Fractional Quantum Hall Effect in an Array of Quantum Wires

Abstract

We demonstrate the emergence of the quantum Hall (QH) hierarchy in a 2D model of coupled quantum wires in a perpendicular magnetic field. At commensurate values of the magnetic field, the system can develop instabilities to appropriate inter-wire electron hopping processes that drive the system into a variety of QH states. Some of the QH states are not included in the Haldane-Halperin hierarchy. In addition, we find operators allowed at any field that lead to novel crystals of Laughlin quasiparticles. We demonstrate that any QH state is the groundstate of a Hamiltonian that we explicitly construct. [7]. These have lead to a deep understanding of the excitation spectrum of QH states and of the structure of the QH hierarchy.The purpose of this paper is to develop a new formalism, which reproduces the QH hierarchy in a model consisting of a two-dimensional array of quantum wires in a perpendicular magnetic field B. The model could be relevant for semiconductor quantum wires, ropes of carbon nanotubes, and for stripes that arise in QH systems in the higher Landau systems [8]. Aside from the direct relevance of the model, our calculations provide a novel, and in many ways a simpler, approach to describe the fractional QH effect. Given its success in treating the Fractional QH effect, it is likely that our technique will prove useful for understanding other strongly correlated states.We use the bosonization technique [9], developed for one-dimensional systems, and relate the QH effect to coupled Luttinger liquids. It has been shown recently [10][11][12] that for a range of interwire charge and current interactions, there is a phase in which interwire Josephson, charge-and spin-density-wave, and single-particle couplings are irrelevant. This sliding Luttinger liquid (SLL) or smectic metal phase is the quantum analog of the sliding phase found recently in DNA-lipid complexes and stacked XY models [13,14]. The SLL resembles a Luttinger liquid, with transport properties that exhibit power-law singularities as a function of temperature. It has also been demonstrated [15] that in a magnetic field the phase space of SLL expands considerably.We will show that at commensurate magnetic fields the SLL phase can be unstable to inter-wire electron hopping processes that lead the formation of an energy gap and the QH effect. The presence of the ν = 1 QH state in such a model was first noted by Sondhi and Yang [15]. Our calculation builds on this work. We systematically classify all electron hopping operators and identify those that lead to QH states. This construction leads to QH states which are not in the Haldane-Halperin hierarchy [3]. We also show that any QH state is the groundstate of a Hamiltonian that we explicitly construct.

Chiral currents in one-dimensional fractional quantum Hall states

“…This gap opening is just the two wire version of the wire-construction of the quantum Hall effect 8 . One pair of modes is gapped out, and one pair remains gapless in analogy with the chiral edge states in the integer quantum Hall effect.…”

“…The states generated by this operator correspond to the Laughlin FQH phases as constructed by coupled wires 8 . The scaling dimension of O p is given by…”

“…They reported an observation of the chiral currents flowing around the ladder due to the effective magnetic field 1,7 . Motivated by this experimental ability, the coupled wire realization of the bosonic Laughling ν = 1/2 fractional quantum Hall effect (FQHE) introduced by Kane et al 8 , was recently suggested in two-leg ladders for strong on-site interactions 9,10 .…”

“…By FQH in 1D, we refer to the coupled wire construction introduced by Kane et al 8 . While the 2D limit of this construction corresponds to the robust quantum Hall liquid, we herein study the thin-stripe regime, in which the system is sensitive to small perturbations, and hence finding detectable signatures is more demanding.…”

Coupled-wire construction of chiral spin liquids

“…The CWC for topologically ordered quantum states of matter originates from the pioneering work by Kane and collaborators on deriving a scenario of FQH states from suitably chosen many-particle couplings in a set of coupled quantum wires [32]. Important preceding work has been on sliding Luttinger liquid phases, which already installed the notion of using the magnetic field as a way to favorably tune the scaling dimension of inter-wire couplings [33].…”