polarizer analyzer rubbed plates 4 N EMATICS IN T WO D IMENSIONS : - - PowerPoint PPT Presentation

SMECTICS, SYMMETRY BREAKING AND SURFACES

Gareth Alexander

Bryan Gin-ge Chen Elisabetta Matsumoto Randall Kamien University of Virginia, March 25th 2010

Photo by Michi Nakata

Department of Physics & Astronomy University of Pennsylvania

1

LIQUID CRYSTAL MESOPHASES

cool or increase concentration Isotropic Nematic Smectic-A

uniaxial directional order

• ne-dimensional

positional order

2

http://commons.wikimedia.org/wiki/File:Nematische_Phase_Schlierentextur.jpg

NEMATICS IN TWO DIMENSIONS

3

http://commons.wikimedia.org/wiki/File:Nematische_Phase_Schlierentextur.jpg

NEMATICS IN TWO DIMENSIONS

3

http://commons.wikimedia.org/wiki/File:Nematische_Phase_Schlierentextur.jpg

NEMATICS IN TWO DIMENSIONS

3

“polarizer” “analyzer”

rubbed plates

NEMATICS IN TWO DIMENSIONS: WHAT ARE WE SEEING?

4

P A

NEMATICS IN TWO DIMENSIONS: WHAT ARE WE SEEING?

5

P A

the brushes are the preimages of the polarizer and analyzer direction

NEMATICS IN TWO DIMENSIONS: WHAT ARE WE SEEING?

5

http://commons.wikimedia.org/wiki/File:Nematische_Phase_Schlierentextur.jpg

±1 ±¹/₂

Maps from R2\{0} → RP 1

NEMATICS IN TWO DIMENSIONS

6

Lavrentovich & Natishin, EPL 12, 135 (1990)

HIGHER CHARGES?

7

Maps from R2\{0} → S1

DISLOCATIONS: DEFECTS IN THE TRANSLATIONAL ORDER

8

DISCLINATIONS: DEFECTS IN THE ORIENTATIONAL ORDER

Maps from

9

φ

n a

π

GROUND STATE MANIFOLD: FUNDAMENTAL GROUP

10

φ

n a

π

GROUND STATE MANIFOLD: FUNDAMENTAL GROUP

10

φ

n a

π F S

GROUND STATE MANIFOLD: FUNDAMENTAL GROUP

10

φ

n a

π

Maps from R2\{0} →S, F|FS−1F −1 = S

F S

GROUND STATE MANIFOLD: FUNDAMENTAL GROUP

10

Ground State Manifold

B

Sample

T

defects

DEFECTS AND HOMOTOPY: QUICK REVIEW

11

Ground State Manifold

B

Sample

T Maps fromπ1(B) → π1(T)

defects

DEFECTS AND HOMOTOPY: QUICK REVIEW

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B T

fix conjugacy class in B free homotopy on T

DEFECTS AND HOMOTOPY: QUICK REVIEW

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B T

fix conjugacy class in B free homotopy on T

Maps fromB → Cl(α), α ∈ π1(T)

DEFECTS AND HOMOTOPY: QUICK REVIEW

12

Maps fromB → Cl(α), α ∈ π1(T)

F FS2

DEFECTS AND HOMOTOPY: QUICK REVIEW

13

S(FS2)S−1 = SFS = F F FS2

DEFECTS AND HOMOTOPY: QUICK REVIEW

13

• N. D. Mermin, Rev. Mod. Phys. 51, 591-648 (1979); V. Poénaru, Commun. Math. Phys. 80, 127-136 (1981)

128

• V. Poenaru

( n

• singularity)
• Fig. 1. Measured foliations (2 dimensional "smectics")
• Fig. 2. Non measured foliation

mathematical tool which we

w i l l

be putting to use is the "theory of foliations". We

w i l l g i v e

now a very sketchy idea of what this is all about more mathematical details are to be found in the appendix at the end of this paper. Assume, for simplicity, that the physical space M is an open region of the euclidean space R

n and that V is the set of all /c dimensional linear subvarieties of

R

n (this is called the Grassman manifold G

n). Assume, also, that the order

parameter associates to every peM — such a /c dimensional linear subvariety (p) passing through p. One

w i l l

say that defines a foliation if M — can be completely covered by two by two disjoint, /c dimensional smooth, connected layers such that (p) is the tangent space of the (unique) layer passing through the point p. If k= 1, such layers always exist, because one just has to integrate an ordinary differential equation in order to get them. But if k ^ 2, a field of /c dimensional planes very seldom defines a foliation. The condition for this to be the case is a non linear "integrability condition" involving the first order derivations of the

map . If the condition is satisfied, we say that is "integrable" (and if this is so,

then a foliation # " is defined by ). A very important class of foliations are the so called "measured foliations", for which the layers (or rather "leaves" as they are usually called) are all equidistant.

