Interaction effects in Aharonov-Bohm–Kondo rings

We study the conductance through an Aharonov-Bohm ring, containing a quantum dot in the Kondo regime in one arm, at finite temperature and arbitrary electronic density. We develop a general method for this calculation based on changing the basis to the screening and nonscreening channels. We show that an unusual term appears in the conductance, involving the connected four-point Green's function of the conduction electrons. However, this term and the terms quadratic in the $\mathrm{T}$ matrix can be eliminated at sufficiently low temperatures, leading to an expression for the conductance linear in the Kondo T matrix. Explicit results are given for temperatures that are high compared to the Kondo temperature.

Figure 1

The model of a small ABK ring considered in this paper. The reference arm is a direct link between sites $-$1 and 1. The quantum dot, denoted by $d$, consists of a single level, with an on-site Coulomb repulsion, tunnel-coupled to the ring. The ring encloses a flux $h\phi /e$ which is included symmetrically in the phases of the tunneling amplitudes of the dot in the gauge chosen here.

Figure 2

Feynman diagram representation of Kubo calculation of the conductance. Solid and dashed lines represent the free Green's function of the $\psi$ and $\varphi$ electrons, respectively, and are diagonal in the initial and final momenta, i.e., proportional to $2\pi \delta (k_1-k_2)$. The hashed circle connecting two solid lines denotes the exact Green's function of $\psi$ electrons. (a) Diagrammatic representation of Eq. (3.27). A decomposition of the exact $\psi$ Green's function into its free and T-matrix parts is implied in all the other diagrams, as well. Therefore, diagrams (b)–(e) each partially contribute to the background conductance. Diagrams (c)–(e) each partially contribute to the linear in the T-matrix part of the conductance. Diagram (e) contributes to the quadratic in the T-matrix part of the conductance. (f) The connected part of the conductance.

Figure 3

The required contour deformation for summing over Matsubara frequencies in Eq. (3.42). The hatched regions mark the vertical position of singularities of $P(z,z+i{\omega }_p)$.

Figure 4

Feynman diagrams for the T matrix of the Kondo model to order $J^2$. The cross indicates potential scattering and the double dashed line is the “propagator” of the impurity spin. The latter is shown as a loop to emphasize that we trace over the impurity spin. The corresponding hole diagrams are obtained by changing the direction of the arrows.

Figure 5

The amputated connected function to order $J^2$. Note that in contrast to some previous works,[30] here we trace over the impurity spin.

Figure 6

(a)–(c) Kondo-type corrections to the conductance versus flux due to the presence of the dot for $T_K\ll T\ll t$ for various values of $p_F$ (proportional to density) and background tunneling amplitude $\tau =t^{\prime }/t$. We have chosen $\gamma =1$ and adjusted $(t_L^2+t_R^2)/\left(Ut\right)$ to a different value for each curve, given by (4.24), so that ${\nu }_{p_F}{J}_{p_F{p}_F}$ has the value $.217$ at $\phi =0$ for each curve.