Nonabelian Vortices Energetics vs Topology

Nonabelian Vortices : Energetics vs Topology
Amit Kr. Bhattacharjee
IMSc, Chennai
April 7, 2009
Amit Kr. Bhattacharjee (IMSc, Chennai) Soft matter journal club April 7, 2009 1 / 17

Central Concepts
A study of energetics of nonabelian vortices in two dimensions is
presented (Kobdaj and Thomas, Nuclear Physics B, 413, 689
(1994)), where the rotational symmetry of the system is been
spontaneously broken to a discrete group.
Vortices are characterized by group of quarternions as
π1(SO3/D2) = Q8.
An ansatz is proposed for isolated vortices.
Energetic implications were soughted out for different class of
vortices which has far reaching consequences in condesed matter
systems.

Organization
Nematic liquid crytal basics.
Topological classiﬁcation of defects.
Model of isolated nonabelian vortex.
Minimum energy vortex ansatz.
System of multi-vortices : selection rules.
Connection to biaxial liquid crystalline phase : topological rigidity ?
References.

Background
nematic mesophases
(biaxial phase)
Alignment tensor order have ﬁve degrees of freedom, 2 degrees of
order and 3 angles to specify principal direction.

Topological classification of defects : General scheme
Defining the order parameter φ(x, t) of the phase.
Identifying the map between physical space to the abstract order
parameter space R.
Defining the homotopy group πi(R), i = dsp − ddef − 1.
uniaxial nematic defects are characterized through
π1(S2/Z2) = Z2, having unstable integer and stable half integer
charged defects.
biaxial nematic defects are characterized through
π1(S3/D2) = Q8, having stable integer and half integer charged
defects. The abelian group D2 has four elements denoted by I,i,j,p
with the relations i2 = I, ij = ji = p. Non-abelian group of
quaternions form five conjugacy classes
C0 = {I}, ¯C0 = {J}, Cx = {±i}, Cy = {±j} and Cz = {±k} with
the multiplication rule, ij = −ji = p, jp = −pj = i, pi = −ip =
j, JJ = I, ii = jj = pp = J, ijp = J. The C0 class correspond to 4π
defect which is topologically unstable whereas the other four
classes correspond to four distinct stable defects.

Defects in nematics
Figure: Uniaxial and Biaxial defects

Digression to singular vortex model
Energy density ε = 1
2 Tr[gab(∂aφ)(∂bφ)] − V(φ), where
◮ V(φ) = −1
2 µ2
0Trφ2
+ 1
3 ρTrφ3
+ 1
4 λTrφ4
+ 1
4 λ′
(Trφ2
)2
.
◮ gab
is 2d ﬂat euclidean metric tensor.
◮ a, b stands for the 2d polar coordinates (r, θ)
In a coordinate system aligned with the principal axes,
φdiag =


ϕ1 0 0
0 ϕ2 0
0 0 −(ϕ1 + ϕ2)


For ρ = 0 but different values of λ, λ′, the extrema of the potential
correspond to ellipses in φ1 − φ2 plane,
(λ + λ′)(ϕ2
1 + ϕ1ϕ2 + ϕ2
2) − 1 = 0.
For ρ = 0, V(φ) has three minima on this ellipses at the points
ϕ2 = aϕ1, a = 1, −1
2 and −2.

Contd...
These 6 cut-points designate unbroken U(1) symmetry,
corresponding to abelian vortices.
For ρ = 0, all other points correspond to non-abelian vortices
because of the factorization of the symmetry group SO(3) with D2.

Ansatz for vortex
φ(r, θ) = G(θ)φdiag(r)G−1(θ) with the constraint G(θ + 2π) = G(θ)h
where G(θ) is the group element of SO(3) and h that of D2
corresponding to discrete rotations by 180 degrees about x,y or z
axes.
With the constraint, the group elements that minimizes the vortex
energy correspond to discrete rotations by θ/2 denoted as
Gi, Gj, Gk , G−1
i , G−1
j and G−1
k corresponding to non-abelian vortices
and anti-vortices of Cx , Cy and Cz(because of φ → −φ symmetry),
whereas rotations by θ correspond to ¯C0.

