Order-parameter space

An order-parameter space is the set of allowed equilibrium values of an order parameter after accounting for the relevant symmetry breaking. When a symmetry group $G$ leaves a selected state invariant only under a subgroup $H$, the distinct symmetry-related states are often represented by the quotient $G/H$.

For a planar magnet with a fixed nonzero magnetization magnitude and free direction, the space is a circle. For a nematic orientation, opposite directions represent the same state, giving a different space. Its topology constrains possible topological defects; the physical identification of states matters as much as the number of coordinates.