Theoretical Analysis of the Transmission Phase Shift of a Quantum Dot in the Presence of Kondo Correlations

We study the effects of Kondo correlations on the transmission phase shift of a quantum dot coupled to two leads in comparison with the experimental determinations made by Aharonov-Bohm (AB) quantum interferometry. We propose here a theoretical interpretation of these results based on scattering theory combined with Bethe ansatz calculations. We show that there is a factor of 2 difference between the phase of the $S$-matrix responsible for the shift in the AB oscillations and the one controlling the conductance. Quantitative agreement is obtained with experimental results for two different values of the coupling to the leads.

Figure 1

Experimental conductance $G_{\exp }(V_G)$ and phase shift $\phi (V_G)$ as a function of $V_G$ (values taken from Ref. 4; cf. text). Comparison is made with the curve $G(V_G)={sin}^2[\phi (V_G)/2]$.

Figure 2

(a) Theoretical results from the BA calculations for the occupation number $n_0$ as a function of the normalized energy $(-{\epsilon }_0/U+1/2)$ at different values of $\Gamma /U$. Note that $n_0=1$ at the symmetric limit ${\epsilon }_0=-U/2$ and the existence of a plateau in its vicinity when $\Gamma /U\le 0.25$. (b) The same quantity as a function of the renormalized energy ${\epsilon }^{*}/\Gamma$ at different values of $\Gamma /U$.

Figure 3

Fit of the experimental data for the $V_G$ dependence of the phase shift with the BA results for $\delta =\pi n_0$ as a function of $-{\epsilon }_0^{*}/\Gamma$. Making use of the electron-hole symmetry, experimental results both below and above the symmetric limit are reported using the same scales. They are represented by triangles pointing down and up, respectively, in the unitary limit, and by circles and squares, respectively, in the Kondo regime (incorporating a shift in the $\delta$ scale; cf. text). The best fit is obtained for $\Gamma /U=0.5$ in the unitary limit (both below and above the symmetric limit), and for $\Gamma /U=0.07$ or 0.05 in the Kondo regime (below or above the symmetric limit, respectively).

Figure 4

Phase shift as a function of the gate voltage $V_G$. (a) Unitary limit. Theoretical results from Bethe ansatz calculations at $\Gamma /U=0.5$ compared to the experimental data for $\phi /\pi ={\delta }_{\exp }/\pi +0.29$ (triangles pointing down and up). (b) Kondo regime. Same thing with $\Gamma /U=0.04$ and 0.07, $\phi /\pi ={\delta }_{\exp }/\pi +0.01$ (circles and squares).