Energy bonds as correlators for long-range symmetry-protected topological models and models with long-range topological order

Topological properties are associated with nonlocal entanglement and global properties. On the other hand, in some topological systems, it has been shown that quasilocal operators detect topological transitions. We show in this work that in the case of noninteracting long-range symmetry-protected topological models, energy bonds signal topological transitions. Interestingly, we also find that they display some signatures at the Berezinskii-Kosterlitz-Thouless transition that occurs in a spin chain with first and second neighbor interactions, if we consider the first excited state instead of the ground state of the system. Moreover, we show that the ground-state topological transition in the spin Kitaev model in a honeycomb lattice, which displays topological long-range order, is also detected by the energy bond correlator. Despite the model being interacting, in the ground state it reduces to a noninteracting model. Even for the spin-liquid phase of the two-dimensional Heisenberg model with first and second neighbor interactions, where the system has true long-range entanglement and topological order, the local bonds do signal the topological phase transition.