Imprint of topological degeneracy in quasi-one-dimensional fractional quantum Hall states

Abstract

We consider an annular superconductor-insulator-superconductor Josephson-junction, with the insulator being a double layer of electron and holes at Abelian fractional quantum Hall states of identical fillings. When the two superconductors gap out the edge modes, the system has a topological ground state degeneracy in the thermodynamic limit akin to the fractional quantum Hall degeneracy on a torus. In the quasi-one-dimensional limit, where the width of the insulator becomes small, the ground state energies are split. We discuss several implications of the topological degeneracy that survive the crossover to the quasi-one-dimensional limit. In particular, the Josephson effect shows a $2\pi d$-periodicity, where $d$ is the ground state degeneracy in the 2 dimensional limit. We find that at special values of the relative phase between the two superconductors there are protected crossing points in which the degeneracy is not completely lifted. These features occur also if the insulator is a time-reversal-invariant fractional topological insulator. We describe the latter using a construction based on coupled wires. Furthermore, when the superconductors are replaced by systems with an appropriate magnetic order that gap the edges via a spin-flipping backscattering, the Josephson effect is replaced by a spin Josephson effect.

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• General scatterings and electronic states in the quantum-wire network of moiré systems
• Unification of parton and coupled-wire approaches to quantum magnetism in two dimensions
• Generalized parafermions and non-local Josephson effect in multi-layer systems
• Microscopic Study of the Coupled-Wire Construction and Plausible Realization in Spin-Dependent Optical Lattices
• Fractional second-order topological insulator from a three-dimensional coupled-wires construction
• Fractional Spin Quantum Hall Effect in Weakly Coupled Spin Chain Arrays
• Bridging three-dimensional coupled-wire models and cellular topological states: Solvable models for topological and fracton orders
• Dimensional reduction of the Luttinger-Ward functional for spin-degenerate $D$-dimensional electron gases
• A Theorem on Extensive Spectral Degeneracy for Systems with Higher Symmetries in General Dimensions
• Dihedral twist liquid models from emergent Majorana fermions
• Manifestation of edge-bulk incompatibility in fractional quantum Hall platform
• Fractional chiral second-order topological insulator from a three-dimensional array of coupled wires
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders