On the equivalence between n-state spin and vertex models on the square lattice

In this paper we investigate a correspondence among spin and vertex models with the same number of local states on the square lattice with toroidal boundary conditions. We argue that the partition functions of an arbitrary $n$-state spin model and of a certain specific $n$-state vertex model coincide for finite lattice sizes. The equivalent vertex model has $n^3$ non-null Boltzmann weights and their relationship with the edge weights of the spin model is explicitly presented. In particular, the Ising model in a magnetic field is mapped to an eight-vertex model whose weights configurations combine both even and odd number of incoming and outcoming arrows at a vertex. We have studied the Yang-Baxter algebra for such mixed eight-vertex model when the weights are invariant under arrows reversing. We find that while the Lax operator lie on the same elliptic curve of the even eight-vertex model the respective $\mathrm{R}$-matrix can not be presented in terms of the difference of two rapidities. We also argue that the spin-vertex equivalence may be used to imbed an integrable spin model in the realm of the quantum inverse scattering framework. As an example, we show how to determine the $\mathrm{R}$-matrix of the 27-vertex model equivalent to a three-state spin model devised by Fateev and Zamolodchikov.


Introduction
In statistical mechanics systems are sometimes modeled by lattice models in which the set of possible microstates are specified by placing local spin variables along the interacting sites of the lattice.An energy interaction or equivalently a Boltzmann weight is then assigned to each possible microstate configuration according to the specific lattice model.In this paper we focus on a square lattice of size L × L with periodic boundary conditions in the horizontal and vertical directions.
Here we shall also consider that the microstates are described by discrete spin variables assuming n possible values.One of the simplest type of such model is when the spin variable σ i,j sits on the lattice site (i, j) and the energy interactions involve only nearest neighbors sites configurations.
These systems are called spin models and the simplest prototype is a two state system known as the Ising model [1,2] whose partition function is, (J h σ i,j σ i,j+1 + J v σ i,j σ i+1,j + Hσ i,j ) where the sum < σ i,j > is over all the two-state spins variables σ i,j = ± of the lattice and β = 1 k B T is the thermal factor.The couplings J h and J v correspond to the interaction energies in the horizontal and vertical lattice directions and H represents an external magnetic field.This model in the absence of a magnetic field was solved by Onsager who has computed exactly the respective free energy [3].
Another important family of lattice models are vertex models in which the spin variables sit on the four links of given site of the square lattice.They have emerged in the context of the residual entropy of the ice and in certain phase transition exhibited by hydrogen-bonded crystals [4,5].The statistical configurations are characterized by the two possible positions of the hydrogens which usually are indicated by incoming and outcoming arrows placed along the links [6].One may also use an alternative description of the statistical configurations since there exists a direct correspondence between arrow configurations and a two state spin variable denoted here by the states ±.This gives rise to a two state vertex model and if no restriction is imposed to the hydrogen atoms positions we have on the square lattice sixteen possible vertex weights configurations.These vertex configurations are shown in Fig. (1) using both the arrow configurations and the spin ± variables.
We observe that weights can be organized in terms of two distinct families of eight vertex states according to the even or odd number arrows orientations at a vertex.The standard eight vertex model [7] corresponds to the system in which the number of in and out arrows are even and the Figure 1: The sixteen-vertex configurations of the general two state vertex model.In the top row we showed the even eight energy weights w i and in the down row we have indicated the odd energy weights by v i .
respective weights are denoted by w 1 , . . ., w 8 .The vertex model with an odd number of in and out arrows has been denominated odd eight-vertex model which encompasses the classical Ashkin-Teller model [8].The corresponding vertex weights are indicated here by v 1 , . . ., v 8 .We remark that our notation for the even vertex weights is the most common choice in the literature [6] while the representation of the odd vertex weights is done by reversing the up vertical states.The partition function of the full sixteen-vertex model can be defined by the following sum, where the summation is extended over all possible arrow configurations, the integers n i and m j indicate respectively the total number of weights w i and v j in given lattice configuration.We assume periodic boundary conditions for arrows configurations in both lattice directions.
Over the years certain mapping relations between two state spin models and the eight-vertex and the sixteen-vertex models on the square lattice have been introduced in the literature.There is a mapping between the Ising model with zero magnetic field and the eight-vertex model with even weights in which the spin variables lie on the faces of the square lattice [9][10][11].This is however a two-to-one correspondence and the partition function of the Ising model turns out to be twice of that of the equivalent even eight-vertex model.In another mapping one introduces the spin variables on the medial points of the square lattice and the configurations for the four spins around a vertex give rises to sixteen-vertex weights [12].By using this equivalence the Ising model in the presence of a magnetic field can be mapped to a specific sixteen-vertex model on the square lattice [6,13].
We observe that in such correspondence the spin model has twice as many lattice points as the vertex model and in the thermodynamic limit the free-energy of the Ising model in a magnetic field is half of that of the equivalent sixteen-vertex model.However, there is a more direct formulation of the isotropic Ising model with non-zero magnetic field as a sixteen-vertex problem which uses as intermediate step a mapping to a lattice gas on the square lattice [6].The interesting feature of this equivalence is that it is valid for toroidal square lattice with a finite number of lattice points.
