Classical Phase Transitions of Geometrically Constrained O($N$) Spin Systems

We study the phase transition between the high temperature algebraic liquid phase and the low temperature ordered phase in several different types of locally constrained O(N) spin systems, using a unified constrained Ginzburg-Landau formalism. The models we will study include: 1, O(N) spin-ice model with cubic symmetry; 2, O(N) spin-ice model with easy-plane and easy-axis anisotropy; 3, a novel O(N) "spin-plaquette" model, with a very different local constraint from the spin-ice. We calculate the renormalization group equations and critical exponents using a systematic \epsilon = 4 - d expansion with constant N, stable fixed points are found for large enough N. In the end we will also study the situation with softened constraints, the defects of the constraints will destroy the algebraic phase and play an important role at all the transitions.