Six-vertex model on a finite lattice: integral representations for nonlocal correlation functions

We consider the problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions. To this aim, we formulate the model as a scalar product of off-shell Bethe states, and, by applying the quantum inverse scattering method, we derive three different integral representations for these states. By suitably combining such representations, and using certain antisymmetrization relation in two sets of variables, it is possible to derive integral representations for various correlation functions. In particular, focusing on the emptiness formation probability, besides reproducing the known result, obtained by other means elsewhere, we provide a new one. By construction, the two representations differ in the number of integrations and their equivalence is related to a hierarchy of highly nontrivial identities.


Introduction
The theory of quantum integrable models can be seen as developing mainly in two directions (see, e.g., [1]).One is related to various (algebraic, analytic and geometric) aspects of integrability (such as Bethe ansatz, Yang-Baxter relation, Baxter T-Q equation, Quantum Groups, etc) of quantum models and mostly deals with the problem of diagonalization of quantum integrals of motion (transfer matrices).The other is related to applications (in gauge and string theories, condensed matters, algebraic combinatorics, probability, etc.) and deals mostly with the problem of evaluation of correlation functions.The latter uses heavily results from the former, but despite significant progress of the theory in general, it still provides numerous challenging problems.One of them concerns the calculation of correlation functions of models with broken translational invariance (e.g., due to the boundary conditions), possibly in cases where keeping finite the size of the system is of importance for extracting interesting (physical or mathematical) information.
A prototypical example is provided by the six-vertex model with domain wall boundary conditions.Current interest in the model is mostly motivated by the occurrence of phase separation [2][3][4][5], which recently triggered a number of numerical studies [6][7][8] and analytical results [9][10][11][12].The model is also of relevance for quantum quenches in the closely related Heisenberg XXZ quantum spin chain [13][14][15][16], and for N = 4 super-Yang-Mills theory [17][18][19].As for correlation functions of the six-vertex model in the bulk (or with periodic boundary conditions), many notable important results were obtained for the XXZ chain (see, among others, papers [20][21][22][23][24][25][26][27][28] and references therein).Although these results cannot be directly applied to the case of domain wall boundary conditions, some aspects, such as multiple integral representations, indeed do prove useful.
Historically, correlation functions of the six-vertex model with domain wall boundary conditions were first studied close to the boundary, where the problem notably simplifies [29][30][31][32][33].Some results and techniques developed in these studies allowed to evaluate a bulk correlation function, the so-called emptiness formation probability, as a multiple integral [34,35].Subsequent study of such integral representation made it possible to derive an exact analytic expression for the spatial curve separating ferroelectric order from disorder (the so-called 'arctic' curve) [36][37][38][39].
In [40], in order to extend previous progress with the emptiness formation probability to other correlation functions, a method for their systematic calculation was proposed.A specific nonlocal correlation function, named row configuration probability, was introduced, in terms of which various other (nonlocal and local) correlation functions can in principle be obtained by suitable summations over position parameters.The row configuration probability can be viewed as a product of two factors, which are in fact components of off-shell Bethe states.These states are complementary to each other in the sense that they are build over different pseudovacuum states (all-spins-up and all-spins-down); their scalar product is exactly the partition function.The main observation in [40] is that these two factors can be represented as multiple integrals involving the same number of integrations.
The main difficulty of the proposed method concerns the possibility of performing suitable summations and integrations to simplify the resulting expressions.As already noted in [40], to reproduce the previously obtained integral representation for the emptiness formation probability, one has to deal with a nontrivial problem of antisymmetrization over two sets of integration variables.In this respect, the existence of some useful antisymmetrization identity allows to show that indeed the proposed approach can be useful in practice [41].
In the present paper, we overview the method of [40,41] and provide some improvements which allow us to obtain further results.We start by formulating the model as a scalar product of off-shell Bethe states; applying the Quantum Inverse Scattering Method (QISM) [42] (see also [1] and references therein), we derive three constructively different integral representations for the components of the off-shell Bethe states.
Next, we show how such representations can be combined to build various local or nonlocal correlation functions.In particular, focusing on the emptiness formation probability, we combine two out of the three available representations and show how the problem of evaluating some intricate multiple sums and integrals can be tackled so that one can recover the multiple integral representation first derived in [34].The antisymmetrization relation proven in [41] plays a crucial role in this alternative derivation.
Finally, by combining a different pair of representations for the components of the off-shell Bethe states, we derive an alternative multiple integral representation for the emptiness formation probability.The existence of two essentially different integral representations for the same object leads to a hierarchy of identities involving the (generating function of the) one-point boundary correlation function.
Further study of such identities is required for a full understanding of their implications.
The paper is organized as follows.In the next section, after giving some definitions and notations, we sketch the strategy of our derivation, and set up QISM in the context of the considered problem.The core calculation of the components of the off-shell Bethe states for the inhomogeneous model is contained in section 3.In section 4 we perform the homogeneous limit, and obtain representations in terms of multiple integrals.In section 5, starting from a suitable combination of two such representations, we reproduce the previously obtained integral representation for the emptiness probability.In section 6 we show that the same procedure, when applied to a different combination of components of off-shell Bethe states, leads to an alternative and essentially different integral representation for the emptiness formation probability.

The model
In this section we define the model, introduce some nonlocal correlation functions of interest, and formulate the problem in the framework the QISM. . . .
Figure 1.The model: (a) The six possible vertex configurations, and their weights; (b) The N × N square lattice with domain wall boundary conditions; also shown is the assigment of parameters λ 1 , . . ., λ N and ν 1 , . . ., ν N to the vertical and horizontal lines, respectively.

