JHEP01(2016)138
Both φ and ψ fields give rise to excitations with mass m. The first order kinetic terms
mean that there are parti c l es , but no anti-particle s. The φ excitations have spin −1/2k,
while the ψ excitations have spin 1/2−1/2k. In what follows we will someti m es refer to the
φ excitations as “bosons” and the ψ excitations as “fermions”, despite the fact that both
are in general anyonic. The action (2.1) also admits a relevant def orm ati on consistent with
supersymmetry which, as discussed in [
19], drives the system to a quantum Hall state. We
will not turn on this deformation here.
The quartic potential term s describe a delta-function contact interaction between
anyons. The potential between bosons depends on the sign
3
of k: it is repulsive for k > 0
and attract i ve for k < 0. This distinction will play an important role in this paper and the
physics of anyons is rather different in the two cases.
The Chern -Si mon s coupling in (
2.1) ens ur e s that any particle excitation is accompanied
by magnetic flux. This is imposed through Gauss’ law which, classically, reads
B =
2π
k
φ
i
φ
†
i
− ψ
i
ψ
†
i
(2.2)
At the quantum level, this means that operators φ
i
and ψ
i
must be dressed with flux
to make them gauge invariant. This is easily achieved in the N
c
= 1 Abelian theory
as explained, for example, in [6]. We introduce the dual photon σ, normalised to have
periodicity 2π. The Chern-Sim ons term ensures that e
iσ
has charge −k under the U(1)
gauge symmetry and composite, gauge invar i ant operators can be defined as
Φ
i
= e
−iσ/k
φ
i
, Ψ
i
= e
−iσ/k
ψ
i
(2.3)
For t he non-Abelian U(N
c
) theories, it is more difficult to write ex pl i c i t expressions for the
local gauge invariant operator s. (One can easily construct non-local operators through the
addition of a Wilson line. ) Nonetheless, one expects that such objects exist, transforming
in the N
c
representation of the global part of the gauge group (and in the N
f
representation
of the flavour group).
At this point, we pause to note that the operators (
2.3) involve fractional monopole op-
erators. These are acceptable in our non-relativistic theory, and also in gappe d relativistic
theories. They would not, however, be allowed in a rel ati v i s ti c Chern-Simons matter theory
at the conformal poi nt.
4
(See, for ex ampl e , [15–18].) One way to see this is to note that, in
relativistic theories, the state-operator map takes monopole operators inserted in the plane
to magnet i c flux over the two-sphe r e , where standard Dirac quantisation ensures that it
is integer valued. In contrast, in non-relativistic theories the operators are transformed to
flux in a harmonic trap, where there is no such quantisation requirement. The upshot is
that in conformal relativistic theories, one must work with operators of the schematic form
e
−iσ
φ
k
. For us, however, Φ and Ψ defined in (
2.3) are sensible operators.
3
There is a second class of theories, related to (
2.1) by a parity transformation. This flips the sign of
the Chern-Simons term and the Pauli term, leaving the quartic potential invariant.
4
We thank Kimyeong Lee for a discussion on this point.
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