One can think of a measured foliation as being a very rough mathematical model

• f a smectic liquid crystal. Figure 1 below shows some examples of such measured

foliations, with singularities, in dimension 2 (n = 2, k—i). By contrast, the foliation in Fig. 2 is not measured.

Now, with respect to the standard homotopy theory, here comes a new fact. If

• ur ordered medium is modeled by a (measured) foliation with V— G

n>k, although

every individual value (p)eV is acceptable, a global map M —

• V is not

necessarily acceptable. All this is very much in line with Mermin's critique.

The first two paragraphs of this paper will give instances of the following two

basic facts (in this framework of ordered media defined by foliations): (i) N ot every (homotopy class of) defect(s) predicted by pure homotopy theory is necessarily realized. In particular, we show that for a punctual defect of a two dimensional smectic the index of the corresponding plane field takes only the

128

• V. Poenaru

(no singularity)

• Fig. 1. Measured foliations (2 dimensional "smectics")
• Fig. 2. Non measured foliation

mathematical tool which we will be putting to use is the "theory of foliations". We

will give now a very sketchy idea of what this is all about more mathematical

details are to be found in the appendix at the end of this paper. Assume, for simplicity, that the physical space M is an open region of the euclidean space R

n and that V is the set of all /c dimensional linear subvarieties of

R

n (this is called the Grassman manifold G

n). Assume, also, that the order

parameter associates to every peM — such a /c dimensional linear subvariety (p) passing through p. One will say that defines a foliation if M — can be completely covered by two by two disjoint, /c dimensional smooth, connected layers such that (p) is the tangent space of the (unique) layer passing through the point p. If k= 1, such layers always exist, because one just has to integrate an ordinary differential equation in order to get them. But if k ^ 2, a field of /c dimensional planes very seldom defines a foliation. The condition for this to be the case is a non linear "integrability condition" involving the first order derivations of the

map . If the condition is satisfied, we say that is "integrable" (and if this is so,

then a foliation # " is defined by ).

A very important class of foliations are the so called "measured foliations", for which the layers (or rather "leaves" as they are usually called) are all equidistant.

One can think of a measured foliation as being a very rough mathematical model

• f a smectic liquid crystal. Figure 1 below shows some examples of such measured

foliations, with singularities, in dimension 2 (n = 2, k—i). By contrast, the foliation in Fig. 2 is not measured.

Now, with respect to the standard homotopy theory, here comes a new fact. If

• ur ordered medium is modeled by a (measured) foliation with V— G

n>k, although

every individual value (p)eV is acceptable, a global map M —

• V is not

necessarily acceptable. All this is very much in line with Mermin's critique.

The first two paragraphs of this paper will give instances of the following two

basic facts (in this framework of ordered media defined by foliations): (i) N ot every (homotopy class of) defect(s) predicted by pure homotopy theory is necessarily realized. In particular, we show that for a punctual defect of a two dimensional smectic the index of the corresponding plane field takes only the

Theorem (Po´ enaru) Let n be a field of directors [a line field] in R2 with an isolated singularity at 0, defining a measured

• foliation. Then I(n) ≤ 1. In particular, a vector

field ξ on R2, with an isolated singularity at 0, such that ∇ × ξ = 0, has the property that I(ξ) ≤ 1.

