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arXiv:cond-mat/0607255 · Physical Review Letters

Quantum Criticality in a Double-Quantum-Dot System

Gergely Zaránd, Chung‐Hou Chung, Pascal Simon, Matthias Vojta C

Published Oct 17, 2006• 105 citations
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Abstract

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.

DOI
10.1103/physrevlett.97.166802

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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 propose a realization of the two-impurity Anderson model in a double quantum-dot device. When charge transfer between the dots is suppressed, the system exhibits a quantum phase transition, which is controlled by a surface of non-Fermi-liquid fixed points parameterized by the charge valences of the dots. Employing conformal field theory techniques, we identify the scaling exponents that govern transport and thermodynamics close to criticality. We also determine the dynamical exponents that set the time scale for the buildup of the non-Fermi-liquid state after the system is suddenly shifted into the critical region, e.g., by a change of a nearby gate voltage.
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 develop a theory of electron transport in a double quantum dot device recently proposed in [Y. Oreg and D. Goldhaber-Gordon, Phys. Rev. Lett. 90, 136602 (2003)] for the observation of the two-channel Kondo effect. Our theory provides a strategy for tuning the device to the non-Fermi-liquid fixed point, which is a quantum critical point in the space of device parameters. We explore the corresponding quantum phase transition, and make explicit predictions for behavior of the differential conductance in the vicinity of the quantum critical point.
Over recent decades, a growing number of systems, many of them quantum critical, have been shown to exhibit non-Fermi-liquid behavior, but a full analytic understanding of such systems out of equilibrium is still lacking. In this paper, we provide a distinct example with broad applications in correlated mesoscopic systems to address this issue---a two-channel Kondo-Luttinger model where a Kondo impurity couples to two voltage-biased interacting electron leads, experimentally realizable in a dissipative quantum dot. Therein, an exotic quantum phase transition has been known to exist since the 1990s from the one-channel to two-channel Kondo ground states by enhancing electron interactions in the leads, but a controlled analytic approach to this quantum critical point has not yet been established due to the breakdown of weak-coupling perturbation theory near this strong-coupling critical point. We present a controlled method to this long-standing problem by mapping the system in the strong-coupling regime to an effective spin-boson-fermion Hamiltonian. Another type of non-Fermi-liquid quantum critical point is discovered with a distinct logarithmic-in-temperature and -voltage dependence in transport. We further obtain an analytical form for the universal differential conductance out of equilibrium near the transition. Our approach can be further generalized to study nonequilibrium physics of other strong-coupling low-dimensional non-Fermi-liquid fixed points. The relevance of our results for recent experiments is discussed.
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.
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.