Kelland, Stewart Beaton

### Description

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...[Show more] 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.

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