the temperature σ on the results. Our method is compared to the baselines (with and without) dimension reduction. We can see a 2.25×, 3.55× and 4.53× average improvement for aR@{1, 5, 20} by only using the Clip_RN50×4 features. We provide a few qualitative results in Fig. 4 from the test set. The second case shows that none of the previous neither our method is able to find the right image, however our method was able to capture the sharp hairstyle and gla-cial aspect of the picture. The first and last ones show the limitation of purely semantic features for visual retrieval, while our method leads to more diverse results. On the last example, it can be noticed that while a semantically biased features extractor will firstly output cats when querying a cat, our method finds image from very different classes, still sharing connection on the dark eyes surrounding.

Comparative study. Datasets scales and contents influence the results of each learning approach, which made us question the importance of the pre-trained network chosen for the features extraction module. In the Table 1 in annex we compare Residual networks of different depths : ResNet18, ResNet50, ResNet101, ResNet152, [START_REF] He | Deep residual learning for image recognition[END_REF], the self-supervised DINO framework pre-trained with Re-sNet50 [START_REF] Caron | Emerging properties in self-supervised vision transformers[END_REF], FaceNet pre-trained on VGGFace2 [START_REF] Schroff | FaceNet : A unified embedding for face recognition and clustering[END_REF], BiT-M of different sizes : ResNet-50x1, ResNet-50x3, ResNet-101x1, ResNet-101x3, and ResNet-152x4 [START_REF] Kolesnikov | Large scale learning of general visual representations for transfer[END_REF], Barlow Twins Self-Supervised model pre-trained on ResNet50 [START_REF] Zbontar | Barlow twins : Self-supervised learning via redundancy reduction[END_REF] and CLIP, a framework that works on learning visual concepts from natural language supervision [START_REF] Radford | Learning transferable visual models from natural language supervision[END_REF].

In [START_REF] Zhuang | Unsupervised neural network models of the ventral visual stream[END_REF], they found that the unsupervised models as well as self-supervised ones were the best at modelling and approximating the ventral visual stream, though their similarity task is different from ours. We compared different learning types to confirm that self-supervised models are the best ones for approximating the visual ventral system, which is coherent with their findings.

Discussions

In this article, we focus on the specific task of learning visual image similarities. Inspired by previous works on neural features adaptation to psycho-cognitive representations, we proposed a method to adapt semantic neural representations to visual ones. In average, our method improves the retrieval score up to 4.53×. We observed as well a qualitative improvement on the returned candidates. In addition, we saw that the CLIP framework was the best in our case, and we assume that it is due to two things : (1) First, such model has been trained on a dataset which consists of 400M images gathered from the internet ; (2) Second, because of the robustness of the CLIP model and its ability to generalize the concepts as well as its flexibility when it comes to semantic bias. Despite those improvements, some retrieval results are still not meaningful. Accordingly to [START_REF] Scott | Can machine learning account for human visual object shape similarity judgments ?[END_REF], in contrary to humans, neural features may be less suited to achieve viewpoints invariance. Moreover, as for several pairs the shared visual similarity relies on few pixels, this problem could be tackled in a finegrained framework. Since pairs in the TLL dataset are not equally sampled from the meta-features of an image (color, texture, layout, etc.), identifying those different metafeatures used by human to judge about the visual similarity between images could help to re-balance learning. Further investigations are ongoing.

Annexe A

  left , a right ) (a) Image representations (b) Adaptation function learning

FIGURE 2 -

 2 FIGURE 2 -Pipeline to learn an adaptation function able to compute visual similarity from image pairs.

FIGURE 3 -

 3 FIGURE 3 -Comparison of Asymmetric Recall curves at different ranks between baselines with σ = 15. Dashed line correspond to performances before adaptation, plain ones correspond to the adapted ones.

TABLE 1 -

 1 Asymmetric Recall scores of each model before and after adaptation (our method). We only extracted features from one of the last layer in models marked with a * . Scores are averaged on 20 runs and reported with the standard deviation.

