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arXiv:2211.06487 · Journal of Statistical Mechanics Theory and Experiment

On the partition function of the Sp(2n) integrable vertex model

G A P Ribeiro

Published Apr 1, 2023• 3 citations
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Abstract

Abstract We study the partition function per site of the integrable S p ( 2 n ) vertex model on the square lattice. We establish a set of transfer matrix fusion relations for this model. The solution of these functional relations in the thermodynamic limit allows us to compute the partition function per site of the fundamental S p ( 2 n ) representation of the vertex model. In addition, we also obtain the partition function of vertex models mixing the fundamental with other representations.

DOI
10.1088/1742-5468/acc730

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This thesis deals with two two-dimensional lattice models in statistical mechanics. The first of these is the zero-field 20-vertex model on a triangular lattice. This model is the analogue to the ice-type, zero-field six-vertex model on a square lattice. We solve this problem exactly for a restricted set of vertex weights. The other model is the q-component Potts model on a square lattice. This model is equivalent to a staggered ice-type model on a square lattice. Using a variational method we estimate the critical exponent ß for several values of q. In part I we outline the fundamental statistical mechanical ideas needed, including the definition of the partition function and its relation to the thermodynamic quantities. Mention is also made of the existence and uniqueness of the thermodynamic limit for these models. The 20-vertex model is discussed in part II. The transfer matrix approach is adopted and a Bethe ansatz is used for the eigenvector. Unlike the square lattice ice-type model, the triangular lattice 20-vertex model is only soluble by this method for a restricted set of vertex weights. In particular, the ten parameters determining the vertex weights must be expressed in terms of four independent parameters. This results in only one type of non-trivial temperature independent solution. This solution exhibits critical behaviour similar to the square lattice KDP model. The general solution has three states: disordered, antiferroelectric and ferroelectric. Which state the model is in is determined by one of the above four parameters. The transition from the disordered to the antiferroelectric state is of infinite-order (i.e the free energy has an essential singularity), and from the disordered to the ferroelectric state is of finite-order. We also compare our results to the free-fermion solution of the same model and note where the two methods overlap. In part III the Potts model is considered. The model is defined and shown to be equivalent to a staggered ice-type model. A variational principle for the eigenvalues of the transfer matrix of the staggered ice model is determined. Using the zero-temperature solution as a ^uide a simple form for a sequence of approximations to the eigenvector, below T , is developed. By using the variational principle we arrive at a set of matrix equations determining the eigenvector at a given level of approximation. By noting some of the symmetries of the transfer matrix we are able to separate some of the dependences of the matrix elements and consequently simplify the original set of equations. We are also able to express the percolation probability ( the analogue of the spontaneous magnetization for general values of q ) of the Potts model as a simple function of these matrix elements. At each level of approximation there is a set of algebraic equations to solve. We have used an electronic computer to solve these equations for the first five levels of approximation, and for several values of q between zero and four. With our results we estimate the critical exponent 3 for the Potts model. These estimates agree reasonably well with previous work and indicate 3=0.145 ±.017 for q=l.

