Statistical Mechanics of Low Angle Grain Boundaries in Two Dimensions

We explore order in low angle grain boundaries (LAGBs) embedded in a two-dimensional crystal at thermal equilibrium. Symmetric LAGBs subject to a periodic Peierls potential undergo, with increasing temperatures, a thermal depinning transition, above which the potential is irrelevant at long wavelengths and the LAGB exhibits transverse fluctuations that grow logarithmically with inter-dislocation distance. Longitudinal fluctuations lead to a series of melting transitions marked by the sequential disappearance of diverging algebraic Bragg peaks with universal critical exponents. Aspects of our theory are checked by a mapping onto random matrix theory.