Multiple integral representation for the trigonometric SOS model with domain wall boundaries
MMultiple integral representation for thetrigonometric SOS model withdomain wall boundaries
W. Galleas
ARC Centre of Excellence for the Mathematicsand Statistics of Complex Systems,The University of MelbourneVIC 3010, Australia [email protected]
Abstract
Using the dynamical Yang-Baxter algebra we derive a functional equa-tion for the partition function of the trigonometric SOS model withdomain wall boundary conditions. The solution of the equation is givenin terms of a multiple contour integral.
PACS numbers: 05.50+q, 02.30.IKKeywords: Dynamical Yang-Baxter Equation, Functional equations,Domain wall boundariesNovember 2011 a r X i v : . [ m a t h - ph ] J a n ontents su (2) 19B Polynomial structure and asymptotic behaviour 20C Special Zeroes 21D Z θ as a symmetric function 23E Uniqueness 24F Solution for L = 1 25 The study of two-dimensional lattice models in Statistical Mechanics advanced dramat-ically with the advent of Baxter’s concept of commuting transfer matrices [1]. Thismethod introduced the concept of integrability in Statistical Mechanics and paved theway for the development of a variety of exact methods exploring the aforementionedcommutativity. As examples of those methods we have Baxter’s T − Q relation [1], thealgebraic Bethe ansatz [2], the inversion trick [3], etc. Also a variety of models can betackled by those same techniques such as vertex models, solid-on-solid models and hardsquare models.Nevertheless, the implementation of those methods depends drastically on the bound-ary conditions chosen and the case of domain wall boundaries deserves special attention.This kind of boundary condition for the six vertex model was introduced by Korepin in [4]who also obtained a recurrence relation determining the model partition function. Thisrecurrence relation was later on solved by Izergin in terms of a determinant [5]. More-over, the exact solution of this model raised the issue of the sensitivity of the six vertexmodel bulk properties with respect to the boundary conditions in the thermodynamicallimit [6]. 2 = 123......... NN + 1 j = 1 2 3 . . . . . . . . . . . . M M + 1 Figure 1:
Two-dimensional lattice with M × N retangular cells. A natural question that emerges in this scenario is how this sensitivity with respectto boundary conditions extends to the eight vertex model. In that case, although thepartition function has been evaluated in [7–9], the lack of manageable expressions haveeluded the analysis of the thermodynamical limit except for a particular value of theanisotropy parameter [10].Motivated by this scenario and keeping in mind the relation between Baxter’s eightvertex model and solid-on-solid models, also refereed to as SOS models, here we demon-strate that the algebraic-functional method introduced in [11,12] can also be used in thatcase. Using that method we derive a multiple integral formula for the partition functionof the trigonometric SOS model with domain wall boundaries.This paper is organised as follows. In the Sec. 2 we briefly describe SOS models inStatistical Mechanics with emphasis on the case of domain wall boundary conditions. Inthe Sec. 3 we present the dynamical Yang-Baxter algebra and demonstrate how it can beexplored in order to obtain a functional equation for the partition function of the SOSmodel with domain wall boundaries. This functional equation is analised in Sec. 4 andthe solution of the equation is given in Sec. 5 as a multiple contour integral. Technicaldetails are presented in App. A through App. F.
We consider a two-dimesional lattice formed by retangular cells juxtaposed as in Fig. 1.For a lattice with N + 1 rows and M + 1 columns we have M × N retangular cellsand we associate the Boltzmann weight w ij (cid:18) l i +1 ,j l i +1 ,j +1 l i,j l i,j +1 (cid:19) to the cell enclosed by thecartesian coordinates ( i, j ), ( i, j + 1), ( i + 1 , j ) and ( i + 1 , j + 1). Each retangular cellis simply referred as face and its configuration is characterised by the set of variables3 ij (cid:18) l i +1 ,j l i +1 ,j +1 l i,j l i,j +1 (cid:19) (cid:39) l i,j l i,j +1 l i +1 ,j +1 l i +1 ,j Figure 2:
Face at coordinate ( i, j ) and its Boltzmann weight. T l i +1 , ,...,l i +1 ,M +1 l i, ,...,l i,M +1 (cid:39) l i, l i, . . . . . . . . . l i,M +1 l i +1 , l i +1 , . . . . . . . . . l i +1 ,M +1 Figure 3:
Face monodromy matrix. { l i,j , l i,j +1 , l i +1 ,j , l i +1 ,j +1 } . Diagrammatically this association is depicted in Fig. 2 and thepartition function of the system is then given by Z = (cid:88) { l i,j } N (cid:89) i =1 M (cid:89) j =1 w ij (cid:18) l i +1 ,j l i +1 ,j +1 l i,j l i,j +1 (cid:19) . (2.1) Face monodromy matrix.
Let us define a matrix T with components T l i +1 , ,...,l i +1 ,M +1 l i, ,...,l i,M +1 = M (cid:89) j =1 w ij (cid:18) l i +1 ,j l i +1 ,j +1 l i,j l i,j +1 (cid:19) . (2.2)We shall refer to T as face monodromy matrix and it is diagrammatically representedin the Fig. 3. For simplicity of notation we also refer to the components of the facemonodromy matrix defined by (2.2) as T i +1 i . Employing this notation the partitionfunction Z reads Z = (cid:88) { l i,j } T T . . . T N +1 N , (2.3)and the lattice in Fig. 1 can be entirely built as the product of elements T i +1 i representingeach horizontal layer. Boundary conditions and integrability.
Following Baxter [13] we consider the facemonodromy matrix (2.2) with periodic boundary conditions in the horizontal direction.This corresponds to setting l i,M +1 = l i, , and under these conditions it is convenient todefine an operator T with components T l i +1 , ,...,l i +1 ,M l i, ,...,l i,M = w i, (cid:18) l i +1 , l i +1 , l i, l i, (cid:19) w i, (cid:18) l i +1 , l i +1 , l i, l i, (cid:19) . . . w i,M (cid:18) l i +1 ,M l i +1 , l i,M l i, (cid:19) . (2.4)4 + v uv l l l l l l l (cid:39) vu u + v l l l l l l l Figure 4:
Graphical representation of the Yang-Baxter relation for SOS models.
The operator T is commonly denominated (face) transfer matrix and it plays an im-portant role in estabilishing the integrability of two-dimensional lattice models [13]. Weproceed by defining a second transfer matrix T (cid:48) similarly to (2.4) but with Boltzmannweights w (cid:48) instead of w . Next we look for conditions on w and w (cid:48) such that the transfermatrices T and T (cid:48) form a commutative family, i.e. [ T, T (cid:48) ] = 0. This requirement leadsto the following relation (cid:88) l w v (cid:18) l l l l (cid:19) w u + v (cid:18) l l l l (cid:19) w u (cid:18) l l l l (cid:19) = (cid:88) l w u (cid:18) l l l l (cid:19) w u + v (cid:18) l l l l (cid:19) w v (cid:18) l l l l (cid:19) , (2.5)where u and v are complex variables parameterising the manifold where the transfermatrices form a commutative family. For a detailed derivation of (2.5) we refer to [14].The Eq. (2.5) is usually referred to as Yang-Baxter relation [15], or simply Hexagonidentity , and the local equivalence transformation described by (2.5) is depicted in Fig. 4.Moreover, if we also consider periodic boundary conditions in the vertical direction, i.e. l N +1 ,j = l ,j , the partition function (2.1) becomes simply Z = Tr (cid:0) T N (cid:1) , (2.6)and its evaluation is translated into an eigenvalue problem [16, 17]. Throughout thispaper we shall consider a different class of boundary conditions where (2.6) does notapply, although the bulk model is still governed by statistical weights satisfying (2.5). The trigonometric SOS model.