We remark that this type of correspondence has been further elaborated in [14] yet restricted to the case of isotropic J v = J h interactions.For later comparison with our results we present below the expressions of the weights of the equivalent sixteen-vertex model.It turns out that the partition functions of the isotropic Ising model in a magnetic field (1) and of the sixteen-vertex model (2) coincide provided that the vertex weights are given by [6], We next mention that equivalences between spin and vertex models with arbitrary number of local states have also been pursued in the literature [15,16].These correspondences involve somehow either a different number of states for the spin and vertex models or if the number of the states are equal the respective mapping is of multiplicity two.One of the purpose of this paper is to associate to any n-state spin model with next-neighbor interactions an equivalent n-state vertex model both defined on the square lattice.Our mapping is in the sense of the last mentioned equivalence among the isotropic Ising model in a nonzero magnetic field and the sixteen-vertex model.This means that our correspondence implies that the partition functions of the n-state spin and vertex models coincide where the summation is similar to the one already defined for the general sixteen vertex model.
In this paper we argue that the partition functions of the Ising model in a magnetic field (1) and that of the mixed eight-vertex model ( 4) are the same for particular choices of the vertex weights.
The dependence of the vertex weights with the Ising model edge interactions is exhibited in Fig. (2).
We emphasize that such relationship is valid for an anisotropic Ising model with generic couplings J h and J v in the presence of an external magnetic field.
e β(−J h −Jv) e β(−J h −Jv) We have organized this paper as follows.In next section we introduce the n-state spin and vertex models and formulate their partition functions in terms of the transfer matrix concept.In section 3 we describe two possible correspondences of the among the n-state spin model and a nstate vertex model with n 3 non-null weights.We use the Hamiltonian limit to built an anstaz for the main structure of the weights of the equivalent n-state vertex model.The match of the partition functions is done by comparing the respective transfer matrices operators for n ≤ 4 and L ≤ 6 and we conjecture that our mappings should be valid for general n and L. In section 4 we study the Yang-Baxter algebra for the mixed eight-vertex model on the subspace of symmetrical weights.This manifold encodes the Ising model without an external magnetic field.We find a solution to the Yang-Baxter relation in which the respective R-matrix is not expressible in terms of the difference of the spectral variables parameterizing the respective Lax operator.The corresponding spin chain is shown to be related to that of the XY model in a transverse magnetic field with a Dzyaloshinky-Moriya interaction.This has motived us in section 5 to investigate possible mapping among the mixed eight-vertex model and the even eight-vertex model with weights satisfying the free-fermion condition.As a byproduct of this analysis we present novel correspondences among the Ising model in absence of magnetic field and the free-fermion even eight-vertex model such that their partition functions are exactly the same for toroidal finite lattice.In section 6 we discuss the possibility of embedding an integrable spin model on the context of the quantum inverse scattering framework.
In particular, we have determined the underlying R-matrix of the 27-vertex model equivalent to the N = 3 Fateev-Zamolodchikov spin model.In the appendices we summarize some of technical details we have omitted in the main text. 2 The n-state spin and vertex models The n-state spin lattice model with nearest neighbors interactions can be built out of n 2 horizontal and n 2 vertical edge interactions weights [7,17].Let us denote by σ i,j the state variables at the site (i, j) of the square lattice of size L. We then associate local horizontal W h (σ i,j , σ i,j+1 ) and vertical W v (σ i,j , σ i+1,j ) Boltzmann weights to characterize the energy interactions among two neighboring spins.These edge weights are schematically shown in Fig. (3).
) and the vertical W v (σ i,j , σ i+1,j ) local Boltzmann weights of spin models.The spin variables σ i,j can take n possible values.
The respective partition function is the sum of the product of all the local Boltzmann weights which can be written as follows, where sum is over all allowed spin σ i,j configurations on the lattice.Periodic boundary conditions are imposed considering the identifications σ i,L+1 = σ i,1 and σ L+1,j = σ 1,j .
It is well known that the partition function ( 5) can be obtained as the trace of successive matrix multiplications of an operator called transfer matrix [18].For the spin model it is sometimes convenient to consider the diagonal-to-diagonal transfer matrix T dia (L) which is built on the lattice states along the diagonals [19,20].We recall that this operator plays an important role on the construction of integrable spin models derived from a family of commuting transfer matrices [7,17].Considering periodic conditions in the horizontal the elements of the diagonal-to-diagonal transfer matrix are, where b L+1 = b 1 .We observe that the transfer matrix is defined in the so-called quantum space ⊗C n and the partition function can be obtained as follows, In the vertex models the local configurations are defined by the spin variables attached to the four links of the square lattice joining together at the vertex [7,17].To a given vertex at site (i, j) we associate a Boltzmann weight w (α ij , α ij+1 |γ ij , γ i+1j ) as exhibited in Fig. (4).Here we assume that the horizontal and vertical spin variables α ij and γ ij take values on the same finite set constituted of n states.This means that the total number of weights defining this vertex model is therefore n 4 .The partition function associated to the such n-state vertex models on the square lattice can be written by the following expression, where sum is over all allowed horizontal α ij and vertical γ ij spin configurations on the lattice.Periodic boundary conditions are imposed considering the identifications α iL+1 = α i1 and γ L+1j = γ 1j .
The vertex models has the advantage of having an underlying tensor structure.This plays an important role in connection with quantum spin chains and to the formulation of a quantum version of the inverse scattering method [21].This property allows one to represent the partition function under toroidal boundary conditions as the trace over two spaces associated to the horizontal and vertical degrees of freedom of the vertex weights.To this end we define a set of matrices named Lax operators as follows, where e i 1 ,i 2 denotes the n × n matrix with only one non-vanishing entry with value 1 at row i 1 and column i 2 .This is the basis of the space of the Lax operator denominated auxiliary or horizontal space A = C n .The vertical degrees of freedom gives rise to the quantum space basis e (j) (10) where I n is the n × n identity matrix.
By virtue of the periodic boundary condition on the horizontal direction the row-to-row transfer matrix T ver (L) associated to the vertex models can be written in a compact form as the trace over the auxiliary space of an ordered product of Lax operators, where T ver (L) is again an operator belonging to the quantum space Once again the partition function of the vertex model can be obtained by multiplying layers of transfer matrices and for toroidal boundary conditions we obtain, In next section we shall argue that the partitions functions Z spin (L) and Z ver (L) coincide for a suitable choice of the vertex weights w(i 1 , i 2 |i 3 , i 4 ).