FUNDAMENTAL GROUP: NOT THE WHOLE STORY

14

• N. D. Mermin, Rev. Mod. Phys. 51, 591-648 (1979); V. Poénaru, Commun. Math. Phys. 80, 127-136 (1981)

128

• V. Poenaru

( n

• singularity)
• Fig. 1. Measured foliations (2 dimensional "smectics")
• Fig. 2. Non measured foliation

mathematical tool which we

w i l l

be putting to use is the "theory of foliations". We

w i l l g i v e

now a very sketchy idea of what this is all about more mathematical details are to be found in the appendix at the end of this paper. Assume, for simplicity, that the physical space M is an open region of the euclidean space R

n and that V is the set of all /c dimensional linear subvarieties of

R

n (this is called the Grassman manifold G

n). Assume, also, that the order

parameter associates to every peM — such a /c dimensional linear subvariety (p) passing through p. One

w i l l

say that defines a foliation if M — can be completely covered by two by two disjoint, /c dimensional smooth, connected layers such that (p) is the tangent space of the (unique) layer passing through the point p. If k= 1, such layers always exist, because one just has to integrate an ordinary differential equation in order to get them. But if k ^ 2, a field of /c dimensional planes very seldom defines a foliation. The condition for this to be the case is a non linear "integrability condition" involving the first order derivations of the

map . If the condition is satisfied, we say that is "integrable" (and if this is so,

then a foliation # " is defined by ). A very important class of foliations are the so called "measured foliations", for which the layers (or rather "leaves" as they are usually called) are all equidistant.

One can think of a measured foliation as being a very rough mathematical model

• f a smectic liquid crystal. Figure 1 below shows some examples of such measured

foliations, with singularities, in dimension 2 (n = 2, k—i). By contrast, the foliation in Fig. 2 is not measured.

Now, with respect to the standard homotopy theory, here comes a new fact. If

• ur ordered medium is modeled by a (measured) foliation with V— G

n>k, although

every individual value (p)eV is acceptable, a global map M —

• V is not

necessarily acceptable. All this is very much in line with Mermin's critique.

The first two paragraphs of this paper will give instances of the following two

basic facts (in this framework of ordered media defined by foliations): (i) N ot every (homotopy class of) defect(s) predicted by pure homotopy theory is necessarily realized. In particular, we show that for a punctual defect of a two dimensional smectic the index of the corresponding plane field takes only the

128

• V. Poenaru

(no singularity)

• Fig. 1. Measured foliations (2 dimensional "smectics")
• Fig. 2. Non measured foliation

mathematical tool which we will be putting to use is the "theory of foliations". We

will give now a very sketchy idea of what this is all about more mathematical

details are to be found in the appendix at the end of this paper. Assume, for simplicity, that the physical space M is an open region of the euclidean space R

n and that V is the set of all /c dimensional linear subvarieties of

R

n (this is called the Grassman manifold G

n). Assume, also, that the order

parameter associates to every peM — such a /c dimensional linear subvariety (p) passing through p. One will say that defines a foliation if M — can be completely covered by two by two disjoint, /c dimensional smooth, connected layers such that (p) is the tangent space of the (unique) layer passing through the point p. If k= 1, such layers always exist, because one just has to integrate an ordinary differential equation in order to get them. But if k ^ 2, a field of /c dimensional planes very seldom defines a foliation. The condition for this to be the case is a non linear "integrability condition" involving the first order derivations of the

map . If the condition is satisfied, we say that is "integrable" (and if this is so,

then a foliation # " is defined by ).

A very important class of foliations are the so called "measured foliations", for which the layers (or rather "leaves" as they are usually called) are all equidistant.

One can think of a measured foliation as being a very rough mathematical model

• f a smectic liquid crystal. Figure 1 below shows some examples of such measured

foliations, with singularities, in dimension 2 (n = 2, k—i). By contrast, the foliation in Fig. 2 is not measured.

Now, with respect to the standard homotopy theory, here comes a new fact. If

• ur ordered medium is modeled by a (measured) foliation with V— G

n>k, although

every individual value (p)eV is acceptable, a global map M —

• V is not

necessarily acceptable. All this is very much in line with Mermin's critique.

The first two paragraphs of this paper will give instances of the following two

basic facts (in this framework of ordered media defined by foliations): (i) N ot every (homotopy class of) defect(s) predicted by pure homotopy theory is necessarily realized. In particular, we show that for a punctual defect of a two dimensional smectic the index of the corresponding plane field takes only the

Theorem (Po´ enaru) Let n be a field of directors [a line field] in R2 with an isolated singularity at 0, defining a measured

• foliation. Then I(n) ≤ 1. In particular, a vector

field ξ on R2, with an isolated singularity at 0, such that ∇ × ξ = 0, has the property that I(ξ) ≤ 1.

