  The two-level model for a double quantum dot coupled to two leads, which is
ubiquitously used to describe charge oscillations, transmission-phase lapses
and correlation-induced resonances, is considered in its general form. The
model features arbitrary tunnelling matrix elements among the two levels and
the leads and between the levels themselves (including the effect of
Aharonov-Bohm fluxes), as well as inter-level repulsive interactions. We show
that this model is exactly mapped onto a generalized Anderson model of a single
impurity, where the electrons acquire a pseudo-spin degree of freedom, which is
conserved by the tunnelling but not within the dot. Focusing on the
local-moment regime where the dot is singly occupied, we show that the
effective low-energy Hamiltonian is that of the anisotropic Kondo model in the
presence of a tilted magnetic field. For moderate values of the (renormalized)
field, the Bethe ansatz solution of the isotropic Kondo model allows us to
derive accurate expressions for the dot occupation numbers, and henceforth its
zero-temperature transmission. Our results are in excellent agreement with
those obtained from the Bethe ansatz for the isotropic Anderson model, and with
the functional and numerical renormalization-group calculations of Meden and
Marquardt [Phys. Rev. Lett. 96, 146801 (2006)], which are valid for the general
anisotropic case. In addition we present highly accurate estimates for the
validity of the Schrieffer-Wolff transformation (which maps the Anderson
Hamiltonian onto the low-energy Kondo model) at both the high- and low-magnetic
field limits. Perhaps most importantly, we provide a single coherent picture
for the host of phenomena to which this model has been applied.
