We consider strongly correlated regimes which emerge at low temperature in coupled quantum dot (or magnetic impurity) systems. In strongly correlated systems a single particle description fails to explain the observed behaviour, so we resort to many body methods. We describe our system using a 2-impurity Anderson model and develop a numerical renormalisation group procedure which provides non-perturbative insight into the low energy behaviour, through calculation of dynamic quantities. We combine this approach with renormalised perturbation theory, thus acquiring a picture of how the Hamiltonian and interactions change at low energies. These approaches are first used to study the emergence of a Kondo effect with an SU(4) symmetry in capacitively-coupled double quantum dot systems. We classify the 'types' of SU(4) symmetry which can emerge and show how an experimentalist might achieve such emergence through tuning their system. We provide a way of distinguishing between the SU(2) and SU(4) Kondo regimes by considering the conductance. We also study a quantum critical point which occurs in the Heisenberg coupled quantum dot/impurity model. There is an anomalous entropy contributed by the impurities in this regime which is indicative of an uncoupled Majorana Fermion. We calculate dynamic quantities in regimes with different symmetries and establish correspondence with the 2-channel Kondo model. We formulate possible pictures of the underlying mechanisms of the critical point and construct a Majorana fermion model for the case with particle-hole symmetry, which explains the non-Fermi liquid energy levels and degeneracies obtained. We conjecture that a Majorana zero mode is present, and that this is responsible for the anomalous entropy.
We consider strongly correlated regimes which emerge at low temperature in coupled quantum dot (or magnetic impurity) systems. In strongly correlated systems a single particle description fails to explain the observed behaviour, so we resort to many body methods. We describe our system using a 2-impurity Anderson model and develop a numerical renormalisation group procedure which provides non-perturbative insight into the low energy behaviour, through calculation of dynamic quantities. We combine this approach with renormalised perturbation theory, thus acquiring a picture of how the Hamiltonian and interactions change at low energies. These approaches are first used to study the emergence of a Kondo effect with an SU(4) symmetry in capacitively-coupled double quantum dot systems. We classify the 'types' of SU(4) symmetry which can emerge and show how an experimentalist might achieve such emergence through tuning their system. We provide a way of distinguishing between the SU(2) and SU(4) Kondo regimes by considering the conductance. We also study a quantum critical point which occurs in the Heisenberg coupled quantum dot/impurity model. There is an anomalous entropy contributed by the impurities in this regime which is indicative of an uncoupled Majorana Fermion. We calculate dynamic quantities in regimes with different symmetries and establish correspondence with the 2-channel Kondo model. We formulate possible pictures of the underlying mechanisms of the critical point and construct a Majorana fermion model for the case with particle-hole symmetry, which explains the non-Fermi liquid energy levels and degeneracies obtained. We conjecture that a Majorana zero mode is present, and that this is responsible for the anomalous entropy.
The numerical renormalization group is employed to study a double quantum (DQD) dot system consisting of two equivalent single-level dots, each coupled to its own lead and with a mutual capacitive coupling embodied in an interdot interaction U', in addition to the intradot Coulomb interaction U. We focus on the regime with two electrons on the DQD, and the evolution of the system on increasing U'/U. The spin-Kondo effect arising for U'=0 (SU(2) x SU(2)) is found to persist robustly with increasing U'/U, before a rapid but continuous crossover to (a) the SU(4) point U'=U where charge and spin degrees of freedom are entangled and the Kondo scale strongly enhanced; and then (b) a charge-Kondo state, in which a charge-pseudospin is quenched on coupling to the leads/conduction channels. A quantum phase transition of Kosterlitz-Thouless type then occurs from this Fermi liquid, strong coupling (SC) phase, to a broken symmetry, non-Fermi liquid charge ordered (CO) phase at a critical U'_c. Our emphasis in this paper is on the structure, stability and flows between the underlying RG fixed points, on the overall phase diagram in the (U,U')-plane and evolution of the characteristic low-energy Kondo scale inherent to the SC phase; and on static physical properties such as spin- and charge-susceptibilities (staggered and uniform), including universality and scaling behaviour in the strongly correlated regime. Some exact results for associated Wilson ratios are also obtained.
We consider a double dot system of equivalent, capacitively coupled semiconducting quantum dots, each coupled to its own lead, in a regime where there are two electrons on the double dot. Employing the numerical renormalization group, we focus here on single-particle dynamics and the zero-bias conductance, considering in particular the rich range of behaviour arising as the interdot coupling is progressively increased through the strong coupling (SC) phase, from the spin-Kondo regime, across the SU(4) point to the charge-Kondo regime; and then towards and through the quantum phase transition to a charge-ordered (CO) phase. We first consider the two-self-energy description required to describe the broken symmetry CO phase, and implications thereof for the non-Fermi liquid nature of this phase. Numerical results for single-particle dynamics on all frequency scales are then considered, with particular emphasis on universality and scaling of low-energy dynamics throughout the SC phase. The role of symmetry breaking perturbations is also briefly discussed.
