Research Interests
 Integrable systems
 Mathematical Physics
 Statistical Mechanics
 Exact solutions of lattice models
 Critical Phenomena (phase transitions)
 Conformal/Quantum Field theory
Research Groups
Extra Information
The main thrust of my current research is exactly solvable twodimensional 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 ADE lattice models and their generalizations. The ADE lattice models are constructed from the data of the classical simplylaced ADE Lie algebras. They include some well known models such as the Ising model, tricritical Ising model and 3state Potts model as special cases. These models are integrable because the local Boltzmann face weights satisfy the celebrated YangBaxter 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, Tsystems, Ysystems) 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 YangBaxter 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 offcritical (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
Name 
Thesis title 
Alessandra VITTORINI ORGEAS 
"Exact solution of nonunitary lattice models in two dimensional statistical mechanics" 
Past Postgraduate Supervision
Name 
Thesis title 
Simon VILLANI 
"Aspects of loop models including polymers and percolation" 
Past MSc Students
Name 
Project title 
Hannah FITZGERALD 

Joel GILL 

Adam ONG 

Daniel SCHEPISI 

Elena TARTAGLIA 

Subject(s) Currently Teaching
Code 
Subject 
MAST90060 
Mathematical Statistical Mechanics (Semester 1) 
Recent Grant History
Year(s) 
Source 
Type 
Title 
2008  2011 
ARC 
Discovery 
Exact solution of generalized models of polymers and percolation in two dimensions 
2006  2008 
ARC 
Discovery 
Quantum Spectra 
Responsibilities
 Academic Board member
 AMSI representative
 Chair of Math. Sciences Library Users Committee
 Science Faculty Library Users Committee member
Committees
 Math. Sciences Library Users Committee
 Strategic Planning Committee