The spin-vertex correspondence
We start by noticing that the edge weights of the n-state spin model can be seen as coordinates of the product of two projective spaces while the Boltzmann weights of a n-state vertex model may be interpreted as points of a single projective space.Here we would like to discuss a map ϕ among these two projective spaces of the form, where m ≤ n 4 − 1 since some of the vertex weights may be zero.
In order to shed some light on the structure of the vertex weights we investigate the Hamiltonian limit of both spin and vertex models.For the spin model the underlying spin chain Hamiltonian is obtained by expanding the edge weights around a point in which the diagonal-to-diagonal transfer matrix reduces to the identity matrix, see for instance [22][23][24].To this end we start our analysis by assuming that the edge weights can be expanded in terms of some spectral parameter denoted here by ε.We next consider that the expansion of the edge weights around ε = 0 up to the first order is given by, where Ẇh (i 1 , i 2 ) and Ẇv (i 1 , i 2 ) denote the edge weights expansion coefficients.
At this point we note that at ε = 0 the diagonal-to-diagonal transfer matrix (6) indeed becomes the identity matrix I d with dimension d = n L .The Hamiltonian limit is obtained by expanding the transfer matrix (6) about the parameter ε and up to the first order we have, where the two-body Hamiltonian H (spin) j,j+1 is given by, For the vertex model the Hamiltonian limit is considered by expanding the logarithm of the row-to-row transfer matrix (11) around a point in which the respective Lax operator reduces to the permutator defined on the tensor product C n ⊗ C n [25].We next assume that the expansion of the vertex weights giving the permutator at zero order is as follows, where ẇ(i 1 , i 2 |i 3 , i 4 ) is the vertex weights first order expansion coefficients.By expanding the logarithm of the transfer matrix about ε we obtain to the first order, where the two-body Hamiltonian H (ver) j,j+1 is, The next step is to compare the expressions for the spin and vertex two-body Hamiltonians (16,19).We find that they can be matched up to the first order in ε once we have the following relationships, Motivated by the last identification we propose our ansatz for the mapping among the edge and vertex weights, namely and therefore the equivalent vertex model has n 3 non-null weights.
We now start presenting our evidences that the ansatz (21) should imply that the partition functions of the spin and vertex models coincide.To this end we turn our attention to the computation of the vertex model row-to-row transfer matrix (11) for some values of n and L. For L = 1 the transfer matrix is just sum of matrices associated to the diagonal partitions of the Lax operators.By using the mapping relation ( 21) its matrix elements can be computed for arbitrary n to be, For L = 2 we have to compute the product of partitions of two Lax operators and this calculation is in general involved for an arbitrary n-state vertex model.In our case however the equivalent vertex model has a number of suitable null weights and we have find the matrices elements of such products of partitions are single monomials constituted by the product of two vertex weights.We have carried out these computations explicitly for n ≤ 4 and after using the proposal (21) we found that the matrix elements of the row-row transfer matrix can be organized as follows, We observe that results for L = 1, 2 have indeed a simple pattern very similar to that of diagonalto-diagonal transfer matrix, see Eq.( 6).This fact permits us to guess what could be the structure of the matrix elements of the row-to-row transfer matrix for arbitrary L, namely where b L+1 = b 1 .With the help of symbolic algebra packages we have verified for n ≤ 4 and L ≤ 6 that the formulae (24) indeed produce the matrix elements of the row-to-row transfer matrix (11) with the vertex weights (21).In the case of two-state models we managed to check this result up to L = 10.Based on this checking we conjecture that such result should be valid for arbitrary n and L but a systematic proof of that has eluded us so far.
We finally note that the matrices elements ( 24) is same of the diagonal-to-diagonal transfer matrix (6) by interchanging the horizontal and vertical edge weights.This is equivalent to rotation of the lattice by 90 0 degrees and certainly the partition function is invariant under such transformation.
Therefore, under the assumption of the validity of the expression (24) for arbitrary n and L we can write the partition function of the spin model as, provided that the vertex and the edge weights are related by Eq.( 21).
This leads us to conjecture that the partition functions of the spin and vertex models coincide when the respective Boltzmann weights satisfy the relation given by Eq.( 21).We stress that this result does not depend on the fact that the spin and vertex models have necessarily an underlying Hamiltonian limit.

