Download Topological Theory of Defects: An Introduction

Topological Theory of Defects: An Introduction
Shreyas Patankar
Chennai Mathematical Institute
Order Parameter Spaces
I
Consider an ordered medium in real space R3 .
I
That is, every point in that medium is associated with a
“parameter” f
I
f can, depending on the medium be a scalar, vector, dyad etc.
Examples
1. Planar magnetisation
2. Ordinary magnetisation
3. Ordinary nematics
4. Biaxial nematics
Planar magnetisation
I
Consider ferromagnetic crystal such that magnetisation at
each point is restricted to lie in a plane.
I
Think of this as a 2-D unit vector assigned to each point in
R3 .
Here, the order parameter space is the unit circle S 1 in R2 (or C)
Ordinary magnetisation
I
Consider usual ferromagnet such that magnetisation at each
point can point in any direction in space.
I
That is, we can assign to each point in R3 a unit 3-D vector.
Thus, the order parameter space here is the unit sphere S 2
Nematic liquid crystals
When liquid crytals are in nematic phase behave essentially like a
fluid of elongated crystals.
I
Order parameter for nematic liquid crystals is the unit sphere,
except with diametrically opposite points identified.
I
This can be written as the dyad f (r) = M(r) = n̂(r)n̂(r), that
is, Mij (r) = ni (r)nj (r) (n̂ is a unit vector)
Biaxial nematics
I
Biaxial nematics are liquid crystals with elongations along two
or three orthogonal axes.
I
Although conjectured long ago (Chandreshekhar, 1984), they
were shown to exist only very recently (Madsen et al (2004)).
I
The order parameter for these possess the rectangular
symmetry, that is diametrically opposite points along all three
axis are identified.
I
We shall see important properties of this order parameter
space.
Usefulness of topological representation
I
Topology of order parameter space used to study defects
(discontinuities) in ordered media (crystals etc.)
I
Defect is ‘topologically stable’ if given a perturbation that
respects the constraints on the system, the defect does not
change.
I
Consider a two dimensional system with planar spin as the
order parameter.
I
Suppose a defect is formed by either of the configurations
shown.
I
Both these defects are topologically stable.
I
This is observed because, as shall be shown, the order
parameter space for planar spins is not simply connected.
I
This is in contrast to the system with ordinary spins as the
order parameter space (for line defects!).
I
There, the order parameter space S 2 is simply connected, and
hence, all line defects are unstable.
I
Using allowed perturbations only, we can make line defects
vanish.
Topology preliminaries
For more rigorous discussions, refer to Munkres, Allen Hatcher, etc.
I
Let f1 , f2 be two continuous paths in a path connected space
X , that have the same starting and ending points.
I
Then, f1 , f2 are said to be homotopic if f1 can be “smoothly
deformed” to f2 (See image below).
I
The deformation function is itself called a homotopy.
I
It can be shown that homotopy is an equivalence relation
I
The (homotopic) equivalence class [f ] of a path f is defined
as the set of all paths that are homotopic to f .
I
A path that is homotopic to a single point is called
nullhomotopic.
I
Let x0 be a point in a path-connected space X , let f , g be
continuous paths that start and end at x0 .
I
We can form a new path by going along f first and then along
g.
I
This new path is said to be the composition path f ◦ g (not to
be confused with composite function!)
I
Set of all equivalence classess of paths that start and end at
x0 is a group under the operation of path compostion.
I
This is called the fundamental group of X at the point x0 ,
written as π1 (X , x0 ).
Examples
1. The fundamental group of the unit circle S 1 is collection of
paths with different number of windings. Observe that this is
isomorphic to the group of integers under addition, written as
π1 (S 1 ) ∼
=Z
2. On the unit sphere, any loop can be smoothly deformed to a
point. Thus, all loops on the sphere are nullhomotopic, and
the fundamental group is trivial π1 (S 2 ) ∼
=0
Group quotient
I
Let H be a subgroup of a group G
I
Define the set Hx = {hx|h ∈ H} for all x ∈ G
I
All x1 , x2 such that Hx1 = Hx2 are said to lie in the same
right coset of H
I
The collection of all right cosets of H is called the group
quotient G /H
Groups in order parameter space
I
A group G that is also a topological space is said to be a
topological group if the group operation and the inversion
function are both continuous.
I
The set of all points of G that can be connected by a
continuous path to the identity e is called the connected
component G0 containing the identity.
