Phase transitions through excited-state level crossings and topological indicators: the case of the XXZ chain with staggered Ising interaction
We combine two ways of determining the phase diagram of the spin-$1/2$ XXZ chain with a staggered Ising interaction and uniform transverse exchange, based on exact diagonalization. The model realizes a competition between N\'eel order and bond-dimerized phases generated by the alternating Ising interaction. The simplest approach to determine the phase boundaries is to use topological indicators based on generalized position operators (GPOs). We show that in general, the bosonized and numerical results for the topological indicators agree. The second is the method of crossings of excited energy levels (MCEL), which is justified by conformal field theory. Despite the partial loss of translational symmetry induced by the alternating Ising interaction, we show that, with the aid of the GPO to identify the relevant level crossings, the MCEL provides an accurate determination of the phase boundary between the N\'eel and dimerized phases. While the jumps of a topological indicator based on a GPO provide a qualitatively correct phase diagram, its accuracy is affected when the gap is very small (or the correlation length very large) at one side of the transition, as we show using field-theoretical arguments. The combination of both methods provides a more efficient way of calculating phase diagrams for correlated one-dimensional models than other widely used conventional approaches.