#
Potts model

## The q-state Potts model was the subject of my thesis.
It was introduced to study magnetic critical phenomena
and is defined on a lattice in d-dimensions.
It contains as special case also
percolation (q -> 1)
as shown in a very elegant proof by Kasteleyn and Fortuin.
The figure shows a sketch of a generalization of this proof
to the anisotropic case out of my thesis
which as you can see was written before the invention of TeX so that
everything was drawn by hand. Using (at that time)
large scale Monte Carlo simulations I was able to show as
published
in Z. Phys. B, Vol.35, p.171-175 (1979) that in three dimensions the
3-state Potts model has a weak first order phase transition.

##
With Eschbach and Stauffer we showed that
in two dimensions Monte Carlo renormalization
gave good results for q=1, i.e. percolation, but failed to reproduce
the at that time predicted and in the meantime established
correlation length exponent on 2/3 for the 4-state Potts model as
published
in Phys. Rev. B, Vol.23, p.422-425 (1981).
Using the description of a two-dimensional classical model
by a one-dimensional quantum Hamiltonian and solving exactly
for finite chains using a transfer matrix technique I showed as
published
in Z. Phys. B, Vol.43, p.55-60 (1981) the reason for the above
mentioned problem: the 4-state Potts model has a logarithmic
prefactor.

## Again using one-dimensional quantum chains we found with
Hector Martin an essential singularity at the critical
point of the 3-state Potts model on the square lattice having
ferromagnetic interactions in one direction and antiferromagnetic
ones in the other direction as published
in J. Phys. A, Vol.17, p.657-665 (1984).

## With Klaus Fesser we studied a model which is very similar
to the Potts model, namely the clock model.
In particular we established the phase diagramm of the
asymmetric clock model on a Cayley tree which is very rich
since it has commensurate and incommensurate phases as
published
in J. Phys. A, Vol.17, p.1493-1507 (1984).