The partition function.
The six-vertex model with domain wall boundary conditions is a model of arrows lying on the edges of a square lattice with N horizontal and N vertical lines.Arrow configurations are constrained by the 'icerule', requiring each vertex to have the same number of incoming and outgoing arrows [43].Boltzmann weights are assigned to the six possible vertex configurations of arrows allowed by the ice-rule.With no loss of generality, we require the model to be invariant under reversal of all arrows.We thus have three distinct Boltzmann weights, denoted a, b, c.The domain wall boundary conditions are defined as follows: all arrows on the left and right boundaries are outgoing while all arrows on the top and bottom boundaries are incoming, see figure 1.
To use QISM for calculations we will consider the inhomogeneous version of the model, in which the weights of the vertex being at the intersection of kth horizontal line and αth vertical line are [44] where and we enumerate vertical lines (labelled by Greek indices) from right to left, and horizontal lines (labelled by Latin indices) from top to bottom.The parameters λ 1 , . . ., λ N are assumed to be all different; the same is assumed about ν 1 , . . ., ν N .
To obtain the homogeneous model, after applying QISM, we set these parameters equal within each set, λ α = λ and ν k = ν where, with no loss of generality, we can choose ν = 0.The partition function of the inhomogeneous model is defined as the sum over all possible configurations, each configuration being assigned its Boltzmann weight, which is the product of all vertex weights over the lattice, Here w αk (C) takes values w αk (C) = a αk , b αk , c αk , depending on the configuration C. Because of (2.1), Z N = Z N (λ 1 , . . ., λ N ; ν 1 , . . ., ν N ) where the λ's and ν's are regarded as 'variables'; η is regarded as a parameter (having the meaning of a 'coupling constant') and it is often omitted in notations.In QISM the dependence on λ's and ν's play an important role (in particular, Z N is invariant under permutations within each set of variables).

