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).