2Vortices

III Classical and Quantum Solitons



2.6 Jackiw–Pi vortices
So far, we have been thinking about electromagnetism, using the abelian Higgs
model. There is a different system that is useful in condensed matter physics. We
look at vortices in Chern–Simons–Higgs theory. This has a different Lagrangian
which is not Lorentz invariant instead of having the Maxwell curvature term,
we have the Chern–Simons Lagrangian term. We again work in two space
dimensions, with the Lagrangian density given by
L =
κ
4
ε
µνλ
A
µ
F
νλ
(iΦ, D
0
Φ)
1
2
||
2
+
1
2κ
|Φ|
4
,
where κ is a constant,
F
νλ
=
ν
A
λ
λ
A
ν
is the electromagnetic field and, as before,
D
0
Φ =
Φ
t
iA
0
Φ.
The first term is the Chern–Simons term, while the rest is the Schr¨odinger
Lagrangian density with a |Φ|
4
potential term.
Varying with respect to Φ, the Euler–Lagrange equation gives the Schr¨odinger
equation
iD
0
Φ +
1
2
D
2
j
Φ +
1
κ
|Φ|
2
Φ = 0.
If we take the variation with respect to A
0
instead, then we have
κB + |Φ|
2
= 0.
This is unusual it looks more like a constraint than an evolution equation,
and is a characteristic feature of Chern–Simons theories.
The other equations give conditions like
1
A
0
=
0
A
1
+
1
κ
(iΦ, D
2
Φ)
2
A
0
=
0
A
2
1
κ
(iΦ, D
1
Φ).
This is peculiar compared to Maxwell theory the equations relate the current
directly to the electromagnetic field, rather than its derivative.
For static solutions, we need
i
A
0
=
1
κ
ε
ij
(iΦ, D
j
Φ).
To solve this, we assume the field again satisfies the covariant anti-holomorphicity
condition
D
j
Φ =
jk
D
k
Φ = 0.
Then we can write
1
A
0
= +
1
κ
(iΦ, D
2
Φ) =
1
κ
1
|Φ|
2
2
,
and similarly for the derivative in the second coordinate direction. We can then
integrate these to obtain
A
0
=
|Φ|
2
2κ
.
We can then look at the other two equations, and see how we can solve those.
For static fields, the Schr¨odinger equation becomes
A
0
Φ +
1
2
D
2
j
Φ +
1
κ
|Φ|
2
Φ = 0.
Substituting in A
0
, we obtain
D
2
j
Φ =
|Φ|
2
κ
Let’s then see if this makes sense. We need to see whether this can be consistent
with the holomorphicity condition. The answer is yes, if we have the equation
κB + |Φ|
2
= 0. We calculate
D
2
j
Φ = D
2
1
Φ + D
2
2
Φ = i(D
1
D
2
D
2
D
1
)Φ = +BΦ =
1
κ
|Φ|
2
Φ,
exactly what we wanted.
So the conclusion (check as an exercise) is that we can generate vortex
solutions by solving
D
j
Φ
jk
D
k
Φ = 0
κB + |Φ|
2
= 0.
As in Taubes’ theorem there is a reduction to a scalar equation, which is in this
case solvable explicitly:
log |Φ| =
1
κ
|Φ|
2
.
Setting ρ = |Φ|
2
, this becomes Liouville’s equation
log ρ =
2
κ
ρ,
which can in fact be solved in terms of rational functions see for example
Chapter 5 of the book Solitons in Field Theory and Nonlinear Analysis by Y.
Yang. (As in Taubes’ theorem, there is a corresponding statement providing
solutions via holomorphic rather than anti-holomorphic conditions.)