In the Moore–Read (Pfaffian) candidate description of the fractional quantum Hall state at filling factor ν=5/2\nu=5/2, each charge-e/4e/4 quasihole carries the Ising-anyon label σ\sigma, and an even number of them can fuse into either of two topologically distinct combined states, I\mathbb{I} or ψ\psi. Bonderson, Kitaev, and Shtengel proposed a two-point-contact interferometer with a central antidot (a depleted, gate-defined island hosting a controlled number nn of quasiholes) in which the predicted even–odd interference pattern — no interference for odd nn, a phase that alternates with the fusion channel for even nn — would reveal non-Abelian statistics, provided quasiholes added to the antidot reproducibly occupy the same fusion channel. They left open the quantitative determination of the energy splitting between the two fusion channels of an even number of quasiholes localized on the same antidot: only a naive scaling ∼e2/R\sim e^2/R with unknown sign and prefactor was available, because the Moore–Read wave functions are not exact ground states of any realistic interaction. Later work has fixed the sign and order of magnitude of the splitting for idealized small quasihole clusters in finite systems and shown that the preferred channel is a non-universal, interaction- and screening-dependent quantity, but no quantitative, device-faithful determination for the antidot geometry exists, and no experiment has yet observed the predicted even-nn or 4Φ04\Phi_0 fusion-channel signature. The problem — compute the magnitude, sign, and device-parameter dependence of this splitting and establish whether adiabatic charging reproducibly selects a fusion channel — remains open.

Background

The fractional quantum Hall effect (FQHE) arises in a two-dimensional electron gas in a strong perpendicular magnetic field, where electron–electron interactions produce incompressible liquid states with fractionalized quasiparticles. At filling factor ν=5/2\nu=5/2 (half-filling of the second Landau level, i.e. the second group of highly degenerate single-electron orbitals), the leading candidate ground state is the Moore–Read (MR) or Pfaffian state: a paired composite-fermion condensate whose elementary charge-e/4e/4 quasihole excitations are non-Abelian anyons. Non-Abelian here means that exchanging two identical quasiparticles acts as a unitary matrix on a multi-dimensional degenerate ground-state subspace rather than as a mere phase.

The anyon content of the MR state is described by the Ising anyon model fused with an Abelian charge sector. The Ising model has three particle types, I\mathbb{I} (vacuum), ψ\psi (a neutral fermion), and σ\sigma, with fusion rules σ×σ=I+ψ\sigma\times\sigma=\mathbb{I}+\psi: combining two σ\sigma particles yields a direct sum of two possible total charges, which is the origin of the topological degeneracy. Each MR quasihole carries charge e/4e/4 together with the Ising label σ\sigma, so a composite of nn quasiholes has Ising label Qn=σQ_n=\sigma when nn is odd, but Qn=IQ_n=\mathbb{I} or ψ\psi when nn is even. For quasiholes far apart the two even-nn fusion channels are exactly degenerate — this is topological protection. The question is whether this degeneracy survives when the quasiholes are forced to overlap by confining them on the same small region.

Bonderson, Kitaev, and Shtengel (Phys. Rev. Lett. 96, 016803 (2006)) proposed a two-point-contact interferometer: a Hall bar with two front gates forming constrictions (quantum point contacts) on either side of a central antidot, a gate-depleted island in the bulk whose potential is tuned so that a controllable number nn of quasiholes is localized on it. Edge quasiparticles tunnel between opposite edges at the two contacts (weak-backscattering regime, i.e. small tunneling amplitudes t1,t2t_1,t_2, so that the dominant process involves a single charge-e/4e/4 quasihole), and the two paths enclose the antidot. Winding one quasihole around the nn quasiholes on the antidot applies the monodromy operator MnM_n; the interference term in the longitudinal resistance is ⟨Ψ∣Mn∣Ψ⟩\langle\Psi|M_n|\Psi\rangle. Ising-model braiding rules give: for odd nn, the interference vanishes; for even nn, the interference term is ∝(−1)Nψ\propto(-1)^{N_\psi}, where Nψ=1N_\psi=1 if the nn confined quasiholes fuse to ψ\psi and 00 if they fuse to I\mathbb{I}. Sweeping the magnetic field BB at fixed filling should then show Aharonov–Bohm oscillations of period 4Φ04\Phi_0 (Φ0=h/e\Phi_0=h/e being the flux quantum) for even nn and none for odd nn — the even–odd effect, a clean signature of non-Abelian statistics. Crucially, when nn is even the electrostatic environment effectively measures the fusion channel, collapsing the antidot state into either I\mathbb{I} or ψ\psi.

