- Integrable systems
- Mathematical Physics
- Statistical Mechanics
- Exact solutions of lattice models
- Critical Phenomena (phase transitions)
- Conformal/Quantum Field theory
The main thrust of my current research is exactly solvable two-dimensional lattice models in statistical mechanics and their connections, in the continuum scaling limit, with Conformal (CFT) and Quantum Field Theories (QFT). For the most part, I study A-D-E lattice models and their generalizations. The A-D-E lattice models are constructed from the data of the classical simply-laced A-D-E Lie algebras. They include some well known models such as the Ising model, tricritical Ising model and 3-state Potts model as special cases. These models are integrable because the local Boltzmann face weights satisfy the celebrated Yang-Baxter equation which ensures the existence of commuting transfer matrices. These transfer matrices invariably satisfy special functional equations (in the form of fusion hierarchies, bilinear Hirota equations, T-systems, Y-systems) which can be solved for the spectra of the model. Remarkably, all of these statements remain true in the presence of a boundary provided only that the boundary weights satisfy local boundary Yang-Baxter equations. Consequently, it is possible to calculate many physical quantities of interest such as bulk free energies, boundary free energies, correlation lengths, interfacial tensions and order parameters including the associated critical exponents.
In the continuum scaling limit, when the lattice spacing shrinks to zero, these integrable statistical models carry over to continuum counterparts in Conformal Field Theory (CFT) or Quantum Field Theory (QFT) depending on whether the lattice model is critical (trigonometric Boltzmann weights) or off-critical (elliptic Boltzmann weights) respectively. It is thus possible to calculate quantities of interest for these theories from the lattice. For CFTs it is possible to calculate the central charges, conformal weights, finitized characters, finitized partition functions as well as the underlying fusion (Verlinde, graph, Pasquier, Ocneanu) algebras. A remarkable fact that emerges is that for each conformal boundary condition there exists an integrable boundary condition on the lattice (constructed using a lattice fusion procedure) which reproduces the conformal boundary condition in the continuum scaling limit. For QFTs, it is possible to obtain the Renormalization Group flow between conformal fixed points. This includes massive and massless bulk RG flows induced by perturbing with a thermal or magnetic bulk field as well as boundary RG flows induced by perturbing with a thermal or magnetic boundary field.
Current Postgraduate Supervision
|Alessandra VITTORINI ORGEAS
||"Exact solution of nonunitary lattice models in two dimensional statistical mechanics"
Past Postgraduate Supervision
||"Aspects of loop models including polymers and percolation"
Past MSc Students
Recent Grant History
|2008 - 2011
||Exact solution of generalized models of polymers and percolation in two dimensions
- Academic Board member
- AMSI representative
- Chair of Math. Sciences Library Users Committee
- Science Faculty Library Users Committee member
- Math. Sciences Library Users Committee
- Strategic Planning Committee