	Our aR@1 σ = 15	0.2065 ±0.0045	0.1826 ±0.0044	0.2035 ±0.0057	0.2435 ±0.0040	0.2009 ±0.0048	0.2012 ±0.0047	0.1173 ±0.0038	0.1321 ±0.0045	0.1684 ±0.0044	0.1469 ±0.0047	0.1913 ±0.0034	0.1686 ±0.0046	0.1892 ±0.0040	0.1552 ±0.0032	0.1387 ±0.0034	0.1602 ±0.0042	0.1410 ±0.0046	0.1570 ±0.0035	0.2332 ±0.0042	0.2659 ±0.0052	0.2509 ±0.0055	0.2437 ±0.0039	0.2732 ±0.0049	0.2540 ±0.0052	0.1995 ±0.0051	0.1550 ±0.0040	0.2191 ±0.0038	0.1986 ±0.0049	0.2005 ±0.0041	0.1917 ±0.0041	0.2061 ±0.0047	0.2295 ±0.0052	0.2318 ±0.0046	0.3825 ±0.0046	0.3607 ±0.0047	0.3939 ±0.0071	0.3540 ±0.0046
	Model Learning type Dataset Number of images (M) Concatenated Size PCA Variance Sum Concatenated Score PCA Score Our aR@1 σ = 1	barlow self-supervised ImageNet-1k 1.28 3840 0.7555 0.1389 ±0.0106 0.1357 0.1781 ±0.0041	deits16 * self-supervised ImageNet-1k 1.28 384 0.9352 0.1392 ±0.0107 0.1362 0.1456 ±0.0044	deits8 * self-supervised ImageNet-1k 1.28 384 0.9451 0.149 ±0.0109 0.1477 0.1623 ±0.0057	dino_resnet50 self-supervised ImageNet-1k 1.28 3840 0.7732 0.136 ±0.0089 0.1288 0.1947 ±0.0034	dino_vitb16 * self-supervised ImageNet-1k 1.28 768 0.8505 0.1456 ±0.0100 0.1394 0.1745 ±0.0039	dino_vitb8 * self-supervised ImageNet-1k 1.28 768 0.8199 0.1334 ±0.0122 0.1314 0.1763 ±0.0041	facenet * supervised VGGFace2 3.31 1792 0.9722 0.0533 ±0.0050 0.0611 0.0739 ±0.0035	alexnet * supervised ImageNet-1k 1.28 256 1.0000 0.1023 ±0.0080 0.1041 0.1061 ±0.0043	resnet18 supervised ImageNet-1k 1.28 960 0.9118 0.1283 ±0.0097 0.1259 0.1323 ±0.0035	resnet34 supervised ImageNet-1k 1.28 960 0.9011 0.1236 ±0.0091 0.1304 0.1426 ±0.0037	resnet50 supervised ImageNet-1k 1.28 3840 0.8832 0.1211 ±0.0073 0.1258 0.1340 ±0.0051	resnet101 supervised ImageNet-1k 1.28 3840 0.8974 0.1209 ±0.0073 0.1212 0.1557 ±0.0042	resnet152 supervised ImageNet-1k 1.28 3840 0.8971 0.125 ±0.0094 0.1214 0.1472 ±0.0044	efficientnet_b0 * supervised ImageNet-1k 1.28 1280 0.7725 0.1164 ±0.0093 0.1149 0.1232 ±0.0039	efficientnet_b1 * supervised ImageNet-1k 1.28 1280 0.7630 0.1243 ±0.0067 0.1242 0.1384 ±0.0033	efficientnet_b2 * supervised ImageNet-1k 1.28 1408 0.7423 0.1165 ±0.0071 0.1170 0.1205 ±0.0029	efficientnet_b3 * supervised ImageNet-1k 1.28 1536 0.7188 0.1162 ±0.0090 0.1126 0.1205 ±0.0033	ViT * supervised ImageNet-1k 1.28 768 0.8001 0.121 ±0.0084 0.1173 0.1377 ±0.0032	BiT-M-R50x1 supervised ImageNet-21k 14.19 3840 0.9212 0.1165 ±0.0076 0.1142 0.1722 ±0.0031	BiT-M-R50x3 supervised ImageNet-21k 14.19 11520 0.8768 0.1165 ±0.0099 0.1143 0.2049 ±0.0046	BiT-M-R101x1 supervised ImageNet-21k 14.19 3840 0.9174 0.1114 ±0.0087 0.1054 0.2030 ±0.0051	BiT-M-R101x3 supervised ImageNet-21k 14.19 11520 0.8789 0.1099 ±0.0087 0.1075 0.1800 ±0.0040	BiT-M-R152x2 supervised ImageNet-21k 14.19 7680 0.8877 0.1171 ±0.0058 0.1101 0.2215 ±0.0060	BiT-M-R152x4 supervised ImageNet-21k 14.19 15360 0.8033 0.1174 ±0.0102 0.1143 0.2145 ±0.0024	resnet50_ssl semi-supervised ImageNet-1k 1.28 3840 0.7973 0.1396 ±0.0075 0.1401 0.1883 ±0.0039	resnext101_32x8d_ssl semi-supervised ImageNet-1k 1.28 3840 0.8539 0.1255 ±0.0089 0.1276 0.1589 ±0.0033	resnet50_swsl weakly semi-supervised ImageNet-1k 1.28 3840 0.7912 0.1548 ±0.0099 0.1506 0.1929 ±0.0035	resnext101_32x16d_swsl weakly semi-supervised ImageNet-1k 1.28 3840 0.8729 0.1213 ±0.0068 0.1270 0.1414 ±0.0031	resnext101_32x8d_swsl weakly semi-supervised ImageNet-1k 1.28 3840 0.8702 0.1295 ±0.0067 0.1297 0.1544 ±0.0029	resnext101_32x8d_wsl weakly supervised ImageNet-1k 1.28 3840 0.7861 0.1297 ±0.0094 0.1192 0.1642 ±0.0035	resnext101_32x16d_wsl weakly supervised ImageNet-1k 1.28 3840 0.7731 0.1292 ±0.0079 0.1204 0.1670 ±0.0035	resnext101_32x32d_wsl weakly supervised ImageNet-1k 1.28 3840 0.7651 0.1406 ±0.0091 0.1323 0.1639 ±0.0047	resnext101_32x48d_wsl weakly supervised ImageNet-1k 1.28 3840 0.7629 0.1458 ±0.0085 0.1419 0.1696 ±0.0037	clip_rn101 natural language supervision Internet 400.0 4352 0.8426 0.1674 ±0.0104 0.1658 0.2814 ±0.0050	clip_rn50 natural language supervision Internet 400.0 4864 0.8679 0.1575 ±0.0102 0.1606 0.2834 ±0.0059	clip_rn50x4 natural language supervision Internet 400.0 5440 0.8199 0.1743 ±0.0070 0.1700 0.3282 ±0.0041	clip_vit32_b * natural language supervision Internet 400.0 768 0.8600 0.1714 ±0.0074 0.1797 0.2655 ±0.0068


