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Statistical mechanics of self-avoiding crystalline membranes and topological defect formation.
This thesis investigates two important topics in modern statistical mechanics. The first three chapters are devoted to the statistical mechanics of random surfaces. We present an analysis of extensive large-scale Monte Carlo simulations of Self-avoiding
fixed-connectivity membranes for sizes (number of faces) ranging from 512 to 17672 (triangular) plaquettes. Self-avoidance is implemented via impenetrable plaquettes. We simulate the impenetrable plaquettes model in both three and four bulk dimensions. In both cases we find the membrane to be flat for all temperatures: The size exponent in three dimensions is ν = 0.95(5). The single flat phase appears, furthermore, to be equivalent to the large bending rigidity phase of phantom membranes; the roughness exponent in three dimensions is ζ = 0.63(4) and its Poisson ratio σ = −0.37(6). These results suggest that there is a unique universality class for flat crystalline membranes without attractive interaction. The last chapter deals with the Formation and Dynamics of Topological Strings in a U(1) Linear Sigma model in three spatial dimensions. For over-damped Langevin dynamics we find that defect production is suppressed by an interaction between correlated domains that reduce the effective spatial variation of the phase of the order field. The degree of suppression is sensitive to the quench rate. A detailed description of the numerical methods used to analyze the model is reported.