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arXiv:1009.0860 · Physical Review B

Prospect for observing the quantum critical point in double quantum dot systems

Justin Malecki, Eran Sela, Ian K. Affleck

Published Nov 23, 2010• 17 citations
arXivHTMLPDF

Abstract

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 ${T}_{K}$ 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.

DOI
10.1103/physrevb.82.205327

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The observation of the quantum critical point in a series double quantum dot system depends on the distinct separation of two scales, TK⪢T\ensuremath∗{T}_{K}⪢{T}^{\ensuremath{\ast}}TK​⪢T\ensuremath∗, where TK{T}_{K}TK​ is the Kondo temperature and T\ensuremath∗{T}^{\ensuremath{\ast}}T\ensuremath∗ 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∗{T}^{\ensuremath{\ast}}T\ensuremath∗ 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 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.
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.
We discuss the realization of the quantum-critical non-Fermi-liquid state, originally discovered within the two-impurity Kondo model, in double-quantum-dot systems. Contrary to common belief, the corresponding fixed point is robust against particle-hole and various other asymmetries and is unstable only to charge transfer between the two dots. We propose an experimental setup where such charge transfer processes are suppressed, allowing a controlled approach to the quantum-critical state. We also discuss transport and scaling properties in the vicinity of the critical point.
We consider the non-Fermi-liquid quantum critical state of the spin-S two-impurity Kondo model and its potential realization in a quantum dot device. Using conformal field theory and the numerical renormalization group, we show the critical point to be identical to that of the two-channel Kondo model with additional potential scattering, for any spin S. Distinct conductance signatures are shown to arise as a function of device asymmetry, with the square-root behavior commonly believed to arise at low-energies dominant only in certain regimes.
We study finite-frequency transport properties of the double dot system recently constructed to observe the two-channel Kondo effect [R. M. Potok et al., Nature 446, 167 (2007)]. We derive an analytical expression for the frequency-dependent linear conductance of this device in the Kondo regime. We show how the features characteristic of the two-channel Kondo quantum critical point emerge in this quantity, which we compute using the results of conformal field theory as well as numerical renormalization group methods. We determine the universal crossover functions describing non-Fermi-liquid vs Fermi-liquid crossovers and also investigate the effects of a finite magnetic field.