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.