  In this paper we carry out a low-temperature scan of the phase diagram of
dense two-color QCD with $N_f=2$ quarks. The study is conducted using lattice
simulation with rooted staggered quarks. At small chemical potential we observe
the hadronic phase, where the theory is in a confining state, chiral symmetry
is broken, the baryon density is zero and there is no diquark condensate. At
the critical point $\mu = m_{\pi}/2$ we observe the expected second order
transition to Bose-Einstein condensation of scalar diquarks. In this phase the
system is still in confinement in conjunction with non-zero baryon density, but
the chiral symmetry is restored in the chiral limit. We have also found that in
the first two phases the system is well described by chiral perturbation
theory. For larger values of the chemical potential the system turns into
another phase, where the relevant degrees of freedom are fermions residing
inside the Fermi sphere, and the diquark condensation takes place on the Fermi
surface. In this phase the system is still in confinement, chiral symmetry is
restored and the system is very similar to the quarkyonic state predicted by
SU($N_c$) theory at large $N_c$.
