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 Oct 3 at 23:12
Bohan Lu
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up vote
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For example, are there order parameter space that is homeomorphic to a Klein bottle?
statistical-mechanics condensed-matter topology
asked Oct 3 at 23:12
Bohan Lu
4815
add a comment |Â
up vote
6
down vote
favorite
up vote
6
down vote
favorite
For example, are there order parameter space that is homeomorphic to a Klein bottle?
statistical-mechanics condensed-matter topology
asked Oct 3 at 23:12
Bohan Lu
4815
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 Oct 3 at 23:12
Bohan Lu
4815
asked Oct 3 at 23:12
Bohan Lu
4815
asked Oct 3 at 23:12
Bohan Lu
4815
asked Oct 3 at 23:12
Bohan Lu
4815
asked Oct 3 at 23:12
Bohan Lu
4815
4815
add a comment |Â
add a comment |Â
1 Answer
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up vote
7
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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 Oct 3 at 23:48
Ruben Verresen
3,5921331
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
Oct 4 at 0:47
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
Oct 4 at 0:51
1
@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 $mathbbRP^2 cong SO(3)/mathbb Z_2$. 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 at 2:52
(to see the above mathematically, see example 2.6 and remark 2.15 in math.stanford.edu/~conrad/diffgeomPage/handouts/qtmanifold.pdf )
– Ruben Verresen
Oct 4 at 2:53
1
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 $mathbbS^2 / x sim -x $.
– Max Lein
Oct 4 at 8:16
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
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 Oct 3 at 23:48
Ruben Verresen
3,5921331
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
Oct 4 at 0:47
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
Oct 4 at 0:51
1
@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 $mathbbRP^2 cong SO(3)/mathbb Z_2$. 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 at 2:52
(to see the above mathematically, see example 2.6 and remark 2.15 in math.stanford.edu/~conrad/diffgeomPage/handouts/qtmanifold.pdf )
– Ruben Verresen
Oct 4 at 2:53
1
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 $mathbbS^2 / x sim -x $.
– Max Lein
Oct 4 at 8:16
add a comment |Â
up vote
7
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 Oct 3 at 23:48
Ruben Verresen
3,5921331
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
Oct 4 at 0:47
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
Oct 4 at 0:51
1
@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 $mathbbRP^2 cong SO(3)/mathbb Z_2$. 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 at 2:52
(to see the above mathematically, see example 2.6 and remark 2.15 in math.stanford.edu/~conrad/diffgeomPage/handouts/qtmanifold.pdf )
– Ruben Verresen
Oct 4 at 2:53
1
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 $mathbbS^2 / x sim -x $.
– Max Lein
Oct 4 at 8:16
add a comment |Â
up vote
7
down vote
accepted
up vote
7
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 Oct 3 at 23:48
Ruben Verresen
3,5921331
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 Oct 3 at 23:48
Ruben Verresen
3,5921331
answered Oct 3 at 23:48
Ruben Verresen
3,5921331
answered Oct 3 at 23:48
Ruben Verresen
3,5921331
answered Oct 3 at 23:48
Ruben Verresen
3,5921331
3,5921331
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
Oct 4 at 0:47
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
Oct 4 at 0:51
1
@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 $mathbbRP^2 cong SO(3)/mathbb Z_2$. 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 at 2:52
(to see the above mathematically, see example 2.6 and remark 2.15 in math.stanford.edu/~conrad/diffgeomPage/handouts/qtmanifold.pdf )
– Ruben Verresen
Oct 4 at 2:53
1
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 $mathbbS^2 / x sim -x $.
– Max Lein
Oct 4 at 8:16
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
Oct 4 at 0:47
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
Oct 4 at 0:51
1
@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 $mathbbRP^2 cong SO(3)/mathbb Z_2$. 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 at 2:52
(to see the above mathematically, see example 2.6 and remark 2.15 in math.stanford.edu/~conrad/diffgeomPage/handouts/qtmanifold.pdf )
– Ruben Verresen
Oct 4 at 2:53
1
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 $mathbbS^2 / x sim -x $.
– Max Lein
Oct 4 at 8:16
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
Oct 4 at 0:47
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
Oct 4 at 0:47
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
Oct 4 at 0:51
Sir, I also looked at your website and it was really cool! Your research interest aligns quite well with mine.
– Bohan Lu
Oct 4 at 0:51
1
1
@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 $mathbbRP^2 cong SO(3)/mathbb Z_2$. 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 at 2:52
@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 $mathbbRP^2 cong SO(3)/mathbb Z_2$. 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 at 2:52
(to see the above mathematically, see example 2.6 and remark 2.15 in math.stanford.edu/~conrad/diffgeomPage/handouts/qtmanifold.pdf )
– Ruben Verresen
Oct 4 at 2:53
(to see the above mathematically, see example 2.6 and remark 2.15 in math.stanford.edu/~conrad/diffgeomPage/handouts/qtmanifold.pdf )
– Ruben Verresen
Oct 4 at 2:53
1
1
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 $mathbbS^2 / x sim -x $.
– Max Lein
Oct 4 at 8:16
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 $mathbbS^2 / x sim -x $.
– Max Lein
Oct 4 at 8:16
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