2.2.
Off-shell Bethe states and correlation functions.An interesting property of the domain-wall six-vertex model, directly following from its peculiar boundary conditions and from the ice-rule, is that on the sth row (i.e., on the N vertical edges between the sth and the (s + 1)th horizontal lines, counted from the top, in our conventions) there are exactly s arrows pointing up.It is thus natural to describe configurations on the sth row in terms of the positions r 1 , . . ., r s , of such s up arrows (counted from the right, and with 1 2).Note that these configurations can equivalently be described in terms of the complementary set of integers r1 , . . ., rN−s , denoting the position of the N − s down arrows.
Let us now suppose we are assigned a given sth-row configuration on the N ×N , described by the positions r 1 , . . ., r s of the s up arrows, and let us imagine cutting all the vertical edges between the sth and (s + 1)th horizontal lines of the N × N lattice, thus separating it into two smaller lattices, see figure 2; as a result, we obtain an upper lattice with s horizontal and N vertical lines, and a lower lattice with N − s horizontal and N vertical lines.The boundary conditions on these two lattices are naturally inherited from the depicted procedure, being of domain wall type on three sides, and with s up arrows at positions r 1 , . . ., r s on the fourth side.We will denote by Z top r1,...,rs and Z bot r1,...,rs the partition functions of the six-vertex model on the upper and lower sublattices, respectively.
The partition functions Z top r1,...,rs and Z bot r1,...,rs can be viewed as components of off-shell Bethe states in the context of the algebraic Bethe ansatz.Besides being of relevance on their own right, they find application in further investigations of the limit shape of the model [48], and in combinatorics [49].But our main interest here is in the fact that Z top r1,...,rs and Z bot r1,...,rs can be used, if suitably combined, as building blocks to compute some useful correlation functions.
Our goal in the present paper is therefore twofold.First, to derive some convenient representations for the partition functions Z top r1,...,rs and Z bot r1,...,rs .Second, to devise how they should be combined to evaluate more sophisticate correlation functions.
As for the first goal, we will rely on the QISM [42], and on some additional ingredients, developed in [32,34], which allows to obtain multiple integral representations for these quantities.Note that some representation for, say, Z top r1,...,rs , has been known for quite a long time: we are referring to the 'coordinate wavefunction' representation, that follows from the equivalence of the algebraic and coordinate versions of the Bethe ansatz [50], see appendix A for details.However, for reasons that will become clear below, in relation to our second goal, such 'coordinate wavefunction' representation is not sufficient for our purposes.In the following we will work out two additional representations for the components of the off-shell Bethe states.
We recall that the crossing symmetry is the symmetry of the Boltzmann weights under reflection of the vertex (together with the orientation of the arrows on its edges) with respect to the vertical (or horizontal) line, with the simultaneous interchange of a αk and b αk .This interchange can be implemented by the substitution ..,rs .However, the dependence of these two representations on the row configuration is in terms of two complementary set of integers (r's and r's), while to combine them conveniently to work out representations of correlation functions, we need the two representations for Z top r1,...,rs and Z bot r1,...,rs to be expressed both in terms of the same set of integers.
In view of these considerations, in the following we will work out two distinct and essentially different representations for the two partition functions, which cannot be obtained one from the other simply by applying the crossing symmetry relation (2.8).As a matter of fact, these two distinct representations comes out essentially from two slightly different ways of applying the QISM machinery.They both appear to differ significantly from the longly known 'coordinate wavefunction' representation.
We turn now to the second main goal of the present paper.As mentioned above, Z top r1,...,rs and Z bot r1,...,rs , if suitably combined, can be used as building block to construct other correlation functions.As a first, simple, example, let us mention the row configuration probability, that is the probability of observing a given configuration of arrows on a given row of the lattice.More specifically, we denote by H (r1,...,rs) N,s , the probability of observing on the sth row of the N × N lattice a configuration with s up arrows at positions r 1 , . . ., r s .This nonlocal correlation function was introduced in [40], and, independently, in [51], in the context of combinatorics.It is clear that the row configuration probability can be written as thus reconducting its evaluation to that of Z top r1,...,rs and Z bot r1,...,rs .If we specialize the row configuration probability to the first line, by setting s = 1, we obtain the probability of observing the sole reversed arrow of the first row exactly on the rth vertical edge.This quantity, called one-point boundary correlation function, was introduced and calculated using QISM in [30].
Besides being interesting on its own right, the row configuration probability can be employed to build useful correlation functions, such as, for instance, the onepoint correlation function, or polarization.Let us denote by G (r,s) N the polarization at point (r, s), that is the probability of observing an up arrow on the rth vertical edge of the sth row of the lattice.Such up arrow may be any of the s up arrows occuring in the sth row.Let us suppose it is the lth one, l = 1, . . ., s.We thus have to sum over the positions of the l − 1 up arrows to its right, 1 r 1 < • • • < r l−1 < r, and over the positions of the s − l up arrows to its left, r < r l+1 < • • • < r s N .The result must then be summed over all possible values of l.We can thus write: ..,r l−1 ,r,r l+1 ,...,rs) N . (2.10) Clearly, one has still to work out some convenient procedure to perform the multiple sums.As a training ground to tackle such problem, we will turn our attention towards some slightly simpler, although nonlocal correlation function, namely, the emptiness formation probability.The emptiness formation probability, denoted by F (r,s) N , describes the probability of having on all edges within a top-left rectangular region of size (N − r) × s arrows pointing down or left, see figure 3a.It was first introduced in [34], where it was shown to satisfy a recurrence relation in r, s and N .Such recurrence relation can be solved, allowing to build an s-fold multiple integral representation.
The domain wall boundary conditions, together with the ice rule, imply that the emptiness formation probability may be equivalently defined as the probability of observing the last N − r arrows between the sth and (s + 1)th horizontal lines to be all pointing down, see figure 3b.It can thus be expressed in terms of the row configuration probability, as a sum over 1 r 1 < . . .< r s r, see figure 3c: (2.11) The problem of performing the multiple sum appears here to be simpler with respect to the above mentioned case of polarization, see (2.10).It will be addressed in One more definition of emptiness formation probability: (a) The same as in figure 3a, but reflected with respect to the SW-NE diagonal and with all arrows reversed; (b) The same as in figure 3b, but now, between the rth and (r + 1)th horizontal lines, among the r up arrows, s of them must be at positions 1, . . ., s; (c) In terms of row configuration probability, as a sum over the positions of the remaining n = r − s up arrows s + 1 r s+1 < . . .< r n+s N , where the dashed line now shows that all positions at 1, . . ., s are occupied.section 5, where, basing on the results of [40] and [41], we will recover by different methods the s-fold multiple integral representation worked out in [34].
For later use, let us discuss here an alternative way to express the emptiness formation probability in terms of the row configuration probability, and of the partition functions Z top r1,...,rs and Z bot r1,...,rs .Recalling that the emptiness formation probability F (r,s) N vanishes for s > r, it is convenient to introduce the lattice coordinate n := r − s, with n = 0, 1, . . ., N − s, giving the distance from the antidiagonal of the lattice.Then, by definition, F (s+n,s) N is a weighted sum over all configurations of the domain-wall N × N lattice, conditioned to have an (N − s − n) × s frozen rectangle in the top-left corner.Due to diagonal symmetry, and symmetry under reversal of all arrows, we can equivalently sum over all configurations conditioned to have a frozen rectangular region of size s × (N − s − n) in the bottom right corner, see figure 4a.
Let us now focus on the configurations of arrows on the N vertical edges of the (s + n)th row, and denote by r 1 , . . ., r s+n the position of the s + n up arrows.Due to the condition on the considered configurations, the first s up arrows (counting from the right) will certainly occur at position 1, . . ., s, that is r j = j, j = 1, . . ., s, see figure 4b.It can thus be expressed in terms of the row configuration probability as a sum over the position of the remaining n = r − s arrows, s + 1 r s+1 < • • • < r s+n N , see figure 4c: (2.12)This expression will be our starting point in section 6, where, resorting once more to the technique developed in [40,41], we will obtain an n-fold (rather than s-fold) multiple integral representation for the emptiness formation probability F (r,s) N .
2.3.Quantum Inverse Scattering Method formulation.We now define the main objects of QISM in relation to the model.First, let us consider vector space C 2 and denote its basis vectors as the spin-up and spin-down states To each horizontal and vertical line of the lattice we associate a copy of the vector space C 2 .We also use the convention that an upward or right arrow corresponds to a spin-up state while a downward or left arrow corresponds to a spin-down state.
Next, we introduce the quantum L-operator, which can be defined as a matrix of the Boltzmann weights.Namely, to each vertex being intersection of the αth vertical line and the kth horizontal line we associate the operator L α,k (λ α , ν k ) which acts in the direct product of two vector spaces C 2 : in the 'horizontal' space H k = C 2 (associated with the kth horizontal line) and in the 'vertical' space V α = C 2 (associated with the αth vertical line).Referring to the scattering matrix picture, we regard arrow states on the top and right edges of the vertex as 'in' indices of the L-operator while those on the bottom and left edges as 'out' ones.Explicitly, the L-operator reads . Here τ 's (σ's) are Pauli matrices of the corresponding vertical (horizontal) vector spaces.
Further, we introduce the monodromy matrix, which is an ordered product of L-operators.We define the monodromy matrix here as a product of L-operators along a vertical line, regarding the corresponding vertical space V α as an 'auxiliary' space, and the tensor product of the N horizontal spaces, H = ⊗ N k=1 H k , as the quantum space.In defining the monodromy matrix it is convenient to think of L-operator as acting in V α ⊗ H and, moreover, writing it as 2-by-2 matrix in V α , with the entries being quantum operators (acting in H), .
Here the subscript indicates that this is a matrix in V α and σ l k (l = +, −, z) denote quantum operators in H acting as Pauli matrices in H k and identically elsewhere.
The monodromy matrix is defined as The operators A(λ) = A(λ; ν 1 , . . ., ν N ), etc, act in H. Operators A(λ), B(λ), C(λ), and D(λ), admit simple graphical interpretation as vertical lines of the lattice, with top and bottom arrows fixed.Let us introduce 'all spins down' and 'all spins up' states where |↑ k and |↓ k are basis vectors of H k .In the case of domain wall boundary conditions each vertical line corresponds to an operator B(λ α ) (where α is the number of the horizontl line) while vectors (2.13) describe states on the right and left boundaries; the partition function reads: To fit the row configuration probability H (r1,...,rs) N,s into the framework of QISM, we consider the following decomposition of the monodromy matrix, where T top (λ) is defined as a product of the s first L-operators and T bot (λ) as the product of the remaining N − s ones: We correspondingly decompose the quantum space H into a 'top' quantum space Correspondingly, we introduce the operators A top (λ), . . ., D top (λ), and A bot (λ), . . ., D bot (λ), as operator valued entries of the corresponding monodromy matrices T top (λ) and T bot (λ), respectively.Such a decomposition was originally introduced in the context of the so-called 'two-site model' [1,52].
It is useful to consider the corresponding decomposition of the 'all spins up' and 'all spin down' vectors.For example, we have |⇑ = |⇑ top ⊗|⇑ bot , where, to fit (2.16), we set and an analogous decomposition for the 'all spins down' vector.It is easy to verify that the above defined vectors are eigenvectors of A top (λ), D top (λ), and A bot (λ), D bot (λ), respectively.In particular, we have: Using the notation introduced above, the partition functions on the upper, N × s, lattice can be written, in the spirit of representation (2.14), Similarly, for the partition function on the lower, N × (N − s), lattice we have

