Renormalization group study of capacitively coupled double quantum dots

Abstract

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.

Phase diagram and critical points of a double quantum dot

“…U c 12 → U . These results are in agreement with those given earlier by Galpin et al 9 . The phase diagram for the model with finite J and U 12 , where both types of transitions occur, is shown in Fig.4 for the value U/π∆ = 5.…”

“…In the strong coupling regime U/π∆ ≫ 1, the model corresponds to two Kondo models and with U 12 = 0 it has been much studied as a model of a system with two coupled magnetic impurities 7,[14][15][16] . The model with J = 0 has also been used to study two capacitatively coupled quantum dots 9 . For the numerical renormalization group calculations (NRG) the conduction electron states are transformed to a basis corresponding to a tight-binding chain such that the model takes the form illustrated in Fig 1. The densities of states for the conduction electron baths (halfbandwidth D) are discretized with a parameter Λ(> 1) as in the original calculation of Wilson 17 so that the couplings along the chain fall off as Λ −N/2 , for large N , where N is the N th site along the chain from the dot.…”

Spin-orbital and spin Kondo effects in parallel coupled quantum dots

“…In addition to tunneling also central to the design circuits for logic and quantum information processing based on double dots is an examination of capacitive coupling and interdot exchange. These topics have been discussed both theoretically [8,9,[64][65][66][67][68][69][70][71][72][73][74][75][76][77][78] and experimentally [79][80][81][82][83], but only for SU(2) symmetry. Another theme that has received considerable attention is the role of polarization of electrodes in weakening of Kondo correlations.…”

Capacitively coupled double quantum dot system in the Kondo regime

“…Only recently has this longrange interaction attracted more widespread attention of experimental and theory groups alike. [38][39][40][41][42][43] The T-shape configuration, where two QDs are mutually coupled via a tunneling matrix element, while only one of them is coupled to metallic contacts, has attracted considerable interest as it allows the study of the two stage Kondo (TSK) effect, and interference phenomena as the Fano effect 23,[44][45][46][47] by fine control of the inter-dot tunnel coupling. This effect results from the progressive screening of the localized spin of the electron in each QD as the system is cooled down to very low temperatures.…”