  We calculate the low-lying spectrum of charmed baryons in lattice QCD on the
$32^3\times64$, $N_f=2+1$ PACS-CS gauge configurations at the almost physical
pion mass of $\sim 156$ MeV/c$^2$. By employing a set of interpolating
operators with different Dirac structures and quark-field smearings for the
variational analysis, we extract the ground and first few excited states of the
spin-$1/2$ and spin-$3/2$, singly-, doubly-, and triply-charmed baryons.
Additionally, we study the $\Xi_c$-$\Xi_c^\prime$ mixing and the operator
dependence of the excited states in a variational approach. We identify several
states that lie close to the experimentally observed excited states of the
$\Sigma_c$, $\Xi_c$ and $\Omega_c$ baryons, including some of the $\Xi_c$
states recently reported by LHCb. Our results for the doubly- and
triply-charmed baryons are suggestive for future experiments.
