THE INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS ...

21
QUANTIZATION OF PHYSICAL PARAMETERS
The dimensional reduction of the 3+1 system with Fermi points brings the
anomaly to the (2+1)-dimensional systems with fully gapped fermionic spectrum.
The most pronounced phenomena in these systems are related to the quantization
of physical parameters, and to the fermionic charges of the topological
objects – skyrmions. Here we consider both these effects. They are determined
by the momentum space topological invariant Ñ3 in eqn (11.1). While its ancestor
N3 describes topological defects (singularities of the fermionic propagator) in
the 4-momentum space, Ñ3 describes systems without momentum space defects
and it characterizes the global topology of the fermionic propagator in the whole
3-momentum space (px,py,p0). Ñ3 is thus responsible for the global properties
of the fermionic vacuum, and it enters the linear response of the vacuum state
to some special perturbations.
21.1 Spin and statistics of skyrmions in 2+1 systems
21.1.1 Chern–Simons term as Hopf invariant
Let us start with a thin film of 3He-A. If the thickness a of the film is finite, the
transverse motion of fermions – along the normal ˆz to the film – is quantized.
As a result the fermionic propagator G not only is the matrix in the spin and
Bogoliubov–Nambu spin spaces, but also acquires the indices of the transverse
levels. This allows us to obtain different values of the invariant Ñ3 in eqn (11.1)
by varying the thickness of the film. The Chern–Simons action, which is responsible
for the spin and statistics of skyrmions in the ˆ d-field in 3He-A film, is the
following functional of ˆ d (Volovik and Yakovenko 1989):
SCS{ ˆ d} = Ñ3
¯h
d
64π
2 xdte µνλ AµFνλ . (21.1)
Here Aµ is the auxiliary gauge field whose field strength is expressed through
the ˆ d-vector in the following way:
Fνλ = ∂νAλ − ∂λAν = ˆ
d · ∂ν ˆ d × ∂λ ˆ
d . (21.2)
The field strength Fνλ is related to the density of the topological invariant in
the coordinate spacetime which describes the skyrmions. The topological charge
of the ˆ d-skyrmion is (compare with eqn (16.12))
n2{ ˆ d} = 1
4π
d 2 xF12 = 1
4π
dx dy ˆ
d · ∂x ˆ d × ∂y ˆ
d
. (21.3)