The theory of integrable discrete systems also extends into the quantum domain, and it is here that the true richness of integrability is most visible (quantum mechanics being in essence a theory of discrete and algebraic objects). The current research concentrates on formulating a proper quantum theory for discrete mappings and integrable systems living on the space-time lattice. Exploiting the exactness of the models and integrable structures (R-matrices, quantum Lax pairs and determinants) the investigation is set to produce rigorous and analytic answers to questions which for other models are only accessible through numerical and perturbative methods. As such, quantum integrable discrete models form a paradigm for the development of new approaches in the quantum regime which have potential implications to areas such as string and conformal field theory, as well as in the theory of random matrices in mesoscopic physics, quantum computing and nanotechnology.
The Integrable Systems group in Leeds, one of the leading groups worldwide working on integrable systems, has been strengthened by three recent appointments, one at a senior level. Integrable Systems are systems that, albeit highly nontrivial and nonlinear, are amenable to exact and rigorous techniques for their solvability. They can take many shapes or forms: nonlinear evolution equations, partial and ordinary differential equations and difference equations, Hamiltonian many-body systems, quantum systems and spin models in statistical mechanics. A large number of mathematical techniques have been developed to unravel the rich structures behind these systems. The six permanent members of staff work with five postdocs and postgraduate students.
Leeds is a major centre for the theory of discrete and quantum integrable systems. This group represents a wide range of research activities into integrable nonlinear systems, their symmetries, solution techniques and the underlying mathematical structures, as well as more mathematical aspects of physical systems, for example quantum systems. The models comprise ordinary and partial differential and difference equations, dynamical mappings, discrete Painleve equations, Hamiltonian and many-particle systems and systems of hydrodynamic type. The theory and its specific models have wide-ranging applications, for example, in nonlinear optics, theory of water waves, integrable quantum field theory, statistical mechanics and combinatorics random matrix theory and nanotechnology.