New determinants in the 8VSOS model with domain-wall boundaries
W. Galleas









 













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    ]New determinants in the 8VSOS model 
             with domain-wall boundariesInstitut für Theoretische Physik, Eidgenössische Technische Hochschule Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerlandgalleasw@phys.ethz.ch[2010]82B23; 39B32The work of W.G. is partially supported by the Swiss National Science Foundation through the NCCR SwissMAP and by grant no. 615203 from the European Research Council under the FP7.
            
            In this letter we show the partition function of the 8VSOS model with domain-wall boundaries satisfies the same type of functional equations as its six-vertex model counterpart. We then use these refined functional equations to obtain novel determinantal representations for the aforementioned partition function.
            
            [
                W. Galleas
                
            ==============
            
            
            
            
            § INTRODUCTION
            
            
            Exactly solvable models of Statistical Mechanics usually have a rich mathematical structure on display and some of these structures are not limited to a single model. 
            For instance, Baxter's t-q relations can be found in several integrable two-dimensional lattice models with toroidal boundary conditions <cit.>; and for such models we also find their transfer matrices' spectrum parameterized by solutions of Bethe ansatz equations <cit.>.
            Moreover, some special determinants have also made a striking appearance in the computation of correlation functions of integrable models <cit.>; and it is worth remarking this feature goes back to Kaufman and Onsager's works on the two-dimensional Ising model <cit.>. See also <cit.> and references therein for a review on the subject.
            
            Correlation functions of certain integrable systems have also been shown to be closely related to partition functions of lattice models with
            particular boundary conditions. For instance, Korepin has introduced in <cit.> the six-vertex model with domain-wall boundary conditions as a building block of correlation functions of the xxz spin-chain. Interestingly, the partition function of such six-vertex model can also be expressed as a determinant as shown in <cit.>. 
            This scenario is not limited to the six-vertex model and it has also been established for the 8VSOS model, also known as elliptic 𝔤𝔩_2 solid-on-solid model, under certain constraints <cit.>. In that case the relevant partition function with domain-wall boundaries has been studied in <cit.> where representations in terms of contour integrals and determinants have been obtained. As far as the above-mentioned  determinantal formulae are concerned, the partition function of the 8VSOS model with domain-wall boundary conditions has been written as a sum of Frobenius determinants in <cit.>. Alternatively, the partition function of the 8VSOS model has also been written as a formal single determinant in <cit.> and as a continuous family of determinants in <cit.>. 
            In particular, it is important to remark the latter determinantal formulae originate constructively from the direct resolution of certain functional equations satisfied by the model's partition function.
            
            Both partition functions of the six-vertex and the 8VSOS models with domain-wall boundaries were shown in <cit.> to satisfy certain functional equations originated from the Yang-Baxter algebra and/or its dynamical version.
            However, although the six-vertex model can be regarded as a special limit of the more general 8VSOS model, the structure of the functional equations obtained for these models in <cit.> differs considerably.
            This is precisely one of the points we intend to address in the present work. More exactly, here we intend to show the partition function of the 8VSOS model with domain-wall boundary conditions also satisfies the same type of functional equations as its six-vertex model counterpart.
            
            The functional equation in discussion simply reads 
            
                ∑_i=0^n M_i  ℱ(x_0, x_1, … , x_i, … , x_n) = 0
            
            with ℱ an unknown n-variable symmetric function and coefficients M_i= M_i (x_0, x_1 , … , x_n) on (n+1)-variables x_i ∈.
            One important aspect of such equation seems to be its versatility in accommodating several quantities associated to integrable systems.
            For instance, it can be used to describe partition functions with special boundary conditions <cit.> and scalar products of Bethe vectors <cit.>.
            Moreover, our functional equation seems to be closely related to the theory of integrable differential equations and it was shown in <cit.> to correspond to an analogue of the auxiliar linear problem leading to Lax equation<cit.>. In this way, the above-mentioned functional equation can be regarded as a common structure shared by several integrable systems. Moreover, determinantal solutions emerge naturally from such type of equations as shown in <cit.>; and this is the feature we intend to exploit in this work for the 8VSOS model with domain-wall boundaries.
            
            This paper is organized as follows. In sec:CONV we describe briefly the quantities of interest for this work and introduce conventions employed throughout this paper. We then refer the reader to <cit.> for a more detailed discussion on those quantities. In sec:CONV we also present the functional equations satisfied by the partition function of interest; which have been previously derived with the aid of the dynamical Yang-Baxter algebra. sec:FUN is then devoted to presenting modified versions of the original functional equations and also to showing how they can be brought to the common structure above discussed. We then use sec:DET to discuss the resolution of such equations and concluding remarks are presented in sec:REM.
            
            
            
            
            
            § PARTITION FUNCTION
            
            
            The solid-on-solid model of interest here consists of the juxtaposition of L× L square plaquettes to which statistical weights are assigned. More precisely, we consider coordinates (i,j) ∈ℒ×ℒ with ℒ{ 1, 2, …, L+1 } and write 
            w_i,j for the statistical weight associated to the plaquette enclosed by coordinates (i,j), (i,j+1), (i+1,j) and (i+1,j+1).
            Statistical fluctuations of a given plaquette w_i,j are then characterized by the set of variables { h_i,j, h_i,j+1, h_i+1,j , h_i+1,j+1} with h_i,j usually referred to as height function.
            Moreover, here we will be considering h_i,j = τ + n_i,jγ  with τ, γ∈ and n_i,j∈; and also restrict the height functions in such a way that h_i,j and h_i',j' at neighboring sites only differ by ±γ.
            
            
            We then want to associate a partition function to such statistical system and for that we still need to assign explicit weights w_i,j for allowed configurations of plaquettes; and declare the boundary conditions under consideration. As for Baxter's 8VSOS model we have
            
            w_ij[ τ±γ   τ;   τ τ∓γ ]   =    [x_i - μ_j + γ]  
            
            w_ij[   τ±γ τ± 2γ;     τ   τ±γ ]   =    [τ±γ] [x_i - μ_j] [τ]^-1
             
            w_ij[ τ±γ   τ;   τ τ±γ ]   =     [τ± x_i ∓μ_j] [γ] [τ]^-1
            with [x] corresponding to a Jacobi theta function. More precisely, 
            
                [x] 1/2∑_n = -∞^+ ∞ (-1)^n-1/2 p^(n+1/2)^2 e^-(2n+1) x
            
            with x_i, μ_j ∈ℂ and fixed elliptic nome  0 < p < 1. According to the conventions of <cit.>, the function
            [x] equals the Jacobi theta-function Θ_1 ( x, ν) with p = e^πν. Lastly, we need to declare the model's boundary conditions and here we will be considering domain-walls with h_1,j = h_j,1 = τ + (L+1-j)γ and h_L+1,j = h_j,L+1 = τ + (j-1)γ.
            In this way, the partition function of the inhomogeneous 8VSOS model with domain-wall boundaries reads
            
                𝒵_τ (x_1, x_2, … , x_L |μ_1, μ_2, … , μ_L) ∑_{ h_i,j}∏_i,j=1^L w_ij[   h_i+1,j h_i+1,j+1;     h_i,j   h_i,j+1 ]  .
            
            Although both parameters x_i and μ_j correspond to lattice inhomogeneities, in the literature they are usually referred to as spectral and inhomogeneity parameters respectively. Moreover, τ is usually referred to as dynamical parameter whilst γ receives the name anisotropy parameter. The latter is fixed in our analysis and here we omit its dependence in the LHS of (<ref>).
            
            The partition function (<ref>) enjoys remarkable properties due to the choice of statistical weights (<ref>) as discussed in <cit.>. In particular, it is a symmetric function with respect to the variables x_i and, in order to simplify our notation, it is convenient to introduce the following conventions. 
            
            Let X{ x_1, x_2 , … , x_L } and additionally write X_i^αX∪{ x_α}\{ x_i } and X_i,j^α, βX∪{x_α, x_β}\{x_i, x_j }.
            
            Taking into account Definition <ref>, we shall then simply write 𝒵_τ ( X ) = 𝒵_τ (x_1, x_2, … , x_L |μ_1, μ_2, … , μ_L) for the partition function (<ref>).
            
            
            Now, given the above described conventions, in what follows we shall present the functional equations satisfied by 𝒵_τ. The latter equations have been previously derived in <cit.>.
            
            Equation type A. In our previous works we have used the Algebraic-Functional framework put forward in <cit.> to derive functional equations governing the partition function (<ref>). One of them is referred to as equation type A and it reads
            
                M_0^(𝒜) 𝒵_τ - γ (X) + ∑_i = 0^L N_i^(𝒜) 𝒵_τ (X_i^0) = 0
            
            with coefficients given by
            
            M_0^(𝒜)      [τ]/[τ + L γ]∏_j=1^L [x_0 - μ_j] 
            
            N_0^(𝒜)       -[τ + γ]/[τ + (L+1)γ]∏_j=1^L [x_0 - μ_j + γ] ∏_j=1^L[x_j - x_0 + γ]/[x_j - x_0]
            
            N_i^(𝒜)      [τ + γ + x_0 - x_i] [τ - γ]/[x_i - x_0][τ + (L+1)γ]∏_j=1^L [x_i - μ_j + γ] ∏_j ≠ ij=1^L [x_j - x_i + γ]/[x_j - x_i]     i = 1,2, … , L   . 
            Equation type D. Equation (<ref>) is not the only functional equation describing  𝒵_τ which can be derived along the lines of the Algebraic-Functional method. In addition to (<ref>), here we shall also consider the equation type D reading
            
                M_0^(𝒟) 𝒵_τ + γ (X) + ∑_i = 0^L N_i^(𝒟) 𝒵_τ (X_i^0) = 0   .
            