I
The quotient group G /G0 is sometimes called the zeroth
homotopy group of G .
Almost all the order parameter spaces (R) we see in physics have
an associated group of linear transformations G , that is each
g ∈ G is a continuous linear transform g : R → R
In general, a transformation taking f1 ∈ R to f2 ∈ R need not be
unique.
Consider the set Hf of transforms in G such that for a given f ∈ R,
each member of Hf leaves f unchanged, that is gf = f ∀g ∈ Hf .
Now, for all fi , the subgroups Hfi are isomorphic. Thus we can
denote just one global subgroup H, called the isotropy subgroup
of G .
We can now give a mathematically rigorous description for the
order parameter space.
We shall establsih a one-to-one correspondence between the order
parameter space and the quotient space G /H of the
transformation group and the isotropy subgroup.
This is done simply by mapping each f ∈ R to the corresponding
isotropy subgroup Hf . Showing that this gives a bijection is left as
an excercise.
We shall look at some examples to demonstrate this representation.
Planar magnetisation
I
Choose G to be SO(2), the group of two-dimensional
orthogonal rotations.
I
Then, the isotropy subgroup is simply the identity, as any
other transformation would give a distinct member of R.
I
Thus, the order parameter space is isomorphic to SO(2)
itself.
I
Note that the choice of G here is not unique.
I
Writing unit vectors as n̂ = cos θx̂ + sin θŷ , the natural
choice for G is T (1), the group of all one-dimensional
translations.
I
H here is the group of all translations that are integral
multiples of 2π
Ordinary Spins
I
Choose G = SO(3) the group of all 3-D rotaions.
I
If ẑ is chosen as reference, then SO(2) forms an isotropy
subgroup. Thus, the order parameter space can be written as
SO(3)/SO(2)
I
For convenience we write this using a different choice of G ;
G = SU(2), the group of all unit quaternions
I
Then H can be shown to be H = e iθσz =
I
That is, H is isomorphic to U(1)
iθ
e
0
0 e −iθ
Fundamental group of order parameter space
Theorem
I
Let G be a path-connected, simply connected topological
group and let H be any subgroup.
I
Let H0 be the path connected component of H containing the
identity.
Then, π1 (G /H) ∼
= π0 (H) = H/H0
I
This result is often written as π1 (G /H) ∼
= π0 (H)
Examples
Planar Spins
We have already seen that the order parameter space can be
represented as G /H where G = T (1), H = T2πn .
H is a discrete group and hence the connected component H0 is
just the identity.
Thus, π1 (G /H) = π0 (H) = H/H0 = H Obviously, H is isomorphic
to the set of integers Z, and hence π1 (R) = Z
We can picture this as each homotopy class being given by its
“winding number”, the number of times the loop winds around the
singularity.
Ordinary Spins
I
Recall that order parameter space is given by SU(2)/U(1).
I
Thus, putting π1 (SU(2)/U(1)) = π0 (U(1)) = 0, the trivial
group. Thus, the order parameter space is simply connected.
I
In other words, an ordinary spin medium in three dimensions
cannot have a topologically stable line defect.
I
Notice that we can make this statement just by looking at the
topology of the order parameter space!
Nematics
I
Isotropy subgroup is the group of rotations about a fixed axis
(∼
= U(1)), and π rotations along a perpendicular axis, that is,
0
e iθ/2
v̂ = iσy û =
−e iθ/2
0
I
H is not path-connected.
I
H/H0 is just the two element group isomorphic to Z2 (the
group of integers modulo 2)
I
Thus, there is precisely one class of stable line defects in three
dimensions in a nematic medium.
Biaxial nematics
The only transformation that leave a biaxial nematic invariant are
±π rotations about either of three axes, given by
H = {±1, ±iσx , ±iσy , ±iσz }
This is sometimes called the quaternion group denoted by Q
As Q is discrete, H/H0 is just Q, and hence, π1 (SU(2)/Q) ∼
=Q
There are four distinct classess of topological defects in a biaxial
nematic medium, characterised by
C 0 = {−1}, Cx = {±iσx }, Cy = {±iσy }, Cz = {±iσz }
References
1. Mermin, N. D., 1979, The topological theory of defects in
ordered media, Reviews of Modern Physics, Vol. 51
2. Beach, J. A., W. D. Blair, 1996, Abstract Algebra (2nd ed.),
Waveland Press
3. Herstein, I. N., 1975, Topics in Algebra