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

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Yes.

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.

In fact, see the top answer on this page for a nice explanation and a bunch of good references: https://mathoverflow.net/questions/45832/are-there-examples-of-non-orientable-manifolds-in-nature

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  • $\begingroup$ OK, I forgot that the Mobius strip is just the real projective space minus a disk, my bad. I guess my general curiosity lies in whether order parameter space homeomorphic to each classified closed surface (sphere, n-connected torus, m-connected RP, etc) can correspond to physical systems. What I learned from J. Sethna's books only include examples of the simplest spaces. $\endgroup$ – Bohan Lu Oct 4 '18 at 0:47
  • $\begingroup$ Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine. $\endgroup$ – Bohan Lu Oct 4 '18 at 0:51
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    $\begingroup$ @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... $\endgroup$ – Ruben Verresen Oct 4 '18 at 2:52
  • $\begingroup$ (to see the above mathematically, see example 2.6 and remark 2.15 in math.stanford.edu/~conrad/diffgeomPage/handouts/qtmanifold.pdf ) $\endgroup$ – Ruben Verresen Oct 4 '18 at 2:53
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    $\begingroup$ 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 $\mathbb{S}^2 / \{ x \sim -x \}$. $\endgroup$ – Max Lein Oct 4 '18 at 8:16

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