            The coefficients in (<ref>) are then given by
            
            M_0^(𝒟)      ∏_j=1^L [x_0 - μ_j + γ] 
            
            N_0^(𝒟)       - ∏_j=1^L [x_0 - μ_j] ∏_j=1^L[x_0 - x_j + γ]/[x_0 - x_j]
            
            N_i^(𝒟)      [γ] [τ + (L+1)γ + x_0 - x_i]/[x_0 - x_i] [τ + (L+1)γ]∏_j=1^L [x_i - μ_j] ∏_j=1 
             j ≠ i^L [x_i - x_j + γ]/[x_i - x_j]     i = 1,2, … , L   . 
            
            
            Both equations (<ref>) and (<ref>) are able to characterize 𝒵_τ in closed form as shown in <cit.>. However, the dependence of such equations on the dynamical parameter τ creates some difficulties for obtaining determinant representations for our partition function directly from (<ref>) and (<ref>). In the next section we shall then discuss reformulations of the equations type A and D which will help us circumventing that problem. 
            
            
            
            
            
            
            
            § MODIFIED FUNCTIONAL RELATIONS
            
            
            In this section we want to find modified versions of (<ref>) and (<ref>) where the dynamical parameter τ no longer plays the role of variable. In order to obtain such equations we shall then first proceed along the lines described in <cit.>. 
            
            
            The partition function (<ref>) satisfies the functional relation
            
                ℳ_0^(𝒜) 𝒵_τ (X) + ∑_i=1^L 𝒩_i^(𝒜) 𝒵_τ (X_i^0) + ∑_i=1^L 𝒩̅_i^(𝒜) 𝒵_τ (X_i^0̅) = 0
            
            with coefficients defined as
            ℳ_0^(𝒜)      ∏_j=1^L [x_0̅ - μ_j + γ][x_j - x_0̅ + γ]/[x_0̅ - μ_j][x_j - x_0̅] - ∏_j=1^L [x_0 - μ_j + γ][x_j - x_0 + γ]/[x_0 - μ_j][x_j - x_0]
            𝒩_i^(𝒜)      [γ][τ + γ + x_0 - x_i]/[τ + γ] [x_i - x_0]∏_j=1^L [x_i - μ_j + γ]/[x_0 - μ_j]∏_j = 1 
             j ≠ i^L [x_j - x_i + γ]/[x_j - x_i]
            𝒩̅_i^(𝒜)      [γ][τ + γ + x_0̅ - x_i]/[τ + γ] [x_0̅ - x_i]∏_j=1^L [x_i - μ_j + γ]/[x_0̅ - μ_j]∏_j = 1 
             j ≠ i^L [x_j - x_i + γ]/[x_j - x_i]  . 
            
            The functional equation 
            
                ℳ_0^(𝒟) 𝒵_τ (X) + ∑_i=1^L 𝒩_i^(𝒟) 𝒵_τ (X_i^0) + ∑_i=1^L 𝒩̅_i^(𝒟) 𝒵_τ (X_i^0̅) = 0
            
            with coefficients reading
            ℳ_0^(𝒟)      ∏_j=1^L [x_0̅ - μ_j][x_0̅ - x_j + γ]/[x_0̅ - μ_j + γ][x_0̅ - x_j] - ∏_j=1^L [x_0 - μ_j][x_0 - x_j + γ]/[x_0 - μ_j + γ][x_0 - x_j]
            𝒩_i^(𝒟)      [γ][τ + (L+1)γ + x_0 - x_i]/[τ + (L+1)γ][x_0 - x_i]∏_j=1^L [x_i - μ_j]/[x_0 - μ_j + γ]∏_j = 1 
             j ≠ i^L [x_i - x_j + γ]/[x_i - x_j]
            𝒩̅_i^(𝒟)      [γ][τ + (L+1)γ + x_0̅ - x_i]/[τ + (L+1)γ][x_i - x_0̅]∏_j=1^L [x_i - μ_j]/[x_0̅ - μ_j + γ]∏_j = 1 
             j ≠ i^L [x_i - x_j + γ]/[x_i - x_j]
            
            is fulfilled by the partition function (<ref>).
            
            The proof of Lemma <ref> has been already presented in <cit.>, and the proof of Lemma <ref> is analogous. 
            Therefore, we shall describe here the proof of Lemma <ref> only for completeness reasons. We start our analysis by rewriting 
            (<ref>) as
            
                𝒵_τ - γ (X) = - ∑_i = 0^L N_i^(𝒜) (x_0, x_1, … , x_L)/M_0^(𝒜) (x_0, x_1, … , x_L) 𝒵_τ (X_i^0)
            
            with M_0^(𝒜)  = M_0^(𝒜) (x_0, x_1, … , x_L) and N_i^(𝒜)  = N_i^(𝒜) (x_0, x_1, … , x_L) in order to emphasize the dependence of the coefficients on the relevant set of variables. Next we notice the LHS of (<ref>) is independent of x_0. Consequently, one can write
            
                ∑_i = 0^L N_i^(𝒜) (x_0, x_1, … , x_L)/M_0^(𝒜) (x_0, x_1, … , x_L) 𝒵_τ (X_i^0) = ∑_i = 0^L N_i^(𝒜) (x_0̅, x_1, … , x_L)/M_0^(𝒜) (x_0̅, x_1, … , x_L) 𝒵_τ (X_i̅^0̅)
            
            for generic variables x_0, x_0̅∈ and i̅ = i for 1 ≤ i ≤ L. Equation (<ref>) is then identified with (<ref>).
            