domain: sde phys

text:
Nadia Ouaadi

Ludovic Villard

Saïd Khabba

Pierre Louis Frison

Adnane Chakir

Jamal Ezzahar

Pascal Fanise

Mohamed Kasbani

Valérie Le Dantec

Lionel Jarlan

Investigation of the diurnal cycles of interferometric coherence and backscattering coefficient on irrigated wheat in the South Mediterranean

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Investigation of the diurnal cycles of interferometric coherence and backscattering coefficient on irrigated wheat in the South Mediterranean N. Ouaadi 1 , L. Villard 1 , S. Khabba 2,3 , P.L. Frison 4 , A. Chakir 2,4 , J. Ezzahar 3,5 , P. Fanise 1 , M. Kasbani 1 , V. Le Dantec 1 , L. Jarlan 1 1 CESBIO, University of Toulouse, IRD/CNRS/UPS/CNES, Toulouse, France. 2 LMME, Department of Physics, Faculty of Science Semlalia, Cadi Ayyad University, Marrakech, Morocco. 3 CRSA, Mohammed VI Polytechnic University UM6P, Ben Guerir, Morocco. 4 LaSTIG, Gustave Eiffel University, Paris, France. 5 MTI, ENSA Safi, Cadi Ayyad University, Morocco. 6 ProcEDE, Department of Applied Physique, Cadi Ayyad University, Marrakech, Morocco.

