  Frustration can introduce more-than-two minima in a spin dispersion relation
even in a fully polarized magnet under high magnetic field. We generally
discuss, on the fully polarized phase, the possibility of the chiral symmetry
breaking where the number of magnons pumped by finite temperature deviates to
one side of minima. We study this phase by constructing the Ginzburg-Landau
energy which is controlled by the external magnetic field and interactions
between magnons near the dispersion minima. This chirality breaking phase
accompanies, not a magnetization perpendicular to the external field, but the
vector chirality $S_m \times S_n$. We also discuss the possibility of the
chirality breaking phase on LiCuVO4 slightly above the saturation field.
