  We study possible topological features of Kondo spin liquid phase in terms of
the one- and two-dimensional Kondo necklace models within the frame work of
quantum O(N) non-liner sigma model (NLSM). In the one-dimensional case, it is
found that the bulk properties of the Kodno spin liquid phase are similar to
the well-known Haldane phase at strong coupling fixed point. The difference
between them mainly comes from their boundaries due to the effect of the
topological term. In the two-dimensional case, the system can be mapped onto an
O(4)-like NLSM with some O(3) anisotropy. Interestingly, we find that if
hedgehog-like point defects are included together with the restoration of the
full O(4) symmetry, our model is identical to a kind of SU(2)
symmetry-protected topological (SPT) state. Additionally, if the system has the
O(5) symmetry instead, the effective NLSM with Wess-Zumino-Witten term is just
a description of the surface modes of a three-dimensional SPT state, though
such O(5) NLSM could not be a proper description of Kondo spin liquid phase due
to its gaplessness. We expect that the discussions might provide useful threads
to identify certain microscopic bilayer antiferromagnet models (and related
materials), which can support the desirable SPT states.
