# Do there exist phases of matter where the order parameter space is non-orientable?

For example, are there order parameter space that is homeomorphic to a Klein bottle?

For example, in a nematic phase of matter, the order breaks rotation symmetry in that it fixes an axis, but it does not prefer/fix a direction. The conceptual picture of is of little rods. Hence, antipodal points along the sphere are identified, such that the order parameter space is $$S^2/ \{ \pm 1 \} \cong \mathbb {RP}^2$$, the projective plane! This manifold is indeed non-orientable.
• @BohanLu Well, on first sight I think you can get any manifold which can be expressed as a quotient of two symmetry groups, $G/H$ (physically: imagine spontaneously breaking $G$ down to the subgroup $H$, then the order parameter space is naturally the quotient). The above case is $\mathbb{RP}^2 \cong SO(3)/\mathbb Z_2$. One can similarly write the Klein bottle as $K \cong \left( U(1) \times U(1) \right) / \mathbb Z_2$. Hence there is a phase of matter that has the Klein bottle as its order parameter space. Can't give you a concrete realization on the spot... – Ruben Verresen Oct 4 '18 at 2:52
• Just to add: if you are staring at a LCD monitor right now, you are literally looking at a realization of Ruben's example: the rod-like molecules in liquid crystals do not have a direction and their parameter space is $\mathbb{S}^2 / \{ x \sim -x \}$. – Max Lein Oct 4 '18 at 8:16