  We explore the effects of long term secular perturbations on the distribution
of dark matter particles around Supermassive Black Hole (BH) binaries. We show
that in the hierarchical (in separation) three-body problem, one of the BHs and
a dark matter particle form an inner binary. Gravitational perturbations from
the BH companion, on a much wider orbit, can cause the dark matter particle to
reach extremely high eccentricities and even get accreted onto the BH, by what
is known as the Eccentric Kozai-Lidov (EKL) mechanism. We show that this may
produce a torus-like configuration for the dark matter distribution around the
less massive member of the BH binary. We first consider an Intermediate BH
(IMBH) in the vicinity of our Galactic Center, which may be a relic of a past
minor merger. We show that if the IMBH is close enough (i.e., near the stellar
disk) the EKL mechanism is very efficient in exciting the eccentricity of dark
matter particles in near-polar configurations to extremely high values where
they are accreted by the IMBH. We show that this mechanism is even more
effective if the central BH grows in mass, where we have assumed adiabatic
growth. Since near-polar configurations are disrupted, a torus-like shape is
formed. We also show that this behavior is also likely to be relevant for
Supermassive BH binaries. We suggest that if the BHs are spinning, the accreted
dark matter particles may linger in the ergosphere and thereby may generate
self-annihilations and produce an indirect signature of potential interest.
