*"I got really fascinated by these (1+1)-dimensional models that are solved by the
Bethe Ansatz and how mysteriously they jump out at you and work and you don't know
why. I am trying to understand all this better."*

Richard Feynman (1988)

As usual, Feynman had his finger on the pulse!

**Here is an inkling of what we've been up to recently:**

Over the last several decades the subject of exactly solved (integrable) models
in statistical mechanics has continued to inspire a number of profound
developments in mathematics. There are a number of families of integrable models,
the analysis of which has developed to an extraordinary level of sophistication.
In general we believe that exactly solved models will play a crucial role in
physics, particularly in understanding low dimensions, where quantum effects
are often most pronounced.

In particular, a series of brilliant recent experiments, described as virtuoso triumphs in
nano-engineering, have opened the
way to fabricating low-dimensional quantum gases of trapped atoms. A review
of these ultracold quantum gases in optical lattices has been given by Immanuel Bloch
in the first edition of *Nature Physics*.

**Now here is the thing**. The quasi-1D quantum gases are experimental
realisations of integrable mathematical models of interacting systems of particles
found in the 1960's and early 1970's! Some of these models -- all solved by the
Bethe Ansatz -- have lain fallow and are
not as well understood as they clearly deserve to be. That is where we come in.

Prior to embarking on the investigation of interacting bosons and fermions
we completed detailed calculations of the physical properties of low-dimensional
quantum systems as alluded
to above, namely quantum spin ladders and quantum spin-1 chains.
This work included direct comparison with experimental results
on real compounds. A brief summary of the results for quantum spin ladders
can be found here.
Our review article on this topic has been published in
*Advances in Physics* **56** 465-543 (2007).
Importantly, the mathematical methods and techniques
used for the quantum spin systems are applicable to the quantum gases.

Links to our work on integrable boson and fermion models can be found in
Research Interests.

This work has led us to study the remarkable properties of **anyons**.
Read how Frank Wilczek explains
that anyons do not fit into the usual categories of fermions
and bosons and how they may lead to a realisation of topological quantum computers
and the new field of "anyonics".

In two dimensions, anyons are particles that interpolate between
fermions and bosons. In our recent
work,
we have considered an exactly
solved one-dimensional model of interacting anyons proposed by Anjan Kundu in 1999.
Up until recently, Kundu's work received very little attention.
We have calculated the low-energy properties of the exactly solved model and made an explicit
connection with generalised exclusion statistics proposed earlier by Haldane. In this way,
we showed that the statistics of the 1D interacting anyons interpolates between Bose and
Fermi statistics. In a certain limit the model also suggests that there may be statistics beyond
Fermi statistics. The model also exhibits an interesting resonance effect and connects in a
natural way with the recently proposed *super* Tonks-Girardeau gas.
Our further work on this model can be found
here, with implications for fermionization and
fractional statistics in the 1D Bose gas
here.

Our results should inspire further work on these models
which may possibly be tested in future experiments.
**Now here is another thing**. The most recent advances in quantum atom optics
are such that a number of low-dimensional quantum spin systems may be realised in
the ultraclean environment of optical lattices. This opens up a new testbed for
studying subtle and technologically significant effects in quantum many-body systems.

**Update:** (19 Sep 09) The super Tonks-Girardeau gas has now been realized in experiments [E. Haller et al., Science **325** (2009) 1224].
See, in particular, the reference to our
paper *Evidence for the super Tonks-Girardeau gas*
based on the integrable model of interacting bosons in the attractive regime.

The review
of Bill Sutherland's book
*"Beautiful Models: 70 Years of Exactly Solved
Quantum Many-Body Problems"*.

See also the feature article
*"The Bethe Ansatz after 75 years"*
in the January 2007 issue of *Physics Today* [**60** (2007) 36-40].