Another equivalence
In the transfer matrix method we have first to define the respective layers of the system.As such we can choose the layers based on the configurations of rows of spins as originally devised for the Ising model [18,32].This gives rise to the row-to-row transfer matrix T row (L) in which we can separate the horizontal and vertical interactions by writing where the matrix elements of T v (L) and T h (L) are given by, and for periodic boundary conditions we have a L+1 = a 1 and b L+1 = b 1 .
As before we can express the partition function of the spin model as a trace of a product of transfer matrices, Inspired by our earlier analysis we investigate whether or not the transfer matrix of the vertex model with weights satisfying the condition, can somehow be related to the spin row-to-row transfer matrix by suitable pairing of horizontal and vertical edge weights.
As before we have performed this analysis for models up to four states per site and with L ≤ 6.
It turns out that we find that it is possible to match these transfer matrices, namely provided that the vertex model weights satisfy the following relation, We remark that specifically for n = 2 the above result has been verified up to L = 10.These verifications seems robust enough to conjecture the validity of the expressions (30,31) for arbitrary n and L.

Application to the Ising model
We start by presenting the corresponding edge weights for the Ising model in a magnetic field.
Considering the spin basis the horizontal and vertical edge weights are given by, We now consider the equivalent vertex model and recall that the respective Boltzmann weights satisfy the property w(i 1 , i 2 |i 3 , i 4 ) = 0 for i 1 = i 4 .This means that we have only eight non-null weights and by using the graphical representation given in Fig.
(1) of the Lax operator (9) we have, Considering the first equivalence between spin and vertex model ( 21) it follows from Eq. (32,33) that the Boltzmann weights of the mixed vertex model are, as have been illustrated in Figure (2).
On the other hand the second equivalence between spin and vertex model (31) tell us that the corresponding Boltzmann weights of the mixed vertex model are, which is now illustrated in Fig. (5).

