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arXiv:cond-mat/0311244 · Journal of Physics A Mathematical and General

The packing of two species of polygons on the square lattice

David Dei Cont, Bernard Nienhuis

Published Feb 18, 2004• 13 citations
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Abstract

We decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons. We end up with a 24-vertex model which is known in the literature as the fully packed double loop model (FPL 2 ). In the particular case in which the fugacities of the polygons are the same, the model admits an exact solution. The solution is obtained using coordinate Bethe ansatz and provides a closed expression for the free energy. In particular, we find the free energy of the four-colouring model and the double Hamiltonian walk and recover the known entropy of the Ice model. When both fugacities are set equal to 2 the model undergoes an infinite-order phase transition.

DOI
10.1088/0305-4470/37/9/002

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The ice and Potts models in statistical mechanics

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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
We decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons. We end up with a 24-vertex model which is known in the literature as the fully packed double loop model (FPL 2 ). In the particular case in which the fugacities of the polygons are the same, the model admits an exact solution. The solution is obtained using coordinate Bethe ansatz and provides a closed expression for the free energy. In particular, we find the free energy of the four-colouring model and the double Hamiltonian walk and recover the known entropy of the Ice model. When both fugacities are set equal to 2 the model undergoes an infinite-order phase transition.
We present an investigation of the completely packed O(n) loop model on the square lattice by means of the transfer-matrix method and finite-size scaling. We investigate the model for a number of n values covering a wide range. This model is known to be equivalent with the q-state Potts model with q = n 2 , but here we also investigate the range n 2, the system is in an ordered phase, where the majority of the loops cover part of the elementary faces of the lattice in one of two checkerboard patterns that are in phase coexistence. Furthermore, we find that the numerical results for the free energy density are in agreement with the expressions obtained from the exact analysis of the equivalent six-vertex model. This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu’s 80th birthday.
We derive the nested Bethe ansatz solution of the fully packed O(n)\mathrm{O}(n)O(n) loop model on the honeycomb lattice. From this solution we derive the bulk free energy per site along with the central charge and geometric scaling dimensions describing the critical behavior. In the n=0n=0n=0 limit we obtain the exact compact exponents $\ensuremath{\gamma}=1$ and $\ensuremath{\nu}=\frac{1}{2}$ for Hamiltonian walks, along with the exact value ${\ensuremath{\kappa}}^{2}=3\sqrt{\frac{3}{4}}$ for the connective constant (entropy). Although having sets of scaling dimensions in common, our results indicate that Hamiltonian walks on the honeycomb and Manhattan lattices lie in different universality classes.
Starting from the Bethe ansatz solution we derive a set of coupled non-linear integral equations for the fully packed double loop model (FPL^2) on the square lattice. As an application we find exact expressions for the central charge and for the scaling dimension corresponding to the simplest charge excitation. We study numerically the low-lying excitations corresponding to more general perturbations of the ground state and discover that the corresponding scaling dimensions are well described by the Cartan matrix of sl_4.
Fully packed loop (FPL) models on the square and the honeycomb lattice constitute new classes of critical behaviour, distinct from those of the low-temperature O( n ) model. A simple symmetry argument suggests that such compact phases are only possible when the underlying lattice is bipartite. Motivated by the hope of identifying further compact universality classes we therefore study the FPL model on the square-octagon lattice. Surprisingly, this model is only critical for loop weights n <1.88, and its scaling limit coincides with the dense phase of the O( n ) model. For n = 2 it is exactly equivalent to the self-dual 9-state Potts model. These analytical predictions are confirmed by numerical transfer matrix results. Our conclusions extend to a large class of bipartite decorated lattices.
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.