Understanding the charging of exceptionally narrow levels in quantum dots in the presence of interactions remains a challenge within mesoscopic physics. We address this fundamental question in the generic model of a narrow level capacitively coupled to a broad one. Using bosonization we show that for arbitrary capacitive coupling charging can be described by an analogy to the magnetization in the anisotropic Kondo model, featuring a low-energy crossover scale that depends in a power-law fashion on the tunneling amplitude to the level. Explicit analytical expressions for the exponent are derived and confirmed by detailed numerical and functional renormalization-group calculations.
Understanding the charging of exceptionally narrow levels in quantum dots in the presence of interactions remains a challenge within mesoscopic physics. We address this fundamental question in the generic model of a narrow level capacitively coupled to a broad one. Using bosonization we show that for arbitrary capacitive coupling charging can be described by an analogy to the magnetization in the anisotropic Kondo model, featuring a low-energy crossover scale that depends in a power-law fashion on the tunneling amplitude to the level. Explicit analytical expressions for the exponent are derived and confirmed by detailed numerical and functional renormalization-group calculations.
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 case of a quantum dot connected to two superconducting leads is studied by using the narrow-band limit to describe the superconducting degrees of freedom. The model provides a simple theoretical framework, almost analytical, to analyze the interplay between the Kondo effect, superconductivity, and finite temperature. In the quantum dot Kondo regime, the model is completely characterized by the ratio $\ensuremath{\Delta}/J$, with $\ensuremath{\Delta}$ the superconducting gap and J an effective antiferromagnetic exchange coupling between the dot and the leads. The model allows us to calculate, at any temperature T, the equilibrium Josephson current through the dot in a very straightforward way as a function of $\ensuremath{\Delta}/J$. The behavior of the current allows us to distinguish the four types of hybrid junctions: 0, ${0}^{\ensuremath{'}},{\ensuremath{\pi}}^{\ensuremath{'}}$, and $\ensuremath{\pi}$. The presence of the 0- and ${0}^{\ensuremath{'}}$-junction configurations are intrinsically linked to the Kondo effect in the quantum dot, while the $\ensuremath{\pi}$- and ${\ensuremath{\pi}}^{\ensuremath{'}}$-junction configurations are driven by the superconductivity in the leads. The Josephson critical current has a non-monotonic behavior with temperature, that may be used for the experimental characterization of the fundamental $0\ensuremath{-}\ensuremath{\pi}$ transition. The model allows us to obtain easily a phase diagram $\ensuremath{\Delta}/J$ vs temperature, from where we can obtain an overall picture on the stability of the different types of junctions. From the explicit analytical expressions for the ground-state, low-energy excitations, free energy, and Josephson current, it is easy to understand the physical nature of the main features of the critical current and the phase diagram. The results, obtained with a minimum of numerical effort, are in a good qualitative agreement with more demanding calculational approaches aimed to solve the full model.
We investigate two equivalent, capacitively coupled semiconducting quantum dots, each coupled to its own lead, in a regime where there are two electrons on the double dot. With increasing interdot coupling, a rich range of behavior is uncovered: first a crossover from spin- to charge-Kondo physics, via an intermediate SU(4) state with entangled spin and charge degrees of freedom, followed by a quantum phase transition of Kosterlitz-Thouless type to a non-Fermi-liquid "charge-ordered" phase with finite residual entropy and anomalous transport properties. Physical arguments and numerical renormalization group methods are employed to obtain a detailed understanding of the problem.
Quantum phase transitions out of equilibrium are outstanding emergent subjects in condensed matter physics with great fundamental importance and challenges. We theoretically investigate here the nonequilibrium quantum phase transition in a generic nano-setup: the pseudogap Kondo model where a Kondo quantum dot couples to two---left (L) and right (R)---voltage-biased fermionic leads with power-law density of states (DOS) with respect to their Fermi levels ${\ensuremath{\mu}}_{L/R}$, ${\ensuremath{\rho}}_{c,L(R)}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}\ensuremath{-}{\ensuremath{\mu}}_{L(R)}|}^{r}$ with 0<r<1. In equilibrium (${\ensuremath{\mu}}_{L}\ensuremath{-}{\ensuremath{\mu}}_{R}=0$) and for 0<r<1/2, with increasing Kondo correlations this model exhibits a quantum phase transition from a unscreened local moment (LM) phase to the Kondo screened phase. At finite bias voltages and near criticality, we discover new nonequilibrium universal scaling behaviors in conductance, conduction electron T matrix, and local spin susceptibility via a controlled frequency-dependent renormalization group (RG) approach. The current-induced decoherence is key to understanding these distinct universal nonequilibrium quantum critical regimes. The relevance of our results to experiment is discussed.
We consider the simplest example of a double quantum dot system: two equivalent, capacitively coupled single-level dots, each coupled to its own lead. Focussing on a regime of two electrons on the double dot, a rich range of behaviour is found on increasing the interdot coupling: first a crossover from spin- to charge-Kondo physics, via an intermediate SU(4) state with entangled spin and charge degrees of freedom; followed by a quantum phase transition of Kosterlitz-Thouless type to a non-Fermi liquid 'charge-ordered' phase with finite residual entropy and anomalous transport properties. The numerical renormalization group method, coupled with simple physical arguments, is employed to obtain a detailed understanding of the problem.