Integrable Manifold for the Mixed Vertex Model
In two spatial dimensions a lattice model of statistical mechanics is called integrable when the respective transfer matrix commutes for distinct set of Boltzmann weights [7].For vertex models Figure 5: Alternative equivalence of the Ising model in a magnetic field with a mixed eight-vertex model on the square lattice with toroidal boundary conditions.
a sufficient condition for commuting transfer matrices is the existence of an invertible R-matrix satisfying the following Yang-Baxter algebra, where w ′ and w ′′ denote two different sets of vertex weights and R(ω For the mixed eight-vertex model we may order the basis as |+, + , |+, − , |−, + , |−, − and the respective Lax operator can be represented by the following matrix, In this section we study solutions of the Yang-Baxter equation ( 36) when the vertex weights are unchanged by reversing the arrows which is equivalent to the spin reversal + ↔ − symmetry.As a result we have only four distinct weights due to the following identifications, and we note that such symmetric manifold encodes the Ising model without an external magnetic field, see Eqs. (34,35).
In what follows we also assume that the underlying R-matrix has the same matrix form of the Lax operator.Indicating its matrix elements by bold letters we have, where as in the case of the Lax operator we assume w 2 = w 1 , w 6 = w 5 , v 2 = v 1 and v 6 = v 5 .
We obtain the functional equations constraining the R-matrix elements and the vertex weights by substituting the proposals (37,39) in the Yang-Baxter equation ( 36).In the case of the symmetric manifold (38) we have twelve independent relations which can be subdivided in terms of their number of monomials.We have four simple relations involving only two monomials given by, The eight remaining equations contain four monomials and their expressions are, We consider the solution of Eqs.(40-51) as a system of homogeneous polynomials relations for the R-matrix elements w 1 , w 5 , v 1 and v 5 .We first observe that the determinant of the pair of equations (40,41) and (42,43) are the same and such determinant must vanish.This condition assures that R-matrix entries are not all zero and as result we have the constraint, where ∆ 1 is a free constant.We now can solve Eqs.(40,42) for the R-matrix entries w 1 and v 1 to obtain, By substituting the above results into Eqs.(44-47)we find that they provide us a single functional relation.For instance, we can solve Eqs.(44-47) for v 5 and after some simplifications we obtain, and consequently the R-matrix entries have the same common factor w 5 .
At this point we are left to solve Eqs.(48-51).After using the previous results one can show that these relations are proportional to each other.We find that their solution gives rise to a second algebraic invariant, (w where ∆ 2 is a free constant. Interesting enough, the algebraic invariants (52,55) have a direct one-to-one correspondence with those associated to the even eight-vertex model with symmetric weights.This equivalence is summarized in Appendix A and as result one concludes that the vertex weights w 1 , w 5 , v 1 , v 5 sit on the same elliptic curve satisfied by the vertex weights of the even eight-vertex model [7].Following Baxter monograph [7] the vertex weights of the symmetric mixed eight-vertex model can be uniformized in terms of Jacobi elliptic functions, where x is the spectral parameter, λ is a free parameter and sn[x, k] represents the elliptic Jacobi function of modulus k.The dependence of the invariants ∆ 1 and ∆ 2 with the modulus and λ is given by, where cn[x, k] and dn[x, k] denote the other two elliptic Jacobi functions.
However, contrary to what happens with the symmetric even eight-vertex model, the R-matrix associated to the mixed symmetric eight-vertex model is not given as a function of the difference of spectral parameters.We can see that computing explicitly the R-matrix elements (53,54) in terms of the spectral parameters x 1 and x 2 associated to the uniformization of the set of weights { w , respectively.Taking into account the uniformization (56) and with the help of addition identities of elliptic functions we obtain, and we observe that not all the matrix elements can be represented solely as functions of the difference of spectral parameters.This means that the R-matrix of the mixed symmetric eight-vertex model lie on a surface rather than a curve such is the case of the R-matrix of the symmetric even eight-vertex model.See appendix A for a discussion about the geometry of the later R-matrix.
If we choose the overall normalization w 5 = In addition to that, we have also verified that such R-matrix fulfills the Yang-Baxter equation, being a sufficient condition for the associativity of the Yang-Baxter algebra.
Our last remark concerns with the commutativity of the transfer matrix of the Ising model with zero magnetic field.This spin model is encoded in the symmetric mixed eight-vertex and considering the first spin-vertex equivalence with H = 0 we have, At this point we recall the transfer matrix of the mixed eight-vertex model commutes when the weights satisfy the restrictions (52,55).By substituting the weights (61) in the relation (52) we find that the invariant ∆ 1 has is fixed to the unity.The second invariant ∆ 2 provides us a relation among the horizontal and vertical spin couplings given by, reproducing the celebrated condition for the commutativity of the diagonal-to-diagonal transfer matrix of the Ising model with zero magnetic field [19].Recall here that such relation has been previously derived in the context of the even eight-vertex which can be regarded as a two next nearest-neighbour Ising model at some decoupling point [7].By way of contrast our derivation is in the context of the Ising model originally solved by Onsager [3].
It turns out that similar analysis can also be carried for the second spin-vertex correspondence (35) which for H = 0 reads, Now after substituting the weights (63) in the expressions of the invariants (52,55) we obtain, and therefore the row-to-row transfer matrix of the Ising model does not yields a one parameter family of commuting operators.