            The coefficients (<ref>) differ from the ones presented in <cit.> only by an overall multiplicative common factor.
            
            
            As far as the structure of the equations (<ref>) and (<ref>) is concerned, it is important to remark it differs significantly from the original equations (<ref>) and (<ref>). For instance, in (<ref>) and (<ref>) the dynamical parameter τ plays the role of variable whilst it can be fixed in equations (<ref>) and (<ref>). 
            On the other hand, (<ref>) and (<ref>) also depends on an extra spectral variable, namely x_0̅, in addition to the set 
            {x_0, x_1, …, x_L } present in the original equations (<ref>) and (<ref>).
            Next we remind the reader our goal here is to find an equation describing 𝒵_τ with the same structure as the one describing the partition function of the six-vertex model with domain-wall boundaries. Such equation for the six-vertex model can be read off from (<ref>) and (<ref>) simply by taking the trigonometric limit of (<ref>) followed  by the limit τ→∞. At first sight it seems we would achieve such structure by setting x_0̅ = x_0
            in (<ref>) and (<ref>). However, the inspection of the coefficients (<ref>) and (<ref>) shows this is not the case. We shall then describe an alternative route for unveiling such equation and for that it is useful to introduce the following conventions.
            
            [Permutation] 
            Let Fun(^L+2) be the space of meromorphic functions on ^L+2 and 𝔖_L+2 the symmetric group acting by permutations on (x_0 , x_1 , … , x_L, x_0̅) ∈^L+2. We then write Π_i,j∈𝒮_L+2 for the 2-cycle permuting variables x_i and x_j.
            
            
            
            Next we proceed by inspecting the action of Π_i,j on equations (<ref>) and (<ref>). As for 1 ≤ i , j ≤ L we find our equations are invariant under the action of Π_i,j. On the other hand, the action of Π_0̅,m∘Π_0,l for 0 ≤ l < m ≤ L produces new independent equations as shown in <cit.>. More precisely, the action of Π_0̅,m∘Π_0,l on (<ref>) and 
            (<ref>) unfold each one of those equations into the system
            𝒫_0^(l,m) 𝒵_τ (X) + ∑_i=1^L 𝒬_i^(l,m) 𝒵_τ (X_i^0) + ∑_i=1^L 𝒬̅_i^(l,m) 𝒵_τ (X_i^0̅) + ∑_1 ≤ i, j ≤ Lℛ_ij^(l,m) 𝒵_τ ( X_i,j^0, 0̅ ) = 0  
            
            with indexes l and m on the interval 0 ≤ l < m ≤ L. The coefficients appearing in (<ref>) require some clarifications but, before discussing them, we remark (<ref>) contains one term 𝒵_τ (X), L terms of the form 𝒵_τ (X_i^0), another L terms of the form 𝒵_τ (X_i^0̅); and L(L-1)/2 terms 𝒵_τ ( X_i,j^0, 0̅ ). As for the coefficients in (<ref>), they read
            
                𝒫_0^(0, m)   Π_0̅, m 𝒩̅_m^(𝒯)   𝒬̅_j^(0, m)   Π_0̅, m ℳ_0^(𝒯)     j=m 
                Π_0̅, m 𝒩̅_j^(𝒯)    
                𝒬_j^(0, m)   Π_0̅, m 𝒩_m^(𝒯)     j=m 
                 0            ℛ_ij^(0, m)   Π_0̅, m 𝒩_i^(𝒯)     j=m 
                Π_0̅, m 𝒩_j^(𝒯)     i=m 
                 0
            
            for 1 ≤ m ≤ L and 
            
                𝒫_0^(l, m)    0    𝒬̅_j^(l, m)   Π_0̅, m∘Π_0, l 𝒩̅_l^(𝒯)     j=l 
                Π_0̅, m∘Π_0, l 𝒩_l^(𝒯)      j=m 
                 0               
                𝒬_j^(l, m)   Π_0̅, m∘Π_0, l 𝒩̅_m^(𝒯)     j=l 
                Π_0̅, m∘Π_0, l 𝒩_m^(𝒯)     j=m 
                 0                  ℛ_ij^(l, m)   Π_0̅, m∘Π_0, l ℳ_0^(𝒯)     i=l, j=m 
                Π_0̅, m∘Π_0, l 𝒩̅_j^(𝒯)      i=l, j ≠ m 
                Π_0̅, m∘Π_0, l 𝒩_j^(𝒯)      i=m 
                Π_0̅, m∘Π_0, l 𝒩̅_i^(𝒯)      j=l 
                Π_0̅, m∘Π_0, l 𝒩_i^(𝒯)      j=m, i ≠ l 
                 0                   .
            
            for 1 ≤ l < m ≤ L. Also, in (<ref>) and (<ref>) we have introduced the label 𝒯∈{𝒜 , 𝒟}
            in order to capture the implementation of our procedure for both equations (<ref>) and (<ref>). Moreover, we remark the original equations (<ref>) and (<ref>) are also included in the structure (<ref>); and they will be identified by the indexes l=0 and m = 0̅. In this way, we write
            
                𝒫_0^(0, 0)   ℳ_0^(𝒯)   𝒬̅_j^(0, 0)   𝒩̅_j^(𝒯)
                𝒬_j^(0, 0)   𝒩_j^(𝒯)   ℛ_ij^(0, 0)    0    .
            
