Scaling and Decoherence in the Nonequilibrium Kondo Model

We study the Kondo effect in quantum dots in an out-of-equilibrium state due to an applied dc-voltage bias. Using the method of infinitesimal unitary transformations (“flow equations”), we develop a perturbative scaling picture that naturally contains both equilibrium coherence and nonequilibrium decoherence effects. This framework allows one to study the competition between Kondo effect and current-induced decoherence, and it establishes a large regime dominated by single-channel Kondo physics for asymmetrically coupled quantum dots.

Figure 1

Left: Conventional scaling picture where states are integrated out. Right: Flow equation approach. Here all scattering processes with energy transfer $|\Delta E|\lesssim B^{-1/2}$ are retained in $H(B)$. Both pictures depict a late phase of the flow where $\Lambda$ and $B^{-1/2}$ are smaller than the voltage bias. Scattering processes between all states are taken into account in the initial phase of the flow (not depicted here).

Figure 2

Universal curves for the flow of the coupling constants $g_l$ (solid lines), $g_t$ (dashed lines) and $g_r$ (dotted lines) for symmetrically coupled quantum dots [(a) with ${\Gamma }_l/{\Gamma }_r=1$] and for an asymmetrically coupled quantum dot [(b) with ${\Gamma }_l/{\Gamma }_r=4$]. Results are shown for various ratios $V/T_{\mathrm{K}}$ labeling the curves from top to bottom. $g_l$ and $g_r$ coincide in the symmetric case, therefore only $g_l$ is shown in a.

Figure 3

Phase diagram of the nonequilibrium Kondo model as a function of asymmetry ${\Gamma }_l/{\Gamma }_r$ and voltage bias. The solid line separates a weak-coupling regime (defined by all couplings remaining smaller than 0.5 during the entire flow) from a strong-coupling regime. The contour lines (dashed) show the ratio $\alpha =g_l(B_l){\Gamma }_r/g_r(B_r){\Gamma }_l$; see text.