A. Zee, Quantum Field Theory in a Nutshell · Problem VI.2.2

Field Theory Redux

Physics — QFT

Problem

For a challenge, derive the effective field theory for Hall fluids with filling factor

\nu = m/k

with

k

an odd integer, such as

\nu = \frac{2}{5}

. [Hint: You have to introduce

m

gauge potentials

a_{I\lambda}

and generalize (2) to

J^\mu = (1/2\pi) \varepsilon^{\mu\nu\lambda} \partial_\nu \sum_{I=1}^m a_{I\lambda}

. The effective theory turns out to be

\mathcal{L} = \frac{1}{4\pi} \sum_{I,J=1}^m a_I K_{IJ} \partial a_J + \sum_{I=1}^m a_{I\mu} \tilde{j}_I^\mu + \cdots

with the integer

k

replaced by a matrix

K

. Compare with (13).]