            
            Our next step is to solve the system (<ref>), including the equation (l,m) = (0, 0̅), for the terms 𝒵_τ (X), 
            𝒵_τ (X_i^0̅) (1 ≤ i ≤ L) and 𝒵_τ ( X_i,j^0, 0̅ ) (1 ≤ i < j ≤ L). 
            We clearly have enough equations for that and this procedure allows us to express each one of such functions in terms of 𝒵_τ (X_i^0) (1 ≤ i ≤ L). Using Cramer's method we then find neat expressions for the above-mentioned functions 
            and, in particular, the solution for 𝒵_τ (X) can be written as the equation
            
                ∑_i=0^L ℋ_i^(𝒯) 𝒵_τ (X_i^0) = 0
            
            with coefficients ℋ_i^(𝒯) expressed in terms of determinants. More precisely, ℋ_i^(𝒯)  = det(Ω_i^(𝒯) ) and in order to present the explicit form of the matrices Ω_i^(𝒯), it is convenient to introduce the labels m_i,j, n_k,l×→ defined as
            
            m_i,j       L  i - i(i+1)/2 + j +1  
            
            n_k,l       L (k-1) - k(k+1)/2 + l  . 
            
            Hence, we have 
            
                Ω_0^(𝒯)  = [ 𝒫_0   𝒬   ℛ ]
            
            with 𝒫_0 a L(L+1)+2/2× 1 matrix with entries (𝒫_0)_m_i,j,1𝒫_0^(i, j) for i=j=0 and 0 ≤ i < j ≤ L. The matrix 𝒬, in its turn, has dimension L(L+1)+2/2× L and entries
            𝒬_m_i,j,k𝒬̅_k^(i, j) for i=j=0, 0 ≤ i < j ≤ L and 1 ≤ k ≤ L.
            Lastly, we have the entries ℛ_m_i,j, n_k,lℛ_kl^(i, j) for i=j=0, 0 ≤ i < j ≤ L and 1 ≤ k < l ≤ L forming the L(L+1)+2/2×L(L-1)/2-dimensional matrix ℛ.
            The remaining matrices Ω_k^(𝒯) then read
            
                Ω_k^(𝒯) = [ 𝒫_k   𝒬   ℛ ]
            
            with (𝒫_k)_m_i,j,1𝒬_k^(i, j) for indexes i=j=0 and 0 ≤ i < j ≤ L. In this way, the matrix
            𝒫_k is also a L(L+1)+2/2× 1-dimensional matrix. 
            
            
            
            Equation (<ref>) can now be recognized as the fundamental functional equation discussed in sec:INTRO. Although such equation has been shown to describe several quantities of interest, the coefficients of (<ref>) still exhibit additional features that were not present in the equations previously analyzed. For instance, the coefficients ℋ_i^(𝒯) depend on the additional spectral parameter x_0̅ which does not appear in the arguments of the functions 𝒵_τ. Therefore, (<ref>) corresponds to a family of fundamental functional equations with the same structure but different coefficients. The latter could then be unveiled by the series expansion of ℋ_i^(𝒯) in x_0̅. Alternatively, one could also produce new fundamental equations by simply replacing x_0̅↦ z_l in each coefficient ℋ_i^(𝒯).
            This unusual dependence on an extra parameter could also be used for solving (<ref>) along the lines described in <cit.> but, before that, one would still need to clarify the number of linearly independent equations obtained through such procedures. 
            In this way, here we will simply consider the approach used in <cit.> for the six-vertex model which makes use of the symmetric group action for 
            enlarging (<ref>) to a system of linear functional equations. The latter can then be straightforwardly used to derive determinantal solutions.
            
            
            
            
            
            
            § DETERMINANTAL SOLUTIONS
            
            
            This section is devoted to finding the unique solution of (<ref>) and the consequent determination of the partition function 𝒵_τ. As for that we shall proceed along the lines described in <cit.> for the six-vertex model and first consider the action of Π_0,j with 1 ≤ j ≤Ł on (<ref>). This procedure then uncovers the set of relations 
            
                ∑_i=0^L ℋ_i^(𝒯,   j) 𝒵_τ (X_i^0) = 0          1 ≤ j ≤ L
            
            with coefficients
            ℋ_i^(𝒯,   j)Π_0,jℋ_j^(𝒯)     i=0 
            Π_0,jℋ_0^(𝒯)     i=j 
            Π_0,jℋ_i^(𝒯)     i≠ 0, j
              . 
            
