Realizing the parity-based diagnostics of the non-Abelian $\nu=5/2$ quantum Hall state in realistic devices: quasiparticle-parity control, capacitance, stability, and fluctuation constraints

The even-denominator fractional quantum Hall state at filling factor $\nu=5/2$ is conjectured to host non-Abelian quasiparticles of charge $e/4$. Stern and Halperin (Phys. Rev. Lett. 96, 016802 (2006)) proposed two mesoscopic diagnostics whose outcome is controlled by the parity of the number $n_{\mathrm{is}}$ of localized $e/4$ quasiparticles in a $\nu=5/2$ island: in an open interferometer, the interference visibility of backscattered $e/4$ quasiparticles should vanish for odd $n_{\mathrm{is}}$ and reappear for even $n_{\mathrm{is}}$; in a closed island, the even-odd modulation of Coulomb-blockade peak spacings should be switched off for odd $n_{\mathrm{is}}$. The task is to realize and validate these parity-based diagnostics in realistic devices: controlled incrementation and knowledge of $n_{\mathrm{is}}$ and its parity despite disorder and density nonuniformities; determination of the effective island capacitance $C$ (coupled to edges, exterior bulk, and gates) needed for charge stability; satisfaction of the temperature and relaxation-time constraints $T\ll\delta_0$ and $\tau=C/G$ long compared to the measurement; suppression or characterization of temporal parity fluctuations; and characterization of the statistics with which quasiparticles enter the island under magnetic-field increments (on average one $e/4$ quasiparticle per $\Phi_0/10$). As of 2026, controlled coherent interference at even-denominator states has been achieved in bilayer-graphene Fabry-Perot interferometers, and gate-controlled antidots now allow controlled localization of enclosed anyons, but the non-Abelian, parity-dependent signature at $\nu=5/2$ itself has not been observed, so the experimental core of the problem remains open.

Background

The fractional quantum Hall (FQH) effect arises in a two-dimensional electron gas at low temperature in a strong perpendicular magnetic field, where kinetic energy is quenched into Landau levels and interactions produce incompressible liquids with fractionally charged excitations. At filling factor $\nu=5/2$ (two full Landau levels plus a half-filled second Landau level), the leading theoretical candidates — the Moore-Read Pfaffian state and its particle-hole conjugate, the anti-Pfaffian — are $p$-wave paired states of composite fermions (electrons bound to an even number of flux quanta). In the Moore-Read description (G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991)), the half-filled level forms a paired condensate whose vortices each bind a Majorana zero mode — a zero-energy, self-conjugate fermionic excitation, two of which pair into a single ordinary fermion — while the edge carries, besides the charged mode, a neutral Majorana mode; the vortices carry charge $e/4$ and obey non-Abelian (Ising-type) exchange statistics, meaning that exchanging two quasiparticles acts as a non-commuting rotation within a degenerate ground-state manifold rather than a mere phase. Establishing or ruling out this non-Abelian order is the central open question of the $5/2$ state and the physical basis for proposed topological qubits (S. Das Sarma, M. Freedman, C. Nayak, Phys. Rev. Lett. 94, 166802 (2005)).

