  Topological Kondo insulator (TKI) is an essential example of interacting
topological insulator, where electron's correlation effect plays a key role.
However, most of our understanding on this timely issue comes from numerical
simulations, (particularly in one-spatial dimension) which exactly includes
correlation effect but is black box for extracting underlying physics. In this
work, we use a non-uniform antiferromagnetic mean-field (nAFM) theory to
understand the underlying physics in a TKI model, the $1D$ $p-$wave periodic
Anderson model ($p$-PAM). Comparing with numerically exact quantum Monte Carlo
simulation, we find that nAFM theory is an excellent approximation for
ground-state properties when onsite Hubbard interaction is weak. This
emphasizes the dominating antiferromagnetic correlation in this system and
local antiferromagnetic picture captures the qualitative nature of interacting
many-body ground state. Adding extra conduction electron band to $p$-PAM leads
to a quantum phase transition from Haldane phase into topological trivial
phase. We believe these results may be helpful for understanding novel physics
in interacting TKI materials such as SmB$_{6}$ and other related compounds.
