GEOMETRICAL APPROACH TO PHASE TRANSITIONS
Institut für Theoretische Physik, Universität LeipzigUsing percolation theory as a paradigm, a geometrical approach to phase transitions is developed. The theory is worked out explicitly for the two-dimensional Ising model---one of the simplest statistical models exhibiting non-trivial critical behavior. Other systems considered include Bose-Einstein condensates (BEC) and superfluid He-4. It is shown that the fractal dimensions of the relevant geometrical objects (Peierls domain walls in the Ising model, worldlines in BEC, vortex loops in superfluid He) encode the critical exponents. The lectures are physically intuitive and non-technical in nature.
See also: A. Schakel. Entangled Vortices: Onsager's geometrical picture of superfluid phase transitions. cond-mat/0208067
Wolfhard Janke, Adriaan M. J. Schakel. Geometrical vs. Fortuin-Kasteleyn Clusters in the Two-Dimensional $q$-State Potts Model. cond-mat/0311624