Date: 28 April 2025 - 30 April 2025
The emergence of long range order at low temperatures in atomistic systems with continuous symmetry is a fundamental, yet poorly understood phenomenon in physics. To address this challenge I will introduce a discrete microscopic model for an elastic crystal with dislocations in three dimensions, originally proposed by Ariza and Ortiz. The model is rich enough to support some realistic features of three-dimensional dislocation theory, most notably grains and the Read-Shockley law for grain boundaries, which I will review and show how to derive microscopically in the context of the Ariza-Ortiz model, at least in a simple, explicit geometry. I will also explain how to analyze the model at positive temperatures, in terms of a Gibbs distribution with energy function given by the Ariza–Ortiz Hamiltonian plus a contribution from the dislocation cores. The main result is that the model exhibits Long Range Positional Order (LRPO) at low temperatures. Its proof is based on the tools of discrete exterior calculus, together with cluster expansion techniques. In this mini-course I will introduce these methods and explain how to combine them in order to prove existence of LRPO. Time permitting, I will discuss some perspectives about the extension of these ideas and methods to two dimensions. Based on joint work with Florian Theil.