The prospect of using semiconductor quantum dots as an experimental tool to distinguish Majorana zero modes (MZMs) from other zero-energy excitations such as Kondo resonances has brought up the fundamental question of whether topological superconductivity and the Kondo effect can coexist in these systems. Here, we study the Kondo effect in a quantum dot coupled to a metallic contact and to a pair of MZMs. We consider a situation in which the MZMs are spin polarized in opposite directions. By using numerical renormalization group calculations and scaling analysis of the renormalization group equations, we show that the Kondo effect takes place at low temperatures, regardless of the coupling to the MZMs. Interestingly, we find that the Kondo singlet essentially decouples from the MZMs such that the residual impurity entropy can show local non-Fermi-liquid properties characteristic of the single Majorana excitations. This offers the possibility of tuning between Fermi-liquid and non-Fermi-liquid regimes simply by changing the quantum-dot-MZM couplings.
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 Coulomb interaction U12 and a spin 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=U12c 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.
We investigate systems comprising chains and rings of quantum dots, coupled to two metallic leads. Such systems allow to study the competition between orbital and spin degrees of freedom in a nanodevice, and the effect this subtle interplay has on two-channel Kondo (2CK) physics. We demonstrate that a rich range of strongly correlated electron behaviour results, with non-Fermi liquid 2CK phases and non-trivial phase transitions accessible. We employ physical arguments and the numerical renormalization group (NRG) technique to analyse these systems in detail, examining in particular both thermodynamic and dynamical properties. When leads are coupled to either end of a chain of dots, we show that the resulting behaviour on low temperature/energy scales can be understood in terms of simpler paradigmatic quantum `impurity' models. An effective low-energy single-spin 2CK model is derived for all odd-length chains, while the behaviour of even-length chains is related fundamentally to that of the classic `two-impurity Kondo' model. In particular, for small interdot coupling, we show that an effective coupling mediated though incipient single-channel Kondo states drives all odd chains to the 2CK fixed point (FP) on the lowest temperature/energy scales. A theory is also developed to describe a phase transition in even chains. We derive an effective channel-anisotropic 2CK model, which indicates that the critical FP of such models must be the 2CK FP. This physical picture is confirmed using NRG for various chain systems. We also examine the effect of local frustration on 2CK physics in mirror-symmetric ring systems. The importance of geometry and symmetry is demonstrated clearly in the markedly different physical behaviour that arises in systems where two leads are either connected to the same dot, or to neighbouring dots. In the latter case, we show for all odd-membered rings that two distinct 2CK phases, with different ground state parities, arise on tuning the interdot couplings. A frustration-induced phase transition thus occurs, the 2CK phases being separated by a novel critical point for which an effective low-energy model is derived. Precisely at the transition, parity mixing of the quasidegenerate local trimer states acts to destabilise the 2CK FPs, and the critical FP is shown to consist of a free pseudospin together with effective single-channel spin quenching. While connecting both leads to the same dot again results in two parity-distinct phases, a simple level-crossing transition now results due to the symmetry of the setup. The proposed geometry also allows access to a novel ferromagnetically-coupled two-channel local moment phase. Driven by varying the interdot couplings and occurring at the point of inherent magnetic frustration, such transitions in ring structures provide a striking example of the subtle interplay between internal spin and orbital degrees of freedom in coupled quantum dot systems, and the resulting effect on Kondo physics.
The system of several (N) quantum dots coupled in parallel to the same single-mode conduction channel can be modeled as a single-channel N-impurity Anderson model. Using the generalized Schrieffer-Wolff transformation we show that near the particle-hole symmetric point, the effective Hamiltonian in the local moment regime is the N-impurity S=1∕2 Kondo model. The conduction-band-mediated RKKY exchange interaction between the dots is ferromagnetic and at intermediate temperatures locks the moments into a maximal spin S=N∕2 ground state. We provide an analytical estimate for the RKKY interaction. At low temperatures the spin is partially screened by the conduction electrons to $N∕2\ensuremath{-}1∕2$ due to the Kondo effect. By comparing accurate numerical renormalization group results for magnetic susceptibility of the N-impurity Anderson model to the exact Bethe ansatz results of a S=N∕2 SU(2) Kondo system we show that at low-temperature the quantum dots can be described by the effective S=N∕2 Kondo model. Moreover, the Kondo temperature is independent of the number of impurities N. We demonstrate the robustness of the spin N∕2 ground state as well as of the associated S=N∕2 Kondo effect by studying the stability of the system with respect to various experimentally relevant perturbations. We finally explore various quantum phase transitions driven by these perturbations.