Bulletin of the Australian Mathematical Society•Oct 1, 1976•1 citation
Abstract We study the partition function per site of the integrable S p ( 2 n ) vertex model on the square lattice. We establish a set of transfer matrix fusion relations for this model. The solution of these functional relations in the thermodynamic limit allows us to compute the partition function per site of the fundamental S p ( 2 n ) representation of the vertex model. In addition, we also obtain the partition function of vertex models mixing the fundamental with other representations.
Abstract In this paper we investigate certain fusion relations associated to an integrable vertex model on the square lattice which is invariant under Sp (4) symmetry. We establish a set of functional relations which include a transfer matrix inversion identity. The solution of these relations in the thermodynamic limit allows us to compute the partition function per site of the fundamental Sp (4) representation of the vertex model. As a byproduct we also obtain the partition function per site of a vertex model mixing the four and five dimensional representations of the Sp (4) symmetry.
The six-vertex model on an N × N square lattice with domain wall boundary conditions is considered. A Fredholm determinant representation for the partition function of the model is given. The kernel of the corresponding integral operator is of the so-called integrable type, and involves classical orthogonal polynomials. From this representation, a ‘reconstruction’ formula is proposed, which expresses the partition function as the trace of a suitably chosen quantum operator, in the spirit of corner transfer matrix and vertex operator approaches to integrable spin models.
It is shown that an application of the techniques of division and integration of lattice sites reduces Ising models to equivalent vertex models. When the Ising model is on a two-dimensional lattice and no crossing occurs for any two of line segments connecting two lattice sites between which an interaction exists, the equivalent vertex model is shown to be a free-fermion vertex model. An argument is also given to show that the present method provides a method of expressing the partition function of a general Ising model on the square lattice, with noncrossing interactions within a square plaquette, in terms of a determinant of a 4×4 matrix.
We introduce and study the domain wall boundary partition function of the integrable six-vertex model with triangular boundary. We first formulate the domain wall boundary partition function with triangular boundary by using the Uq(sl2)U_q(sl_2)Uq​(sl2​) RRR-matrix and a special class of the triangular KKK-matrix. By using its graphical representation, we make the Izergin-Korepin analysis with the help of the Yang-Baxter relation and the reflection equation to give a characterization of the partition function. The explict form of the symmetric function representing the partition function is presented by showing that it satisfies all the required properties.
This thesis deals with two two-dimensional lattice models in statistical mechanics. The first of these is the zero-field 20-vertex model on a triangular lattice. This model is the analogue to the ice-type, zero-field six-vertex model on a square lattice. We solve this problem exactly for a restricted set of vertex weights. The other model is the q-component Potts model on a square lattice. This model is equivalent to a staggered ice-type model on a square lattice. Using a variational method we estimate the critical exponent ß for several values of q. In part I we outline the fundamental statistical mechanical ideas needed, including the definition of the partition function and its relation to the thermodynamic quantities. Mention is also made of the existence and uniqueness of the thermodynamic limit for these models. The 20-vertex model is discussed in part II. The transfer matrix approach is adopted and a Bethe ansatz is used for the eigenvector. Unlike the square lattice ice-type model, the triangular lattice 20-vertex model is only soluble by this method for a restricted set of vertex weights. In particular, the ten parameters determining the vertex weights must be expressed in terms of four independent parameters. This results in only one type of non-trivial temperature independent solution. This solution exhibits critical behaviour similar to the square lattice KDP model. The general solution has three states: disordered, antiferroelectric and ferroelectric. Which state the model is in is determined by one of the above four parameters. The transition from the disordered to the antiferroelectric state is of infinite-order (i.e the free energy has an essential singularity), and from the disordered to the ferroelectric state is of finite-order. We also compare our results to the free-fermion solution of the same model and note where the two methods overlap. In part III the Potts model is considered. The model is defined and shown to be equivalent to a staggered ice-type model. A variational principle for the eigenvalues of the transfer matrix of the staggered ice model is determined. Using the zero-temperature solution as a ^uide a simple form for a sequence of approximations to the eigenvector, below T , is developed. By using the variational principle we arrive at a set of matrix equations determining the eigenvector at a given level of approximation. By noting some of the symmetries of the transfer matrix we are able to separate some of the dependences of the matrix elements and consequently simplify the original set of equations. We are also able to express the percolation probability ( the analogue of the spontaneous magnetization for general values of q ) of the Potts model as a simple function of these matrix elements. At each level of approximation there is a set of algebraic equations to solve. We have used an electronic computer to solve these equations for the first five levels of approximation, and for several values of q between zero and four. With our results we estimate the critical exponent 3 for the Potts model. These estimates agree reasonably well with previous work and indicate 3=0.145 ±.017 for q=l.