Principal Chiral Model in Correlated Electron Systems

We discuss noncollinear magnetic phenomena whose local order parameter is characterized by more than one spin vector. By focusing on the simple cases of 2D triangular and 3D pyrochlore lattices, we demonstrate that their low-energy theories can be described by a one-parametric class of sigma models continuously interpolating between the classical Heisenberg model and the principal chiral model ${\rm Tr} ,\left( \partial_a U \partial_a U^\dagger\right)$forall$U\in $\text{SU}$(2)$. The target space can be viewed as a U$(1)$ fibration over the $CP(1)$ space. The 3D version of our model is further generalized to break spatial and spin rotation symmetry $$\text{SO}$(3) \times $\text{SO}$(3) \to $\text{SO}$(3)$.