Chiral Potts model --
The field of solvable models in statistical mechanics has ramifications in
many areas, such as integrable models, quantum field theory, random matrix
models, knot theory, quantum groups, graph theory and various problems in
combinatorics. The chiral Potts model continues to present fascinating
challenges: in particular two expressions for the free energy were obtained
in 1988 and until now it has not been at all obvious that they agree.
This problem has now been resolved: Baxter has explicitly shown that they are
completely equivalent.
Zamolodchikov model --
The Zamolodchikov model is a solvable three-dimensional model. It shares
some intriguing symmetries with the simple "free-fermion"
three-dimensional model, which makes one speculate that the methods
of the latter may apply to the former. These ideas are easy to
test for the two-layer case, this has been done by Baxter and
Bazhanov and a number of simple and interesting properties have
been observed.
Mangazeev and Boos (Bonn) studied the three-layer case of the Zamolodchikov model. They obtained density functions for a distribution of bethe roots for the ground state and solved functional equations for the transfer-matrices in thermodynamic limit. Mangazeev studied a three-dimensional model with continuously distributed spin variables. He obtained continuous analogs of star-triangle and star-star relations when the quantum parameter is not a root of unity.
Spin ladders --
Batchelor and Maslen initiated a new direction of work on
exactly solvable quantum spin ladders. Such quantum spin chains
coupled together to form experimentally realisable ladder-like
structures exhibit remarkable low-dimensional quantum properties. A
family of solvable n-leg ladder models was constructed with an
underlying su(2n) symmetry and shown to be extensions of
previously solved models.
Another ladder model was shown to satisfy the Hecke algebra. Yet other models were constructed which are related to the orthogonal and symplectic algebras.
In work with de Gier the 3-leg model was further generalised to include the 3-leg ladder and a spin-tube as special cases. The phase diagram of this general model was then obtained from the solution in terms of the Bethe Ansatz equations. de Gier and Batchelor subsequently examined the properties of this model in the presence of a magnetic field. This provided valuable exact results as benchmarks for future studies. The phase diagram exhibited magnetisation plateaus of relevance to the pure Heisenberg spin ladders.
Dilute AL models --
Batchelor and Seaton (La Trobe) continued their work on the
family of dilute AL lattice models. Explicit calculations were
performed for the transfer matrix excitation spectrum of the dilute
A4 model. The results obtained verified previous conjectures for
the connection with the E7 mass spectrum.
Stromatolites --
Batchelor, Burne (Geology), Henry (UNSW) and Watt (UNSW) began
work on mathematical models for the growth of stromatolites.
Briefly, stromatolites are laminated structures produced as
a result of the environmental interactions of benthic
microbial communities. They are the only macroscopic evidence
of life prior to the evolution of macroscopic plants and animals.
Our work aims to provide mathematical insight to the often heated
discussion of the criteria that might be useful for
distinguishing organically produced stromatolites from similar
accretions produced by inorganic means. As a first step we have
used the deterministic variant of the Kardar-Parisi-Zhang
equation for the evolution of a growing interface to model
patterning produced by successive laminations in certain
stromatolites.
Perk-Schulz models --
The Corner Transfer matrices, invented by Baxter, are an important
tool for investigating solvable lattice models, particularly so
since the discovery of the connections with representation theory
(of which Davies is one of the originators). Davies is currently
working on extension to models connected with quantum affine
super-algebras; particularly the Perk-Schulz models and associated
spin chains which are of interest in the study of high Tc
superconductivity.
Chiral Potts model and algebraic geometry --
Together with Davies, Neeman studied the
algebraic geometry of the Yang-Baxter equations for the chiral
Potts model. This statistical mechanics model gives rise to
problems in algebraic geometry, sometimes quite interesting ones.
Further progress has been made in 1999, particularly in relation to
the three-state model, for which the Jacobian is a product of
elliptic curves; a complete description has been obtained. The
long-term objective of this work is to tackle outstanding problems
which have proved intractable using "classical" mathematics.
Categories and subgroups --
Together with Christensen (John Hopkins) and Keller (UFR de Mathematiques),
Neeman found a counterexample. Adams showed that every generalised
homology theory, on the category of finite spaces (compact
manifolds, for example) is represented by a spectrum. We show, by
our counterexample, that this is false for the derived categories
of general enough rings.
Neeman found a short proof of a theorem of Martin (Sydney), about restricting representations of surface groups to free subgroups. The proof is shorter and simpler, and proves a stronger result.
Quantum field theory --
Bazhanov, Lukyanov (Rutgers) and Zamolodchikov (Rutgers) studied various
integrability structures in quantum field theory. They found a new
and fascinating connection between spectral properties of Baxter's
Q-operators in conformal field theory and the spectral properties
of the one-dimensional Schrödinger equation. Using this connection,
it has been possible to prove the strong-weak barrier duality
relation for the non-linear mobility related to the impurity
transport in a non-equilibrium Luttinger liquid.
The functional relations for the commuting transfer matrices for models with higher symmetries were studied algebraically by Bazhanov, Hibberd (Theoretical Physics, IAS) and Khoroshkin (ITEP, Moscow). A quantum Boussinesq theory has been developed that describes the integrabilty structure of the conformal field theory with extended W-symmetry.
Software Development --
Software for investigating, experimenting with, and visualising
representative systems in non-linear dynamics continues to be under
active development and testing. The first version has been in use
in TLTSU laboratories since July.