When a fractional quantum Hall (FQH) annulus is threaded by a time-varying flux, theory predicts transient multiple Aharonov-Bohm (MAB) oscillations of basic period qh/eqh/e (filling ν=1/q\nu=1/q), relaxed to h/eh/e periodicity by interedge motion of fractionally charged quasiparticles, with the measured Hall conductance quenched to zero or an integer once quasiparticle tunneling is fast (Thouless and Gefen, 1991; Gefen and Thouless, 1993). The quantitative theory of this effect still rests on phenomenological edge-state occupation assumptions, asymptotic limits and limiting-case solutions of a zero-temperature stochastic charge-transfer model, and restriction to ν=1/q\nu=1/q. The open problem is to construct the quantitative dynamical theory of flux periodicity and conductance quenching in multiply-connected FQH systems, covering the full crossover between quasiparticle- and electron-dominated interedge dynamics. A complete theory would naturally address: (i) a microscopic or rigorously controlled effective theory of edge states in the finite annular geometry with realistic boundary potentials, yielding interedge quasiparticle tunneling matrix elements and boundary-induced level displacements; (ii) a finite-temperature kinetic theory of the quasiparticle-to-electron crossover in the measured conductance and flux periodicity for arbitrary ratios of the three relevant time scales; (iii) a nonperturbative treatment of the stochastic charge-transfer model, including possible hysteretic I-V curves and global stability of steady states; (iv) the resonant modification of the AC response when the driving frequency approaches the edge-level spacing; and (v) falsifiable experimental parameter windows in which these signatures could be observed, motivating comparison with the modern quantum Hall interferometer program.

Background

Fractional quantum Hall effect (FQHE): a two-dimensional electron gas in a strong perpendicular magnetic field at low temperature forms incompressible liquid states at fractional Landau-level filling ν\nu (the ratio of electron density to magnetic flux density in units of the flux quantum). At ν=1/q\nu=1/q (qq odd) the Laughlin state — the prototypical incompressible FQH liquid — is gapped, and its excitations carry fractional charge e∗=e/qe^*=e/q and, for q>1q>1, anyonic exchange statistics: in two dimensions, exchanging two identical quasiparticles can multiply the many-body wave function by a phase not restricted to the bosonic or fermionic values. Multiply-connected geometry and the annulus setup: confine the FQH liquid to an annulus with inner radius R1R_1 and outer radius R2R_2, threaded by a solenoid whose flux Φ\Phi is varied continuously in time. Faraday's law produces an azimuthal electric field that drives a Hall current from one edge to the other; in the idealized Gefen-Thouless setup each edge connects through a small-capacitance tunnel junction to a reservoir, and an ammeter measures the current between the edges. Byers-Yang theorem and multiple Aharonov-Bohm (MAB) periods: Byers and Yang proved that equilibrium spectra and thermodynamic properties of an electronic system are periodic in flux with the electronic period h/eh/e, independent of interactions. Naively, an observed flux periodicity h/e∗h/e^* would demonstrate transport by fractionally charged quasiparticles; resistance-oscillation measurements in narrow FQH conductors were interpreted this way (Simmons et al., 1989), while a competing Coulomb-blockade explanation left the attribution controversial. Thouless and Gefen (1991) resolved the tension dynamically: as Φ\Phi varies, the ground state repeatedly switches among nearly degenerate Laughlin states related by a q-fold broken symmetry, so a driven system exhibits transient oscillations of basic period qh/eqh/e; relaxation to true h/eh/e periodicity occurs through quasiparticle motion between edges, with a decay time exponential in system size at zero temperature and linear in size at finite temperature. The quenching of the higher period, not its observation, is the signature of free fractionally charged quasiparticles. Time scales and quenching: Gefen and Thouless (1993) organized the dynamics around three time scales — the flux-pumping time (time to change Φ\Phi by one flux quantum), the interedge quasiparticle relaxation time τqp\tau_{qp} (controlled by the longitudinal conductivity σxx\sigma_{xx}, the edge confining potential U0U_0, and the annulus width), and the lead relaxation time (the RCRC time of the tunnel junctions). When quasiparticle tunneling is slow, each flux quantum transfers one quasiparticle across the annulus and the ammeter registers the fraction e∗/ee^*/e per cycle; when it is fast, the measured Hall conductance is quenched to zero or a nontrivial integer. Stochastic charge-transfer model: lacking a microscopic edge theory, the 1993 work models the system by stochastic rate equations for the charge QQ transferred between edges and reservoirs, in analogy with the 'orthodox' Coulomb-blockade theory of small-capacitance tunnel junctions — the standard rate-equation description in which the electrostatic charging energy makes charge transfer a discrete stochastic sequence of tunneling events; its limiting solutions indicate that these effects may be manifested in dc measurements as well, with physics related to that of the Coulomb blockade. Modern context: the descendant experimental program is the quantum Hall Fabry-Pérot interferometer (FPI), in which constrictions define an island in a FQH bar and quasiparticle interference versus flux probes anyonic statistics. The same core questions reappear there: the observed flux period is h/eh/e or h/e∗h/e^* depending on the competition between edge tunneling and bulk quasiparticle occupation (bulk-edge capacitive coupling), and the low-energy edge description is the chiral Luttinger liquid — the standard theory of a FQH edge as a unidirectionally propagating charged density wave, formally a chiral conformal field theory (an exactly solvable one-dimensional quantum field theory). Current FPI theory still parameterizes the electrostatics phenomenologically, and the dynamical crossover theory of the original problem has not been constructed.

