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We show that when there is a sudden transition from a small to a large Fermi surface at a field-induced quantum critical point, similar to what may have been observed in some heavy-fermion compounds, an additional term has to be taken into account in the Luttinger-Friedel sum rule.We calculate this additional term for a local model which has a field-induced quantum critical point (QCP) and show that it changes abruptly at the transition, such that it satisfies a generalized Luttinger-Friedel sum rule on each side of the transition, and characterizes the two Fermi-liquid phases separated by the QCP as a discrete (topological) index., Comment: 5 pages, 6 figures

The Luttinger Theorem, which relates the electron density to the volume of the Fermi surface in an itinerant electron system, is taken to be one of the essential features of a Fermi liquid. The microscopic derivation of this result depends on the vanishing of a certain integral, the Luttinger integral $I_{\rm L}$, which is also the basis of the Friedel sum rule for impurity models, relating the impurity occupation number to the scattering phase shift of the conduction electrons. It is known that non-zero values of $I_{\rm L}$ with $I_{\rm L}=\pm\pi/2$, occur in impurity models in phases with non-analytic low energy scattering, classified as singular Fermi liquids. Here we show the same values, $I_{\rm L}=\pm\pi/2$, occur in an impurity model in phases with regular low energy Fermi liquid behavior. Consequently the Luttinger integral can be taken to characterize these phases, and the quantum critical points separating them interpreted as topological., Comment: 5 pages 7 figures

Studies of the effects of the Hund's rule coupling J_H in multiple orbit impurities or quantum dots using different models have led to quite different predictions for the Kondo temperature T_K as a function of J_H. We show that the differences depend on whether or not the models conserve orbital angular momentum about the impurity site. Using numerical renormalization group (NRG) calculations, we deduce the renormalized parameters for the Fermi liquid regime, and show that, despite the differences between the models, the low energy fixed point in the strong correlation regime is universal with a single energy scale T_K, and just two renormalized interaction parameters, a renormalized single orbital term, U = 4T_K, and renormalized Hund's rule term, J_H = 8T_K/3., Comment: 6 pages, 6 figures

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 $U_{12}$ and an exchange interaction $J$. The model has two types of quantum critical points, one at $J=J_c$ to a local singlet state and one at $U_{12}=U_{12}^c$ 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 $J=J_c$. 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., Comment: 18 pages 31 figures

We find the emergence of strong correlations and universality on the approach to the quantum critical points of a two impurity Anderson model. The two impurities are coupled by an inter-impurity exchange interaction $J$ and direct interaction $U_{12}$ and are hybridized with separate conduction channels.The low energy behavior is described in terms of renormalized parameters, which can be deduced from numerical renormalization group (NRG) calculations. We show that on the approach to the transitions to a local singlet and a local charged ordered state, the quasiparticle weight factor $z\to 0$, and the renormalized parameters can be expressed in terms of a single energy scale $T^*$. The values of the renormalized interaction parameters in terms of $T^*$ can be predicted from the condition of continuity of the spin and charge susceptibilities, and correspond to strong correlation as they are greater than or equal to the effective band width. These predictions are confirmed by the NRG calculations, including the case when the onsite interaction U=0., Comment: 5 pages 5 figures

The triangular triple quantum dot is an interesting system which can demonstrate various types of the Kondo effects, such as the one due to the local spin S=1 moment caused by the Nagaoka ferromagnetic mechanism and the SU(4) Kondo effect. We theoretically study the low-temperature properties and the Kondo energy scale of the triangular triple quantum dot, using the Wilson numerical renormalization group. We have explored a wide parameter region of the electron-filling and distortions which break the symmetry of an equilateral structure. Our results give a comprehensive overview of how the Kondo behavior varies in the different the regions in the wide parameter space of the triangular triple quantum dot., Comment: 18 pages; 21 figures (Figs.17 and 20 are added)

We extend the renormalized perturbation theory for the single impurity Anderson model to the n-channel model with a Hund's rule coupling, and show that the exact results for the spin, orbital and charge susceptibilities, as well as the leading low temperature dependence for the resistivity, are obtained by working to second order in the renormalized couplings. A universal relation is obtained between the renormalized parameters, independent of n, in the Kondo regime. An expression for the dynamic spin susceptibility is also derived by taking into account repeated quasiparticle scattering, which is asymptotically exact in the low frequency regime and satisfies the Korringa-Shiba relation. The renormalized parameters, including the renormalized Hund's rule coupling, are deduced from numerical renormalization group calculations for the model for the case n=2. The results confirm explicitly the universal relations between the parameters in the Kondo regime. Using these results we evaluate the spin, orbital and charge susceptibilities, temperature dependence of the low temperature resistivity and dynamic spin susceptibility for the particle-hole symmetric n=2 model., Comment: 13 pages 14 figures