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?










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










    share|cite|improve this question























      up vote
      6
      down vote

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      up vote
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      down vote

      favorite











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










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







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      asked Oct 3 at 23:12









      Bohan Lu

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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






          share|cite|improve this answer




















          • 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










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          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






          share|cite|improve this answer




















          • 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














          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






          share|cite|improve this answer




















          • 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












          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






          share|cite|improve this answer












          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







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          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
















          • 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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