Topological terms vs topological states in spin models with ladder dimerisation

We analyze the possible existence of topological phases in two-legged spin ladders considering a staggered interaction in both chains. When the staggered interaction in one chain is shifted by one site with respect to the other chain, the model can be mapped, in the continuum limit, into a nonlinear sigma model plus with a so-called topological term which is nonvanishing even if the number of legs is two. This suggests a quantum phase transition that is explored numerically, using a DMRG approach, by considering energy gaps, surface correlations as well as nonlocal correlators and entanglement indicators such as entropy and degeneracy of reduced density matrix levels. This characterisation indicates the appearance of a topological phase (an analogue of Mott-Haldane transition) with genuine multipartite entanglement, and provide insights in the role of topological field-theoretic terms vs topological states in quasi 1d quantum systems.