Fermion Chern Simons Theory of Hierarchical Fractional Quantum Hall States

TL;DR

This paper develops a Chern-Simons effective theory for hierarchical fractional quantum Hall states derived from Jain states, analyzing their stability, edge states, and tunneling conductance scaling behavior.

Contribution

It introduces a new effective bulk theory for fully polarized hierarchical FQH states and examines their stability, edge excitations, and tunneling properties.

Findings

Universal tunneling conductance exponent $eta=1/ u$.

Hierarchical states are stable under proposed criteria.

Edge states are described as unreconstructed and unresolved.

Abstract

We present an effective Chern-Simons theory for the bulk fully polarized fractional quantum Hall (FQH) hierarchical states constructed as daughters of general states of the Jain series, {\it i. e.} as FQH states of the quasi-particles or quasi-holes of Jain states. We discuss the stability of these new states and present two reasonable stability criteria. We discuss the theory of their edge states which follows naturally from this bulk theory. We construct the operators that create elementary excitations, and discuss the scaling behavior of the tunneling conductance in different situations. Under the assumption that the edge states of these fully polarized hierarchical states are unreconstructed and unresolved, we find that the differential conductance $G$ for tunneling of electrons from a Fermi liquid into {\em any} hierarchical Jain FQH states has the scaling behavior $G\sim V^\alpha$ with the universal exponent $\alpha=1/\nu$, where $\nu$ is the filling fraction of the hierarchical state. Finally, we explore alternative ways of constructing FQH states with the same filling fractions as partially polarized states, and conclude that this is not possible within our approach.