Can we construct a Chern-Simons theory provided that the ground state is degenerate and gapped, like the Abelian fractional quantum Hall effect?

Question

I am studying the Chern-Simons approach to fractional quantum Hall effect, which a special focus on the topological order in the context of Abelian fractional quantum Hall effect. To me, the logic to adopt (Maxwell-) Chern-Simons theory as an effective theory is pretty bottom up (says, Wen's book): We can write down a current with fractional Hall conductivity, from which we can introduce the statistical gauge fields as auxiliary fields for the effective action and it turns out to be the Chern-Simons theory. However, it seems to me that the concept of topological order is mainly based on ground state degeneracy and gapped bulk systems.

My questions are:

1. Consider systems satisfy (my prototype example is Laughlin's state at filling 1/m): (i) gapped bulk; (ii) degenerate ground state; (iii) $U(1)$ charge conservation; (iv) only one kind of fractional excitation. What are the extra minimal ingredients needed to derive the corresponding effective (topological) action? Is it possible to derive the Chern-Simons theory directly, only based on these constraints?
2. Also, can the fractional Hall conductivity be derived solely from these constraints?

Answer 1

The answer to your question is basically yes, if we are willing to accept an algebraic description of 2+1D gapped phases. In this algebraic description, one does not resort to a particular field theory. Rather, we abstract out "universal" properties of low-energy localized excitations, the anyons. The mathematical structure is called a modular tensor category (MTC). In this theory, all such excitations can be classified into a few "elementary" types (and their "direct sums"). Two excitations can be fused to form another excitation. MTC also contains information about braidings between anyons and mathematical consistency conditions for fusion and braiding.

The only tricky question is what you mean by "having only one kind of excitation". If that means there is just one type of nontrivial anyon in the MTC, then the internal mathematical consistency of MTC only allows two possibilities: either a semion theory, e.g. $\nu=1/2$ Laughlin state, or Fibonacci anyons (assuming the system is bosonic. For fermions there is one more choice). So this is probably too restricted.

A slightly different way to proceed is to assume that all anyons are Abelian. What it means is that the quantum state of the system is uniquely determined once the locations of the anyons are fixed. It can be shown that if the anyons are Abelian, then the universal properties can be captured by a $U(1)\times U(1)\times\cdots$ Chern-Simons theory. Now if we further assume that all anyons can be "generated" by a fundamental one from fusion, then it must be a simple U(1) Chern-Simons theory.

U(1) symmetry can also be incorporated into this framework abstractly, and one can indeed derive various constraints on what the Hall conductance could be based on very general considerations.