Determining an effective model for the two-stage-Kondo regime: Numerical renormalization group results*

A composite impurity in a metal explores different configurations, where its net magnetic moment may be screened by the electrons in the host.  An interesting example is the two-stage Kondo (TSK) system where screening sets  in with successively smaller energy scales.  In contrast, the impurities  may prefer a local singlet disconnected from the metal.  This competition  is decided by fine-tuning of the couplings in the system, as has been  studied before.  A double quantum dot T-shape geometry, where a  `hanging’ dot is connected to current leads only via another dot,  represents a flexible system in which these different regimes can be  explored experimentally.  It has been difficult, however, to clearly differentiate the two regimes.  Here, we provide a prescription to better identify the regime where  the TSK occurs in such double dot geometry. The TSK regime requires a balance of the ratio t₀₁ /Γ₀ between  the inter-dot coupling (t₀₁) and the coupling of the QD connected  to the Fermi sea Γ₀. Above a certain value of this ratio, the system crosses over to a molecular regime, where the quantum dots form a local singlet, and no Kondo screening occurs. Here, we establish that there is a region in the t₀₁ -Γ₀  parameter space where a pure TSK  regime occurs, i.e., where the properties of the second Kondo stage can be accurately described by a single impurity Anderson model  with effective/renormalized parameters. By examining the magnetic susceptibility of the hanging QD, we show that a single parameter, Γ_eff , can accurately   simulate this susceptibility. This effective model also provides the hanging QD spectral function with great accuracy in a limited range of the t₀₁ -Γ₀ parameter space,  thus defining the region where a true TSK regime occurs. We also  show that in this parameter range, the spin correlations between both  quantum dots show a universal behavior. Our results may guide experimental groups to choose parameter values that will place the system either in the TSK regime or in the  crossover to the molecular regime.