Phase diagram and critical points of a double quantum dot

Abstract

We apply a combination of numerical renormalization group (NRG) and renormalized perturbation theory (RPT) to a model of two quantum dots (impurities) described by two Anderson impurity models hybridized to their respective baths. The dots are coupled via a direct interaction U12 and an exchange interaction J. The model has two types of quantum critical points, one at J = Jc to a local singlet state and one at U12 = U c 12 to a locally charge ordered state. The renormalized parameters which determine the low energy behavior are calculated from the NRG. The results confirm the values predicted from the RPT on the approach to the critical points, which can be expressed in terms of a single energy scale T * in all cases. This includes cases without particle-hole symmetry, and cases with asymmetry between the dots, where there is also a transition at J = Jc. The results give a comprehensive quantitative picture of the behavior of the model in the low energy Fermi liquid regimes, and some of the conclusions regarding the emergence of a single energy scale may apply to a more general class of quantum critical points, such as those observed in some heavy fermion systems.

Analysis of low-energy response and possible emergent SU(4) Kondo state in a double quantum dot

“…We have calculated earlier the leading T 2 corrections to the self-energy for the SU(4) particle-hole symmetric model 25,42 which arise purely from the imaginary part of the self-energy. Le Hur et al 18 have carried out a similar calculation for the non particle-hole symmetric SU(4) model and find a contribution in this case of the same order to the real part of the self-energy, such that the temperature corrections of order T 2 cancel out in the expression for the conductance so the leading contribution in this case is of order T 3 .…”

“…They can be accurately deduced from an analysis of the low energy excitations of an NRG calculation. 24,25 The low energy effective model describes a Fermi liquid in which the quasiparticles interact via the terms,Ũ andŨ 12 . There is a point of 4-fold degeneracy for the effective Hamiltonian when the interaction terms between the quasiparticles set to zero, U =Ũ 12 = 0.…”

“…We have described in earlier work how these can be deduced from an analysis of the low energy fixed point in a numerical renormalization group (NRG) calculation. 24,25 We apply the method to the model being investigated here and describe the results of these calculations in detail in the next section. We use Λ = 6 (this comparatively large value gives accurate estimates for the renormalized parameters as can be checked in the case of a single impurity model) and typically retain 4000 states in our NRG calculations.…”

“…We can hence describe the low energy behavior in terms of well defined quasiparticles and their interactions. We have shown in earlier work 24,25 in the absence of a magnetic field that these quasiparticles can be taken to correspond to renormalized versions of the parameters that specify the bare model,ε d,i,σ ,Γ i ,Ũ i andŨ 12 . The Hamiltonian describing the low energy fixed point and the leading irrelevant correction terms then has the same form as the original model, but with the bare parameters replaced by the renormalized ones.…”

“…These can be determined from an analysis of the low energy fixed point in a numerical renormalization group 22,23 (NRG) calculation. 24,25 Once these have been determined several response functions, such as the spin and charge susceptibilities at zero temperature and the linear conductance through the dots, can be calculated from exact expressions for these quantities in terms of these renormalized parameters. By comparing with exact Bethe ansatz results it has been shown that very accurate numerical results can be obtained from these calculations.…”