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arXiv:1208.0513 · Physical Review B

Phase diagram and critical points of a double quantum dot

Yunori Nishikawa, Daniel J. G. Crow, Alex C. Hewson

Published Sep 26, 2012• 16 citations
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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 Coulomb interaction ${U}_{12}$ and a spin 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.

DOI
10.1103/physrevb.86.125134

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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{U}_{12}U12​ and a spin exchange interaction JJJ. The model has two types of quantum critical points, one at J=JcJ={J}_{c}J=Jc​ to a local singlet state and one at U12=U12c{U}_{12}={U}_{12}^{c}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∗{T}^{*}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=JcJ={J}_{c}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 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 interimpurity 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. We show that on the approach to the transitions to a local singlet and a local charged ordered state, the quasiparticle weight factor z→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. These predictions are confirmed by the numerical renormalization group calculations, including the case when the on site interaction U=0.
The observation of the quantum critical point in a series double quantum dot system depends on the distinct separation of two scales, ${T}_{K}⪢{T}^{\ensuremath{\ast}}$, where TK{T}_{K}TK​ is the Kondo temperature and ${T}^{\ensuremath{\ast}}$ is the scale at which the system renormalizes away from the quantum critical point to a stable Fermi-liquid fixed point. Using the two-impurity Kondo model, we provide a derivation of ${T}^{\ensuremath{\ast}}$ based on the renormalization group (RG) to lowest order. This result is confirmed by a numerical RG (NRG) analysis which supplements the analytic derivation with additional quantitative precision. The form of the low-energy Fermi-liquid fixed point is derived and subsequently confirmed by the NRG. From this analysis, we conclude that the aforementioned separation of scales is satisfied, allowing the possibility that the quantum critical point may be measured in a future experiment on such double quantum dot systems.
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.
Abstract. 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) × 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.
The Anderson Impurity Model is a generic model to describe systems in which one or more energy levels are coupled to an electronic host. On the energy levels the electrons are subject to a Coulomb interaction. Possible applications of this model are magnetic atoms in a metal or quantum dot systems. Already for one impurity it displays very interesting physics including the well known Kondo effect. For two and more impurities the properties of the system in the local moment regime are determined by an interplay of the Kondo effect and the coupling between the impurity spins. Due to the interplay of multiple energy scales, this model poses a serious task for any computational tool. For a single impurity the model can be very efficiently treated using Wilson"s numerical renormalization group (NRG). Since the numerical effort of NRG calculations rises exponentially with the number of impurities, generically it cannot be applied to systems with more than two impurities. The functional renormalization group (fRG) has been very successfully applied to quantum impurity problems in the past. In this work it will be used to calculate the single-particle dynamics for Anderson-like quantum impurity systems in equilibrium. A comparison to NRG in the case of a single impurity will serve as a benchmark before systems with two impurities and quantum dot systems with two and three dots will be studied. The interplay between the Kondo effect and magnetic and orbital ordering will be investigated in these systems. Although the truncation in the presented fRG scheme is motivated by second order perturbation theory, it will be shown that fRG works also in those cases where perturbation theory fails. Generally it will turn out that fRG is capable of treating a broad variety of Anderson impurity systems within the same framework, including asymmetric impurity levels and interaction between multiple impurities.