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arXiv:1902.07722 · Physical Review Letters

Superconducting Correlations Out of Repulsive Interactions on a Fractional Quantum Hall Edge

Jukka I. Väyrynen, Moshe Goldstein, Yuval Gefen

Published Jun 13, 2019• 5 citations
arXivHTMLPDF

Abstract

We consider a fractional quantum Hall bilayer system with an interface between quantum Hall states of filling fractions (ν_{top},ν_{bottom})=(1,1) and (1/3,2), motivated by a recent approach to engineering artificial edges [Y. Ronen et al., Nat. Phys. 14, 411 (2018)NPAHAX1745-247310.1038/s41567-017-0035-2]. We show that random tunneling and strong repulsive interactions within one of the layers will drive the system to a stable fixed point with two counterpropagating charge modes which have attractive interactions. As a result, slowly decaying correlations on the edge become predominantly superconducting. We discuss the resulting observable effects and derive general requirements for electron attraction in Abelian quantum Hall states. The broader interest in fractional quantum Hall edge with quasi-long-range superconducting order lies in the prospects of hosting exotic anyonic boundary excitations, which may serve as a platform for topological quantum computation.

DOI
10.1103/physrevlett.122.236802

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We consider a fractional quantum Hall bilayer system with an interface between quantum Hall states of filling fractions (ν_{top},ν_{bottom})=(1,1) and (1/3,2), motivated by a recent approach to engineering artificial edges [Y. Ronen et al., Nat. Phys. 14, 411 (2018)NPAHAX1745-247310.1038/s41567-017-0035-2]. We show that random tunneling and strong repulsive interactions within one of the layers will drive the system to a stable fixed point with two counterpropagating charge modes which have attractive interactions. As a result, slowly decaying correlations on the edge become predominantly superconducting. We discuss the resulting observable effects and derive general requirements for electron attraction in Abelian quantum Hall states. The broader interest in fractional quantum Hall edge with quasi-long-range superconducting order lies in the prospects of hosting exotic anyonic boundary excitations, which may serve as a platform for topological quantum computation.
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We study continuous quantum phase transitions that can occur in bilayer fractional quantum Hall (FQH) systems as the interlayer tunneling and interlayer repulsion are tuned. We introduce a slave-particle gauge theory description of a series of continuous transitions from the (ppq)(ppq)(ppq) Abelian bilayer states to a set of non-Abelian FQH states, which we dub orbifold FQH states, of which the Z4{Z}_{4}Z4​ parafermion (Read-Rezayi) state is a special case. This provides an example in which Z2{Z}_{2}Z2​ electron fractionalization leads to non-Abelian topological phases. The naive ``ideal'' wave functions and ideal Hamiltonians associated with these orbifold states do not in general correspond to incompressible phases but, instead, lie at a nearby critical point. We discuss this unusual situation from the perspective of the pattern-of-zeros/vertex algebra frameworks and discuss implications for the conceptual foundations of these approaches. Due to the proximity in the phase diagram of these non-Abelian states to the (ppq)(ppq)(ppq) bilayer states, they may be experimentally relevant, both as candidates for describing the plateaus in single-layer systems at filling fractions 8/38/38/3 and 12/512/512/5 and as a way to tune to non-Abelian states in double-layer or wide quantum wells.
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We study proximity coupling between a superconductor and counter-propagating gapless modes arising on the edges of Abelian fractional quantum Hall liquids with filling fraction ν=1/mν=1/mν=1/m (with mmm an odd integer). This setup can be utilized to create non-Abelian parafermion zero-modes if the coupling to the superconductor opens an energy gap in the counter-propagating modes. However, when the coupling to the superconductor is weak an energy gap is opened only in the presence of sufficiently strong attractive interactions between the edge modes, which do not commonly occur in solid state experimental realizations. We therefore investigate the possibility of obtaining a gapped phase by increasing the strength of the proximity coupling to the superconductor. To this end, we use an effective wire construction model for the quantum Hall liquid and employ renormalization group methods to obtain the phase diagram of the system. Surprisingly, at strong proximity coupling we find a gapped phase which is stabilized for sufficiently strong repulsive interactions in the bulk of the quantum Hall fluids. We furthermore identify a duality transformation that maps between the weak coupling and strong coupling regimes, and use it to show that the gapped phases in both regimes are continuously connected through an intermediate proximity coupling regime.
Recent theoretical insights into the possibility of non-Abelian phases in $\ensuremath{\nu}=2/3$ fractional quantum Hall states revived the interest in the numerical phase diagram of the problem. We investigate the effect of various kinds of two-body interlayer couplings on the (330) bilayer state and exactly solve the Hamiltonian for up to 14 electrons on sphere and torus geometries. We consider interlayer tunneling, short-ranged repulsive/attractive pseudopotential interactions, and Coulomb repulsion. We find a 6-fold ground-state degeneracy on the torus when the interlayer hollow-core interaction is dominant. To identify the topological nature of this phase we measure the orbital-cut entanglement spectrum, quasihole counting, topological entanglement entropy, and wave-function overlap. Comparing the numerical results to the theoretical predictions, we interpret this 6-fold ground-state degeneracy phase to be the non-Abelian bilayer Fibonacci state.
Multicomponent quantum Hall systems with internal degrees of freedom provide a fertile ground for the emergence of exotic quantum liquids. Here, we investigate the possibility of non-Abelian topological order in the half-filled fractional quantum Hall (FQH) bilayer system driven by the tunneling effect between two layers. By means of the state-of-the-art density-matrix renormalization group, we unveil ``fingerprint'' evidence of the non-Abelian Moore-Read Pfaffian state emerging in the intermediate-tunneling regime, including the ground-state degeneracy on the torus geometry and the topological entanglement spectroscopy (entanglement spectrum and topological entanglement entropy) on the spherical geometry, respectively. Remarkably, the phase transition from the previously identified Abelian (331) Halperin state to the non-Abelian Moore-Read Pfaffian state is determined to be continuous, which is signaled by the continuous evolution of the universal part of the entanglement spectrum, and discontinuities in the excitation gap and the derivative of the ground-state energy. Our results not only provide a ``proof-of-principle'' demonstration of realizing a non-Abelian state through coupling different degrees of freedom, but also open up a possibility in FQH bilayer systems for detecting different chiral $p\text{\ensuremath{-}}\text{wave}$ pairing states.