Utilizing lagging-phase analysis to identify topological phase transitions in 1D chains

Abstract

The Zak phase serves as a reliable index for pinpointing topological phase transitions within one-dimensional chains, typically ascertained through numerical methods in complex situations. Nonetheless, in proximity to the transition threshold, the numerical Zak phase may become ambiguous. This ambiguity arises from the discrete nature of numerical approaches and the nonlinear relationship between the wave function's phase and the wave vector (k). In response, this paper presents an innovative method aimed at accurately determining the winding number, thereby facilitating the identification of topological phase transitions. Our approach hinges on analyzing the evolution of the phase difference between the projections of the Bloch wave function onto two distinct sublattices. We demonstrate the efficacy of this method through three illustrative examples: the Su–Schrieffer–Heeger model, a magnetic vortex chain, and a trimer lattice chain. The results indicate that our proposed numerical lagging-phase examination method achieves superior precision in identifying topological phase transitions, particularly at critical junctures, compared to the conventional numerical Zak phase approach.