The 'top' and 'bottom' partition functions
In this section we compute the components of the off-shell Bethe states (or partition functions) Z bot r1,...,rs and Z top r1,...,rs using the technique of the commutation relations for the entries of the quantum monodromy matrix (the RTT relation).
3.1.Fundamental commutation relations.One of the most basic relations of QISM is the so-called 'RLL' relation [1,42], which reads Here R αα ′ (λ, λ ′ ), called the R-matrix, is a matrix acting in the direct product of two auxiliary vector spaces, V α ⊗ V α ′ , and it can be conveniently represented as a 4-by-4 matrix (we assume that the first space refers to the 2-by-2 blocks, while the second one to the entries in the blocks): .
It is to be mentioned that here and below we are mainly following notations and conventions of book [1].
The importance of the RLL relation above resides in that it implies the following relation, which, in turn, can be called RTT relation, This relation contains in total 16 commutation relations, between the operators A(λ), B(λ), C(λ), and D(λ).In the following we need only some of these commutation relations, namely Taking into account relation (3.3) and using relation (3.5), one can obtain, in the usual spirit of the algebraic Bethe ansatz calculation (see [1,42]), the relation: Similarly, taking into account (3.2) and using (3.6), one obtains Analogously, relation (3.7) together with (3.3) give Finally, due to (3.4) and (3.8), we have Evidently, decomposition (2.15) for the monodromy matrices implies the existence of RTT relations, analogous to relations (3.1), for the 'top' and 'bottom' quantum spaces.These, in turn, contains all commutation relations between operators A top (λ), . . ., D top (λ), and between operators A bot (λ), . . ., D bot (λ), respectively; below we use commutation relations (3.9)-(3.12)for the 'top' and 'bottom' quantum spaces.
Looking at formulae (2.19) and (2.20), we anticipate that the four relations (3.9)-(3.12)lead to four different representations, two for Z top r1,...,rs and two for Z bot r1,...,rs .In particular, the two resulting representation for, say Z top r1,...,rs , are essentially different, each one being in turn related through crossing symmetry to one of the two representations obtained for Z bot r1,...,rs .
To proceed further, it is convenient to introduce the function where function d(λ, λ ′ ) has been defined in (2.5), and We now reexpress functions f (λ, λ ′ ) and g(λ, λ ′ ) appearing in (3.13) in terms of functions d(λ, λ ′ ) and e(λ, λ ′ ), defined in (2.5) and (3.15).We also substitute the Izergin-Korepin expression (2.3) for the partition function appearing in (3.13).In this lengthy but standard computation we arrive at the formula: Here the function χ(α, β) is defined as while M [α1,...,αs;1,...,s] denotes the (N − s) × (N − s) matrix obtained from the matrix M, see (2.4), by removing rows α 1 , . . ., α s , and the first s columns.Note that, since function v r (λ) vanishes for λ = λ α (α = r + 1, . . ., N ), all the sums appearing in (3.16) can be extended up to the value N .Finally, note that the functions e(λ, µ) appearing in the denominator in last line of (3.16) are exactly compensated by corresponding functions in the numerators, that are hidden in the definition of functions v r (λ).As a consequence, each term of the sum remain regular even in the limit where two λ's differ exactly by 2η.
It is worth to comment that in computing Z bot r1,...,rs here we could have proceeded differently.Namely, starting from representation (2.20), we could have chosen to use commutation relation (3.10) to move all operators A bot (λ) to the left, and make them act on |⇑ bot .As a result, Z bot r1,...,rs would have been expressed as an (N − s)fold sum of minors of order N − s of the matrix M, see (2.4).In this way we would have arrived at an essentially different representation, in comparison with (3.16).As it becomes clear below, it is the combination of these two complementary representations, one for Z bot r1,...,rs and another for Z top r1,...,rs , that may lead to useful representations for the row configuration probability and other correlation functions therefrom.The resulting expression appears to be dual (under crossing symmetry) to the one computed in section 3.2.We report it in appendix B for the sake of completeness.
Here we exploit the second possibility, namely, we commute the B top (λ)'s to the right through the D top (λ)'s.More specifically, we can use commutation relation (3.8) to move all D top (λ)'s to the left, and make them act on ⇓ top |, exploiting relation (2.18).Using s times commutation relation (3.12), acting on the right on the vector |⇑ top , multiplying from the left with the vector Here Z s (λ α1 , . . ., λ αs ; ν 1 , . . ., ν s ) denotes the partition function of the inhomogeneous model on the s × s lattice, with the indicated spectral parameters.Substituting the Izergin-Korepin partition function, one can further rewrite (3.18) as a multiple sum involving s × s determinant; for our purposes below the above representation appears to be sufficient.
As a side comment, it is worth to mention that for s = 1 the formula (3.18) reduces to a sum of r 1 terms.On the other hand, evaluating the weight of the sole possible configuration, one can find that Apparently, the equivalence of the two expressions is due to certain identity; a direct proof of the relevant identity can be found in [53] (see also [30]).A natural question is whether such identity generalizes to values of s > 1.The answer appears to be positive.We report such remarkable identity and discuss some particular cases in appendix C.
To conclude this section, we wish to emphasize that representations (3.13), or equivalently (3.16), for Z bot r1,...,rs , and (3.18) for Z top r1,...,rs , depend on the row configuration through r 1 , . . ., r s denoting the positions of the up arrows.We recall that these representations have been obtained by repeated use of (3.9) and (3.12), respectively.If instead we had worked out analogous derivations starting from relations (3.11) and (3.10), we would have obtained two different representations, depending on the row configuration through the complementary set of integers r1 , . . ., rN−s denoting the position of the down arrows (see appendix B, formula (B.2) for Z top r1,...,rs ).It is clear that the two additional representations are simply related to (3.18) and (3.16) by the crossing symmetry transformation, see (2.8).
Finally, we stress once more that such representations are all essentially different from the so-called 'coordinate wavefunction' representation, that follows from the equivalence of the algebraic and coordinate Bethe ansatz [50].In particular, referring to Z top r1,...,rs for definiteness, representations (3.18), and (B.2) are both different from (A.1).