Measured: Not:

FUNDAMENTAL GROUP: NOT THE WHOLE STORY

14

Chen, Alexander, Kamien, PNAS 106, 15577-15582 (2009)

SMECTIC PHASE FIELD AS A HEIGHT FUNCTION

15

Chen, Alexander, Kamien, PNAS 106, 15577-15582 (2009)

φ(x, y)

y

x

SMECTIC PHASE FIELD AS A HEIGHT FUNCTION

15

φ = 0

SMECTIC PHASE FIELD AS A HEIGHT FUNCTION

Chen, Alexander, Kamien, PNAS 106, 15577-15582 (2009)

16

φ = 1

SMECTIC PHASE FIELD AS A HEIGHT FUNCTION

Chen, Alexander, Kamien, PNAS 106, 15577-15582 (2009)

17

φ = 2

SMECTIC PHASE FIELD AS A HEIGHT FUNCTION

Chen, Alexander, Kamien, PNAS 106, 15577-15582 (2009)

18

φ = 3

SMECTIC PHASE FIELD AS A HEIGHT FUNCTION

Chen, Alexander, Kamien, PNAS 106, 15577-15582 (2009)

19

φ = 4

SMECTIC PHASE FIELD AS A HEIGHT FUNCTION

Chen, Alexander, Kamien, PNAS 106, 15577-15582 (2009)

20

CONTOUR MAPS: SMECTIC DISCLINATIONS

21

Peaks or Basins (+1)

CONTOUR MAPS: SMECTIC DISCLINATIONS

21

Peaks or Basins (+1) Passes (-1)

CONTOUR MAPS: SMECTIC DISCLINATIONS

21

Maps from R2\{0} → S1

EDGE DISLOCATIONS IN TWO DIMENSIONS

22

+2 DISLOCATION

Dislocation is a helicoid!

23

+2 DISLOCATION

Dislocation is a helicoid!

24

+2 DISLOCATION

Dislocation is a helicoid!

25

+2 DISLOCATION

Dislocation is a helicoid!

26

+2 DISLOCATION

Dislocation is a helicoid!

27

+2 DISLOCATION

Dislocation is a helicoid!

28

+2 DISLOCATION

Dislocation is a helicoid!

29

+2 DISLOCATION

Dislocation is a helicoid!

30

+2 DISLOCATION

Dislocation is a helicoid!

31

+2 DISLOCATION

Dislocation is a helicoid!

32

+2 DISLOCATION

Dislocation is a helicoid!

33

+2 DISLOCATION

Dislocation is a helicoid!

34

+2 DISLOCATION

Dislocation is a helicoid!

35

+2 DISLOCATION

Dislocation is a helicoid!

36

+2 DISLOCATION

Dislocation is a helicoid!

37

+2 DISLOCATION

Dislocation is a helicoid!

38

+2 DISLOCATION

Dislocation is a helicoid!

39

+2 DISLOCATION

Dislocation is a helicoid!

40

SMECTIC SYMMETRIES: LAYER OR LAYERS?

density wave: Phase is periodic ... ... and unoriented

41

SMECTIC SYMMETRIES: LAYER OR LAYERS?

density wave: Phase is periodic ... ... and unoriented

41

SMECTIC SYMMETRIES: LAYER OR LAYERS?

density wave:

