  We develop a microscopic theory for the spin Seebeck effect (SSE) in N\'eel
and canted phases of antiferromagnetic insulators. We calculate the DC spin
current tunneling from the antiferromagnet to an attached metal, incorporating
the spin-wave theory and the non-equilibrium Green's function approach. Our
result shows a sign change of the spin current at the spin-flop phase
transition between N\'eel and canted phases, which is in agreement with a
recent experiment for the SSE of $\rm Cr_2O_3$ in a semi-quantitative level.
The sign change can be interpreted from the argument based on the density of
states of up- and down-spin magnons, which is related to the polarized-neutron
scattering spectra. The theory also demonstrates that the spin current in the
N\'eel phase is governed by the magnon correlation, while that in the canted
phase consists of two parts: Contributions from not only the magnon dynamics
but also the static transverse magnetization. This result leads to a prediction
that at sufficiently low temperatures, the spin current non-monotonically
changes as a function of magnetic field in the canted phase. Towards a more
unified understanding of the SSE in antiferromagnets, we further discuss some
missing links of theories of SSE: Interface properties, effects of the
transverse spin moment in the canted phase, the spin-orbit coupling in the
metal, etc. Finally, we compare the SSE of antiferromagnets with those of
different magnetic phases such as ferromagnets, ferrimagnets, an
one-dimensional spin liquid, a spin-nematic liquid, and a spin-Peierls
(dimerized) phase.
