Valence bond fluctuations in the Kitaev spin model

We introduce valence bond fluctuations, or bipartite fluctuations associated to bond-bond correlation functions, to characterize quantum spin liquids and the entanglement properties of them. Using analytical and numerical approaches, we find an identical scaling law between valence bond fluctuations and entanglement entropy in the two-dimensional Kitaev spin model and in one-dimensional chain analogs. We also show how these valence bond fluctuations can locate, via the linear scaling prefactor, the quantum phase transitions between the three gapped and the gapless Majorana semimetal phases in the honeycomb model. We then study the effect of a uniform magnetic field along the [111] direction opening a gap in the intermediate phase which becomes topological. We still obtain a robust signal to characterize the transitions toward the three gapped phases. The area-law behavior of such bipartite fluctuations in two dimensions is also distinguishable from the one in the Néel magnetic state that follows a volume square growth.