Integral representations for the 'top' and 'bottom' partition functions
In this section we derive representations for Z bot r1,...,rs and Z top r1,...,rs in terms of s-fold contour integrals.
4.1.Orthogonal polynomial representation for the 'bottom' partition function.Let us first consider Z bot r1,...,rs .To start with, we evaluate the homogeneous limit for expression (3.16).We resort to the procedure successfully used in [32,34].It is based on the observation that the multiple sum in (3.16) reminds the Laplace expansion of some determinant, since the minor appearing in the last line depends on the summed indices only through their absence.Thus the first step is to rewrite representation (3.16) in a determinant form.For this purpose we set where the ξ's will be sent to zero in the limit (as well as the ν's).Keeping the ξ's nonzero (and different from each other), and using the fact that for a function f (x), regular near x = λ, the relation exp It is to be emphasized that this expression is still for the inhomogeneous model; it represents an equivalent way of writing the multiple sum in (3.16).
We can now perform the homogeneous limit along the lines of [47].Specifically, we send ξ 1 , . . ., ξ N and ν 1 , . . ., ν N to zero.The procedure is explained in full detail in [34].Factoring out the partition function of the entire lattice, see (2.6), we obtain where, in writing the determinant, we have changed the order of columns with respect to (4.1).Here and below, when considering the homogeneous model we use the short notation for the weights, a = a(λ, 0), b = b(λ, 0), and for the partition function, Z N = Z N (λ, . . ., λ; 0, . . ., 0).
In order to rewrite (4.2) in integral form, we first transform the N × N determinant into some more convenient and smaller s × s determinant, given in terms of a set of orthogonal polynomials.These polynomials naturally emerge when the determinant of matrix N entering the homogeneous partition function (2.6) is interpreted as a Gram determinant associated to certain integral measure.
The derivation of the s × s determinant representation from (4.2) is based on the following facts.Let {P n (x)} ∞ n=0 be a set of orthogonal polynomials, where the integration domain is assumed over the real axis.The weight µ(x) is real nonnegative and we choose h n 's such that P n (x) = x n + . . ., i.e., that the leading coefficient of P n (x) is equal to one.Let c n denote the nth moment of the weight µ(x), More generally (see, e.g., [54]), for s = 1, . . ., N , the following formula is valid  In our case, c n = ∂ n λ ϕ(λ), and the integration measure µ(x)dx is given by the Laplace transform of the function ϕ(λ); for explicit expressions, see [3].
Noting that sin(ε we have, in virtue of (4.4), the following orthogonal polynomials representation: This representation is valid for arbitrary values of the parameters of the model, independently of the regime.

Integral representation for the 'bottom' partition function.
Our aim now is to rewrite representation (4.7) as a multiple integral.This can be done using the procedure provided in [34], where it was worked out on the example of the emptiness formation probability.
A special role below is played by the one-point boundary correlation function, denoted H The whole procedure of transforming representation (4.7) into a multiple integral representation is based on the following key identity (see [34] for a proof): Here, f (z) is an arbitrary function regular at the origin, C 0 is a small simple closed counterclockwise contour around the point z = 0, and h N (z) (not to be confused with h n in (4.3)) is the generating function of the one-point boundary correlation function (4.8), Clearly, h N (0) = H 1 N = a 2(N −1) cZ N −1 , and h N (1) = 1.Further, we introduce functions h N,s (z 1 , . . ., z s ), where the second subscript, s = 1, . . ., N , refers to the number of arguments.These functions are defined as The functions h N,s (z 1 , . . ., z s ) are symmetric polynomials of degree N − 1 in each of their variables.Due to the structure of (4.10), it is easy to check the relations which will play a crucial role below.The functions h N,s (z 1 , . . ., z s ) can be viewed as the multi-variable generalizations of h N (z) and turn out to be alternative representations (with respect to the Izergin-Korepin partition function) for the partially inhomogeneous partition functions [34].
To proceed, in representation (4.7) it is convenient to express ω(ε) in terms of ω(ε), by means of the identity where we have used the parametrization Next, resorting to identity (4.9), we obtain the multiple integral representation This representation is the final formula for Z bot r1,...,rs .It is valid for arbitrary values of the parameters of the model, independently of the regime.
Note that the expression (4.13) also implies, through the crossing symmetry transformation, an analogous (N −s)-fold integral representation for Z top r1,...,rs , which depends on the row configuration through the positions r1 , . . ., rN−s of the N − s down arrows, for details, see appendix B, formula (B.4).

