Topological phases in two-legged Heisenberg ladders with alternating interactions

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 NL sigma M plus a topological term which is nonvanishing when the number of legs is two. This implies the existence of a critical point which distinguishes two phases. We perform a numerical analysis of energy levels, parity, and string nonlocal order parameters, correlation functions between x, y, z components of spins at the edges of an open ladder, the degeneracy of the entanglement spectrum, and the entanglement entropy to characterize these two different phases. We identify one phase with a Mott insulator and the other one with a Haldane insulator.