Statistical mechanics & quantum spin-chains
Our principal interests in this area relate to exactly solvable or integrable models. These occur mainly in two dimensions and are usually endowed with a rich underlying mathematical structure which allows one to obtain exact, analytic results. For many purposes one often restricts oneself to a one-dimensional sublattice, called a (quantum) spin-chain as it can be visualized as a string of coupled atoms with magnetic spin. There is therefore a close connection between two-dimensional statistical models and one-dimensional spin systems.
The main challenge in the area of research is to derive analytic expressions which describe the correlations in such systems. These would allow for making predictions concerning electric, magnetic or transport properties of low-dimensional physical systems.
The analytic computation of the aforementioned correlations is a formidable mathematical challenge and as of yet has only been achieved for a few models. In all of these cases algebras and their representation theory play an essential role in the solution alongside other techniques such as the method of commuting transfer matrices or the Bethe ansatz.
Some examples of systems and their associated algebras we have worked on are
- Potts models: Temperley-Lieb algebras
- vertex models: Yangians and quantum groups
- solid-on-solid models: elliptic algebras.
In the continuum limit there is a close connection with integrable quantum field theory. Many of the aforementioned models undergo a phase transition. At the transition point the correlations within the lattice are long-ranging and can effectively described by conformal field theory. In contrast, near the transition point the correlations exponentially decay over a characteristic length scale which in many cases can be identified with the Compton wave length of the lightest particle in a massive quantum field theory.
If you would like to know more about this research topic or contact individual staff members about the possibility of a PhD in this area, please consult the personal websites of the staff members listed below. An elementary introduction to the physical motivation (concentrating, for simplicity on the Temperley-Lieb case) can be found here.
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