Classical Heisenberg and XY models on zigzag ladder lattices with nearest-neighbor bilinear-biquadratic exchange: Exact solution for the ground-state problem

An exact and complete solution of the ground-state problem for the classical Heisenberg and XY models with nearest-neighbor bilinear-biquadratic exchange on two- and three-dimensional lattices composed of isosceles triangles is determined with the use of a cluster method. It is shown how the geometric frustration due to the presence of triangles as structural units leads to the emergence of a rich phase diagram with incommensurate spiral orderings and their collinear limits, as well as canted and noncoplanar (conical) structures. Surprisingly, there are two different spiral phases with both continuous and discontinuous phase transitions between them. One of these phases is degenerate on two-dimensional partially anisotropic triangular lattice. This degeneracy is lifted on three-dimensional lattices. Canted phase is highly degenerate and this degeneracy persists on three-dimensional lattices.