On the role of elasticity and defects in phase transitions

Models for phase transitions involve order parameters (eg., Ising or XY spins), defined over a base space (eg., 2D lattice or R²). Starting with the discovery of graphene and up until the recent experiments involving Bose-Einstein condensation in spherical shells, there has been a lot of interest in understanding the interplay of geometry (elasticity) and topology (defects) on phase transitions. The effect of elasticity can be two-fold ---- as a microscopic degree of freedom through lattice displacements driven purely by temperature, or via macroscopic elastic deformation as the parameter driving the phase transition. While the influence of defects in the order parameter has received much attention, the role of defects in the base space has largely remained unexplored. This work focuses on the effect of elasticity in the presence of topological defects in the base space. We first perform Monte-Carlo simulations in 2D considering only the spin degree of freedom, and varying the density of quenched lattice defects (dislocations and grain boundaries), to isolate the role of lattice topology on phase transitions. Further, the simulations are extended to include (microscopic) lattice displacements as an extra order parameter that couples to the spin degree of freedom. Finally, we explore the role of (macroscopic) bulk elasticity in the presence of topological defects, by treating the elastic deformation as the critical parameter. The simulations are complemented by theoretical calculations.