Integral representation for the 'top' partition function.
Let us now turn to the 'top' partition function, Z top r1,...,rs .Before proceeding, it is worth to mention that a multiple integral representation for such quantity has already been worked out in [40], basing on the well-known 'coordinate wavefunction' representation (A.1), that follows from the equivalence of the algebraic and coordinate Bethe ansatz [50].It reads see appendix A for a derivation.
We will now derive another, significantly different multiple integral representation, which will play a crucial role in the following.Let us turn back to representation (3.18) for Z top r1,...,rs and note that the multiple sum therein can be interpreted as the sum of residues of some function in the s-fold complex plane.Specifically, we can use the following identity where C {λ} := C λ1 ∪ • • • ∪ C λr s is a simple closed counterclockwise contour in the complex plane of the integration variable, enclosing points λ 1 ,. . .,λ rs and no other singularity of the integrand.The function F (ζ 1 , . . ., ζ s ) is a generic analytic function of its variables, regular in each variable within the region delimited by C {λ} .
We now reexpress functions f (λ α , λ β ) and g(λ α , λ β ) appearing in representation (3.18)As a result, we obtain the following multiple integral representation:  It is worth to recall that, besides the already mentioned poles at λ 1 , . . ., λ rs , the only other poles of the integrand within the strip of width π originate from the functions a(ζ j , ν k ) appearing in the denominator.As for Z s , it is a regular function of its spectral parameters.Concerning the functions e(ζ k , ζ j ) in the denominator, the poles they give rise to are only apparent.Indeed, focussing (4.16), a careful comparison with (3.18) shows that the functions e(λ α k , λ αj ) appearing in the denominator have been introduced to conveniently rewrite the products of function e(λ αj , λ βj ) in the numerator.In other words, the pole induced by the functions e(ζ k , ζ j ) in the denominator of (4.17) are exactly cancelled by corresponding zeroes in the numerator, hidden in the products of function e(ζ j , λ βj ).
In representation (4.17) for Z top r1,...,rs , the homogeneous limit in spectral parameters λ 1 , . . ., λ s , can be done in a straightforward way, since no new singularity arises as two λ j 's assume the same value.We just set λ 1 = λ 2 = • • • = λ s = λ, and get where C λ is a small simple closed counterclockwise contour around point λ.This representation can be related to the so-called 'coordinate wavefunction' representation (A.1) by means of a suitable deformation of the integration contours; we refer to last part of appendix A for details.
Next, we perform the homogeneous limit in spectral parameters ν 1 , . . ., ν s .This is straighforward as well, since all expressions appearing in (4.18) are regular separately in this limit.We get To proceed further we need to use the following identity expressing the partially inhomogeneous partition function Z N (λ 1 , . . ., λ N ) ≡ Z N (λ 1 , . . ., λ N ; 0, . . ., 0) in terms of the generating function for the one-point boundary correlation function (see [34] for further details and proof): Here, the function γ(ξ) also depends on λ (and η) as a parameter and reads: Using now the relation (4.19) and changing the integration variables Formula (4.21) is one of our main results here.
Note that our last integral representation (4.21) differs significantly from (4.14), which is based on the 'coordinate wavefunction' representation [50].It appears that these two representations can be related by a suitable deformation of integration contours.Such relation is most easily seen at the level of (4.18), that is for the inhomegeneous version of the model, see appendix A for details.
It is worth mentioning that the expression (4.21) also implies, through crossing symmetry, an analogous (N −s)-fold integral representation for Z bot r1,...,rs , depending on the row configuration through the position r1 , . . ., rN−s of the N −s down arrows.This representation is given in appendix B, see formula (B.5).
Finally, we emphasize that that the two procedures leading from the inhomogeneous representations (3.16) and (3.18) to the multiple integral representations (4.13) and (4.21) for Z bot r1,...,rs and Z top r1,...,rs are quite different and cannot be interchanged.
To recapitulate, we have thus in total three different representations for Z top r1,...,rs and three for Z bot r1,...,rs .Concerning Z top r1,...,rs , we have the two representations (4.21) and (B.4), besides the well-known 'coordinate wavefunction' representation, see (4.14) for its multiple integral form.Corresponding representations for Z bot r1,...,rs , related to (4.21) and (B.4) by crossing symmetry, are given by (B.5) and (4.13), respectively.Each of these representations appears to depend on the row configuration either through the r's or through the r's.

Emptiness formation probability
In this section, we show how the integral representation for the emptiness formation probability derived in [34] can be recovered from the integral representations for Z bot r1,...,rs and Z top r1,...,rs .The alternative derivation presented here relies on certain relation involving antisymmetrization with respect to two set of variables, recently proved in [41].
We discuss here two antisymmetrization relations playing a relevant role in the calculation of integral representations for the emptiness formation probability.
The second antisymmetrization relation we wish to discuss reads [41]: where .
The relation (5.5) can be proven by induction in s, using the symmetries in the involved variables and comparing singularities of both sides, along the lines of the proof of the relation (5.1) given in [20], see appendix C therein.It is to be mentioned that similar relations appear in connection with the theory of symmetric polynomials [56][57][58][59][60]. Relation (5.5) does not seem to be a particular case of any of them, even if sharing the property that its right-hand side is expressible in terms of the Izergin-Korepin partition function (2.3).Instead, it appears to extend to the trigonometric case some antisymmetrization relation originally derived in the rational case by Gaudin, see [53], Appendix B. Also, (5.5) generalizes some antisymmetrization relation given in [61], in the context of the asymmetric simple exclusion process.
It is convenient to introduce the notation W s (x 1 , . . ., x s ; y 1 , . . ., y s ) = s j,k=1 (x j + y k − 2∆x j y k ) The last line is easily evaluated thanks to the recursion relation for the inhomogeneous partition function [45].We have Reexpressing now the right-hand side of (5.11) in terms of variables x 1 , . . ., x j , we finally obtain which will turn out useful below.

