Pairing in Luttinger Liquids and Quantum Hall States

Abstract

We study spinless electrons in a single channel quantum wire interacting through attractive interaction, and the quantum Hall states that may be constructed by an array of such wires. For a single wire the electrons may form two phases, the Luttinger liquid and the strongly paired phase. The Luttinger liquid is gapless to one- and two-electron excitations, while the strongly paired state is gapped to the former and gapless to the latter. In contrast to the case in which the wire is proximity-coupled to an external superconductor, for an isolated wire there is no separate phase of a topological, weakly paired, superconductor. Rather, this phase is adiabatically connected to the Luttinger liquid phase. The properties of the one dimensional topological superconductor emerge when the number of channels in the wire becomes large. The quantum Hall states that may be formed by an array of single-channel wires depend on the Landau level filling factors. For odd-denominator fillings $\nu =1/(2n+1)$, wires at the Luttinger phase form Laughlin states while wires in the strongly paired phase form bosonic fractional quantum Hall state of strongly-bound pairs at a filling of $1/(8n+4)$. The transition between the two is of the universality class of Ising transitions in three dimensions. For even-denominator fractions $\nu =1/2n$ the two single-wire phases translate into four quantum Hall states. Two of those states are bosonic fractional quantum Hall states of weakly- and strongly- bound pairs of electrons. The other two are non-Abelian quantum Hall states, which originate from coupling wires close to their critical point. One of these non-Abelian states is the Moore-Read state. The transition between all these states are of the universality class of Majorana transitions. We point out some of the properties that characterize the different phases and the phase transitions.

16 pages

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Cited by in corpus (57)

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• Pascal conductance series in ballistic one-dimensional LaAlO${}_3$/SrTiO${}_3$ channels
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• Zero-Magnetic Field Fractional Quantum States
• Topological and symmetry-enriched random quantum critical points
• From one-dimensional charge conserving superconductors to the gapless Haldane phase
• The sixteenfold way and the quantum Hall effect at half-integer filling factors
• Quantum Hall hierarchy from coupled wires
• Majorana Spin Liquids, Topology and Superconductivity in Ladders
• Rigorous results on topological superconductivity with particle number conservation
• Non-diagonal anisotropic quantum Hall states
• A model of chiral spin liquids with Abelian and non-Abelian topological phases
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• Coupled wire construction of a topological phase with chiral tricritical Ising edge modes
• From Topological Superconductivity to Quantum Hall States in Coupled Wires
• Coupled Wire Model of Z4 Orbifold Quantum Hall States
• Quantum field theory of nematic transitions in spin orbit coupled spin-1 polar bosons
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• PH-Pfaffian order in a translationally and rotationally invariant system
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• Quantum spin liquids bootstrapped from Ising criticality in Rydberg arrays
• One-dimensional extended Hubbard model with soft-core potential