• sheets cross at the fixed points of these

point symmetries

• only slices at these heights yield

consistent smectics

• critical points are constrained to these

heights Phase is periodic ... ... and unoriented

41

+1/2 DISCLINATION

42

+1/2 DISCLINATION

42

+1/2 DISCLINATION

42

+1/2 DISCLINATION

42

• 1/2 DISCLINATION

43

• 1/2 DISCLINATION

43

• 1/2 DISCLINATION

43

• 1/2 DISCLINATION

43

• 1/2 DISCLINATION

43

PINCH

44

PINCH

44

PINCH

44

PINCH

44

PINCH

44

PINCH

44

THE DISLOCATION

45

THE DISLOCATION

45

THE DISLOCATION

45

THE DISLOCATION

45

THE DISLOCATION

45

THE DISLOCATION

45

THE DISLOCATION

45

THE DISLOCATION

45

THE DISLOCATION

45

DISCLINATION DIPOLE: +1 DISLOCATION

46

DISCLINATION DIPOLE: +1 DISLOCATION

47

DISCLINATION DIPOLE: +1 DISLOCATION

48

DISCLINATION DIPOLE: +1 DISLOCATION

49

DISCLINATION DIPOLE: +1 DISLOCATION

50

DISCLINATION DIPOLE: +1 DISLOCATION

51

DISCLINATION DIPOLE: +1 DISLOCATION

52

DISCLINATION DIPOLE: +1 DISLOCATION

53

DISCLINATION DIPOLE: +1 DISLOCATION

54

DISCLINATION DIPOLE: +1 DISLOCATION

55

DISCLINATION DIPOLE: +1 DISLOCATION

56

DISCLINATION DIPOLE: +1 DISLOCATION

57

DISCLINATION DIPOLE: +1 DISLOCATION

58

FREE ENERGY AND ROTATIONAL INVARIANCE

Linear elasticity:

density wave:

59

FREE ENERGY AND ROTATIONAL INVARIANCE

Linear elasticity:

density wave:

Nonlinear elasticity:

59

Viewing φ as a graph: Equal spacing of curves:

SURFACE ENERGETICS

60

Viewing φ as a graph: Equal spacing of curves:

SURFACE ENERGETICS

Candidate:

“Willmore in a field”

60

EQUAL SPACING

61

K = 0

isometric to the plane

EQUAL SPACING

61

K = 0

isometric to the plane

EQUAL SPACING

61

K = 0

isometric to the plane

EQUAL SPACING

61

FOCAL CONICS

Nastishin, Meyer, and Kléman (2008), C. Williams, from de Gennes & Prost

Friedel, Granjean, Bull. Soc. Fr. Minéral. 33, 409-465 (1910)

62

TWO CONES

Alexander, Chen, Matsumoto, Kamien, (2010)

63

TWO CONES

Alexander, Chen, Matsumoto, Kamien, (2010)

64

TWO CONES

Alexander, Chen, Matsumoto, Kamien, (2010)

65

TWO CONES

Alexander, Chen, Matsumoto, Kamien, (2010)

66

TWO CONES

Alexander, Chen, Matsumoto, Kamien, (2010)

67

SHEDDING LIGHT ON FOCAL CONICS

Alexander, Chen, Matsumoto, Kamien, (2010)

68

SHEDDING LIGHT ON FOCAL CONICS

Alexander, Chen, Matsumoto, Kamien, (2010)

68

SHEDDING LIGHT ON FOCAL CONICS

Alexander, Chen, Matsumoto, Kamien, (2010)

68

SHEDDING LIGHT ON FOCAL CONICS

light cone

Alexander, Chen, Matsumoto, Kamien, (2010)

68

SHEDDING LIGHT ON FOCAL CONICS

light cone

Equal spacing ⇔ Null hypersurface

Alexander, Chen, Matsumoto, Kamien, (2010)

68

SPACE-LIKE SEPARATED EVENTS

Alexander, Chen, Matsumoto, Kamien, (2010)

69

SPACE-LIKE SEPARATED EVENTS

Alexander, Chen, Matsumoto, Kamien, (2010)

69

SPACE-LIKE SEPARATED EVENTS

Alexander, Chen, Matsumoto, Kamien, (2010)

69

SPACE-LIKE SEPARATED EVENTS

Alexander, Chen, Matsumoto, Kamien, (2010)

70

SPACE-LIKE SEPARATED EVENTS

Alexander, Chen, Matsumoto, Kamien, (2010)

70

TIME-LIKE SEPARATED EVENTS

Alexander, Chen, Matsumoto, Kamien, (2010)

71

TIME-LIKE SEPARATED EVENTS

Alexander, Chen, Matsumoto, Kamien, (2010)

72

TIME-LIKE SEPARATED EVENTS

Alexander, Chen, Matsumoto, Kamien, (2010)

72

FOCAL SETS

space-like separated events time-like separated events

Alexander, Chen, Matsumoto, Kamien, (2010)

