  Ground state properties of carbon isotopes, including root-mean-square radii,
neutron separation energies, single particle spectra, and shapes are
systematically studied with the deformed relativistic Hartree-Bogoliubov theory
in continuum. The calculations with the effective interaction PK1 reproduce the
available data reasonably well. The shell evolution in this isotopic chain is
investigated by examining the single particle spectra. The inversion of neutron
orbitals $\nu 2s_{1/2}$ and $\nu 1d_{5/2}$ compared with the order of neutron
orbitals in stable nuclei is revealed in $^{15-20,22}\mathrm{C}$ when these
nuclei are constrained to spherical shape. Neutron halos in $^{15,19,22}$C are
studied in detail and their halo structures are mainly caused by the inversion
of ($\nu 2s_{1/2},\nu 1d_{5/2}$) and deformation effects. In particular,
$^{15}$C and $^{22}$C are deformed halo nuclei with shape decoupling effects in
ground states.
