Frustration, curvature, and defect lines in metallic glasses and the cholesteric blue phase

We develop continuum elastic theories for the blue phases of cholesteric liquid crystals and for the metallic glasses. These theories are frustrated and possess ground states with networks of defect lines; the frustration can be relieved in a space of constant positive curvature (a sphere in four dimensions). The order parameter for the blue phase is a director (in ssR $P^2$), and for the metallic glasses it is a 4×4 rotation matrix [in SO(4)]. The elastic energies are written using covariant derivatives which are zero for the local low-energy configurations. The theories are nonlinear sigma models. We also develop a renormalization-group analysis of disclination cores in models with ssR $P^2$, vector, SO(3), and SO(4) order-parameter spaces. The energy of disclination lines diverges logarithmically as their core size is taken to zero. A total divergence can be added to the energy. Through an unusual combination of energetic and topological effects, it contributes an energy proportional to the product of the length and strengths of the disclinations. We use this total divergence as a counterterm to keep the disclination energies fixed in the continuum limit.