The ice model Introduction The six-vertex model , which is the object of this chapter, is a special case of the eight-vertex model on a two-dimensional square lattice introduced by Fan and Wu (1970) in order to summarize a class of exactly solvable models in classical statistical mechanics. The thermodynamics of the six-vertex model is by now known in its full generality (Yang and Yang, 1966a–d; Lieb, 1967a) which is not the case for the eight-vertex model, but only the self-conjugate one (Baxter, 1971a). The general model can be considered as a two-dimensional idealization of a crystalline system in which pairs of adjacent atoms or radicals on the network are linked by ‘hydrogen bonds’. Of ionic type, this link between two neighbouring electronegative atoms is realized by a proton H + which is located closer to one of the atoms than to the other. On each link of the network there thus exist two equilibrium positions for H + (Pauling, 1960). If the coordination number at each site equals 4, as is the case for the oxygen atom in hexagonal ice, we indeed have eight possible proton configurations on a given site, which give rise to the eight vertices. In contrast, for the physical systems for which this model could be viewed as a valid idealization – ice H 2 O, the ferroelectric PO 4 H 2 K, the antiferroelectric PO 4 H 2 NH 4 – there exist at most two H + next to each site.
This chapter introduces the six-vertex model defined on a two-dimensional square lattice. It describes the model in detail, since it gives an important prototype of many solvable lattice models defined on two-dimensional lattices. The eight-vertex model, which generalizes the six-vertex model directly, may be considered as the most important exactly solvable model in statistical mechanics. The chapter explores some features of the six-vertex model defined on a square lattice. It reviews a method for diagonalizing the transfer matrix, which is called the coordinate Bethe ansatz, and provides the expressions of the free energy per site in the ferroelectric, the antiferroelectric and the disordered phases, respectively. The chapter discusses some mathematical theories associated with integrable models such as the braid group and the quantum groups. It outlines the Ising model, the Potts models and the chiral Potts model and then the eight-vertex model and the interaction-round-a-face models.
Abstract Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and the six-vertex model, are the most relevant and physically interesting, in particular through their connection to the one-dimensional integrable quantum mechanical models and the Bethe ansatz. This chapter introduces power- ful tools to examine vertex models, including the R- and L-matrices to encode the Boltzmann vertex weights and the monodromy and transfer matrices, which encode the integrability of the vertex models (i.e. that transfer matrices of different spectral parameters commute). This integrability is ultimately expressed in the Yang–Baxter relations.
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.
Abstract Any planar set of intersecting straight lines forms a four-coordinated graph, or ‘lattice’, provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagomé lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.
These are introductory lectures on application of the free field representation (bosonization) techniques to the solid-on-\S olid (SOS) and eight-vertex models.We start from the very beginnings, including the physical badcground of lattice models and some basic information on quantum integrability.After definitions of the eight-vertex and SOS models, we describe their relation known as the vertex-face correspondence.Then, skipping the Bethe ansatz solution, wc turn to the problem of calculation of correlation functions by mcaiis of the fit.cficld rcprcscntation.We explain, how the vertex-face correspondence works on the level of vertex operators and bosonization, making it possible to express the correlation functions of the eight-vertex model in terms of the free field representation aimed to describe the SOS model.1 Eight-vertex model arid commuting traxisfer Ilatl
⋅ ice.\'Since we shall speak about some models of physical importance.let us formulate first the general physical framework.Consider a classical system of interacting particles.Let
C be a state of this system, which can run some (generally infinite) set of admissible states.Let
E(C) be the energy of the state C. defined by usual IIamiltonian mechanics.Suppose that the system weakly interacts with a thermal bath of the temperature
T .The most fundamental postulate of statistical mechanics, the Gibbs law, says that the probability of the state
C is given by
w(C)=Z1e−B(C)/T .(1.1) lt is easy to understand, what is the proportionality coefficient
1/Z .As the total probability for thc system to be in any state is unity.we have for the partition functions of the system
Z=∑ce−F(C)/7 .(1.2) Since the space of configurations can be continu$ \mathit{0}\prime 18,$
†,11e\sum may turn out to be an
itιtegra↓↓ion , but we shall not consider this general case in these lectures.The partition function encodes the most fundamental observable thermodynamic function of the model, the free eneryy:
F=E−TS=−Tlog Z.(1.3)Here
E is the total energy of the system,
E=∑E(C)w(c,)=Z(T)dTT∼d,Z(T)=−T2dTTdF=F−TdTdF , while
S=−∑w(c,)logw(C)=−dTdΓ′is the entropy.The first equality in (1.3) is the thermodynamic definition of the free energy, which, in principle, was established before Gibbs, while the second equality provides its statistical interpretation.Let
fi⟨C ) be some functions of the state of the system.The correlation functions are generic expectation values
⟨f1…fr′)=∑c∣ι)(C)fι(C)…fn(C) .