For two quasiholes pinned at the same location, the paper writes explicit variational wave functions in the two channels, Ψ2qh,I=∏jzj ΨGS\Psi_{2qh,\mathbb{I}}=\prod_j z_j\,\Psi_{GS} and Ψ2qh,ψ=∏jzj A[(z1−z2)−1(z3z4)−1⋯ ]Ψpair\Psi_{2qh,\psi}=\prod_j z_j\,\mathcal A[(z_1-z_2)^{-1}(z_3z_4)^{-1}\cdots]\Psi_{\mathrm{pair}}, built on the MR ground state ΨGS\Psi_{GS} (an antisymmetrized product of paired electron coordinates zjz_j, in units of the magnetic length ℓ0=ℏ/eB\ell_0=\sqrt{\hbar/eB}). The authors state that using these as variational functions to estimate the energy difference under realistic electron–electron interactions 'appears hopeless', because ΨGS\Psi_{GS} is not the exact ground state of any realistic interaction (it is exact only for an unphysical threebody Hamiltonian); they call this an open problem, note that a naive argument suggests a splitting of order e2/Re^2/R (with ee the electron charge and RR the antidot radius, and e2/ℓ0e^2/\ell_0 the natural Coulomb energy scale) but with an unknown and possibly small prefactor, and point out the experimental stakes: if adiabatic charging of the antidot reproducibly leaves the system in the same energetically preferred channel (they guess 'probably I\mathbb{I}'), the resistance cycles through four values as even nn increases; if the channel is chosen randomly at each charging event, the even-nn periodicity is washed out and the predicted 4Φ04\Phi_0 test fails. Related acknowledged open issues are the smaller excitation gap at ν=5/2\nu=5/2 compared to Laughlin states, stray quasiparticles trapped by disorder (which change the total parity inside the interferometer), and finite-temperature/finite-coherence-length degradation of the parity-dependent signatures.

Subsequent literature supplies partial answers that frame the open core. Baraban, Zikos, Bonesteel, and Simon (Phys. Rev. Lett. 103, 076801 (2009)) used Monte Carlo evaluation of MR trial states with the first-Landau-level Coulomb interaction: when two of four quasiholes are brought within one magnetic length, the ψ\psi-channel state is lower in energy, with a splitting of order 0.01 e2/ℓ00.01\,e^2/\ell_0 (an explicitly acknowledged upper bound, since mixing with states above the many-body gap is neglected), and the splitting between the putatively degenerate doublet decays exponentially with quasihole separation (decay length ξE≈2.3 ℓ0\xi_E\approx2.3\,\ell_0). Storni and Morf (Phys. Rev. B 83, 195306 (2011)) confirmed the order of magnitude by exact diagonalization (direct diagonalization of the many-body Hamiltonian in a finite Hilbert space) of small systems with pinning potentials interpolating between the MR model interaction and Coulomb, but found the doublet mixes with higher states and stated that further investigation is needed. More recent numerics — studies of clustered MR quasiholes using Jack polynomials (a family of exact polynomial many-body wave functions that serve as a convenient numerical basis for fractional quantum Hall trial states) and exact-diagonalization methods, published in 2025, and infinite-cylinder DMRG (density-matrix renormalization group, a variational method optimized for quasi-one-dimensional geometries such as long cylinders) applied to the gate-screened anti-Pfaffian (the particle–hole conjugate of the Pfaffian, an equally weighted candidate for the ν=5/2\nu=5/2 ground state) by Wang and Zaletel (2026) — show that realistic two-body interactions generically lift the I\mathbb{I}/ψ\psi degeneracy of clustered quasiholes and that the preferred fusion channel depends on the interaction range and on the distance to screening gates (with the ψ\psi channel favored on the hole side and at short gate distance in the anti-Pfaffian, and the non-ψ\psi channel taking over at larger gate distance on the particle side). None of these computations models the actual antidot geometry (a confined island embedded in an interferometer, with a specific confinement potential, well width, Landau-level mixing, and screening), and the charging/selection dynamics has not been treated.

Problem Statement

Determine, quantitatively and for a realistic description of the ν=5/2\nu=5/2 fractional quantum Hall state, the energy splitting between the two combined Ising fusion channels — I\mathbb{I} versus ψ\psi — of an even number of charge-e/4e/4 quasiholes localized together on the same antidot (a gate-confined depleted island of radius RR embedded in a two-point-contact interferometer). Specifically: (i) the magnitude and sign of this splitting as functions of the device and interaction parameters — antidot radius and confinement-potential profile, electron–electron interaction with well width, Landau-level mixing, and screening by nearby gates — including whether the naive e2/Re^2/R scaling holds and what its prefactor is, for the Pfaffian and the anti-Pfaffian, the two leading candidate ground states of ν=5/2\nu=5/2, reporting how the sign and magnitude of the splitting depend on which of the two is assumed; and (ii) whether, in consequence, adiabatic charging of the antidot reproducibly deposits each added quasihole pair into the same energetically preferred fusion channel — so that the interferometer resistance cycles through the four predicted values as the even quasihole number nn increases and shows the 4Φ04\Phi_0 Aharonov–Bohm period at fixed filling — or whether the channel is instead selected essentially at random at each charging event, washing out the even-nn periodicity. The splitting is sought in the antidot geometry with finite-size, above-gap-state-mixing, finite-temperature, and stray-quasiparticle effects taken into account; together, (i) and (ii) determine whether the antidot-interferometry proposal is a reliable, repeatable detector of non-Abelian statistics at ν=5/2\nu=5/2.