Sustainable management of water resources is of critical importance in the South-Mediterranean, which is suffering from water scarcity intensified by the impact of climate change. In this context, early detection of water stress by monitoring the water status of canopies can contribute to an optimal use of irrigation water use. C-band radar data have shown great potential for monitoring the vegetation hydric conditions. Over forests, several studies have observed a diurnal cycle in the backscattering coefficient that can reach up to 1 dB. The objective of this work is to investigate the behavior of radar data over annual crops by collecting in situ measurements of the interferometric coherence (𝜌) and the backscattering coefficient (𝜎 0 ) every 15 minutes using a radar set-up composed of six C-band antennas (full polarization), installed on a 20 m high tower and targeting a winter wheat field in Morocco. The first results obtained from the analysis of these data collected during the 2020 agricultural season reveal the existence of a diurnal cycle of interferometric coherence whose amplitude increases with canopy development. The decrease of coherence at dawn is most likely related to evapotranspiration while dynamic of coherence in the afternoon is associated to wind intensity. Similarly, these preliminary results show that the daily cycle of the backscattering coefficient is in good agreement with the daily evolution of evapotranspiration.

Key-words:

Abstract

The experiment (named hereafter MOCTAR) was conducted in the Haouz plain in Morocco (see figure 1).

The climate in the region is semiarid with a main annual rainfall around 250 mm and a reference evapotranspiration of about 1600.

The experiment is installed over a wheat field of about 1.5 ha selected within a private farm.

The fields is grown in winter wheat during 2020 agricultural seasons (January to June) and the irrigation method used is the drip technique. The field is also equipped with an eddy covariance that allow half-hourly measurements of evapotranspiration, soil moisture and wind speed.

Field campaigns are also carried out to measure soil moisture, surface roughness, vegetation above-ground biomass and cover fraction .

 The analysis of the first observations acquired by a in situ C-band radar over an irrigated crop in the south mediterranean region reveals the existence of a diurnal cycle of the interferometric coherence and of the backscattering coefficient whose amplitudes increase with the development of vegetation.  The low values of ρ observed at the end of the afternoon are related to the strong winds while the drop at dawn is concomitant with the increase in evapotranspiration.  For the backscattering coefficient, a good agreement is observed between the evolutions of its daily average and the evolution of evapotranspiration.  These preliminary results demonstrate the existence of important dependencies between the infra-daily Cband dynamic and the physiological functioning of wheat, which could be exploited in the future for the detection of water stress using radar data acquired at a sub-daily timescale.  Minimum values obtained in the late afternoon correspond to the higher value of wind intensity.

𝛔 𝟎 :

 A daily cycle is observed when the canopy is developed.

 Increase of the signal in the morning concomitant with the increase of LE.

 SSM have a limited impact on ρ while the marked daily cycle of 𝜎 0 when the soil is bare is more likely related to SSM.

  Diurnal cycle; C-band; Backscattering coefficient; Interferometric coherence; Semi-arid Mediterranean region; Wheat.

  funded by the LMI TREMA and TOSCA/CNES project. It has been conducted since January 2020. The experimental setup consists of six C-band antennas installed at the top of a 20 m high tower. It allows the full polarization acquisitions of 𝜌 and 𝜎 0 with a 15 minutes time step.

Figure 2 :

 2 Figure 2: Time series of the 𝜎 0 measured by MOCTAR and Sentinel-1 over the wheat field at 06H30 at VH polarization during the 2020 growing season. The Time series of SSM (surface soil moisture), LE (evapotranspiration) are also plotted

Figure 3 :

 3 Figure 3: Average daily cycles of ρ and 𝜎 0 at VH polarization, SSM, ET, and V (wind speed) represented by box-plots for 4 key periods of the agricultural season.

Figure

  Figure 1: Study area located in Morocco