Known integral representations.
In [34] various representations has been worked out for the emptiness formation probability F (r,s) N .In particular, using the Yang-Baxter commutation relations, and next performing the homogeneous limit, the following representation in terms of the orthogonal polynomials was obtained: ε1,...,εs=0 . (5.13) Here, we use the same notations as in section 4.1.In particular, K n (x) denote the orthogonal polynomials (4.5), associated to the Hankel matrix (2.7), and the functions ω(ε) and ω(ε) are defined in (4.6).
The identity (4.9) when applied to (5.13), yields for the emptiness formation probability the following multiple integral representation: Here, C 0 denotes, as before, a small simple anticlockwise oriented contour around the point z = 0, and the function h N,s (z 1 , . . ., z s ) is defined in (4.10).
Note that the integrand in (5.14) is not symmetric with respect to the permutation of the integration variables.However, the antisymmetrization relation (5.4) allows to write down the essentially equivalent representation Here, u j 's are given in terms of z j 's by (5.3).The representation (5.15) with the symmetric integrand has been proved of importance, for example, in the evaluation of the phase separation curves of the model [37,38].
Given integral representation (5.15) for the emptiness formation probability, a natural question concerns the possibility of deriving it by suitably combining the integral expressions obtained above for Z top r1,...,rs and Z bot r1,...,rs .As we will show below, the answer is affermative.

5.
3. An alternative and simpler derivation.We propose here an alternative derivation of (5.15), with respect to the one originally proposed in [34].Here we start from the relation (2.11), and substitute the integral representations (4.14) and (4.13) for Z top r1,...,rs and Z bot r1,...,rs in the expression for the row configuration probability (2.9).An essential role in the derivation is played by relation (5.5).
For convenience, we change the integration variables z j → x j /t, j = 1, . . ., s in (4.13), that yields (5.16) and also we change w j → 1/(ty j ) in (4.14), that yields Z top r1,...,rs = c s a s(N −1) Then, inserting into (2.9) and (2.11), we have To prove that this representation indeed reduces to (5.15), one has to perform the multiple sum and evaluate s integrations.
First, let us focus on the multiple sum in (5.17).It is clear that, due to the integrations around the points x j = 0, j = 1, . . ., s, one can extend the sum over the values 1 (5.18) Hence, (5.17) simplifies to To perform the integrations, we first observe that the form of the integrand in (5.19) allows us to apply the antisymmetrization relation (5.5).We obtain where we have used the notation (5.6).

