Bradley Landau
How do phase transitions emerge in discrete vector-spin generalisations of the Curie–Weiss model?
As the heart of statistical mechanics, phase transitions describe how interactions lead systems from disorder into order. Whilst the Curie-Weiss (CW) model has provided a solvable framework for understanding these phenomena, its scope is limited to spin-like systems with two distinct states. Extensions such as the Curie-Weiss-Potts (CWP) model have effectively introduced multiple distinct states. But, they are unable to capture mirror-symmetry, often an important feature of magnetic systems. This project will develop a discrete vector-spin model in which spins are constrained to orthogonal axes, allowing for parallel, antiparallel and orthogonal interactions. It focusses on the asymptotic behaviour of the magnetisation density in the thermodynamic limit, identifying whether it concentrates at zero or at a spontaneous magnetisation. Objectives include determining the dimensional dependence of critical values, conditions under which fluctuations follow Gaussian versus non-Gaussian behaviour, and how dimensionality and external fields affect continuity and fluctuation structure. Under the solvable framework that mean-field models provide, this project aims to provide new insight into the role of symmetry and dimensionality in phase transitions, offering direct comparisons to the CWP model and continuous spin systems like XY and Heisenberg.
Bradley Landau
Monash University
Bradley Landau is a student at Monash University who has recently completed his second year of study in mathematics and mechanical engineering. He has developed a strong interest in statistical mechanics, particularly in understanding how phase transitions emerge in discrete vector-spin generalisations of the Curie–Weiss model. He is also interested in probabilistic lattice models, having studied bond percolation and self-avoiding walk models. Outside of mathematics, Bradley enjoys bouldering and experimenting with specialty coffee. From brewing and roasting at home to analysing temperature-time roast curves to improve the flavour.
