Scanning-gate Fabry–Pérot interferometry on a quantum point contact (QPC) in the zero-bias-anomaly regime (B. Brun et al., Phys. Rev. Lett. 116, 136801 (2016)) revealed an abrupt shift of the interference fringes by roughly π at zero source–drain bias and two phase jumps around the conductance peak — signatures attributed to scattering off a Kondo-correlated, spontaneously localized state. However, existing Kondo-phase theory covers only equilibrium quantum dots with externally defined barriers: the scattering phase of a Kondo effect that develops on an interaction-induced, self-consistently localized QPC state has never been calculated, no prediction exists for the Kondo phase shift out of equilibrium, the decoherence of Kondo correlations at finite bias is unmodeled, and the mapping between the measured reflection phase of an asymmetric Fabry–Pérot cavity and the canonical transmission phase of a Kondo scatterer is understood only at the single-particle level. The problem is to construct this missing microscopic nonequilibrium theory — spectral function and scattering phase versus gate voltage, bias, and temperature — and to use it to quantitatively reproduce the measured π-scale zero-bias shift and double phase lapse, thereby linking QPC device electrostatics, the observed Kondo temperature, and the phase behavior in one predictive framework.

Background

A quantum point contact (QPC) is a short, gate-defined constriction connecting two reservoirs of a two-dimensional electron gas; its conductance is quantized in steps of 2e2/h2e^2/h because each occupied one-dimensional subband transmits one conductance quantum. Just below the first plateau, transport shows the long-standing '0.7 anomaly': a shoulder near 0.7×2e2/h0.7\times 2e^2/h that weakens with increasing temperature and splits in a parallel magnetic field, in a way reminiscent of the Kondo effect — the many-body screening of a localized spin by conduction electrons that sets in below a Kondo temperature TKT_K. A closely related signature is the zero-bias anomaly (ZBA), a peak in the differential conductance at zero source–drain bias. Unlike an artificial quantum dot, whose tunnel barriers and bound levels are defined by gates, a QPC near pinch-off has such a low electron density that Coulomb interactions are expected to drive a spontaneous, self-consistent charge localization, producing a quasi-bound state — a magnetic impurity built by the electrons themselves rather than by device architecture. Local-moment formation of this kind was predicted by density-matrix-renormalization-group and related calculations (Hirose, Meir and Wingreen, Phys. Rev. Lett. 90, 026804 (2003); Rejec and Meir, Nature 442, 900 (2006)), and experiments have observed odd–even localized charge states in QPCs (Iqbal et al., Nature 501, 79 (2013); Zhang, Wu and Chang, arXiv:1211.2269). If an odd number of electrons localizes, a Kondo effect should develop at sufficiently low temperature.

For quantum dots in the Kondo regime the equilibrium transmission phase is well understood. The Friedel sum rule — a relation between the scattering phase shift of conduction electrons and the occupation of the impurity orbital — pins the phase at π/2\pi/2 across a Kondo valley, and calculations for the Anderson model (a single level with on-site Coulomb repulsion coupled to two leads) predict, as a function of bias or gate voltage at equilibrium, a characteristic 'Kondo double phase lapse': two sharp jumps by π\pi bracketing the narrow Kondo resonance that sits at the Fermi level (Gerland, von Delft, Costi and Oreg, Phys. Rev. Lett. 84, 3710 (2000)). Later scattering-theory work confirmed and extended the equilibrium results (Carmi, Oreg, Berkooz and Goldhaber-Gordon, Phys. Rev. B 86, 115129 (2012)). Nonequilibrium Kondo theory, by contrast, has been developed mainly for conductance and spectral functions (e.g. Pletyukhov and Schoeller, Phys. Rev. Lett. 108, 260601 (2012)); the scattering phase at finite bias has not been computed. A prior transmission-phase measurement on a QPC (Kobayashi et al., arXiv:1306.6689) found no strong deviation from single-particle behavior.

Brun et al. (2016) performed the first phase-sensitive measurement of the QPC zero-bias anomaly. They used scanning gate microscopy (SGM), in which the charged tip of a scanning probe weakly back-scatters electrons; the tip and the QPC together form a Fabry–Pérot interferometer (an electronic analog of an optical cavity), so the conductance oscillates with tip position and the fringe phase tracks the phase of the QPC reflection amplitude. Whenever the ZBA was present, the fringes shifted abruptly by half a period (a phase shift of order π\pi) within the bias window of the anomaly, and two phase jumps appeared around the conductance peak — behavior that noninteracting models cannot reproduce and that the authors attributed to the scattering phase of a Kondo system. The theoretical situation, however, remained unsatisfactory in two specific ways. First, the measured quantity is a reflection phase in an asymmetric interferometer: the supplement's single-particle analysis shows it lies between zero and twice the canonical transmission phase and depends on the asymmetry of the effective barriers, so relating it to the underlying many-body phase shift requires a systematic treatment that does not exist. Second, all existing Kondo-phase predictions concern equilibrium dots with externally defined barriers. The paper states explicitly that the specific properties of a Kondo effect developing on a self-consistently localized state 'have not been calculated yet', and that 'no theoretical prediction of the Kondo phase shift out of equilibrium exists at the moment', with decoherence of the Kondo correlations at finite bias an additional unsettled ingredient. Reviews of the 0.7 anomaly continue to list its microscopic origin, including the status of the Kondo interpretation, as an open question (Das and Green, arXiv:1806.06239).

Problem Statement

Develop a controlled, microscopic, nonequilibrium theory of the scattering phase of a Kondo-correlated, self-consistently localized state in a quantum point contact, and use it to give a quantitative account of the phase-sensitive measurements of the QPC zero-bias anomaly (B. Brun et al., Phys. Rev. Lett. 116, 136801 (2016)). The theory should, as jointly supporting parts of this single objective: (1) compute, for a realistic QPC potential in which the localized state emerges self-consistently from interactions rather than from externally defined barriers, the many-body spectral function and scattering phase as functions of gate voltage, temperature, and source–drain bias; (2) treat the Kondo correlations out of equilibrium, including their decoherence at finite bias, so as to predict the measured bias-dependent phase behavior — the abrupt reflection-phase shift of order π\pi at zero bias and the double phase lapse around the conductance peak; and (3) establish systematically how a reflection-phase measurement in an asymmetric Fabry–Pérot (scanning-gate) geometry maps onto the canonical transmission phase of a Kondo scatterer, for controlled barrier asymmetry and weak tip back-scattering. A complete answer connects device electrostatics, the measured Kondo temperature TKT_K, and the measured phase behavior in one quantitative and predictive framework: it must reproduce the two named features — the abrupt reflection-phase shift of order π\pi at zero bias and the double phase lapse around the conductance peak — and derive how the spectral function and scattering phase depend on gate voltage, temperature, and bias. Because the source reports these as observed features rather than as a tabulated dataset with fixed device parameters, the completion boundary is agreement with those features and with the derived dependence, not curve-level agreement with an unstated tolerance. Treatments that replace the self-consistent localized state by a static artificial-dot model, or that cover only the equilibrium phase, do not answer the question as posed.

References

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