  We study the Kondo effect in a single-electron transistor device realized in
a single-wall carbon nanotube. The K-K' double orbital degeneracy of a
nanotube, which originates from the peculiar two-dimensional band structure of
graphene, plays the role of a pseudo-spin. Screening of this pseudo-spin,
together with the real spin, can result in an SU(4) Kondo effect at low
temperatures. For such an exotic Kondo effect to arise, it is crucial that this
orbital quantum number is conserved during tunneling. Experimentally, this
conservation is not obvious and some mixing in the orbital channel may occur.
Here we investigate in detail the role of mixing and asymmetry in the tunneling
coupling and analyze how different Kondo effects, from the SU(4) symmetry to a
two-level SU(2) symmetry, emerge depending on the mixing and/or asymmetry. We
use four different theoretical approaches to address both the linear and
non-linear conductance for different values of the external magnetic field. Our
results point out clearly the experimental conditions to observe exclusively
SU(4) Kondo physics. Although we focus on nanotube quantum dots, our results
also apply to vertical quantum dots. We also mention that a finite amount of
orbital mixing corresponds, in the pseudospin language, to having non-collinear
leads with respect to the orbital ''magnetization'' axis which defines the two
pseudospin orientations in the nanotube quantum dot. In this sense, some of our
results are also relevant to the problem of a Kondo quantum dot coupled to
non-collinear ferromagnetic leads.