The variables { l i,j , l i,j +1 , l i +1 ,j , l i +1 ,j +1 } depicted inFig. 2 are also called height functions and they characterise the configuration of theassociated face. The degree of freedom l i,j is also refereed to as the colour of the face atthe position ( i, j ). Furthermore, we can also impose restrictions on l i,j such that onlycertain configurations of colours are allowed in the statistical sum (2.1).In what follows we will be dealing with a lattice formed by coloured faces whereeach adjacent face can not have the same colour. As it was remarked by Baxter in [18],this system can be thought as a system of particles interacting through an infinitelyrepulsive force between nearst neighbors of the same type. For the trigonometric SOSmodel we will have l i,j = θ + ¯ l i,j γ , where ¯ l i,j is an integer variable while θ and γ are5 θ + γθ − γ θ a + θ θ − γθ + γ θ a − θ + γ θθ θ − γ b + θ − γ θθ θ + γ b − θ + γ θθ θ + γ c + θ − γ θθ θ − γ c − Figure 5:
Face-vertex configurations and the associated Boltzmann weight. complex numbers. In the Fig. 5 the height function l i,j characterising the colour of theface is projected into the center of the ( i, j ) face. Interestingly enough, it was remarkedby Lenard an equivalence between this colouring of faces and the configurations of asix vertex model. This equivalence is depicted in Fig. 5 where basically one removesthe outer edges of the four-faces set and places arrows on the internal edges accordingto a certain rule. This rule is as follows: each face of a four-faces set is visited in theanticlockwise direction. If the colour changes by + γ when intersecting an edge, thisedge receives an arrow pointing inwards; and if the colour changes by − γ we place anarrow pointing outwards on that edge. For a particular class of statistical weights thismodel is also called Three-colouring model [18, 20]. In that case the height function l i,j is conveniently labeled by an element of the ring Z . More precisely, the Z structurelabeling the colours of the faces is unveiled by considering the remainder after divisionof ¯ l i,j by 3. The dynamical Yang-Baxter equation.
In [21–23] it was demonstrated that theBoltzmann weights of a face model satisfying (2.5) are encoded in the solutions of thedynamical Yang-Baxter equation. This dynamical version of the quantum Yang-Baxterequation was proposed by Felder in [21] as the quantised form of a modified classicalYang-Baxter equation. In that case this modified classical Yang-Baxter equation arises asthe compatibility condition for the Knizhnik-Zamolodchikov-Bernard equations [24, 25].Previous to that, the dynamical Yang-Baxter equation had appeared in connection tothe Liouville string field theory in [26].Now let V = v + ⊕ v − be a two-dimensional complex vector space and consider theoperator R ( λ, θ ) ∈ End( V ⊗ V ) with λ, θ ∈ C . The dynamical Yang-Baxter equation forthe trigonometric SOS model then reads R ( λ − λ , θ − γ ˆ h ) R ( λ − λ , θ ) R ( λ − λ , θ − γ ˆ h ) = R ( λ − λ , θ ) R ( λ − λ , θ − γ ˆ h ) R ( λ − λ , θ ) (2.7)where ˆ h = diag(1 , − V ⊗ V ⊗ V ) and the action of R ( λ, θ − γ ˆ h ) on v ⊗ v ⊗ v is understood as[ R ( λ, θ − γh ) v ⊗ v ] ⊗ v , (2.8)keeping in mind that ˆ h v = hv . In other words, h is simply the eigenvalue of ˆ h on the See
Note added in proof of [19]. R ( λ, θ ) = a + ( λ, θ ) 0 0 00 b + ( λ, θ ) c + ( λ, θ ) 00 c − ( λ, θ ) b − ( λ, θ ) 00 0 0 a − ( λ, θ ) (2.9)with non-null entries a ± ( λ, θ ) = sinh ( λ + γ ) b ± ( λ, θ ) = sinh ( λ ) sinh ( θ ∓ γ )sinh ( θ ) c ± ( λ, θ ) = sinh ( γ ) sinh ( θ ∓ λ )sinh ( θ ) . (2.10)The solution described by (2.9) and (2.10) consists of a particular limit of the ellipticsolution found in [21, 22]. Also it is important to remark here that such solutions arein correspondence with Baxter’s eight-vertex model after a vertex-face transformation[27, 28]. Moreover, the dynamical R -matrix (2.9) satisfies the ice rule (cid:104) R ab ( λ, θ ) , ˆ h a + ˆ h b (cid:105) = 0 (2.11)which plays an important role in estabilishing an algebra associated to dynamical R -matrices.Now we turn our attention to the relation between solutions of the dynamical Yang-Baxter equation (2.7) and the statistical weights of a SOS model satisfying (2.5). Atthis stage the observations of Lenard [19], the vertex-face transformation introducedby Baxter [28] and Felder’s dynamical Yang-Baxter equation [22] converge to the samepoint. Following [22] we thus have w λ (cid:18) θ − γ θθ θ + γ (cid:19) = a + ( λ, θ ) w λ (cid:18) θ + γ θθ θ − γ (cid:19) = a − ( λ, θ ) w λ (cid:18) θ − γ θ − γθ θ − γ (cid:19) = b + ( λ, θ ) w λ (cid:18) θ + γ θ + 2 γθ θ + γ (cid:19) = b − ( λ, θ ) w λ (cid:18) θ − γ θθ θ − γ (cid:19) = c + ( λ, θ ) w λ (cid:18) θ + γ θθ θ + γ (cid:19) = c − ( λ, θ ) . (2.12)This association is also depicted in Fig. 5. The dynamical monodromy matrix.
Still following [22, 23] we define an inhomo-geneous monodromy matrix T a ( λ, θ ) formed by the ordered product of dynamical R -matrices. More precisely, this dynamical monodromy matrix reads T a ( λ, θ ) = L (cid:89) i =1 R ai ( λ − µ i , θ i ) (2.13)7ith θ i = θ − γ L (cid:88) k = i +1 ˆ h k , and it should not be confused with the face monodromy matrixdefined in (2.2).The dynamical monodromy matrix (2.13) is an operator living in the tensor productspace V a ⊗ V ⊗ · · · ⊗ V L . The space V a will be refereed to as auxiliar space while thetensor product V ⊗ · · · ⊗ V L will be called quantum space. In this way T a ( λ, θ ) canbe regarded as a matrix on the auxiliar space whose entries are matrices living in thequantum space. Here we shall restrict ourselves to the monodromy matrix built out ofthe R -matrix (2.9). Therefore, it can be conveniently denoted as T a ( λ, θ ) = (cid:18) A ( λ, θ ) B ( λ, θ ) C ( λ, θ ) D ( λ, θ ) (cid:19) . (2.14)Due to the dynamical Yang-Baxter equation (2.7) and the ice rule (2.11), one can demon-strate that the monodromy matrix (2.13) satisfies the algebraic relation R ab ( λ − λ , θ − γ H) T a ( λ , θ ) T b ( λ , θ − γ ˆ h a ) = T b ( λ , θ ) T a ( λ , θ − γ ˆ h b ) R ab ( λ − λ , θ )(2.15)where H = L (cid:88) k =1 ˆ h k . With the help of the definition of ˆ h , we can identify H as the Cartanelement of the su (2) algebra on the tensor product space V ⊗ · · · ⊗ V L . Furthermore,in the limit θ → ∞ we can immediately see that (2.9) becomes the standard R -matrixinvariant under the quantum affine algebra U q [ (cid:98) su (2)]. This observation extends to themonodromy matrix (2.13) and the algebra (2.15), which become respectively the standardtrigonometric six vertex model monodromy matrix and Yang-Baxter algebra [29, 2]. Domain wall boundary conditions.
