Abstract
We investigate the appearance of π lapses in the transmission phase θ of a two-level quantum dot with Coulomb interaction U. Using the numerical and functional renormalization group methods we study the entire parameter space for spin-polarized as well as spin-degenerate dots, modelled by spinless or spinful electrons, respectively. We investigate the effect of finite temperatures T. For small T and sufficiently small single-particle spacings δ of the dot levels we find π phase lapses between two transmission peaks in an overwhelming part of the parameter space of the level-lead couplings. For large δ the appearance or not of a phase lapse between resonances depends on the relative sign of the level-lead couplings in analogy to the U = 0 case. We show that this generic scenario is the same for spin-polarized and spin-degenerate dots. We emphasize that in contrast to dots with more levels, for a two-level dot with small δ and generic dot-lead couplings (that is up to cases with special symmetry) the ‘universal’ phase lapse behaviour is already established at U = 0. The most important effect of the Coulomb interaction is to increase the separation of the transmission resonances. The relation of the appearance of phase lapses to the inversion of the population of the dot levels is discussed. For the spin-polarized case and low temperatures we compare our results to recent mean-field studies. For small δ correlations are found to strongly alter the mean-field picture.
Highlights
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT two successive peaks, θ always jumped sharply downward by π
We showed that the universal phase lapse and transmission zero behaviour appearing at small δ can be understood as resulting from a Fano-type interference effect [23] involving transport through two or more effective dot levels, whose positions and widths have been renormalized by the Coulomb interaction and coupling to the leads
We do not recover certain peculiar features of the mean-field results of [10, 11] namely the occurrence, in certain regimes of parameter space, of a phase lapse of less than π, accompanied by the disappearance of the corresponding transmission zero [6, 10]. These features turn out to be artefacts of the mean-field approximation, which misses the rather simple scenario for the phase lapse behaviour of a two-level dot at small δ: for generic level-lead couplings a phase lapse and transmission zero between two transmission peaks is already present at U = 0; increasing the Coulomb interaction the peaks become well-separated while the phase lapse and transmission zero remain in the valley between them
Summary
We introduce our model for the two-level dot. We argue that it is the energy dependent (effective) transmission amplitude t(ω) which one has to compute if one is interested in comparing to the measurements of [17]–[19] of the magnitude of the transmission amplitude and its phase. The amplitude t(ω) can be determined from the matrix elements of the dot’s interacting one-particle Green function. We discuss aspects of the NRG and the fRG specific to our problem
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