Problem Statement

Construct the quantitative dynamical theory of flux periodicity and Hall-conductance quenching for a multiply-connected fractional quantum Hall system, in the lineage of Thouless and Gefen (Phys. Rev. Lett. 66, 806 (1991)) and Gefen and Thouless (Phys. Rev. B 47, 10423 (1993)). The reference system is a FQH annulus at filling ν=1/q\nu=1/q (qq an odd integer) threaded by a flux Φ(t)\Phi(t) varied at a controlled rate, with both edges coupled through small-capacitance tunnel junctions to reservoirs and the interedge current measured; an extension to general odd-denominator fractions is a natural direction but not a binding requirement of this problem. The open target is a dynamical theory of the measured conductance and flux periodicity that goes beyond the asymptotic-limit, zero-temperature stochastic charge-transfer treatment of the source works and covers the full crossover between quasiparticle-dominated and electron-dominated interedge dynamics, for arbitrary ratios of the relevant time scales (flux pumping; interedge quasiparticle relaxation; lead RCRC relaxation). Any answer must be consistent with the established limiting behaviors: transient multiple Aharonov-Bohm oscillations of basic periodicity qh/eqh/e (q>1q>1), relaxing towards h/eh/e periodicity through interedge motion of fractionally charged quasiparticles, with a decay time exponential in the system size at zero temperature and linear in the system size at finite temperature; quenching of the measured FQHE conductivity to zero or to a nontrivial integer when fractional charges become abundant and interedge transport is fast; dc manifestations whose physics is related to that of the Coulomb blockade; and the Byers-Yang theorem for equilibrium properties.

Beyond these limits, a complete dynamical theory would naturally address: (a) a microscopic, or rigorously controlled effective, many-body description of the edge states in the finite annular geometry with realistic boundary potentials, from which the interedge quasiparticle tunneling matrix elements and boundary-induced level displacements are derived rather than assumed; (b) a finite-temperature kinetic theory of the measured Hall conductance and flux periodicity across the full crossover between quasiparticle-dominated and electron-dominated dynamics, valid for arbitrary ratios of the three time scales; (c) a nonperturbative treatment of the stochastic charge-transfer model in the regime of strong quasiparticle-electron competition, including possible hysteretic I-V curves and the global stability of the steady states; (d) the modification of the predicted AC response when the external driving frequency approaches the edge-state level spacing (the resonant regime); and (e) falsifiable predictions of the device parameter windows (geometry, filling, temperature, contact resistance, drive rate and frequency) in which the multiple-period transient, the quenching of the measured Hall conductance, and the dc fractional-charge signatures can be observed. These directions are the contextual scope of the theory, not a set of separately required sub-goals. A resolution must nevertheless cover the crossover regime itself — not only its limiting cases — and remain consistent with the Byers-Yang theorem for equilibrium properties.