We shall now consider the trigonometric SOSmodel on the lattice described in Fig. 1 with N = M = L + 1 and special bound-ary conditions. As for the boundary conditions, we set l ,j = l j, = θ + ( L + 1 − j ) γ and l L +1 ,j = l j,L +1 = θ + ( j − γ . This special boundary conditions is illustrated inFig. 6 together with the corresponding structure of vertices. In the vertices language wecan immediately recognize this special boundary conditions as the case of domain wallboundaries introduced by Korepin in [4]. This observation if of fundamental importanceallowing us to express the model partition function in terms of the entries of the mon-odromy matrix (2.14) analogously to the case of the standard six vertex model [4]. Thediagrammatical interpretation of (2.13) is given in [22, 23] and for a discussion on theconstruction of the partition function (2.1) in terms of the components (2.14) we referto [30]. For the trigonometric SOS model considered here we only have to keep in mindthat the entries of the dynamical monodromy matrix (2.14) also depends on the colourvariable θ governed by the height functions l j, . In this way the partition function (2.1)for the trigonometric SOS model with domain wall boundaries can be written as Z θ = (cid:104) ¯0 | L (cid:89) j =1 B ( λ j , θ + jγ ) | (cid:105) (2.16)8 θ + γθ + 2 γθ + 3 γ θ + γθ + 2 γθ + γθ + 2 γ θ + 2 γθ + γθ + 2 γθ + γ θ + 3 γθ + 2 γθ + γθ ( a ) ( b ) Figure 6: ( a ) SOS model with domain wall boundaries. ( b ) The correspondingstructure of vertices. where | (cid:105) = L (cid:79) i =1 (cid:18) (cid:19) and | ¯0 (cid:105) = L (cid:79) i =1 (cid:18) (cid:19) . (2.17) The dynamical Yang-Baxter algebra (2.15) encodes commutation rules for the entriesof the dynamical monodromy matrix (2.13). In contrast to the standard Yang-Baxteralgebra, the relation (2.15) not only contains the generators A ( λ, θ ), B ( λ, θ ), C ( λ, θ ) and D ( λ, θ ), but also the su (2) Cartan generator H. This indicates that the algebra definedby (2.15) needs to be complemented. The su (2) Cartan generator. The definition of H together with (2.13) and (2.14)allow us to directly compute the commutators between the su (2) Cartan generator Hand the entries of the monodromy matrix (2.14). In the limit θ → ∞ this analysis hasbeen performed in [2, 4] and here we find that there are no significant modifications forarbitrary θ . Nevertheless, this analysis can also be found in App. A and we have[ A ( λ, θ ) , H] = 0 [ B ( λ, θ ) , H] = 2 B ( λ, θ )[ C ( λ, θ ) , H] = − C ( λ, θ ) [ D ( λ, θ ) , H] = 0 . (3.1)From the R -matrix (2.9) we can see that the generator H will appear in (2.15) only as e ± γ H . Thus for convenience we define the operator K = q H with q = e γ , in such a waythat the commutation rules (3.1) imply the following relations: B ( λ, θ )K = q K B ( λ, θ ) [ A ( λ, θ ) , K] = 0 C ( λ, θ )K = q − K C ( λ, θ ) [ D ( λ, θ ) , K] = 0 . (3.2)The commutation rules (3.2) in addition to (2.15) form an extended Yang-Baxteralgebra with generators A ( λ, θ ), B ( λ, θ ), C ( λ, θ ), D ( λ, θ ) and K ± . For our purposes here9e only need to extract a few commutation relations from (2.15). Namely, B ( λ , θ ) B ( λ , θ + γ ) = B ( λ , θ ) B ( λ , θ + γ ) A ( λ , θ + γ ) B ( λ , θ ) = s ( λ − λ + γ ) s ( λ − λ ) s ( θ + γ ) s ( θ + 2 γ ) B ( λ , θ + γ ) A ( λ , θ + 2 γ ) − s ( θ + γ − λ + λ ) s ( λ − λ ) s ( γ ) s ( θ + 2 γ ) B ( λ , θ + γ ) A ( λ , θ + 2 γ ) D ( λ , θ − γ ) B ( λ , θ ) = s ( λ − λ + γ ) s ( λ − λ ) B ( λ , θ − γ ) D ( λ , θ )[ tq K − − t − q − K][ tq K − − t − q − K] − − s ( γ ) s ( λ − λ ) B ( λ , θ − γ ) D ( λ , θ )[ tq ¯ x ¯ x − K − − t − q − ¯ x − ¯ x K][ tq K − − t − q − K] − C ( λ , θ + γ ) B ( λ , θ ) = s ( θ ) s ( θ + γ ) B ( λ , θ + γ ) C ( λ , θ + 2 γ )[ tq K − − t − q − K][ tq K − − t − q − K] − + s ( γ ) s ( θ + γ ) s ( θ + γ + λ − λ ) s ( λ − λ ) A ( λ , θ + γ ) D ( λ , θ )[ tq K − − t − q − K][ tq K − − t − q − K] − − s ( γ ) s ( λ − λ ) A ( λ , θ + γ ) D ( λ , θ )[ tq ¯ x ¯ x − K − − t − q − ¯ x − ¯ x K][ tq K − − t − q − K] − . (3.3)In the above commutation rules we have employed the notation s ( λ ) = sinh( λ ), t = e θ and ¯ x i = e λ i . The algebra formed by the relations (3.2) and (3.3) will be one of the mainingredients in the derivation of a functional relation determining the partition function(2.16).In order to proceed we will also need to consider the action of the generators A ( λ, θ ), B ( λ, θ ), C ( λ, θ ), D ( λ, θ ) and K on the states | (cid:105) and | ¯0 (cid:105) defined in (2.17). Those statesare the su (2) highest and lowest weight states respectively, and from (2.9), (2.13) and(2.14) we readly obtain the relationsK ± | (cid:105) = q ± L | (cid:105) A ( λ, θ ) | (cid:105) = L (cid:89) i =1 s ( λ − µ i + γ ) | (cid:105) D ( λ, θ ) | (cid:105) = s ( θ + γ ) s ( θ − ( L − γ ) L (cid:89) i =1 s ( λ − µ i ) | (cid:105) . (3.4)Moreover, we also obtain the properties B ( λ, θ ) | (cid:105) = † C ( λ, θ ) | (cid:105) = 0 B ( λ, θ ) | ¯0 (cid:105) = 0 C ( λ, θ ) | ¯0 (cid:105) = ‡ , (3.5)10here the symbols † and ‡ stand for non-null values. The relations (3.5) together with(3.1) support considering B ( λ, θ ) and C ( λ, θ ) as creation and annihilation operatorsrespectively with respect to the pseudo-vacuum state | (cid:105) .Altogether the relations (3.2)-(3.5) allow us to derive the following formula, C ( λ , θ + γ ) n (cid:89) i =1 B ( λ i , θ + ( i − γ ) | (cid:105) = n (cid:88) i =1 M i n − (cid:89) j =1 B ( λ r ( i ) j , θ + jγ ) | (cid:105) + n (cid:88) j =2 j − (cid:88) i =1 N ji B ( λ , θ + γ ) n − (cid:89) k =1 B ( λ s ( ij ) k , θ + ( k + 1) γ ) | (cid:105) , (3.6)where r ( i ) j = (cid:40) j ≤ j < ij + 1 i ≤ j ≤ n − s ( ij ) k = k ≤ k < ik + 1 i ≤ k < jk + 2 j ≤ k ≤ n − . (3.7)In their turn the coefficients M i and N ji are given by M i = s ( γ ) s ( λ i − λ ) s ( θ + γ ) s ( θ + nγ ) s ( λ − λ i + θ + (2 n − − L ) γ ) s ( θ + (2 n − − L ) γ ) s ( θ + ( n − L ) γ ) s ( θ + (2 n − L ) γ ) s ( θ + nγ ) s ( θ + ( n − L ) γ ) × L (cid:89) l =1 s ( λ − µ l + γ ) s ( λ i − µ l ) n (cid:89) k =1 k (cid:54) = i s ( λ i − λ k + γ ) s ( λ i − λ k ) s ( λ k − λ + γ ) s ( λ k − λ )+ s ( γ ) s ( λ − λ i ) s ( λ − λ i + θ + γ ) s ( θ + nγ ) s ( θ + ( n − L ) γ ) s ( θ + (2 n − L ) γ ) s ( θ + nγ ) s ( θ + ( n − L ) γ ) × L (cid:89) l =1 s ( λ i − µ l + γ ) s ( λ − µ l ) n (cid:89) k =1 k (cid:54) = i s ( λ − λ k + γ ) s ( λ − λ k ) s ( λ k − λ i + γ ) s ( λ k − λ i ) (3.8) N ji = s ( γ ) s ( λ − λ j ) s ( γ ) s ( λ i − λ ) s ( λ j − λ i + γ ) s ( λ j − λ i ) s ( λ − λ i + θ + γ ) s ( θ + nγ ) s ( λ − λ j + θ + (2 n − − L ) γ ) s ( θ + (2 n − − L ) γ ) × s ( θ + ( n − L ) γ ) s ( θ + (2 n − L ) γ ) s ( θ + nγ ) s ( θ + ( n − L ) γ ) L (cid:89) l =1 s ( λ i − µ l + γ ) s ( λ j − µ l ) × n (cid:89) m =1 m (cid:54) = i,j s ( λ j − λ m + γ ) s ( λ j − λ m ) s ( λ m − λ i + γ ) s ( λ m − λ i ) 11 s ( γ ) s ( λ − λ i ) s ( γ ) s ( λ j − λ ) s ( λ i − λ j + γ ) s ( λ i − λ j ) s ( λ − λ j + θ + γ ) s ( θ + nγ ) s ( λ − λ i + θ + (2 n − − L ) γ ) s ( θ + (2 n − − L ) γ ) × s ( θ + ( n − L ) γ ) s ( θ + (2 n − L ) γ ) s ( θ + nγ ) s ( θ + ( n − L ) γ ) L (cid:89) l =1 s ( λ j − µ l + γ ) s ( λ i − µ l ) × n (cid:89) m =1 m (cid:54) = i,j s ( λ i − λ m + γ ) s ( λ i − λ m ) s ( λ m − λ j + γ ) s ( λ m − λ j ) . (3.9)As expected, the expressions (3.6)-(3.9) recover the ones presented in [4, 11] in the limit θ → ∞ . At this stage we have gathered most of the ingredients required to obtain afunctional equation determining the partition function (2.16). Functional Equation.
The relation (3.6) is valid for the product of an arbitrarynumber of operators B ( λ i , θ ), and following the approach devised in [11], we examinethe quantity (cid:104) ¯0 | C ( λ , θ + γ ) L +1 (cid:89) i =1 B ( λ i , θ + ( i − γ ) | (cid:105) (3.10)under the light of the extended Yang-Baxter algebra formed by (2.15) and (3.2). Thusconsidering (3.6) and the definition (2.16) we immediately obtain the relation (cid:104) ¯0 | C ( λ , θ + γ ) L +1 (cid:89) i =1 B ( λ i , θ + ( i − γ ) | (cid:105) = L +1 (cid:88) i M i Z θ ( λ , . . . , λ i − , λ i +1 , . . . , λ L +1 )+ L +1 (cid:88) j =2 j − (cid:88) i =1 N ji Z θ ( λ , . . . , λ i − , λ i +1 , . . . , λ j − , λ j +1 , . . . , λ L +1 ) , (3.11)where the coefficients M i and N ji are the ones given by (3.8) and (3.9) with n = L + 1.On the other hand, the highest weight representation theory of the su (2) algebra tells usthat the LHS of (3.11) vanishes. This property has been already discussed in [11] but forcompleteness we also include it in the App. A. In this way we are left with the followingfunctional equation for the partition function (2.16), L +1 (cid:88) i =1 M i Z θ ( λ , . . . , λ i − , λ i +1 , . . . , λ L +1 )+ L +1 (cid:88) j =2 j − (cid:88) i =1 N ji Z θ ( λ , . . . , λ i − , λ i +1 , . . . , λ j − , λ j +1 , . . . , λ L +1 ) = 0 . (3.12)Some comments are in order at this stage. The partition function Z θ depends on twosets of variables, { λ i } and { µ i } , as well as parameters γ and θ . Nevertheless, within thisapproach we can see that the set of variables { µ i } can also be regarded as parameters.12he Eq. (3.12) is linear and homogeneous in Z θ and this observation will have importantconsequences for the characterisation of the desired solution. The homogeneity of (3.12)tells us that if Z θ is a solution, so is αZ θ where α is a constant. In fact α only needsto be independent of the variables { λ i } . Therefore, the Eq. (3.12) can determine thepartition function at most up to a constant and the full determination of the partitionfunction will require that we are at least able to compute Z θ for some particular valueof the variables λ i . Moreover, the linearity of (3.12) tells us that if Z (1) θ and Z (2) θ aretwo solutions of (3.12), the linear combination Z (1) θ + Z (2) θ is also a solution. This factsuggests that classifying the classes of unique solutions of (3.12) is an important step inthis framework. The Eq. (3.12) has the same structure of the functional equation derivedin [11], however the coefficients M i and N ji are deformed by the dynamical parameter θ .In what follows these issues will be discussed and the desired solution of (3.12) will bepresented. In this section we shall consider extra properties expected for the partition function(2.16) which will enable us to select the appropriate solution of (3.12). These propertiesinclude the multivariate polynomial structure of the partition function Z θ , as well as itsasymptotic behaviour. In order to avoid an overcrowded section we shall discuss thementioned properties in the App. B and here we present only the required results. Polynomial structure.
The partition function Z θ is of the form, Z θ ( λ , . . . , λ L ) = ¯ Z θ ( x , . . . , x L ) L (cid:89) i =1 x L i , (4.1)where ¯ Z θ ( x , . . . , x L ) is a polynomial of order L in each variable x i = e λ i separately.The same polynomial structure holds if we consider the variables { µ i } instead of { λ i } ,although this property will not be required. Asymptotic behaviour.