            Next we exploit the fact that (<ref>) also exhibits the structure of a linear system of algebraic equations and use the L equations contained in (<ref>) to express each function 𝒵_τ (X_i^0) for 1 ≤ i ≤ L in terms of 𝒵_τ (X). 
            In this way, we find
            
                𝒵_τ (X_i^0) = det(𝒲_i^(𝒯))/det(𝒲_0^(𝒯) )𝒵_τ (X)
            
            where 𝒲_i^(𝒯) and 𝒲_0^(𝒯) are L × L matrices. Their entries are simply defined as
            
            (𝒲_0^(𝒯))_α, β      ℋ_α^(𝒯,  β)     1 ≤α , β≤ L 
            
            (𝒲_i^(𝒯))_α, β      
            -ℋ_0^(𝒯 ,  β)    β = i 
            ℋ_α^(𝒯 ,  β)    otherwise  .
            
            The relation (<ref>) can now be regarded as an one-variable functional equation and separation of variables allows us to conclude 
            𝒵_τ (X_i^0) = det(𝒲_i^(𝒯)) f(x_0, x_1, x_2, … , x_L) and 
            𝒵_τ (X) = det(𝒲_0^(𝒯)) f(x_0, x_1, x_2, … , x_L) for a given function f.
            As for the determination of f we then use the basic conditions
            
                ∂/∂ x_i( det(𝒲_i^(𝒯)) f  ) = 0
            
            for 0 ≤ i ≤ L as discussed in details in <cit.>. In order to present our results in a more compact manner let us also write p_L  L^2 (L+1)/2 and q_L  (L-1)(L^2+2)/2. In this way, by collecting the results so far we have
            𝒵_τ (X_i^0)    =    - ( [τ + γ]/[L γ])^p_L( [(L+1)γ]/[τ])^q_L∏_k=1^L-1( [k γ]/[τ + (k+1)γ])^L∏_x ∈X_i^0∏_j=1^L [x - μ_j + γ] 
                  × ( [∑_l=1^L (x_l - μ_l) - γ]/[∑_l=1^L (x_l - μ_l) + τ + Lγ]) det(𝒲_i^(𝒜))  /. det(𝒲_i^(𝒜))|_τ = - (L+1)γ
            for index i in the range 0 ≤ i ≤ L. Also, here we recall the entries of 𝒲_i^(𝒜) are built out of the coefficients associated to the modified equation A (<ref>).
            Alternatively, one could also have considered 𝒯 = 𝒟 and in that case one obtains the representations
            𝒵_τ (X_i^0)    =    (-1)^L[τ + (L+1)γ]/[τ + γ]( [τ + (L+1)γ]/[L γ])^p_L( [(L+1)γ]/[τ + (L+2) γ])^q_L∏_k=1^L-1( [k γ]/[τ + (k+1)γ])^L
                  × ( [∑_l=1^L (x_l - μ_l) + (L+1)γ]/[∑_l=1^L (x_l - μ_l) + τ + (L+2)γ]) ∏_x ∈X_i^0∏_j=1^L [x - μ_j ]  det(𝒲_i^(𝒟))  /. det(𝒲_i^(𝒟))|_τ = - γ
            
            for 0 ≤ i ≤ L. Here we also remark the matrices 𝒲_i^(𝒟) are built out of the coefficients (<ref>) associated to the modified equation D (<ref>); and explicit examples of matrices 𝒲_0^(𝒯) for small lattice lengths 
            can be found in app:EG.
            
            
            Formulae (<ref>) and (<ref>) for each value of the index i seem to correspond to an independent representation since
            Π_0,i det(𝒲_i^(𝒯)) does not seem to be trivially related to det(𝒲_0^(𝒯)). In this way, each one of the formulae (<ref>) and (<ref>) describes L+1 two-parameters 
            families of continuous representations.
            
            
            
            § CONCLUDING REMARKS
            
            
            In this work we have elaborated on the Algebraic-Functional approach to the partition function of the 8VSOS model with domain-wall boundary conditions. In particular, we have shown the latter partition function obeys the same type of functional equation as its six-vertex model counterpart. The significance of this result is twofold: on the one hand we have enlarged the diversity of quantities governed by such type of linear functional equation; and on the other hand, we have consequently obtained new determinantal representations for the aforementioned partition function. 
            
            Although the functional equation in discussion, namely (<ref>), exhibits the same structure as the equations previously studied in the context of the six-vertex model, it still possesses additional features that were not encountered before. For instance, its coefficients 
            ℋ_i^(𝒯) also depend non-trivially on an extra spectral parameter playing no role in the arguments of the functions
            𝒵_τ. In this way, we can regard (<ref>) as a continuous family of functional equations as discussed in sec:FUN.
            Moreover, one advantage of having some quantity described by equations with the structure of (<ref>) is that determinantal solutions follow naturally with little dependence on the explicit form of the equation's coefficients. In this way, we have used (<ref>) to find novel 
            determinantal representations for the partition function of the 8VSOS model with domain-wall boundaries. 
            
