# How to understand the Chern-Simons effective theory in Fractional Quantum Hall Liquid?

In FQH liquid, the effective Lagrangian of a $\nu=1/m$ Laughlin state is given in Xiao-Gang Wen's Book Quantum Field Theory of Many-Body Systems Chapter 7: $$\mathcal{L} = \frac{m}{4\pi}A_\mu \partial_\nu a_\lambda \epsilon^{\mu\nu\lambda}+ \frac{e}{2\pi}a_\mu\partial_\nu a_\lambda \epsilon^{\mu\nu\lambda}$$ in which $A_\mu$ is the electromagnetic field and $a_\mu$ is an emergent Chern Simons gauge field. But the interacting 2DEG has the following action: $$S[\psi^\dagger,\psi,A_\mu]=\int d^3x \left[\psi^\dagger(D_0-\mu)\psi-\psi^\dagger\frac{D_i^2}{2m}\psi\right] + S_I[\psi^\dagger,\psi]$$ in which $D_\mu = \partial_\mu+ieA_\mu$. So can we derive the effective theory directly from the action of interacting 2DEG?

Besides, in Naoto Nagaosa's Book Quantum Field Theory in Condensed Matter Physics Chapter 6, the author introduced the Chern-Simons gauge theory in another way: 2D fermion theory is equivalent to a 2D boson theory coupled with a Chern-Simons gauge field with an odd quantum flux $$S[\phi^\dagger,\phi,A_\mu,a_\mu]=\int d^3x \left[\phi^\dagger(D_0-\mu)\phi-\phi^\dagger\frac{D_i^2}{2m}\phi\right] + S_I[\phi^\dagger,\phi] + \frac{i}{4\theta}\int d^3x a_\mu\partial_\nu a_\lambda\epsilon^{\mu\nu\lambda}$$ in which $D_\mu = \partial_\mu+ieA_\mu + iea_\mu$. What is the relationship between these two different Chern-Simons gauge theory?

It is an old question but very important so I will answer.

The latter one, is not an effective low energy theory it is an exact model.

However the wen's theory low energy effective theory.

you can rigorously get the first one from latter. The outline of calculation is as following.

1. first make a mean field theory such that the system will be a Bose liquid and you will find a mean field solution.

2. then you have to introduce ansatz for vortex excitation, and expand the action around those ansatz.

3. you have to make a Hubbard transformation to get rid of the non linearity.

4. by varying the phase of the the ansatz you will find a continuity equation, and you can introduce a dual field, due to that constraint. and write the action in terms of this dual field.

5. at the end you will integrate out the $$a$$'s and you will left with exactly wen's low energy topological effective theory.

This method is called composite bosons, and chern simons ginzburg landau theory (CSGL). You can find paper expcility showing those steps. The wen's method is called hydrodynamic theory.

• Thanks for adding to this! Commented Dec 10, 2019 at 2:40
• @physshyp , woluld you please give some reference papers where above mentioned steps are worke out Commented Nov 23, 2021 at 19:52
• @PRakeshKumarDora worldscientific.com/doi/abs/10.1142/S0217979292000037 Commented Nov 24, 2021 at 5:29
• Thanks for the reference. Commented Nov 24, 2021 at 7:09

Wen's theory is 'dropped from sky'.

Upon integrating out gapped fermions (no matter relativistic or non-relativistic), the left theory is Wen's Chern-Simons theory.

Chern-Simons has the correct description of response, degeneracy, quasi-particle braiding e.t.c. for Abelian 2D topological phases.