Series expansion

The introduction of the Chern-Simons field enables us to expand the partition function around a different saddle point. This helps us to identify an appropriate starting point for the series expansion of the partition function in perturbation theory. In this section we will show that in first approximation the partition function of the system described above

$$Z=\int \pmaat{\psi}\pmaat{\psi^{*}}\pmaat{a} \exp(i S[a,\psi])$$ (4.6)

yields incompressible states at Jain's series for the fractional quantum Hall effect. As it stands this partition function includes integration over gauge equivalent configurations. Although it is not explicitly shown, we will fix a gauge when necessary.

In experiments the system is probed with electromagnetic fields (a voltage is applied for example). That's why we introduce a fluctuation ( $\tilde{A}_{\mu}$) on the electromagnetic field, with zero average, which we will use to find the response of the system to such perturbations. The mean electromagnetic field, which is of course very important for the structure of the effective action we will find, will not be very visible in the formalism, because it is a non-dynamical background.

The excitations, which the fermion degrees of freedom represent, are, as we will see, gapped. Furthermore we assume that the excitations created by the electromagnetic probe have a smaller energy scale than then the fermion excitations, hence we want to integrate out the fermion degrees of freedom. The integration of the fermion field is complicated by the interaction term, which is fourth order in the fields. We solve this by introducing a new Bose field.

$$\begin{multline} e^{ -i \intm{z}{3} \intm{z'}{3} \frac{1}{2} \left[ \absp{\psi(... ...2} \intm{z}{3} \intm{z'}{3} \lambda(z) V^{-1}(z-z') \lambda(z') } \end{multline}$$

The quadratic term in $\psi$ now has a factor $\lambda - a_{0}$. We then shift $a_{o} \rightarrow a_{o}+\lambda$ to remove the coupling between $\lambda$ and $\absp{\psi}{2}$. Because $a_{0}$ is also present in the Chern-Simons term the term linear in $\lambda$ receives a contribution, which now is $-(\theta b + \bar{\rho})\lambda$. It is now trivial to integrate out the field $\lambda$ to yield the following action.

$$\begin{multline} S = \intm{z}{3} \left\{ \psi^{\dagger}(z) [i D_0 +\mu]\psi(z) ... ...mbda} (a-\tilde{A})_{\mu} \partial_{\nu} (a-\tilde{A})_{\lambda} \end{multline}$$

Note that we have we shifted $a_{\mu}$ again, this to remove the probe ( $\tilde{A}_{\mu}$) from the covariant derivatives and into the Chern-Simons and interaction terms. This simplifies the dependence of our final effective action on $\tilde{A}_{\mu}$. We now shift the Chern-Simons field, $a_{\mu} \to \overline{a_{\mu}}+a_{\mu}$, and expand the partition function in terms of quantum fluctuations $a_{\mu}$ around the classical configuration $\overline{a_{\mu}}$.

$$\begin{multline} Z[a]=\int \pmaat{\psi^{*}}\pmaat{\psi} \; \bigg\{ 1 + i \int d... ... _{\overline{a}} \\ + .... \bigg\} \exp ( i S[\overline{a}]) \\ \end{multline}$$

We now integrate out the fermion fields yielding expectation values (which we will denote with $\langle ... \rangle$) in a theory of fermions moving in an effective magnetic field, $\mathcal{B}=B+\overline{b}$. If we now write the series that we obtained as an exponential we find an effective action as a series in the field $a_{\mu}$.

$$\begin{multline} S_{eff} = \int dx \; a_{\mu} \Langle \vdiff{S}{a_{\mu}} \Rangl... ...S_{CS}(\overline{a}-\tilde{A}) + S_{int}(\overline{a}-\tilde{A}) \end{multline}$$

By demanding that the action is stationary under small fluctuations,

$$\Langle \vdiff{S}{a_{\mu}}\bigg\vert _{\overline{a}} \Rangle=0,$$ (4.7)

we obtain the classical equations of motion,

As described above the averages are just expectation values for a system of non-interacting fermions in a magnetic field. But such systems we can handle; these are precisely the systems studied in section 2.3. We know that there exists uniform density and currentless ground states for such systems. If we assume that the fermions form such a ground state, the equations (4.13) are solved with

$$\overline{b} =-\frac{\overline{\rho}}{\theta}$$ (4.9)

$$\overline{e} =0.$$ (4.10)

This defines the classical configuration upon which we build our quantum theory. Our theory is thus a theory of excitations on a state with uniform fermion density living in a magnetic field that is the sum of the external field and the Chern-Simons field. We know of course that such a state is characterized by Landau levels (see 2.3) and that for certain values of the magnetic field, when a Landau-level is precisely filled, there exist incompressible states. In other words, if the filling fraction of these effective Landau-levels, $\nu^{*}=\frac{2\pi\overline{\rho}}{B+\overline{b}}$, is an integer ($p$), we will have a quantum Hall effect. In this case the true filling fraction is not an integer, but a fraction,

$$\nu=\frac{1}{\frac{1}{\nu_{*}}+\frac{1}{2\pi\theta}}=\frac{p}{2sp+1}, \;\; p,s \in \mathbbm{Z}.$$ (4.11)

This is precisely Jain's series (see (2.33)). The introduction of the gauge field has enabled us to find a mapping from a system of interacting fermions living in magnetic field $B$ to a system of non-interacting (composite) fermions living in a magnetic field $\mathcal{B}$, analogous to the arguments given by Jain (see section 2.4). This is a theory we can work with, as we will make clear in the coming sections.