Kondo spin liquid in the Kondo necklace model: Classical disordered phase versus symmetry-protected topological state

We have studied Kondo spin liquid phase of Kondo necklace models from the perspective of quantum O(N) non-linear sigma model (NLSM) field theory, particularly we focus on its possible topologically nontrivial phases. In the one-dimensional case, the Kondo spin liquid phase is a usual quantum disordered phase in contrast to the well-known topologically nontrivial Haldane phase due to destructive interference effect of topological θ term. In the two-dimensional case, the system can be mapped onto an O(4)-like NLSM with some O(3) anisotropy. Interestingly, 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, which highlights a possible link between the extended Kondo necklace models and the desirable SPT states. Additionally, if the system has the expanded O(5) symmetry instead, the effective NLSM with the Wess–Zumino–Witten term is just a description of the surface modes of a three-dimensional SPT state. The deviations from fully symmetrical cases are discussed. We expect that the results might provide useful threads to identify certain microscopic bilayer antiferromagnet models (and related materials), which can support the SPT states.