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
|DΦ|
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 = iε
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 − iε
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.)