8Liquid crystals hydrodynamics

III Theoretical Physics of Soft Condensed Matter

8.3 Topological defects in three dimensions

In three dimensions, we also have defects of the above kind lying along a line.

For such line defects, everything we said so far carries through — separated at a

distance R, the interaction force is

˜κ

R

and so and so we similarly have

L(t) ∼ t

1/2

.

However, in three dimensions, the

q

=

±

1

2

defects are the same topologically. In

other words, we can change +

q

to

−q

via continuous, local deformations. This

involves rotating out to the

z

direction, which is not available in two dimensions.

While it is possible to visually understand how this works, it is difficult to draw

on paper, and it is also evident that we should proceed in more formal manners

to ensure we understand exactly how these things work.

To begin, we have a space

M

of order parameters. In our case, this is the

space of all possible orientations of rods.

Example.

In the case of a polar liquid crystal in

d

dimensions, we have

M

=

S

d−1

, the (d − 1)-dimensional unit sphere.

Example.

For nematic liquid crystals in

d

-dimensions, we have

M

=

RP

d−1

,

which is obtained from S

d−1

by identifying antipodal points.

When we discussed the charge of a topological defect, we picked a loop around

the singularity and see what happened when we went around a defect. So we

pick a domain

D

that encloses a defect core, and consider the map

f

:

D → M

that assigns to each point the order parameter at that point. In our cases,

D

is

a circle S

1

, and so f is a loop in M.

We say two mappings

f

1

, f

2

, are homotopic if they can be continuously

deformed into each other. Defects lie in the same homotopy class if maps for

all

D

’s enclosing them are homotopic. The fundamental group

π

1

(

M

) is the set

of all homotopy classes of maps

S

1

→ M

. This encodes the set of all possible

charges.

Since we call it a fundamental group, it had better have a group structures.

If we have two defects, we can put them next to each other, and pick a new

circle that goes around the two defects. This then gives rise to a new homotopy

class S

1

→ M.

More generally, if we consider

d −n

-dimensional defects, then we can enclose

the defect with a sphere of dimension

n −

1. The corresponding classes live in

the higher homotopy groups π

n−1

(M).

Example.

Observe that

RP

1

is actually just

S

1

in disguise, and so

π

1

(

RP

1

) =

Z

.

The generator of π

1

(RP

1

) is the charge

1

2

topological defect.

Example. We can visualize RP

2

as a certain quotient of the disk, namely

where we identify the two arcs in the boundary according to the arrow. Observe

that the two marked points are in fact the same point under the identification.

If we have a path from the first point to the second point, then this would be

considered a loop in RP

2

, and this is the q =

1

2

defect.

Observe that in the two-dimensional case, the

q

=

±

1

2

defects correspond to

going along the top arc and bottom arc from the left to right respectively. In

RP

2

, there is then a homotopy between these two paths by going through the

disk. So in RP

2

, they lie in the same homotopy class.

In general, it is easy to see that

π

1

(

RP

2

) =

Z/

2

Z

, so

q

=

1

2

is the unique

non-trivial defect.

This is particularly interesting, because two

q

=

1

2

defects can merge and

disappear! Similarly, what you would expect to be a

q

= 1 defect could locally

relax to become trivial.

Observe that in our “line defects”, the core can actually form a loop instead.

We can also have point defects that correspond to elements in

π

2

(

M

)

∼

=

Z

. It is

an exercise to draw some pictures yourself to see how these look.