Modulated phases and upsilon points in a spin model with helical ordering

The ground state of a one-dimensional, classical XY model with competing nearest- and next-nearest-neighbor interactions, a sixfold anisotropy, and an external field is studied. The model shows various modulated phases, depending on the values of the model parameters. Evidence is found that the ground-state phase diagram of this model contains upsilon points, multicritical points of a new class recently discussed by Bassler, Sasaki, and Griffiths. The phase diagram has a self-similar structure near these points, filled with multitudes of "mixed phases" whose spin configurations consist of segments of helical and fan structures separated by interfaces between them.