In the limit x i → ∞ , the function ¯ Z θ possesses the asymp-totic behaviour ¯ Z ∼ ( q − q − ) L L [ L ] q ! L (cid:89) n =1 (1 − q n t ) u L n ( x . . . x L ) L , (4.2)where [ n ] q ! = 1(1 + q )(1 + q + q ) . . . (1 + q + · · · + q n ) denotes the q -factorial functionand u i = e µ i .For the moment we leave the properties (4.1) and (4.2) at rest and proceed with amore careful examination of the functional equation (3.12). Firstly we notice that besidesthe set of variables { λ , λ , . . . , λ L } required to characterise its solution, the Eq. (3.12)13lso depends on variables λ and λ L +1 . Thus the variables λ and λ L +1 can be fixed inorder to fulfill our needs. In particular, those variables can be chosen in such a way thatsolving (3.12) under the conditions (4.1) and (4.2) becomes systematic and simple. Forillustrative purposes, let us see how this approach would work in the case L = 2. In thatcase our functional equation reads M Z θ ( λ , λ ) + M Z θ ( λ , λ ) + M Z θ ( λ , λ )+ N Z θ ( λ , λ ) + N Z θ ( λ , λ ) + N Z θ ( λ , λ ) = 0 (4.3)and we set λ = µ and λ = µ − γ . By doing so we find M | λ µ λ = µ − γ = M | λ µ λ = µ − γ = 0 (4.4)and we also define m = M | λ µ λ = µ − γ = − s ( θ + γ ) s ( θ + 3 γ ) s ( γ ) s ( µ − µ + γ ) s ( µ − µ + γ )¯ m = N | λ µ λ = µ − γ = s ( θ + γ − λ + µ ) s ( θ + 3 γ ) s ( γ ) s ( µ − µ + γ ) s ( λ − µ + γ ) × s ( λ − µ ) s ( λ − µ + γ ) s ( λ − λ + γ ) s ( λ − λ )¯ m = N | λ µ λ = µ − γ = s ( θ + γ − λ + µ ) s ( θ + 3 γ ) s ( γ ) s ( µ − µ + γ ) s ( λ − µ + γ ) × s ( λ − µ ) s ( λ − µ + γ ) s ( λ − λ + γ ) s ( λ − λ ) . (4.5)With the above specialisation of the variables λ and λ , the Eq. (4.3) reduces to Z θ ( λ , λ ) = s ( θ + γ − λ + µ ) s ( θ + γ ) s ( λ − µ + γ ) s ( µ − µ + γ ) s ( λ − µ ) s ( λ − µ + γ ) s ( λ − λ + γ ) s ( λ − λ ) Z θ ( µ , λ )+ s ( θ + γ − λ + µ ) s ( θ + γ ) s ( λ − µ + γ ) s ( µ − µ + γ ) s ( λ − µ ) s ( λ − µ + γ ) s ( λ − λ + γ ) s ( λ − λ ) Z θ ( µ , λ ) − (cid:20) N M (cid:21) λ µ λ = µ − γ Z θ ( µ , µ − γ ) . (4.6)In addition to that the Eq. (4.6) reduces to the following identity when λ = µ , (cid:20) N M (cid:21) λ ,λ µ λ = µ − γ Z θ ( µ , µ − γ ) = Z θ ( µ , λ ) − Z θ ( λ , µ ) . (4.7)As demonstrated in the App. D, the functional equation (3.12) admits only symmetricsolutions, i.e. Z θ ( λ , λ ) = Z θ ( λ , λ ), and consequently the RHS of (4.7) vanishes. Asthe quantity (cid:104) N M (cid:105) λ ,λ µ λ = µ − γ is finite we can thus conclude that Z θ ( µ , µ − γ ) = 0. This14roperty simplifies (4.6) and we are left with s ( µ − µ + γ ) s ( θ + γ ) Z θ ( λ , λ ) = s ( θ + γ − λ + µ ) s ( λ − µ + γ ) s ( λ − µ ) s ( λ − µ + γ ) s ( λ − λ + γ ) s ( λ − λ ) Z θ ( µ , λ ) + s ( θ + γ − λ + µ ) s ( λ − µ + γ ) s ( λ − µ ) s ( λ − µ + γ ) s ( λ − λ + γ ) s ( λ − λ ) Z θ ( µ , λ ) . (4.8)The vanishing condition of Z θ above unveiled have a special appeal since we areinterested in the polynomial solution of (4.3) with order dictated by (4.1). For instance,they allow us to write Z θ ( λ , λ ) | λ = µ = s ( λ − µ + γ ) V ( λ ) (4.9)where now V ( λ ) needs to be a polynomial of the same order as Z θ with L = 1 in orderto satisfy (4.1). The expression (4.9) can now be replaced in (4.8) yielding the followingrelation, Z θ ( λ , λ | µ , µ ) = s ( θ + γ − λ + µ ) s ( θ + γ ) s ( λ − µ + γ ) s ( µ − µ + γ ) s ( λ − µ ) s ( λ − λ + γ ) s ( λ − λ ) V ( λ ) + s ( θ + γ − λ + µ ) s ( θ + γ ) s ( λ − µ + γ ) s ( µ − µ + γ ) s ( λ − µ ) s ( λ − λ + γ ) s ( λ − λ ) V ( λ ) , (4.10)which can be substituted back into the original equation (4.3). This step leave us withan equation involving the functions V ( λ ), V ( λ ), V ( λ ) and V ( λ ) for arbitrary valuesof those variables. Moreover, by setting λ = µ we are left with the equation K V ( λ ) + K V ( λ ) + L V ( λ ) = 0 , (4.11)where the coefficients K , K and L coincide respectively with M , M and N obtainedfrom (3.8) and (3.9) with L = 1 ( n = 2), θ → θ + γ and µ → µ . So the function V ( λ )obeys the same equation as Z θ + γ ( λ | µ ).We have now reached an important stage of this approach. Let us suppose that thesolution of (3.12) with polynomial structure (4.1) is unique. In fact, the uniquenessof the polynomial solutions is demonstrated in the App. E. Since V ( λ ) and Z θ ( λ ) arepolynomials of the same order, that is to say V ( λ ) = Ω Z θ + γ ( λ ) where Ω does notdepend on λ . The solution of (3.12) for L = 1 can be found in the App. F and herewe shall only make use of the solution. In this way the results so far can be gatheredand from (4.10) and (F.5) we immediately obtain an explicit solution for Z θ ( λ , λ ). Thesolution is then given by Z θ ( λ , λ ) = F + F (4.12)15here F ij = Ω s ( θ + γ − λ i + µ ) s ( θ + γ ) s ( θ + 2 γ − λ j + µ ) s ( θ + 2 γ ) s ( λ i − µ + γ ) s ( µ − µ + γ ) s ( λ j − µ ) s ( λ j − λ i + γ ) s ( λ j − λ i ) . (4.13)The constant factor Ω can be fixed by the asymptotic behaviour (4.2) and we find Ω = s ( γ ) s ( µ − µ + γ ). Thus we have completely determined the partition function(2.16) for L = 2 using solely the polynomial structure (4.1) and the asymptotic behaviour(4.2), in addition to the functional equation (3.12). In what follows we shall consider thegeneral L case. Special Zeroes and Symmetry.