References

  1. Parsa Bonderson, Alexei Kitaev, Kirill Shtengel, 'Detecting Non-Abelian Statistics in the ν = 5/2 Fractional Quantum Hall State', Phys. Rev. Lett. 96, 016803 (2006). DOI: 10.1103/PhysRevLett.96.016803; arXiv:cond-mat/0508616; https://arxiv.org/abs/cond-mat/0508616
  2. M. Baraban, G. Zikos, N. Bonesteel, S. H. Simon, 'Numerical Analysis of Quasiholes of the Moore-Read Wave Function', Phys. Rev. Lett. 103, 076801 (2009). DOI: 10.1103/PhysRevLett.103.076801; arXiv:0901.3502; https://arxiv.org/abs/0901.3502
  3. M. Storni, R. H. Morf, 'Localized quasiholes and the Majorana fermion in fractional quantum Hall state at ν = 5/2 via direct diagonalization', Phys. Rev. B 83, 195306 (2011). DOI: 10.1103/PhysRevB.83.195306; arXiv:1101.5290; https://arxiv.org/abs/1101.5290
  4. Ha Quang Trung, Qianhui Xu, Bo Yang, 'Long-range Entanglement and Role of Realistic Interaction in Braiding of Non-Abelian Quasiholes in Fractional Quantum Hall Phases', Phys. Rev. B 112, 205101 (2025). DOI: 10.1103/c4ck-8rcx; arXiv:2503.13380; https://arxiv.org/abs/2503.13380
  5. Qianhui Xu, Guangyue Ji, Yuzhu Wang, Ha Quang Trung, Bo Yang, 'Dynamics of Anyon Clusters in Fractional Quantum Hall Fluids', Phys. Rev. B 112, 235112 (2025). DOI: 10.1103/vgz6-z98r; arXiv:2505.20257; https://arxiv.org/abs/2505.20257
  6. Taige Wang, Michael P. Zaletel, 'Anyon molecules in fractional quantum Hall states', arXiv:2604.09798 (2026). DOI: 10.48550/arXiv.2604.09798; https://arxiv.org/abs/2604.09798
  7. R. L. Willett, K. Shtengel, C. Nayak, L. N. Pfeiffer, Y. J. Chung, M. L. Peabody, K. W. Baldwin, K. W. West, 'Interference measurements of non-Abelian e/4 & Abelian e/2 quasiparticle braiding', Phys. Rev. X 13, 011028 (2023). DOI: 10.1103/PhysRevX.13.011028; arXiv:1905.10248; https://arxiv.org/abs/1905.10248
  8. Jehyun Kim, Himanshu Dev, Amit Shaer, Ravi Kumar, Alexey Ilin, André Haug, Shelly Iskoz, Kenji Watanabe, Takashi Taniguchi, David F. Mross, Ady Stern, Yuval Ronen, 'Aharonov-Bohm Interference in Even-Denominator Fractional Quantum Hall States', Nature 649, 323-329 (2026). DOI: 10.1038/s41586-025-09891-2; arXiv:2412.19886; https://arxiv.org/abs/2412.19886
  9. Jehyun Kim, Amit Shaer, Ravi Kumar, Alexey Ilin, Kenji Watanabe, Takashi Taniguchi, Ady Stern, David F. Mross, Yuval Ronen, 'Selective braiding of different anyons in the even-denominator fractional quantum Hall effect', arXiv:2603.11162 (2026). DOI: 10.48550/arXiv.2603.11162; https://arxiv.org/abs/2603.11162
  10. Christina E. Henzinger, James R. Ehrets, Rikuto Fushio, Junkai Dong, Thomas Werkmeister, Marie E. Wesson, Kenji Watanabe, Takashi Taniguchi, Ashvin Vishwanath, Bertrand I. Halperin, Amir Yacoby, Philip Kim, 'Controlled localization of anyons in a graphene quantum Hall interferometer', arXiv:2603.11182 (2026). DOI: 10.48550/arXiv.2603.11182; https://arxiv.org/abs/2603.11182
  11. Mario Di Luca, Emily Hajigeorgiou, Ning Ma, Alexandra Waldherr, Kenji Watanabe, Takashi Taniguchi, Mitali Banerjee, 'Trapping e/4 quasiparticles in bilayer graphene', arXiv:2608.27444 (2026). DOI: 10.48550/arXiv.2608.27444; https://arxiv.org/abs/2608.27444
  12. Roni Ilan, Bernd Rosenow, Ady Stern, 'Signatures of Non-Abelian Statistics in Nonlinear Coulomb Blockade Transport', Phys. Rev. Lett. 106, 136801 (2011). DOI: 10.1103/PhysRevLett.106.136801; arXiv:1005.4772; https://arxiv.org/abs/1005.4772
  13. Parsa Bonderson, Chetan Nayak, Kirill Shtengel, 'Coulomb Blockade Doppelgängers in Quantum Hall States', Phys. Rev. B 81, 165308 (2010). DOI: 10.1103/PhysRevB.81.165308; arXiv:0909.1056; https://arxiv.org/abs/0909.1056