The R-matrix geometry
In order to investigate the geometric properties of the R-matrix of the symmetric mixed eightvertex model we need to determine the form algebraic variety which is satisfied by matrix elements w 1 , w 5 , v 1 and v 5 .To this end we first analyze the behaviour of the left hand side of algebraic invariants (52,55) when the vertex weights are replaced by the respective R-matrix elements.More precisely, want to compute the auxiliary functions F 1 (ω ′ , ω ′′ ) and F 2 (ω ′ , ω ′′ ) such that, We substitute in Eq.( 65) the expressions of the R-matrix elements (53,54) and after a systematic use of the vertex weights invariants (52,55) we find, The fact that functions F 1 (ω ′ , ω ′′ ) and F 2 (ω ′ , ω ′′ ) are not constants but instead vertex weights dependent suggests that the variables w 1 , w 5 , v 1 , v 5 should lie on two-dimensional variety.The algebraic form of such surface can be determined after we eliminate the vertex weights w The geometric properties of surfaces has been studied by a number of algebraic geometers long ago and have culminated in the famous Kodaira-Enriques classification, see for instance [26,27].The crucial point in such classification problem concerns with the resolution of the nature of the surface singularities.The singular points on a surface form a closed subvariety Sing(S) determined by the zeroes of all the partial derivatives of S, namely and we find that Sing(S) is constituted of twelve isolated singular points given by, Here we are in a fortunate situation since the presence of only isolated singularities tell us that S is a normal surface.It turns out that a normal quartic surface can be either a K3 surface, a ruled surface over an elliptic or a genus 3 curves or still a rational surface [28,29].The classification problem of the surface (67) in one of these four possible categories can be done investigating the nature of the underlying singularities.In our case we find that all the twelve singularities are ordinary double points since the Taylor series expansion around the singular points give rise to nondegenerate quadratic forms [30].These type of singularities do not affect the geometric properties of the surface and the minimal resolution of the singularities are non-singular quartics [28,29].It is well known that projective quartic surfaces without singularities are classical examples of the K3 surfaces [26,27] and therefore the birational class of the surface S is characterized as follows, We finally stress that this scenario is very different from that of the even eight-vertex model with symmetry weights.In fact, the R-matrix of the symmetric even eight-vertex model lie on a elliptic curve rather on a surface, see Appendix A.

The Hamiltonian Limit
Here we consider the Hamiltonian limit of the integrable mixed eight-vertex with symmetry weights discussed in the previous section.The two-body Hamiltonian H (mix) j,j+1 is obtained by expanding the respective Lax operator around the permutation operator P and up to the first order we have, where ε is the expansion parameter and the permutator P The expansion of the vertex weights reducing the Lax operator to the permutator at zero order is as follows, and since the vertex weights are require to satisfy the invariants (52,55) we have two constraints among the expansion coefficients, Collecting the above results we find that the two-body Hamiltonian can be represented by the following matrix, The resulting Hamiltonian for a chain of length L can be represented in terms of spin-1 2 Pauli matrices, and by choosing the overall normalization v5 = −2J/∆ 2 we find that the expression of the Hamiltonian up to an additive constant is, where periodic boundary condition σ is assumed.
Since the Pauli matrices are Hermitian the Hamiltonian (76) is non-Hermitian for arbitrary values of the parameters ∆ 1 and ∆ 2 .However, this operator becomes Hermitian if we restrict the parameters to the following subspace, with 0 ≤ θ ≤ π and κ ∈ R. In this situation we can rewrite the Hamiltonian as, where the couplings h and D lie on a circle of radius κ, The first two terms of the Hamiltonian (78) represent the Ising quantum spin chain in a transverse field interaction related to the classical two-dimensional Ising model in the absence of a magnetic field.
The third term resembles the type of exchange interaction devised by Dzyaloshinsky and Moriya [31] to explain the phenomenon of weak ferromagnetism.This can be better seen by considering the following canonical transformation on the Pauli matrices, where the transformation parameters satisfy the relations, (81) By performing the transformation defined by Eqs.(80,81) we find that the Hamiltonian (78) can be rewritten as follows, where the coupling γ We note that the first two terms of Eq.(82) represent the Hamiltonian of the XY model in the presence of a perpendicular magnetic field h.On the other hand, the last antisymmetric term is exactly the interaction we obtain when the Dzyaloshinsky-Moriya exchange vector is projected in the z-direction.
We conclude this section with the following remark.It is well known that the classical statistical model having the XY model with a magnetic field in the z-direction as the underlying spin chain turns out to be the even eight-vertex model with weights satisfying the free-fermion condition [33][34][35].The Hamiltonian limit of this vertex model has an arbitrary choice of one free parameter in the weights expansion which gives rise to the Dzyaloshinsky-Moriya interaction.In fact, for periodic boundary conditions the Dzyaloshinsky-Moriya term commutes with the Hamiltonian of the XY spin chain.
Since the Hamiltonian limits of the mixed eight-vertex and even free-fermion eight-vertex models are somehow related it is natural to ask whether or not such relationship can be extended on the level of the corresponding partition functions.In next section we explore this possibility and argued that this is case provided that the weights of the mixed eight-vertex model satisfy certain constraints.