73

FOCAL SETS

space-like separated events time-like separated events

• F. G. Friedlander, Math. Proc. Camb. Phil. Soc. 43, 360-373 (1947)

Alexander, Chen, Matsumoto, Kamien, (2010)

74

THREE DIMENSIONS

space-like separated events time-like separated events

• F. G. Friedlander, Math. Proc. Camb. Phil. Soc. 43, 360-373 (1947)

Alexander, Chen, Matsumoto, Kamien, (2010)

75

DUPIN CYCLIDES

two one-dimensional focal sets - “confocal conics”

• F. G. Friedlander, Math. Proc. Camb. Phil. Soc. 43, 360-373 (1947)

Alexander, Chen, Matsumoto, Kamien, (2010)

76

DUPIN CYCLIDES

two one-dimensional focal sets - “confocal conics”

• F. G. Friedlander, Math. Proc. Camb. Phil. Soc. 43, 360-373 (1947)

Alexander, Chen, Matsumoto, Kamien, (2010) Nastishin, Meyer, and Kléman (2008)

76

FOCAL CONICS

Photo: C. Williams, from de Gennes & Prost

77

FOCAL CONICS

Photo: C. Williams, from de Gennes & Prost

77

FOCAL CONICS

Photo: C. Williams, from de Gennes & Prost Friedel, Granjean, Bull. Soc. Fr. Minéral. 33, 409-465 (1910)

77

NESTED FOCAL SETS

Many ellipses are organised through common points - view this as a pair of events

Alexander, Chen, Matsumoto, Kamien, (2010)

focal hyperboloid

78

NESTED FOCAL SETS

Many ellipses are organised through common points - view this as a pair of events

Alexander, Chen, Matsumoto, Kamien, (2010)

focal hyperboloid cut out a circle

78

NESTED FOCAL SETS

Many ellipses are organised through common points - view this as a pair of events

Alexander, Chen, Matsumoto, Kamien, (2010)

focal hyperboloid cut out a circle

78

NESTED FOCAL SETS

Many ellipses are organised through common points - view this as a pair of events

Alexander, Chen, Matsumoto, Kamien, (2010)

focal hyperboloid cut out a circle

78

NESTED FOCAL SETS

Many ellipses are organised through common points - view this as a pair of events

Alexander, Chen, Matsumoto, Kamien, (2010)

focal hyperboloid move with Lorentz transformations

78

NESTED FOCAL SETS

Many ellipses are organised through common points - view this as a pair of events

Alexander, Chen, Matsumoto, Kamien, (2010)

focal hyperboloid and rotations

78

NULL SEPARATION - CORRESPONDING CONES

Two circular subsets with a point in common

Alexander, Chen, Matsumoto, Kamien, (2010)

79

NULL SEPARATION - CORRESPONDING CONES

Two circular subsets with a point in common

Alexander, Chen, Matsumoto, Kamien, (2010)

79

NULL SEPARATION - CORRESPONDING CONES

Two circular subsets with a point in common

Mutually tangent iff foci are null separated

Alexander, Chen, Matsumoto, Kamien, (2010)

79

POLYGONAL TEXTURES: TRÉILLIS ET RÉSEAUX

Alexander, Chen, Matsumoto, Kamien, (2010) Photo: C. Williams, from de Gennes & Prost

80

POLYGONAL TEXTURES: TRÉILLIS ET RÉSEAUX

Alexander, Chen, Matsumoto, Kamien, (2010)

• Multiple tangency of ellipses ⇒ Apollonian packing
• “Curvatures” satisfy the hyperbolic Déscartes-Soddy-Gossett

theorem

• Polygonal boundaries correspond to intersections of hyperboloids

Photo: C. Williams, from de Gennes & Prost

80

NSF DMR05-20020 (PENN MRSEC) DMR05-47230 Gifts from L. J. Bernstein and H. H. Coburn

Bryan Gin-ge Chen Elisabetta Matsumoto Randall Kamien

THANKS! FUNDING

We are grateful to Gary Gibbons, Jim Halverson, Juan Maldacena, Carl Modes and Ari Turner for stimulating discussions

Chen, Alexander, Kamien, PNAS 106, 15577-15582 (2009) Alexander, Chen, Matsumoto, Kamien, (2010)

81