Stability of chiral Luttinger liquids and Abelian quantum Hall states.

A criterion is given for topological stability of Abelian quantum Hall states, and of Luttinger liquids at the boundaries between such states; this suggests a selection rule on states in the quantum Hall hierarchy theory. The linear response of Luttinger liquids to electromagnetic fields is described: the Hall conductance is quantized, irrespective of whether edge modes propagate in different directions. PACS numbers: 72.10.-d 73.20.Dx Typeset using REVTEX 1 A key feature of the quantum Hall effect (QHE) is its stability with respect to impurity potentials. A region of incompressible quantum Hall fluid described by a simple Laughlin state [1] has a current-carrying boundary where gapless excitations can be made. The edge is gapless because the current is carried by a single chiral “Luttinger liquid” mode [2]: stability against impurity perturbations may be associated with the maximal chirality of the edge. In contrast to the case of a non-chiral Luttinger liquid [3], there are no states of opposite chirality, so there is no backscattering (which can drive a “mass-generating” instability in which a pair of gapless modes with opposite chirality annihilate each other). In the integer QHE, there are a number of edge modes, but all have the same chirality, so the edge is stable. However, absolute stability (in the sense that there is no backscattering process) can occur even if the edge is not maximally-chiral; in this Letter, I characterize the criterion for this more precisely. The property of edge stability also suggests a related stability property for “bulk” QHE states. Recently, it has been suggested by Kane, Fisher and Polchinski [4] (KFP) that when a “clean” Hall edge is not maximally chiral, the quantization of the Hall conductance generally breaks down (KFP also claim that, in certain but not all circumstances, “randomness at the edge” drives a novel non-mass-generating instability that preempts this). The coupling of non-maximally-chiral Luttinger liquids to electromagnetic fields is examined here, and in contrast to [4], no breakdown of quantization of the Hall conductance of “clean” edges is found. The essential ingredient of the effective low-energy theory of a quantum Hall state is a charged Abelian Chern-Simons (CS) gauge field [5–7] that couples both to the electromagnetic field and any other neutral degrees of freedom of the theory. In a wide class of possible QHE states (Abelian quantum Hall states), such as those of the hierarchy theory [8,9], the neutral degrees of freedom are also Abelian CS fields [10–13], and the Lagrangian density is given by 2πL = ǫ ( 1 2 h̄(aλ,K∂μaν) + eAλ(q, ∂μaν) )