Mapping among eight-vertex models
The partition function of vertex models may be invariant under various kind of transformations between the corresponding Boltzmann weights.One important such symmetry is called gauge transformation which a similarity transformation acting on the horizontal and vertical space of states of the vertex model [36].Here we shall study such transformation for the mixed eight-vertex model which in terms of the respective Lax operator reads, where M 1 and M 2 may be any non-singular 2 × 2 matrices.
We consider that the transformed Lax operator L (tra) (ω) to be in the form of that associated to the even eight-vertex.In order to avoid confusion between weights notations we write the Lax operator of the even eight-vertex model as, where for completeness the vertex configurations are illustrated in Fig. (6).
The gauge transformation (83) leads to eight quadratic polynomial relations among the elements of the matrices M 1 and M 2 which have to be solved before the weights a ± , b ± , c ± , d ± are fixed.These relations are directly associated to the fact that the even eight-vertex models have exactly eight null weights.We find that these equations have a solution provided that the weights of the mixed eight-vertex model satisfy the following constraints, and the corresponding transformation matrices are given by, where z 1 and z 2 are arbitrary non-null free parameters.
The remaining eight relations coming from the gauge transformation are able to determine the weights of the equivalent even eight-vertex model.After some simplifications we obtain, and from the above expressions we can indeed verify that the weights satisfy the free-fermion condition, At this point we recall that for toroidal boundary conditions the vertex weights c ± and d ± always occurs as product combinations c + c − and d + d − in the sums of the partition function [7].This means that the partition function of the equivalent even eight-vertex model does not depend on the free parameters z 1 and z 2 which can be used to set c − = c + and d − = d + .We next note that our mappings between the mixed eight-vertex and the Ising model given by Eqs.(34,35) fulfill the restriction (85) in the absence of a magnetic field.As a consequence of that we can map the Ising model with zero magnetic field onto the even eight-vertex model with weights satisfying the free-fermion condition.
Considering the map defined by Eq.( 34) we find that the respective weights of the free-fermion even eight-vertex model are 1 , while the equivalence given by Eq.( 35) lead us to the following weights We emphasize that the above mappings are valid on a finite toroidal square lattice in which the partition functions of Ising with zero magnetic field and the free-fermion eight-vertex models with weights (89,90) are exactly the same.We point out that the equivalences here differ from the one between the checkerboard Ising model and the even eight-vertex model with the free-fermion condition [37].In this case the mapping is valid only in the thermodynamic limit and the partition function of the Ising model is twice as that of the equivalent even free-fermion eight-vertex model.
The extention of such mapping to toroidal lattice requires to consider four types of Ising model 1 Here we have set β = 1 partition functions combining periodic and antiperiodic boundary conditions [38].Therefore, we believe that our mappings (89,90) are new in the literature since the Ising model and its equivalent free-fermion eight-vertex model are considered on the same toroidal square lattice.
6 Integrable three-state spin model and the equivalent Rmatrix In this section we argue that our first spin-vertex correspondence provides us in principle the means to determine the underlying R-matrix of the equivalent vertex model associated to a given integrable spin model.Recall that the R-matrix is an essential object for the solution of an integrable model by the quantum inverse scattering method.The basic idea is to use the mapping to built the explicit form of the Lax operator and afterwards we are left to solve the Yang-Baxter algebra for the respective R-matrix.The Yang-Baxter algebra with a given Lax operator leads us to solve a set of linear relations for the R-matrix elements avoiding us to deal with functional equations.Here we shall discuss this alternative approach for an integrable three-state spin model and for sake of illustration we choose the one with simplest weight structure.The example is the N = 3 Fateev-Zamolodchikov spin model [39] whose weights are given by, where b(x) = sin( P i 6 −x) sin( P i 6 +x) and where b(x) = sin(x) cos( P i 6 +x) .We now use the spin-vertex correspondence (21) to built the Lax operator of the corresponding 27-vertex model.This operator can be represented by the following matrix, L (27) where the vertex weights are given by, w 1 (x) = 1, w 2 (x) = sin (x) cos P i 6 + x , w 4 (x) = sin P i 6 − x sin P i 6 + x w 3 (x) = w 5 (x) = sin (x) cos P i 6 + x The next step in this approach is to find the R-matrix which solves the Yang-Baxter algebra,