References

  1. Y. Gefen and D. J. Thouless, 'Detection of fractional charge and quenching of the quantum Hall effect', Phys. Rev. B 47, 10423 (1993). DOI: 10.1103/PhysRevB.47.10423; https://doi.org/10.1103/PhysRevB.47.10423.
  2. D. J. Thouless and Y. Gefen, 'Fractional quantum Hall effect and multiple Aharonov-Bohm periods', Phys. Rev. Lett. 66, 806 (1991). DOI: 10.1103/PhysRevLett.66.806; https://doi.org/10.1103/PhysRevLett.66.806.
  3. J. K. Jain, S. A. Kivelson, and D. J. Thouless, 'Proposed measurement of an effective flux quantum in the fractional quantum Hall effect', Phys. Rev. Lett. 71, 3003 (1993). DOI: 10.1103/PhysRevLett.71.3003; https://doi.org/10.1103/PhysRevLett.71.3003.
  4. I. Levkivskyi, J. Fröhlich, and E. Sukhorukov, 'Theory of fractional quantum Hall interferometers', Phys. Rev. B 86, 245105 (2012). DOI: 10.1103/PhysRevB.86.245105; arXiv:1005.5703; https://doi.org/10.1103/PhysRevB.86.245105.
  5. B. Rosenow and A. Stern, 'Flux superperiods and periodicity transitions in quantum Hall interferometers', Phys. Rev. Lett. 124, 106805 (2020). DOI: 10.1103/PhysRevLett.124.106805; arXiv:1907.06267; https://doi.org/10.1103/PhysRevLett.124.106805.
  6. D. E. Feldman and B. I. Halperin, 'Fractional charge and fractional statistics in the quantum Hall effects', Rep. Prog. Phys. 84, 076501 (2021). DOI: 10.1088/1361-6633/ac03aa; arXiv:2102.08998; https://doi.org/10.1088/1361-6633/ac03aa.
  7. J. Nakamura, S. Liang, G. C. Gardner, and M. J. Manfra, 'Impact of bulk-edge coupling on observation of anyonic braiding statistics in quantum Hall interferometers', Nat. Commun. 13, 344 (2022). DOI: 10.1038/s41467-022-27958-w; https://doi.org/10.1038/s41467-022-27958-w.
  8. S. Liang, J. Nakamura, G. C. Gardner, and M. J. Manfra, 'Single electron interference and capacitive edge mode coupling generates ϕ0/2 flux periodicity in Fabry-Pérot interferometers', Nat. Commun. 16, 7586 (2025). DOI: 10.1038/s41467-025-62797-5; arXiv:2502.00124; https://doi.org/10.1038/s41467-025-62797-5.
  9. R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. Mahalu, 'Direct observation of a fractional charge', Physica B 249-251, 395 (1998). DOI: 10.1016/S0921-4526(98)00139-2; arXiv:cond-mat/9707289; https://doi.org/10.1016/S0921-4526(98)00139-2.
  10. M. Carrega, L. Chirolli, S. Heun, and L. Sorba, 'Anyons in quantum Hall interferometry', Nat. Rev. Phys. 3, 698-711 (2021). DOI: 10.1038/s42254-021-00351-0; https://doi.org/10.1038/s42254-021-00351-0.
  11. N. Byers and C. N. Yang, 'Theoretical considerations concerning quantized magnetic flux in superconducting cylinders', Phys. Rev. Lett. 7, 46 (1961). DOI: 10.1103/PhysRevLett.7.46; https://doi.org/10.1103/PhysRevLett.7.46.
  12. J. A. Simmons, H. P. Wei, L. W. Engel, D. C. Tsui, and M. Shayegan, 'Resistance fluctuations in narrow AlGaAs/GaAs heterostructures: Direct evidence of fractional charge in the fractional quantum Hall effect', Phys. Rev. Lett. 63, 1731 (1989). DOI: 10.1103/PhysRevLett.63.1731; https://doi.org/10.1103/PhysRevLett.63.1731.