Topological Theory of Defects: An Introduction

Transcription

1 Topological Theory of Defects: An Introduction Shreyas Patankar Chennai Mathematical Institute

2 Order Parameter Spaces Consider an ordered medium in real space R 3. That is, every point in that medium is associated with a parameter f 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

3 Planar magnetisation Consider ferromagnetic crystal such that magnetisation at each point is restricted to lie in a plane. Think of this as a 2-D unit vector assigned to each point in R 3. Here, the order parameter space is the unit circle S 1 in R 2 (or C) Ordinary magnetisation Consider usual ferromagnet such that magnetisation at each point can point in any direction in space. That is, we can assign to each point in R 3 a unit 3-D vector. Thus, the order parameter space here is the unit sphere S 2

4 Nematic liquid crystals When liquid crytals are in nematic phase behave essentially like a fluid of elongated crystals. Order parameter for nematic liquid crystals is the unit sphere, except with diametrically opposite points identified. This can be written as the dyad f (r) = M(r) = ˆn(r)ˆn(r), that is, M ij (r) = n i (r)n j (r) (ˆn is a unit vector)

5 Biaxial nematics Biaxial nematics are liquid crystals with elongations along two or three orthogonal axes. Although conjectured long ago (Chandreshekhar, 1984), they were shown to exist only very recently (Madsen et al (2004)). The order parameter for these possess the rectangular symmetry, that is diametrically opposite points along all three axis are identified. We shall see important properties of this order parameter space.

6 Usefulness of topological representation Topology of order parameter space used to study defects (discontinuities) in ordered media (crystals etc.) Defect is topologically stable if given a perturbation that respects the constraints on the system, the defect does not change. Consider a two dimensional system with planar spin as the order parameter. Suppose a defect is formed by either of the configurations shown.

7 Both these defects are topologically stable. This is observed because, as shall be shown, the order parameter space for planar spins is not simply connected. This is in contrast to the system with ordinary spins as the order parameter space (for line defects!). There, the order parameter space S 2 is simply connected, and hence, all line defects are unstable. Using allowed perturbations only, we can make line defects vanish.

8 Topology preliminaries For more rigorous discussions, refer to Munkres, Allen Hatcher, etc. Let f 1, f 2 be two continuous paths in a path connected space X, that have the same starting and ending points. Then, f 1, f 2 are said to be homotopic if f 1 can be smoothly deformed to f 2 (See image below). The deformation function is itself called a homotopy.

9 It can be shown that homotopy is an equivalence relation The (homotopic) equivalence class [f ] of a path f is defined as the set of all paths that are homotopic to f. A path that is homotopic to a single point is called nullhomotopic. Let x 0 be a point in a path-connected space X, let f, g be continuous paths that start and end at x 0. We can form a new path by going along f first and then along g. This new path is said to be the composition path f g (not to be confused with composite function!)

10 Set of all equivalence classess of paths that start and end at x 0 is a group under the operation of path compostion. This is called the fundamental group of X at the point x 0, written as π 1 (X, x 0 ). 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 π 1 (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 π 1 (S 2 ) = 0

11 Group quotient Let H be a subgroup of a group G Define the set Hx = {hx h H} for all x G All x 1, x 2 such that Hx 1 = Hx 2 are said to lie in the same right coset of H The collection of all right cosets of H is called the group quotient G/H

12 Groups in order parameter space 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. The set of all points of G that can be connected by a continuous path to the identity e is called the connected component G 0 containing the identity. The quotient group G/G 0 is sometimes called the zeroth homotopy group of G.

13 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 f 1 R to f 2 R need not be unique. Consider the set H f of transforms in G such that for a given f R, each member of H f leaves f unchanged, that is gf = f g H f. Now, for all f i, the subgroups H fi are isomorphic. Thus we can denote just one global subgroup H, called the isotropy subgroup of G.

14 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 H f. Showing that this gives a bijection is left as an excercise. We shall look at some examples to demonstrate this representation.

15 Planar magnetisation Choose G to be SO(2), the group of two-dimensional orthogonal rotations. Then, the isotropy subgroup is simply the identity, as any other transformation would give a distinct member of R. Thus, the order parameter space is isomorphic to SO(2) itself. Note that the choice of G here is not unique. Writing unit vectors as ˆn = cos θˆx + sin θŷ, the natural choice for G is T (1), the group of all one-dimensional translations. H here is the group of all translations that are integral multiples of 2π

16 Ordinary Spins Choose G = SO(3) the group of all 3-D rotaions. 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) For convenience we write this using a different choice of G; G = SU(2), the group of all unit quaternions Then H can be shown to be H = e iθσz = That is, H is isomorphic to U(1) ( e iθ 0 ) 0 e iθ

17 Fundamental group of order parameter space Theorem Let G be a path-connected, simply connected topological group and let H be any subgroup. Let H 0 be the path connected component of H containing the identity. Then, π 1 (G/H) = π 0 (H) = H/H 0 This result is often written as π 1 (G/H) = π 0 (H)

18 Examples Planar Spins We have already seen that the order parameter space can be represented as G/H where G = T (1), H = T 2πn. H is a discrete group and hence the connected component H 0 is just the identity. Thus, π 1 (G/H) = π 0 (H) = H/H 0 = H Obviously, H is isomorphic to the set of integers Z, and hence π 1 (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.

19 Ordinary Spins Recall that order parameter space is given by SU(2)/U(1). Thus, putting π 1 (SU(2)/U(1)) = π 0 (U(1)) = 0, the trivial group. Thus, the order parameter space is simply connected. In other words, an ordinary spin medium in three dimensions cannot have a topologically stable line defect. Notice that we can make this statement just by looking at the topology of the order parameter space!

20 Nematics 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 ˆv = iσ y û = e iθ/2 0 H is not path-connected. H/H 0 is just the two element group isomorphic to Z 2 (the group of integers modulo 2) Thus, there is precisely one class of stable line defects in three dimensions in a nematic medium.

21 Biaxial nematics The only transformation that leave a biaxial nematic invariant are ±π rotations about either of three axes, given by H = {±1, ±iσ x, ±iσ y, ±iσ z } This is sometimes called the quaternion group denoted by Q As Q is discrete, H/H 0 is just Q, and hence, π 1 (SU(2)/Q) = Q There are four distinct classess of topological defects in a biaxial nematic medium, characterised by C 0 = { 1}, C x = {±iσ x }, C y = {±iσ y }, C z = {±iσ z }

22 References 1. Mermin, N. D., 1979, The topological theory of defects in ordered media, Reviews of Modern Physics, Vol Beach, J. A., W. D. Blair, 1996, Abstract Algebra (2nd ed.), Waveland Press 3. Herstein, I. N., 1975, Topics in Algebra