R
In order to determine the structure of the R-matrix we choose two distinct point x and y and solve numerically the relation (94) for a general 9 × 9 R-matrix.By applying this procedure for a number of distinct pair of points we find that many of the R-matrix elements are zero.We conclude that the basic form of the R-matrix is similar to that of the Lax operator, namely R (27)  The first term of the Hamiltonian (99) is quantum spin chain associated to the N = 3 Fateev-Zamolodchikov spin model [22][23][24] while the additional interactions couple the generators of the Z 3 algebra.Recall here that this situation is similar to that found in section 4 for the mixed eight-vertex model in which besides the Ising quantum chain we have the extra Dzyaloshinky-Moriya interaction 2 .It is plausible to believe that the above analysis can be extended to include other integrable spin models such as the Chiral Potts model [23,24].We hope to address to this problem as well as the analysis of the Yang-Baxter algebra of the n-state mixed vertex model with configurations defined by Eq.( 29) in a future work.

Conclusions
In this paper we have presented evidences on the existence of two possible correspondences between n-state spin and vertex models on square lattice with periodic boundary conditions.The equivalences are in the sense that the partition functions of the spin and the vertex model coincide in a toroidal lattice with arbitrary size.Essential to these mappings was to uncover the suitable vertex configurations of the equivalent vertex model which turns out to have only n 3 non-null weights.From the point of view of algebraic geometric such equivalences can be schematically represented by the following maps, In particular, we have argued that the partition function of the Ising model in an external magnetic field can be reformulated as the partition function of a mixed eight-vertex model.We have studied 2 Note that if we use the relation σ y = iσ x σ z the third term in Eq.( 78) can be rewritten as σ x j σ z j σ z j+1 which couples the generators of the Z 2 symmetry.
invariants [40][41][42].In particular, these invariants have been used to locate the critical line of the isotropic Ising model in a non-zero magnetic field [43] exploring its relation to the sixteen-vertex mentioned in the introduction.In this paper we have put forward a much simpler equivalent vertex model which covers the anisotropic Ising model with arbitrary horizontal and vertical ferromagnetic couplings.Therefore, we expect that our mapping together with the approach advocated in ref. [43] could be useful to determine critical frontier of the Ising model in a more generic situation.
for finite size L with toroidal boundary conditions.It turns out that the equivalent n-state vertex model has only n 3 non-null weights and therefore our mapping is more economical than previous equivalences established for the Ising model in the presence of a magnetic field.In fact, instead of a sixteen-vertex model our results leads us to consider a mixed eight-vertex model in which four weights have an even number of arrows while the other four weights have an odd number of arrows at the vertex.More precisely, the partition function of such equivalent mixed eight-vertex model is defined as,

Figure 2 :
Figure 2: The equivalence of the Ising model in a magnetic field with a mixed eight-vertex model on the square lattice with toroidal boundary conditions.

Figure 4 :
Figure 4: The local Boltzmann weights of vertex models at the (i, j) lattice site.Both spin variables α ij and γ ij can take n possible values.

21 (x 2
1 −x 2 ,k) one can show that the R-matrix of the mixed eight-vertex model with symmetric weights satisfies the standard unitarity property, R , x 1 ) = I 4 (59) reducing to the 4 × 4 permutator at the point x 2 = x 1 .

5 of
Eqs.(65,66).The technical details of this computation are summarized in Appendix B and in what follows we present the main result.It turns out that the underlying surface is defined by the following homogeneous quartic polynomial,

Figure 6 :
Figure 6: The configurations of the even eight-vertex model with weights a ± , b ± , c ± and d ± .