Several experiments were proposed to detect the non-Abelian statistics. Bonderson, Kitaev, and Shtengel (Phys. Rev. Lett. 96, 016803 (2006)) analyzed a Fabry-Perot interferometer, a device in which edge quasiparticles tunnel between two quantum point contacts (short constrictions that backscatter edge states) and interfere; the interference phase jumps each time a localized quasiparticle enters the enclosed area. A key complication is that any realistic device contains an uncontrolled number of disorder-localized quasiparticles in its bulk. In their PRL 96, 016802 paper, Stern and Halperin turned this complication into a diagnostic: they considered a $\nu=5/2$ island containing $n_{\mathrm{is}}$ localized $e/4$ quasiparticles and showed that the interference of tunneling $e/4$ quasiparticles is coherent for even $n_{\mathrm{is}}$ but dephased (visibility vanishes) for odd $n_{\mathrm{is}}$, because for odd $n_{\mathrm{is}}$ the backscattering operator anticommutes with the Majorana operator of the interfering quasiparticle. Equivalently, in a closed island (weakly coupled to a reservoir so its electron number is quantized, exhibiting Coulomb-blockade conductance peaks whose voltage spacings measure the energy cost of adding one electron), the pairing of composite fermions produces an even-odd effect in the addition spectrum — alternating peak spacings associated with the energy $\delta_0$ of the lowest subgap edge state — that is present for even $n_{\mathrm{is}}$ and absent for odd $n_{\mathrm{is}}$. Here $\delta_0$ denotes the energy of the lowest state of the sub-gap edge branch (inversely proportional to the island perimeter for even $n_{\mathrm{is}}$, and zero for odd), $\Phi_0=h/e$ is the magnetic flux quantum, $C$ the effective island capacitance, $G$ the conductance coupling the island to the surrounding bulk, and $\tau=C/G$ the relaxation time of island charge fluctuations. The knobs for varying $n_{\mathrm{is}}$ are a side gate (changes island area at fixed filling), a small Coulomb-blockaded antidot (adds or removes $e/4$ quasiparticles one at a time), and the magnetic field (on average one $e/4$ quasiparticle enters per $\Phi_0/10$, with sample-dependent fluctuations).

Later theory sharpened and qualified these diagnostics: Bishara and Nayak (arXiv:0708.2704) showed that both the Pfaffian and the anti-Pfaffian exhibit the same qualitative even-odd effect, so the test probes non-Abelian order rather than a specific candidate state; Bonderson, Nayak, and Shtengel (Phys. Rev. B 81, 165308 (2010)) showed that Coulomb-blockade peak spacings distinguish different topological orders quite poorly ('Coulomb blockade doppelgangers'), so apparent even-odd structure must be discriminated from device electrostatics. The experiments that the proposal demands must therefore operate in regimes where these confounds are excluded, which is precisely why the proposal's own conditions — quasiparticle-parity control, capacitance determination, $T\ll\delta_0$ and long $\tau$, negligible parity-fluctuation rates, and characterized quasiparticle-entrance statistics — define the experimental problem.

Problem Statement

Can the parity-based diagnostics of the non-Abelian $\nu=5/2$ quantum Hall state proposed by Stern and Halperin be realized and validated in a realistic device? Concretely: construct a $\nu=5/2$ quantum Hall island — either an open Fabry-Perot interferometer formed by two quantum point contacts, or a closed, weakly coupled Coulomb-blockaded island — in which the number $n_{\mathrm{is}}$ of localized $e/4$ quasiparticles can be incremented in a controlled and known way, and determine whether the predicted parity dependence is present: (a) in the open system, the interference contribution to the backscattered current of $e/4$ quasiparticles oscillates with island area for even $n_{\mathrm{is}}$ and is dephased to zero visibility for odd $n_{\mathrm{is}}$; (b) in the closed system, the even-odd modulation of Coulomb-blockade peak spacings (period $2/n_0$ in island area rather than $1/n_0$, where $n_0$ is the density of electrons in the half-filled Landau level of the island) is present for even $n_{\mathrm{is}}$ and absent for odd $n_{\mathrm{is}}$. The two device options are alternatives, and the open-interferometer visibility switching (a) and the closed-island even-odd peak-spacing modulation (b) are independent signatures of the same parity effect. A valid demonstration must simultaneously satisfy and verify the device conditions the source identifies as prerequisites: the effective island capacitance $C$ (electrostatically coupled to the edges, the exterior bulk, and gates) must be known and small enough that thermal charge fluctuations are far below $e/4$ (open) or $e$ (closed); the temperature must satisfy $T\ll\delta_0$ with $\delta_0$ the lowest sub-gap edge-state energy; the parity must be stable over the measurement time, i.e. the relaxation time $\tau=C/G$ and the tunneling/thermal-activation rates of parity changes must be slow compared to the measurement bandwidth; and the statistics of quasiparticle entrance events under magnetic-field sweeps — their positions, field values, and fluctuations about the $\Phi_0/10$ average, which depend on sample-specific disorder and island shape — must be characterized rather than assumed. The scientific objective these results jointly serve is to establish, or rule out, the non-Abelian (Ising/Moore-Read-type) order of the $\nu=5/2$ state through its defining parity signature, while excluding Abelian and electrostatic mimics such as $e/2$-quasiparticle interference and Coulomb-blockade doppelgangers.