            Our new determinants also exhibit some unusual features that have not been found previously in the literature to the best of our knowledge. For instance, here we have obtained determinants of L× L matrices whose entries are also determinants. These sub-determinants are in their turn taken over [L(L+1)+2/2] × [L(L+1)+2/2] matrices and at first look this feature suggests Dodgson's condensation method plays some role in our results. However, the existence of a precise relation with the condensation method still remains unclear to us.
            
            
            
            
            alpha
            
            
            
            
            
            
            
            § MATRICES 𝒲_0^(𝒯)
            
             
            In this appendix we give explicit expressions for the matrices 𝒲_0^(𝒯) entering formulae (<ref>) and (<ref>) for small lattice lenghts. As for the case L=1, the matrix 𝒲_0^(𝒜) is a 1 × 1 matrix with single entry reading
            | [x_0 - x_0̅ + γ][x_0̅ - μ_1 + γ]/[x_0 - x_0̅][x_0̅ - μ_1]  - [x_0 - x_1 + γ][x_1 - μ_1 + γ]/[x_0 - x_1][x_1 - μ_1]   [γ] [x_0̅ - x_0 + τ +  γ] [x_0 - μ_1 + γ]/[τ + γ] [x_0̅ - x_0] [x_0̅ - μ_1]
            [γ] [x_0 - x_0̅  + τ +  γ] [x_0̅ - μ_1 + γ]/[τ + γ] [x_0 - x_0̅] [x_0 - μ_1]   [x_0̅ - x_0  + γ][x_0 - μ_1 + γ]/[x_0̅ - x_0][x_0 - μ_1]  - [x_0̅ - x_1 + γ][x_1 - μ_1 + γ]/[x_0̅ - x_1][x_1 - μ_1]|   . 
            
             Similarly, the single entry of 𝒲_0^(𝒟) for L=1 is given by
             | [x_0̅ - x_0  + γ][x_0̅ - μ_1]/[x_0̅ - x_0][x_0̅ - μ_1 + γ]  - [x_1 - x_0 + γ][x_1 - μ_1]/[x_1 - x_0][x_1 - μ_1 + γ]   [γ] [x_0̅ - x_0 + τ +  2γ] [x_0 - μ_1]/[τ + 2γ] [x_0 - x_0̅] [x_0̅ - μ_1 + γ]
            [γ] [x_0 - x_0̅  + τ +  2γ] [x_0̅ - μ_1]/[τ + 2γ] [x_0̅ - x_0] [x_0 - μ_1 + γ]   [x_0 - x_0̅ + γ][x_0 - μ_1]/[x_0 - x_0̅][x_0 - μ_1 + γ]  - [x_1 - x_0̅ + γ][x_1 - μ_1]/[x_1 - x_0̅][x_1 - μ_1 + γ]|   . 
            
             
            Now turning our attention to the case L=2, the matrices 𝒲_0^(𝒜) and 𝒲_0^(𝒟) are 2 × 2 matrices with entries given by 4 × 4 determinants. In order to conveniently present such matrices we then write 
             
                𝒲_0^(𝒯) = [ | 𝒦_1^(𝒯) | | 𝒦_2^(𝒯) |; | 𝒦_3^(𝒯) | | 𝒦_4^(𝒯) | ]
            
             and introduce functions
             𝒰^i,j_k,l      [γ] [x_i - x_k + γ] [x_j - x_k + τ + γ]/[τ + γ][x_i - x_k] [x_j - x_k]∏_t=1^2 [x_k - μ_t + γ]/[x_l - μ_t]
            𝒰̅^i,j_k,l       -[γ] [x_k - x_i + γ] [x_j - x_k + τ + 3γ]/[τ + 3γ][x_k - x_i] [x_j - x_k]∏_t=1^2 [x_k - μ_t]/[x_l - μ_t + γ]
            𝒱^i,j_k,l      ∏_s ∈{k,l}[x_s - x_i + γ]/[x_s - x_i]∏_t=1^2 [x_i - μ_t + γ]/[x_i - μ_t] -  ∏_s ∈{k,l}[x_s - x_j + γ]/[x_s - x_j]∏_t=1^2 [x_j - μ_t + γ]/[x_j - μ_t]
            𝒱̅^i,j_k,l      ∏_s ∈{k,l}[x_i - x_s + γ]/[x_i - x_s]∏_t=1^2 [x_i - μ_t]/[x_i - μ_t + γ] -  ∏_s ∈{k,l}[x_j - x_s + γ]/[x_j - x_s]∏_t=1^2 [x_j - μ_t]/[x_j - μ_t + γ]  .
            