The first step in order to consider the case witharbitrary values of L is to obtain an analogous of the relation (4.9) which can be obtainedby uncovering special zeroes of our partition function. These zeroes have been unveiledin App. C, and in addition to that we shall also make use of the symmetry propertydiscussed in App. D. Thus taking into account (C.13) and (D.5) we can write Z θ ( λ , . . . , λ L ) | λ = µ = L (cid:89) i =2 s ( λ i − µ + γ ) V ( λ , . . . , λ L ) . (4.14)We proceed by setting λ = µ and λ L +1 = µ − γ in the functional equation (3.12). Bydoing so we obtain the expression Z θ ( λ , . . . , λ L ) = L (cid:88) j =1 L (cid:89) k =1 (cid:54) = j s ( λ k − µ + γ ) ¯ m j m L V ( λ , . . . , λ j − , λ j +1 , . . . , λ L ) , (4.15)where m L = s ( θ + γ ) s ( θ + ( L + 1) γ ) ( − L +1 s ( γ ) L (cid:89) j =2 s ( µ − µ j + γ ) s ( µ j − µ + γ )¯ m j = s ( θ + γ − λ j + µ ) s ( θ + ( L + 1) γ ) ( − L s ( γ ) L (cid:89) k =2 s ( µ k − µ + γ ) s ( λ j − µ k + γ ) × L (cid:89) k =1 (cid:54) = j s ( λ k − µ ) s ( λ k − µ + γ ) s ( λ k − λ j + γ ) s ( λ k − λ j ) . (4.16)It is important to remark here that we have also considered (4.14) and the symmetryproperty Z θ ( . . . , λ i , . . . , λ j , . . . ) = Z θ ( . . . , λ j , . . . , λ i , . . . ) discussed in the App. D in orderto obtain (4.15). 16he relation (4.15) can now be substituted back into the Eq. (3.12) and considering λ L +1 = µ we obtain L (cid:88) i =1 K i V ( λ , . . . , λ i − , λ i +1 , . . . , λ L )+ L (cid:88) j =2 j − (cid:88) i =1 L ji V ( λ , . . . , λ i − , λ i +1 , . . . , λ j − , λ j +1 , . . . , λ L ) = 0 . (4.17)In their turn the coefficients K i and L ji appearing in (4.17) correspond respectively tothe coefficients M i and N ji given in (3.8) and (3.9) with L → L − n = L ), θ → θ + γ and µ i → µ i +1 . Moreover, the compatibility between (4.14) and (4.1) tells us that thefunction V is a multivariate polynomial of the same order as the partition function Z θ for a lattice with dimensions ( L − × ( L − V ( λ , . . . , λ n ) = Ω n Z θ + γ ( λ , . . . , λ n ) . (4.18)In this way the relation (4.15) can be iterated using the results obtained in the App. Fas initial condition.By carrying on with this procedure we obtain the following expression for our partitionfunction: Z θ ( λ , . . . , λ L ) = (cid:88) { i ,...,i L }∈S L F i ...i l (4.19)where F i ...i l = Ω L (cid:81) Lk =2 s ( µ − µ k + γ ) L (cid:89) n =1 s ( θ + nγ − λ i n + µ n ) s ( θ + nγ ) L (cid:89) n =1 L (cid:89) j = n +1 s ( λ i n − µ j + γ ) n − (cid:89) j =1 s ( λ i n − µ j ) × L − (cid:89) n =1 L (cid:89) m>n s ( λ i m − λ i n + γ ) s ( λ i m − λ i n ) . (4.20)Here S L denotes the permutation group of order L and the asymptotic behaviour (4.2)implies in Ω L = s ( γ ) L (cid:81) Lk =2 s ( µ − µ k + γ ). The expression (4.19, 4.20) can be converted into a multiple contour integral similarly tothe expression recently found in [30] for the U q [ (cid:98) su (2)] vertex model. As a matter of fact,multiple contour integrals seems to fit naturally into the algebraic-functional frameworkpresented here. We start by noticing that the solution of (3.12) for L = 1 given in (F.5)can be rewritten as Z θ ( λ ) = s ( γ )2 π i (cid:73) s ( w − λ ) s ( θ + γ − w + µ ) s ( θ + γ ) d w , (5.1)17here the integration contour contains the pole at w = λ . Now we look to the Eq. (4.15)considering (4.18) and keeping in mind that for L = 1 we have (5.1). This suggests thatthe iteration procedure described by (4.15) can be mimicked by Cauchy’s residue formula.It turns out that when we look to (4.15) searching for solutions as contour integrals, wefind a factorised formula for the integrand. In this way we end up with the followingexpression for our partition function, Z θ ( λ , . . . , λ L ) = (cid:20) s ( γ )2 π i (cid:21) L (cid:73) . . . (cid:73) (cid:81) Li =1 (cid:81) Lj = i +1 s ( w j − w i + γ ) s ( w j − w i ) (cid:81) Li,j =1 s ( w i − λ j ) L (cid:89) j =1 s ( θ + jγ − w j + µ j ) s ( θ + jγ ) × L (cid:89) i =1 i − (cid:89) j =1 s ( µ j − w i ) L (cid:89) j = i +1 s ( w i − µ j + γ ) d w . . . d w L , (5.2)where the integration countours enclose the poles at w i = λ j . As expected the expression(5.2) coincides with (4.19)-(4.20) when evaluated using Cauchy’s residue formula. More-over, in the limit θ → ∞ the formula (5.2) reduces to the one obtained in [30] after arelabelling of the variables µ j . This relabelling does not affect the solution Z θ ( λ , . . . , λ L )since this partition function is invariant under the exchange of variables µ i ↔ µ j as dis-cusssed in [4]. The main result of this paper is the integral representation (5.2) obtained for the partitionfunction of the trigonometric SOS model with domain wall boundaries. This integralformula has been obtained by solving a functional equation derived from the dynamicalYang-Baxter algebra. This approach has been proposed in [11, 12] and here we alsopresent a more robust formulation of that method.In contrast to the case considered in [11, 12], where the su (2) algebra only appears inthe final stages of the derivation of (3.12), here it plays an important role from the verybeginning. For instance, the derivation of (3.6) requires the repeated use of the relations(3.2) and (3.4).It is important to remark here that the elliptic version of this same partition functionhas been considered previously in [7–9, 31]. In particular, the work [8] discusses the lackof a single determinant expression for this partition function generalising the Izergin-Korepin determinant. For the three-colouring model case, a functional equation forthis partition function was obtained in [32, 33] though a connection with the functionalequation presented here is not obvious at the moment. It is also worth remarking thatthe trigonometric SOS model with one reflecting end, and the remaining boundaries ofdomain wall type has been considered in [34,35]. In that case the dynamical Yang-Baxteralgebra also plays an important role, though it is only responsible for a few out of sixconditions determining uniquely the model partition function. The approach consideredhere makes use of only three conditions and it would be interesting to extend it to thecase considered in [34, 35]. 18oreover, it has been recently discussed in [36] the usefulness of such integral formulasfor computing correlation functions for the case of domain wall boundaries which makesthe representation (5.2) more attractive. The generalisation of our results for the ellipticcase is under investigation and we hope to report on that in a future publication. The author thanks J. de Gier and M. Sorrell for many useful discussions and collaborationin [30] where similar integral formulas for domain wall boundaries have appeared. Mostof the calculations presented here have been perfomed at the Max-Planck-Institut f¨urGravitationsPhysik (AEI) to which the author express his sincere thanks for the excellentworking conditions. Financial support from the Australian Research Council and TheCentre of Excellence for the Mathematics and Statistics of Complex Systems (MASCOS)is also gratefully acknowledged.