References

1. A. Stern and B. I. Halperin, "Proposed Experiments to Probe the Non-Abelian ν=5/2 Quantum Hall State", Phys. Rev. Lett. 96, 016802 (2006). DOI: 10.1103/PhysRevLett.96.016802, arXiv:cond-mat/0508447, https://doi.org/10.1103/PhysRevLett.96.016802
2. G. Moore and N. Read, "Nonabelions in the fractional quantum hall effect", Nucl. Phys. B 360, 362 (1991). DOI: 10.1016/0550-3213(91)90407-O, https://doi.org/10.1016/0550-3213(91)90407-O
3. S. Das Sarma, M. Freedman, and C. Nayak, "Topologically Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State", Phys. Rev. Lett. 94, 166802 (2005). DOI: 10.1103/PhysRevLett.94.166802, https://doi.org/10.1103/PhysRevLett.94.166802
4. P. Bonderson, A. Kitaev, and K. 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, https://doi.org/10.1103/PhysRevLett.96.016803
5. W. Bishara and C. Nayak, "Edge States and Interferometers in the Pfaffian and anti-Pfaffian States", arXiv:0708.2704 (2007). arXiv: 0708.2704, https://arxiv.org/abs/0708.2704
6. P. Bonderson, C. Nayak, and K. Shtengel, "Coulomb blockade doppelgängers in quantum Hall states", Phys. Rev. B 81, 165308 (2010). DOI: 10.1103/PhysRevB.81.165308, https://doi.org/10.1103/PhysRevB.81.165308
7. R. Ilan, B. Rosenow, and A. Stern, "Signatures of non-Abelian statistics in non-linear coulomb blockaded transport", arXiv:1005.4772 (2010). arXiv: 1005.4772, https://arxiv.org/abs/1005.4772
8. R. L. Willett, L. N. Pfeiffer, and K. W. West, "Measurement of filling factor 5/2 quasiparticle interference with observation of charge e/4 and e/2 period oscillations", Proc. Natl. Acad. Sci. USA 106, 8853 (2009). DOI: 10.1073/pnas.0812599106, https://doi.org/10.1073/pnas.0812599106
9. R. L. Willett, L. N. Pfeiffer, and K. W. West, "Alternation and interchange of e/4 and e/2 period interference oscillations consistent with filling factor 5/2 non-Abelian quasiparticles", Phys. Rev. B 82, 205301 (2010). DOI: 10.1103/PhysRevB.82.205301, https://doi.org/10.1103/PhysRevB.82.205301
10. R. L. Willett, C. Nayak, K. Shtengel, L. N. Pfeiffer, and K. W. West, "Magnetic-Field-Tuned Aharonov-Bohm Oscillations and Evidence for Non-Abelian Anyons at ν=5/2", Phys. Rev. Lett. 111, 186401 (2013). DOI: 10.1103/PhysRevLett.111.186401, https://doi.org/10.1103/PhysRevLett.111.186401
11. R. L. Willett, K. Shtengel, C. Nayak, L. N. Pfeiffer, Y. J. Chung, M. L. Peabody, K. W. Baldwin, and K. W. West, "Interference measurements of non-Abelian e/4 & Abelian e/2 quasiparticle braiding", arXiv:1905.10248 (2019). arXiv: 1905.10248, https://arxiv.org/abs/1905.10248