             In this way, we have
             
                𝒦_1^(𝒜) =    [    𝒱^0̅, 1_0,2  𝒰^2, 0̅_0, 0̅  𝒰^0, 0̅_2, 0̅              0;    𝒰^2,0_0̅, 0     𝒱^0,1_0̅,2     𝒰^0̅,0_2,0  - 𝒰^0̅, 1_2,1;   𝒰^0, 2_0̅, 2     𝒰^0̅,2_0,2    𝒱^2,1_0, 0̅  - 𝒰^0̅, 1_0,1;              0   𝒰^0̅, 2_1, 2 - 𝒰^0̅, 0_1, 0     𝒱^2,0_0̅,1 ]   𝒦_2^(𝒜)   = [    -𝒰^0, 1_2,1  𝒰^2, 0̅_0, 0̅  𝒰^0, 0̅_2, 0̅              0;              0     𝒱^0,1_0̅,2     𝒰^0̅,0_2,0  - 𝒰^0̅, 1_2,1;  -𝒰^0, 1_0̅, 1     𝒰^0̅,2_0,2    𝒱^2,1_0, 0̅  - 𝒰^0̅, 1_0,1;  -𝒰^1, 0_0̅, 0   𝒰^0̅, 2_1, 2 - 𝒰^0̅, 0_1, 0     𝒱^2,0_0̅,1 ]
                
                𝒦_3^(𝒜) =    [   - 𝒰^0, 2_1,2  𝒰^0, 0̅_1, 0̅  𝒰^1, 0̅_0, 0̅              0;  - 𝒰^0,2_0̅, 2     𝒱^1,2_0,0̅     𝒰^0̅,1_0,1  - 𝒰^0̅, 2_0,2;              0     𝒰^0̅,0_1,0    𝒱^0,2_0̅, 1  - 𝒰^0̅, 2_1,2;   𝒰^2, 0_0̅, 0   𝒰^0̅, 0_2, 0 - 𝒰^0̅, 1_2, 1     𝒱^0,1_0̅,2 ]   𝒦_4^(𝒜)    = [    𝒱^0̅, 2_0,1  𝒰^0, 0̅_1, 0̅  𝒰^1, 0̅_0, 0̅              0;    𝒰^0,1_0̅, 1     𝒱^1,2_0,0̅     𝒰^0̅,1_0,1  - 𝒰^0̅, 2_0,2;   𝒰^1, 0_0̅, 0     𝒰^0̅,0_1,0     𝒱^0,2_0̅,1  - 𝒰^0̅, 2_1,2;              0   𝒰^0̅, 0_2, 0 - 𝒰^0̅, 1_2, 1     𝒱^0,1_0̅,2 ]  .
            
            The matrices 𝒦_i^(𝒟) in their turn are given by
            
                𝒦_1^(𝒟) =    [    𝒱̅^0̅, 1_0,2  𝒰̅^2, 0̅_0, 0̅  𝒰̅^0, 0̅_2, 0̅               0;    𝒰̅^2,0_0̅, 0     𝒱̅^0,1_0̅,2     𝒰̅^0̅,0_2,0  - 𝒰̅^0̅, 1_2,1;   𝒰̅^0, 2_0̅, 2     𝒰̅^0̅,2_0,2    𝒱̅^2,1_0, 0̅  - 𝒰̅^0̅, 1_0,1;               0   𝒰̅^0̅, 2_1, 2 - 𝒰̅^0̅, 0_1, 0     𝒱̅^2,0_0̅,1 ]   𝒦_2^(𝒟)   = [    -𝒰̅^0, 1_2,1  𝒰̅^2, 0̅_0, 0̅  𝒰̅^0, 0̅_2, 0̅               0;               0     𝒱̅^0,1_0̅,2     𝒰̅^0̅,0_2,0  - 𝒰̅^0̅, 1_2,1;  -𝒰̅^0, 1_0̅, 1     𝒰̅^0̅,2_0,2    𝒱̅^2,1_0, 0̅  - 𝒰̅^0̅, 1_0,1;  -𝒰̅^1, 0_0̅, 0   𝒰̅^0̅, 2_1, 2 - 𝒰̅^0̅, 0_1, 0     𝒱̅^2,0_0̅,1 ]
                
                𝒦_3^(𝒟) =    [   - 𝒰̅^0, 2_1,2  𝒰̅^0, 0̅_1, 0̅  𝒰̅^1, 0̅_0, 0̅               0;  - 𝒰̅^0,2_0̅, 2     𝒱̅^1,2_0,0̅     𝒰̅^0̅,1_0,1  - 𝒰̅^0̅, 2_0,2;               0     𝒰̅^0̅,0_1,0    𝒱̅^0,2_0̅, 1  - 𝒰̅^0̅, 2_1,2;   𝒰̅^2, 0_0̅, 0   𝒰̅^0̅, 0_2, 0 - 𝒰̅^0̅, 1_2, 1     𝒱̅^0,1_0̅,2 ]   𝒦_4^(𝒟)    = [    𝒱̅^0̅, 2_0,1  𝒰̅^0, 0̅_1, 0̅  𝒰̅^1, 0̅_0, 0̅               0;    𝒰̅^0,1_0̅, 1     𝒱̅^1,2_0,0̅     𝒰̅^0̅,1_0,1  - 𝒰̅^0̅, 2_0,2;   𝒰̅^1, 0_0̅, 0     𝒰̅^0̅,0_1,0     𝒱̅^0,2_0̅,1  - 𝒰̅^0̅, 2_1,2;               0   𝒰̅^0̅, 0_2, 0 - 𝒰̅^0̅, 1_2, 1     𝒱̅^0,1_0̅,2 ]  .
             3cm