A Dynamical Yang-Baxter algebra vs. su (2) The analysis performed here will follow the same lines as the one presented in [12]. Weshall consider the su (2) generators E, F and H satisfying the relations[E , F] = H [H , E] = 2E [H , F] = − , (A.1)and compute their commutation relations with the generators of the dynamical Yang-Baxter algebra A ( λ, θ ), B ( λ, θ ), C ( λ, θ ) and D ( λ, θ ). In fact we will only need theircommutation rules with the Cartan generator H whose fundamental representation onthe quantum space is given by H = L (cid:88) i =1 ˆ h i . (A.2)Here ˆ h i consists of the Pauli matrix ˆ h = (cid:18) − (cid:19) (A.3)acting non-trivially on the space V i of the tensor product V ⊗ · · · ⊗ V L .The ice rule (2.11) can be rewritten as [ R aj ( λ, θ j ) , ˆ h j ] = − [ R aj ( λ, θ j ) , ˆ h a ] which im-mediately lead us to the relation[ T a ( λ, θ ) , H] = − [ T a ( λ, θ ) , ˆ h a ] (A.4)due to the definition (2.2). In terms of the monodromy matrix entries (2.14), the relation(A.4) explicitly reads[ A ( λ, θ ) , H] = 0 [ B ( λ, θ ) , H] = 2 B ( λ, θ )[ C ( λ, θ ) , H] = − C ( λ, θ ) [ D ( λ, θ ) , H] = 0 , (A.5)19hich allows us to exploit the representation theory of the su (2) algebra in order to gaininsight into the dynamical Yang-Baxter algebra generators.For instance, the su (2) highest and lowest weight states | (cid:105) and | ¯0 (cid:105) defined in (2.17)obey the relations H | (cid:105) = L | (cid:105) and H | ¯0 (cid:105) = − L | ¯0 (cid:105) . These properties together with(A.5) allow us to obtain the relationH n (cid:89) i =1 B ( λ i , θ + ( i − γ ) | (cid:105) = ( L − n ) n (cid:89) i =1 B ( λ i , θ + ( i − γ ) | (cid:105) (A.6)which is valid for any number n of operators B ( λ, θ ). Now the relation (A.6) put us inposition to use the su (2) representation theory to draw conclusions about the generator B ( λ, θ ). For the case n = L the expression (A.6) tells us that (cid:81) Li =1 B ( λ i , θ + ( i − γ ) | (cid:105) is an eigenvector of H with eigenvalue − L . On the other hand this is the same eigenvalueassociated with the state | ¯0 (cid:105) . Since this eigenvalue is not degenerated we can concludethat L (cid:89) i =1 B ( λ i , θ + ( i − γ ) | (cid:105) ∼ | ¯0 (cid:105) , (A.7)and from (3.5) we immediately have that L +1 (cid:89) i =1 B ( λ i , θ + ( i − γ ) | (cid:105) = 0 . (A.8)The property (A.8) is an important ingredient for the derivation of the functional equa-tion (3.12). B Polynomial structure and asymptotic behaviour
In order to analyse the dependence of Z θ with the set of variables { λ i } we first considerthe following change of variables: x i = e λ i u i = e µ i q = e γ ¯ x i = e λ i ¯ u i = e µ i t = e θ . (B.1)In terms of the above defined variables, the R -matrix given by (2.9) and (2.10) can bewritten as R = 18 q ¯ x ( x U + V) , (B.2)whereU = (3 q + 1)1 ⊗ q − ⊗ H + ( q − t + 1)( t −
1) 1 ⊗ H − ( q − t + 1)( t −
1) H ⊗
1+ 4(1 − q )( t −
1) E ⊗ F + 4 t ( q − t −
1) F ⊗ EV = − (3 + q )1 ⊗ q − ⊗ H − ( q − t + 1)( t −
1) 1 ⊗ H + ( q − t + 1)( t −
1) H ⊗
1+ 4 t ( q − t −
1) E ⊗ F + 4(1 − q )( t −
1) F ⊗ E . (B.3)20n (B.3) the generators E, F and H are the su (2) generators satisfying (A.1), and con-sidering (B.2), (2.13) and (2.14) we readly obtain the expansion B ( λ i , θ ) = 1¯ x Li (cid:104) f ( i ) L x Li + f ( i ) L − x L − i + · · · + f ( i )0 (cid:105) . (B.4)Now looking to the product (cid:81) Lj =1 B ( λ j , θ + jγ ) appearing in the definition (2.16), wecan conclude that Z θ ( λ , . . . , λ L ) = ¯ Z θ ( x , . . . , x L ) L (cid:89) i =1 ¯ x Li , (B.5)where ¯ Z θ ( x , . . . , x L ) is a polynomial of order L in each variable x i .Also from (B.2) we can see that in the limit x → ∞ only the operator U contributes forthe partition function Z θ . In (B.3) the operator U is written in terms of su(2) generatorswhich allows us to follow the same analysis of [11]. Without significant modifications wefind that ¯ Z ∼ ( q − q − ) L L [ L ] q ! L (cid:89) n =1 (1 − q n t )¯ u Ln ( x . . . x L ) L as x i → ∞ . (B.6)Here the q -factorial function is defined as[ n ] q ! = 1(1 + q )(1 + q + q ) . . . (1 + q + · · · + q n ) . (B.7) C Special Zeroes
One important ingredient for solving the Eq. (3.12) under the conditions (4.1) and(4.2) is the localisation of some special zeroes of our partition function. Since we areinterested in the polynomial solution of (3.12), those zeroes will play an important rolein the characterisation of our solution. We shall start by looking to particular values of L for illustrative purposes and then we proceed to the general case. • L = 2: We set λ = µ and λ = µ − γ in such a way that the functions M , M , N , N and N vanish. For these particular values of λ and λ we are thus left with M | λ ,λ Z θ ( µ − γ, µ ) = 0 . (C.1)Since M | λ ,λ is finite we can conclude that Z θ ( µ − γ, µ ) = 0 . • L = 3:
21y setting λ = µ and λ = µ − γ we obtain (cid:88) i =0 P i Z θ ( µ , µ − γ, λ i ) = 0 (C.2)where P = N | λ ,λ , P = M | λ ,λ and P = M | λ ,λ . (C.3)In terms of the variables x i , the functions P i are rational functions and thus ∃ λ i : P i = 0.Besides the above specialisation of the variables λ and λ , we also choose λ i | P i = 0for i = 1 ,
2. Thus we are left with P | λ ,λ Z θ ( µ , µ − γ, λ ) = 0 , (C.4)and since P | λ ,λ is finite we can conclude that Z θ ( µ , µ − γ, λ ) = 0. • L = 4: For the case L ≥ λ = µ and λ = µ − γ similarly to the previous cases. Under this specialisation theEq. (3.12) reduces to M | λ ,λ Z θ ( λ , λ , µ − γ, µ ) + M | λ ,λ Z θ ( λ , λ , µ − γ, µ ) + M | λ ,λ Z θ ( λ , λ , µ − γ, µ ) + N | λ ,λ Z θ ( λ , λ , µ − γ, µ ) + N | λ ,λ Z θ ( λ , λ , µ − γ, µ ) + N | λ ,λ Z θ ( λ , λ , µ − γ, µ ) = 0 . (C.5)Next we set λ = µ and λ = µ − γ . The Eq. (C.5) does not suffer significantsimplifications and we then proceed by setting λ = µ using the following properties:lim λ → µ M N (cid:12)(cid:12)(cid:12)(cid:12) λ ,λ ,λ ,λ = − λ → µ M N (cid:12)(cid:12)(cid:12)(cid:12) λ ,λ ,λ ,λ = − . (C.6)By doing so we end up with the relation M | λ ,λ ,λ ,λ ,λ Z θ ( µ − γ, λ , µ − γ, µ ) = N | λ ,λ ,λ ,λ ,λ Z θ ( µ , µ , µ − γ, µ ) , (C.7)which is further simplified to N | λ ,λ ,λ ,λ ,λ ,λ Z θ ( µ , µ , µ − γ, µ ) = 0 (C.8)with λ = µ − γ . The quantity N | λ ,λ ,λ ,λ ,λ ,λ is finite which allow us to concludethat Z θ ( µ , µ , µ − γ, µ ) = 0 . (C.9)22ow we move backwards considering the consequences of (C.9) to the previous equa-tions. From (C.8) and (C.9) we have that Z θ ( µ − γ, λ , µ − γ, µ ) = 0 , (C.10)which can be reintroduced in (C.5) with the above mentioned specialisation of λ and λ . This yields the following expression Z θ ( λ , λ , µ − γ, µ ) = − N M (cid:12)(cid:12)(cid:12)(cid:12) λ ,λ ,λ ,λ Z θ ( µ , λ , µ − γ, µ ) − N M (cid:12)(cid:12)(cid:12)(cid:12) λ ,λ ,λ ,λ Z θ ( µ , λ , µ − γ, µ ) . (C.11)Now we replace (C.11) back into (C.5) to obtain the expression (cid:88) i =0 Q i Z θ ( µ , λ i , µ − γ, µ ) = 0 . (C.12)The explicit form of Q i is not enlightening and shall not be presented here. Nevertheless,using similar arguements as for the cases L = 2 , Z θ ( µ , λ, µ − γ, µ ) = 0. Thus from (C.11) we obtain the vanishing condition Z θ ( λ , λ , µ − γ, µ ) = 0 . (C.13) • General L : For arbitrary values of L we initially set λ L +1 = µ and λ L = µ − γ in the functionalequation (3.12), followed by the specialisation λ = µ and λ = µ − γ . We collectthe results at each one of the steps and then start fixing the variables λ L − = µ , λ L − = µ − γ , λ L − = µ and so on until we exhaust all the variables. Then theconsistency condition of each step with the previous ones allow us to conclude that Z θ ( µ , µ − γ, λ , . . . , λ L ) = 0 . (C.14)Together with the symmetry property discussed in App. D, the relation (C.14) plays animportant role for solving (3.12). D Z θ as a symmetric function In this appendix we intend to show that Eq. (3.12) admits only analytic solutions whichare symmetric under the exchange of variables λ i ↔ λ j . This is an expected property ofour partition function (2.16) due to the commutation relations (3.3). Nevertheless, weshall demonstrate that this property is not an extra input required to solve Eq. (3.12).23e start by integrating the Eq. (3.12) over the contour C j containing only thevariable λ j . For a given j , the coefficients M j and N kl ( k, l (cid:54) = j ) do not contain poleswhen λ → λ j . Moreover, we also have the following identities between the coefficientslim λ → λ j s ( λ − λ j ) M k = − lim λ → λ j s ( λ − λ j ) N jk k < j lim λ → λ j s ( λ − λ j ) M k = − lim λ → λ j s ( λ − λ j ) N kj k > j (D.1)for j = 1 , . . . , L . Thus after the integration of (3.12) over the contour C j , we are leftwith the relation L (cid:88) i =1 (cid:54) = j ˇ M i ˇ M L +1 [ Z θ ( λ , . . . , λ i − , λ i +1 , . . . , λ L +1 ) − Z θ ( λ j , λ , . . . , λ i − , λ i +1 , . . . , λ L +1 )] = Z θ ( λ j , λ , . . . , λ L ) − Z θ ( λ , . . . , λ L ) (D.2)where ˇ M k = lim λ → λ j s ( λ − λ j ) M k . (D.3)Two observations are important at this stage. Firstly, the relation (D.2) is valid for j = 1 , . . . , L and thus it provides us with a total of L equations. Secondly, we noticethat the RHS of (D.2) does not depend on the variable λ L +1 . In fact this variable canbe adjusted, together with the results obtained for the ( j − Z θ ( λ , . . . , λ L ) = Z θ ( λ j , λ , . . . , λ j − , λ j +1 , . . . , λ L ) (D.4)for j = 1 , . . . , L and consequently we have the desired symmetry relation Z θ ( λ , . . . , λ i , . . . , λ j , . . . , λ L ) = Z θ ( λ , . . . , λ j , . . . , λ i , . . . , λ L ) . (D.5) E Uniqueness
In this appendix we prove the uniqueness of the multivariate polynomial solution of theEq. (3.12). In order to start we first need to introduce the modified coefficients¯ M i = L +1 (cid:89) j =1 (cid:54) = i x − L j M i and ¯ N ji = L +1 (cid:89) k =0 (cid:54) = i,j x − L j N ji , (E.1)with x i = e λ i . In this way the Eq. (3.12) is given by L +1 (cid:88) i =1 ¯ M i ¯ Z θ ( λ , . . . , λ i − , λ i +1 , . . . , λ L +1 )+ L +1 (cid:88) j =2 j − (cid:88) i =1 ¯ N ji ¯ Z θ ( λ , . . . , λ i − , λ i +1 , . . . , λ j − , λ j +1 , . . . , λ L +1 ) = 0 , (E.2)24n terms of ¯ Z θ which is a polynomial of order L in each variable x i according to (4.1).We shall now explore the linearity of the Eq. (E.2). More precisely that means thefollowing: if ¯ Z and ¯ Z are two multivariate polynomials of type (4.1) satisfying (E.2),then ¯ Z = α ¯ Z − β ¯ Z (E.3)is also a solution for any constants α and β . Polynomials are characterised by the locationof their zeroes and we can express ¯ Z , ¯ Z and ¯ Z as¯ Z ∼ L (cid:89) i =1 ( x − r i ) ¯ Z ∼ L (cid:89) i =1 ( x − s i ) ¯ Z ∼ L (cid:89) i =1 ( x − t i ) (E.4)where x can represent any of the variables x i . Since the constants α and β in (E.3) arearbitrary they can always be fine tunned in order to ensure that ¯ Z is also of order L .Next we set x = r j in (E.3) and from (E.4) we obtain α L (cid:89) i =1 ( r j − s i ) ∼ β L (cid:89) i =1 ( r j − t i ) . (E.5)The relation (E.5) allows us to make important conclusions. For instance, if we assumethat { r i } (cid:54) = { s i } then (E.5) implies that { s i } = { t i } since α and β can always beadjusted to compensate an overall factor. This implies that ¯ Z and ¯ Z are proportionalto each other and so is ¯ Z due to (E.3). This consequence clearly contradicts the initialassumption { r i } (cid:54) = { s i } . The remaining option is allowing { r i } = { s i } and thus ¯ Z and ¯ Z only differ by a constant. This fact together with (E.3) tell us that ¯ Z is alsoproportional to ¯ Z . In summary this analysis shows that if we have two polynomials ofthe same order solving (E.2), they are essentially the same polynomial. This proves theuniqueness of the polynomial solution of (3.12). F Solution for L = 1 Here we shall present the solution of the Eq. (3.12) for the case L = 1 which is offundamental importance in order to derive the solution for general L . For the case L = 1the Eq. (3.12) reads M Z θ ( λ ) + M Z θ ( λ ) + N Z θ ( λ ) = 0 . (F.1)At first look the condition λ i = λ does not seem helpful in finding the solution of (F.1).However, a closer look reveals that the coefficients M , M and N contain poles whenthe variables λ i coincide. As we shall see this fact will be of fundamental importance.We can compute the limit λ i = λ of the Eq. (F.1) using L’Hopital’s rule and we areleft with the following second order differential equation P ¯ Z θ + P d ¯ Z θ dx + P d ¯ Z θ dx = 0 , (F.2)25iven in terms of variables x = e λ and u = e µ . The coefficients in (F.2) are given by P = ( − q + 2 q t + 2 q t ) u + (2 q + 2 q − q t ) xP = ( − q t + 2 q t + 2 q t ) u + (4 q − q t ) xu + ( − q − q + 4 q t ) x P = (1 + q − q t + q t + q t ) xu + ( − q − q t + 5 q t + 5 q t − q t − q t ) ux + ( q + q − q t + q t + q t ) x , (F.3)and by standard methods we find the general solution¯ Z θ ( x ) = C ( x − q t u ) + C ( x − q t u ) (cid:90) e − (cid:82) x P x (cid:48) ) P x (cid:48) ) dx (cid:48) ( x − q t u ) dx , (F.4)where C and C are two arbitrary integration constants. Now the polynomial structure(4.1) asks for C = 0, while the asymptotic behaviour (4.2) implies in C = ( q − q − )2 (1 − t q ) − u − . Thus our partition function for L = 1 is given by Z θ ( λ ) = s ( γ ) s ( θ + γ − λ + µ ) s ( θ + γ ) . (F.5) References [1] R. J. Baxter. Partition function of 8-vertex lattice model.
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