Another representation for the emptiness formation probability
In this section we derive an alternative representation for the emptiness formation probability, which supplements the already known representation (5.15).Crucial ingredients in such derivation are: i) the use of relation (2.12) rather than (2.11) to express the emptiness formation probability in terms of Z top r1,...,rs and Z bot r1,...,rs , and ii) the use of representation (4.21), rather than (4.14), for Z top r1,...,rs .Although the starting point is quite different, the derivation has some similarities with that of section 5.3, in particular, we will again use the antisymmetrization relation (5.5).
where we have used the notation (5.7).
Turning now to the integration with respect to variables z 1 , . . ., z n , we observe that the corresponding integration contours C 0 can be deformed into new contours C {1/w} := C 1/w1 ∪ • • • ∪ C 1/wn , enclosing the poles at 1/w j , j = 1, . . ., n, induced by the factor 1 j,k n (1 − w j z k ) in the denominator of (6.4), without changing the result.
The crucial ingredient permitting such deformation of contours is that the poles induced by the double product in the second line of (6.4) give a vanishing contribution to the integral.This should not come as a surprise, since such double product is the remnant of analogous double products appearing in (3.16) and (4.17).As already commented thereafter, the poles associated to such double products are exactly compensated by corresponding zeroes.However, once the homogeneous limit is performed, one loose track of this fact, and the mechanism of cancellation becomes quite subtle.A direct and detailed description is therefore appropriate.
To start with, let us focus on variable z n .The poles induced by the double product in the second line of (6.4) can be divided into a first set of n − 1 poles at positions (2∆z j − 1)/z j , j = 1, . . ., n − 1, and a second set of n − 1 poles at positions 1/(2∆−z j ), j = 1, . . ., n−1.Concerning the first set of poles, recalling the symmetry of the integrand under interchange of the z j 's, let us focus for definiteness on the pole corresponding to j = 1, and show that the residue of the integrand in (6.4) at z n = (2∆z 1 − 1)/z 1 vanishes upon integration with respect to variable z 1 over contour C 0 .
Indeed, let us inspect the small z 1 behaviour of the residue of the integrand at z n = (2∆z 1 − 1)/z 1 .Such behaviour results from the contribution of four terms.
Let us analyze them in turn.For the first relevant term, it is easily checked that The next term, which contains the considered pole, requires a careful but neverthless straightforward calculation, that gives lim As for the third relevant term, recalling that P n (w 1 , . . ., w n ; z 1 , . . ., z n ) is a polynomial of degree n − 1 in each of its variables, it follows that Finally, concerning the contribution of h N,n (z 1 /t, . . ., z n /t), its calculation is nontrivial.However we can resort to the following property which follows from the determinantal structure of the Izergin-Korepin partition function, and has been proven in [35], see appendix therein.In all, it follows that the residue of the integrand in (6.4) at z n = (2∆z 1 − 1)/z 1 is O(1) as z 1 → 0, and thus vanishes upon integration with respect to variable z 1 over contour C 0 .Due to the simmetry of the integrand under interchange of the z j 's, the same holds for variables z 2 , . . ., z n−1 .The integration contour for z n can thus be deformed from C 0 to a new contour C ′ enclosing the origin and the poles at z n = (2∆z j − 1)/z j , j = 1, . . ., n − 1.
Inspecting now the large |z n | behaviour of the integrand, we observe that the double product in the second line is O(1).We also have As a result, the whole integrand is O(1/z 2 n ) as |z n | → ∞, and the integration over z n along a large contour vanishes.It follows that we can deform the integration contour over variable z n from C ′ , defined in the previous paragraph to a new contour C ′′ enclosing the poles at z n = 1/(2∆−z j ), j = 1, . . ., n−1, and the poles at z n = 1/w l , l = 1, . . ., n, induced by the factor 1 j,k n (1 − w j z k ) in the denominator of (6.4).
Implementing the same procedure for other integration variables as well, we conclude that for each z j , the corresponding integration contour can be deformed into a contour C ′′ enclosing only the poles at z j = 1/w l , l = 1, . . ., n, induced by the denominator in the last line of (6.4), and the poles at z j = 1/(2∆ − z l ), l = 1, . . ., j − 1.Now, concerning the poles at z n = 1/(2∆ − z j ), j = 1, . . ., n, it appears that their contributions vanishes as well.Indeed, focussing again, for definiteness, on the pole corresponding to j = 1, z n = 1/(2∆ − z 1 ), consider the following crucial property of the multivariate polynomial P n (w 1 , . . ., w n ; z 1 , . . ., z n ), P n (w 1 , . . ., w n ; z 1 , . . ., z n ) whose proof goes along the lines of (6.5).This property implies that, after evaluation of the residue of the integrand in (6.4) at z n = 1/(2∆ − z 1 ), the poles at z 1 = 1/w k , k = 1, . . ., n, are all cancelled.It other words, the residue of the integrand at z n = 1/(2∆ − z j ), when subsequently integrated with respect to z j over contour C {1/w} , gives a vanishing contribution.It follows that we can deform the integration contour over variable z n from C ′′ to shrink it down to C {1/w} .Implementing the same procedure for other integration variables as well, we have eventually shown that we can deform each of the integration contours for variables z 1 , . . ., z n from C 0 to a new contour C {1/w} enclosing the poles at 1/w k , k = 1, . . ., n.In other words, (6.4) can be rewritten as 1 (1 − tw j )(w j z j ) N −s × n j,k=1 j =k (w k − w j )(z k − z j ) (w j w k − 2∆w j + 1)(z j z k − 2∆z j + 1) The main benefit of this last expression is evident: the integrations with respect to the variables z 1 , . . ., z n involve now only residues at simple poles.where we have shrunk each of the n integration contours from a very large one down to a small contour enclosing only the pole at z j = 1, j = 1, . . ., n, thus ignoring the contribution of the poles in the double product.Once again, it can be shown that the total contribution of such poles indeed vanishes.
We consider the multiple integral representation (6.7) as one of the main results of the present paper.At variance with the previously known representation (5.15), the number of integrations is n = r − s, that is the lattice distance of the point (r, s) from the antidiagonal, rather than s, the lattice distance from the top boundary.This could be useful to investigate the behaviour of the emptiness formation probability in the so-called Hamiltonian limit, of relevance in connection with quantum quenches of the XXZ quantum spin chain [13][14][15][16].
The primary ingredients in the derivation of the multiple integral representation (6.7) are: i) the evaluation of two alternative representations for the components of the off-shell Bethe states Z top r1,...,rs and Z bot r1,...,rs , that are essentially different from the longly known 'coordinate wavefunction' representation, and ii) the symmetrization relation (5.5).Hopefully, such ingredients could turn useful in the derivation of integral representations for more advanced correlation functions, such as polarization.
The availaibility of two distinct integral representations for the same quantity F .Evaluating it by means of (5.15), with r = 2, s = 1, or (6.7), with s = 1, n = 1, and equating the two results, we get: )]h N (0) relating the first derivatives of function h N (z), h N −1 (z), evaluated at z = 0, and z = 1.Similarly, considering F , one obtain an identity relating the second derivatives of function h N (z), h N −1 (z), h N −2 (z), evaluated at z = 0, and z = 1.Remarkably, the coefficients appearing in such relations do not depend on N .The game can be played further, but the calculations become quite bulky very soon.The existence of such hierachy of identities hints at some nontrivial functional identity for h N (z), which is essentially Izergin-Korepin partition function with just one inhomogeneity.

Figure 2 .
Figure 2. (a) A possible sth-row configuration, with s up arrows at positions r 1 , . . ., r s (here, N = 7, s = 2, r 1 = 2, r 2 = 4); (b) The corresponding top and bottom portions resulting from splitting the original lattice in correspondence of the sth row.

Figure 3 .
Figure 3. Emptiness formation probability: (a) Basic definition, as the probability of having on all edges within a rectangular region of size (N − r) × s in the top-left corner of the lattice, arrows pointing down or left (here is shown the case s = 2, r = 4 and N = 7); (b) Equivalent definition, as the probability of having, between the sth and (s + 1)th horizontal lines, N − r down arrows at positions N − r + 1, . . ., N ; (c) In terms of the row configuration probability, as a sum over 1 r 1 < . . .< r s r, where the dashed line shows the border which cannot be passed by the positions r 1 , . . ., r s of the up arrows in the summation.

3. 3 .
Application to the 'top' partition function.Let us turn to the partition function on the upper sublattice.Having in mind representation (2.19) for Z top r1,...,rs , we can use the fundamental RTT relations to move all B top (λ)'s on one side, and all D top (λ)'s on the other.As already outlined on the example of Z bot r1,...,rs , one can implement this procedure in two different ways.The first possibility is to use commutation relation (3.11) to commute each of the D top (λ)'s to the right, through B top (λ)'s.