Charge2e/3Superconductivity and Topological Degeneracies without Localized Zero Modes in Bilayer Fractional Quantum Hall States

Abstract

It has been recently shown that non-Abelian defects with localized parafermion zero modes can arise in conventional Abelian fractional quantum Hall (FQH) states. Here we propose an alternate route to creating, manipulating, and measuring topologically protected degeneracies in bilayer FQH states coupled to superconductors, without the creation of localized parafermion zero modes. We focus mainly on electron-hole bilayers, with a ±1/3 Laughlin FQH state in each layer, with boundaries that are proximity-coupled to a superconductor. We show that the superconductor induces charge 2e/3 quasiparticle-pair condensation at each boundary of the FQH state, and that this leads to (1) topologically protected degeneracies that can be measured through charge sensing experiments and (2) a fractional charge 2e/3 AC Josephson effect. We demonstrate that an analog of non-Abelian braiding is possible, despite the absence of a localized zero mode. We discuss several practical advantages of this proposal over previous work, and also extensions to electron-electron bilayers at ν = 2/3 + 1/3 coupled to superconductivity, and to electron-hole bilayers with only interlayer tunneling.A profound feature of topologically ordered states of matter is the existence of topologically protected degeneracies [1]. These are degeneracies in the ground state spectrum of a given Hamiltonian for which no local operator can distinguish the topologically distinct states, thus rendering them robust to local perturbations. Topological degeneracies are predicted to occur when a topological phase is defined on a space with nontrivial topology, such as a torus [2,3], when the system hosts non-Abelian quasiparticle excitations [4][5][6][7], and in topological superconductors when Majorana zero modes are bound to cores of vortices or the ends of one-dimensional wires [8][9][10][11][12][13][14][15].It has recently been shown theoretically that a wide class of non-Abelian defects can be realized in conventional Abelian fractional quantum Hall (FQH) states, such as the Laughlin FQH states, by suitably controlling the tunneling processes along the edge modes. In particular, it was shown that in bilayer FQH states, domain walls between regions of interlayer and intra-layer tunneling can give rise to a novel type of non-Abelian defect that generalizes Majorana zero modes by yielding m states per pair of defects, where the integer m depends on the filling fraction ν of the FQH state [16]. These non-Abelian defects were subsequently shown to also arise as lattice defects in certain lattice spin models [17], extending [18]. They were then also shown to arise in FQH systems coupled to superconductors, at domain walls between regions of normal and Andreev backscattering at the edge [19][20][21], where they were referred to as parafermion [22], or fractionalized Majorana, zero modes. The general theory of these defects [23][24][25][26][27], a number of experimental proposals in FQH systems [28][29][30], other exotic phenomena at the boundary of FQH states and superconduct...

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