Group elements
Gi =


1 0 0
0 cos(θ/2) sin(θ/2)
0 −sin(θ/2) cos(θ/2)

 G−1
i =


1 0 0
0 cos(θ/2) −sin(θ/2)
0 sin(θ/2) cos(θ/2)


Gj =


cos(θ/2) 0 sin(θ/2)
0 1 0
−sin(θ/2) 0 cos(θ/2)

 G−1
j =


cos(θ/2) 0 −sin(θ/2)
0 1 0
sin(θ/2) 0 cos(θ/2)


Gk =


cos(θ/2) sin(θ/2) 0
−sin(θ/2) cos(θ/2) 0
0 0 1

 G−1
k =


cos(θ/2) −sin(θ/2) 0
sin(θ/2) cos(θ/2) 0
0 0 1


G−I =


1 0 0
0 cos(θ) sin(θ)
0 −sin(θ) cos(θ)

 G−I =


cos(θ) 0 sin(θ)
0 1 0
−sin(θ) 0 cos(θ)


G−I =


cos(θ) sin(θ) 0
−sin(θ) cos(θ) 0
0 0 1

 GI =


1 0 0
0 1 0
0 0 1



Energetics of vortex
Energy of class α vortex, Eα =
∞
0 drr
2π
0 dθ{1
2(∂r ϕ1)2 +
1
2(∂r ϕ2)2 + 1
2(∂r (ϕ1 + ϕ2))2 + 1
2r2 Pα(ϕ1, ϕ2) − V(ϕ1, ϕ2)},
◮ Pi = 1
2 (ϕ1 + 2ϕ2)2
,
◮ Pj = 1
2 (2ϕ1 + ϕ2)2
,
◮ Pj = 1
2 (ϕ1 − ϕ2)2
Minimization of Eα gives second order differential euation in ϕ1
and ϕ2.
◮ 2∂rr ϕ1 + 2
r ∂r ϕ1 + ∂rr ϕ2 + 1
r ∂r ϕ2 − 1
2r2
∂Pα
∂ϕ1
+ ∂V
∂ϕ1
= 0.
◮ 2∂rr ϕ2 + 2
r ∂r ϕ2 + ∂rr ϕ1 + 1
r ∂r ϕ1 − 1
2r2
∂Pα
∂ϕ2
+ ∂V
∂ϕ2
= 0.
Trial solution ϕ1 = 1√
λ(1+a+a2)
( rµ0
rµ0+1), ϕ2 = aϕ1.
Energy of vortex Eα in a 2D circular box,
Eα = Ec
α + π < Pα > ln(R
rc
), (α = i, j, k), R = radius of box.
Core energy Ec
α = π < Pα >
rc
0
dr
r ( r
r+1)2 + f(< ϕ1 >, < ϕ2 >, r)
=π < Pα > {ln(rc + 1) + 1
rc+1 − 1} + f(< ϕ1 >, < ϕ2 >, r)

System of multi-vortices
Energy of a vortex-antivortex pair,
Eα(z1, z2) = 2Ec
α + 2Eself
α + Eint
α (z1, z2),
◮ Eint
α (z1, z2) = 2π < Pα > ln|z1−z2
R |
As E
self,(α)
−I (= 4Eself
α ) > 2Eself
α , -I will dissociate into a pair of i, j, or
k type.
If Eself
k > Eself
i + Eself
j , k-type vortex dissociates into i and j-type,
◮ Condition for that, −2 < a < −1
2 .
If Eself
i > Eself
j + Eself
k , k-type vortex dissociates into i and j-type,
◮ Condition for that, −2 < a < 1.
If Eself
j > Eself
i + Eself
k , k-type vortex dissociates into i and j-type,
◮ Condition for that, −1
2 < a < 1.

Contd...

Topological rigidity in biaxial nematics ?
The underlying symmetry group of biaxial phase is non-abelian in
nature, so all the discussed theory applies directly to biaxial
defects.
Results from Goldenfeld et. al. support the theory of existence of
only two classes of defects at the late stages.
Poenaru et al conjectured long back that due to the non-abelian
nature, the defects from different class cannot cross by each other
(unlike the abelian case, e.g. uniaxial nematics), slowing the
overall phase ordering dynamics!

Contd...
These conjectures are still not settled, as biaxial nematic phase of
thermotropic systems were not found for several decades until
recent experimental reports by three group.

References
Poenaru and Toulouse, J.Phys.Colloq. 8, 887 (1977).
Zapotocky, Goldbart and Goldenfeld, Phys. Rev. E 51, 1216
(1995).
Priezjev and Pelcovits, Phys. Rev. E 66, 051705 (2002).
Madsen et al, Phys. Rev. Lett. 92, 145505 (2004).
Acharya et al, Phys. Rev. Lett. 92, 145506 (2004).
Merkel et al, Phys. Rev. Lett. 93, 237801 (2004).

Thanks for your attention

Nonabelian Vortices Energetics vs Topology

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