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Do there exist phases of matter where the order parameter space is non-orientable?
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For example, are there order parameter space that is homeomorphic to a Klein bottle?
statistical-mechanics condensed-matter topology
asked 1 hour ago
Bohan Lu
3315
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up vote
3
down vote
favorite
For example, are there order parameter space that is homeomorphic to a Klein bottle?
statistical-mechanics condensed-matter topology
asked 1 hour ago
Bohan Lu
3315
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
For example, are there order parameter space that is homeomorphic to a Klein bottle?
statistical-mechanics condensed-matter topology
asked 1 hour ago
Bohan Lu
3315
For example, are there order parameter space that is homeomorphic to a Klein bottle?
statistical-mechanics condensed-matter topology
statistical-mechanics condensed-matter topology
asked 1 hour ago
Bohan Lu
3315
asked 1 hour ago
Bohan Lu
3315
asked 1 hour ago
Bohan Lu
3315
asked 1 hour ago
Bohan Lu
3315
asked 1 hour ago
Bohan Lu
3315
3315
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
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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
answered 1 hour ago
Ruben Verresen
3,5621331
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.
– Bohan Lu
10 mins ago
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
5 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
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
answered 1 hour ago
Ruben Verresen
3,5621331
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.
– Bohan Lu
10 mins ago
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
5 mins ago
add a comment |Â
up vote
4
down vote
accepted
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
answered 1 hour ago
Ruben Verresen
3,5621331
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.
– Bohan Lu
10 mins ago
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
5 mins ago
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
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
answered 1 hour ago
Ruben Verresen
3,5621331
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
answered 1 hour ago
Ruben Verresen
3,5621331
answered 1 hour ago
Ruben Verresen
3,5621331
answered 1 hour ago
Ruben Verresen
3,5621331
answered 1 hour ago
Ruben Verresen
3,5621331
3,5621331
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.
– Bohan Lu
10 mins ago
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
5 mins ago
add a comment |Â
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.
– Bohan Lu
10 mins ago
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
5 mins ago
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.
– Bohan Lu
10 mins ago
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.
– Bohan Lu
10 mins ago
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
5 mins ago
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
5 mins ago
add a comment |Â
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