12. G. Ben-Shach, C. R. Laumann, I. Neder, A. Yacoby, and B. I. Halperin, "Detecting Non-Abelian Anyons by Charging Spectroscopy", arXiv:1212.1163 (2012). arXiv: 1212.1163, https://arxiv.org/abs/1212.1163
13. M. Yutushui, A. Stern, and D. F. Mross, "Identifying the ν=5/2 Topological Order through Charge Transport Measurements", Phys. Rev. Lett. 128, 016401 (2022). DOI: 10.1103/PhysRevLett.128.016401, https://doi.org/10.1103/PhysRevLett.128.016401
14. M. Carrega, L. Chirolli, S. Heun, and L. Sorba, "Anyons in quantum Hall interferometry", Nat. Rev. Phys. 3, 698 (2021). DOI: 10.1038/s42254-021-00351-0, https://doi.org/10.1038/s42254-021-00351-0
15. A. Demir, N. Staley, S. Aronson, S. Tomarken, K. West, K. Baldwin, L. Pfeiffer, and R. Ashoori, "Correlated Double-Electron Additions at the Edge of a Two-Dimensional Electronic System", Phys. Rev. Lett. 126, 256802 (2021). DOI: 10.1103/PhysRevLett.126.256802, https://doi.org/10.1103/PhysRevLett.126.256802
16. J. Kim, H. Dev, R. Kumar, A. Ilin, A. Haug, V. Bhardwaj, C. Hong, K. Watanabe, T. Taniguchi, A. Stern, and Y. Ronen, "Aharonov-Bohm interference and statistical phase-jump evolution in fractional quantum Hall states in bilayer graphene", Nat. Nanotechnol. 19, 1619 (2024). DOI: 10.1038/s41565-024-01751-w, https://doi.org/10.1038/s41565-024-01751-w
17. T. Werkmeister, J. R. Ehrets, M. E. Wesson, D. Najafabadi, K. Watanabe, T. Taniguchi, B. Halperin, A. Yacoby, and P. Kim, "Anyon braiding and telegraph noise in a graphene interferometer", Science 388, 730 (2025). DOI: 10.1126/science.adp5015, https://doi.org/10.1126/science.adp5015
18. J. Kim, H. Dev, A. Shaer, R. Kumar, A. Ilin, A. Haug, S. Iskoz, K. Watanabe, T. Taniguchi, D. F. Mross, A. Stern, and Y. Ronen, "Aharonov-Bohm interference in even-denominator fractional quantum Hall states", Nature 649, 323-329 (2026). DOI: 10.1038/s41586-025-09891-2, https://doi.org/10.1038/s41586-025-09891-2
19. J. Kim, A. Shaer, R. Kumar, A. Ilin, K. Watanabe, T. Taniguchi, A. Stern, D. F. Mross, and Y. Ronen, "Selective braiding of different anyons in the even-denominator fractional quantum Hall effect", arXiv:2603.11162 (2026). arXiv: 2603.11162, https://arxiv.org/abs/2603.11162
20. C. E. Henzinger, J. R. Ehrets, R. Fushio, J. Dong, T. Werkmeister, M. E. Wesson, K. Watanabe, T. Taniguchi, A. Vishwanath, B. I. Halperin, A. Yacoby, and P. Kim, "Controlled localization of anyons in a graphene quantum Hall interferometer", arXiv:2603.11182 (2026). arXiv: 2603.11182, https://arxiv.org/abs/2603.11182
21. E. Sela and M. Banerjee, "Entropy of Non-Abelian Anyons from Slow Quasiparticle Dynamics in Quantum Hall Interferometers", arXiv:2607.02188 (2026). arXiv: 2607.02188, https://arxiv.org/abs/2607.02188