Network topology: detecting topological phase transitions in the Kitaev chain and the rotor plane

We propose a novel network measure of topological invariants, called small-worldness, for identifying topological phase transitions of quantum and classical spin models. Small-worldness is usually defined in the study of social networks based on the best known discovery that one can find a short chain of acquaintances connecting almost any two people on the planet. Here we demonstrate that the small-world effect provides a useful description to distinguish topologically trivial and non-trivial phases in the Kitaev chain and accurately capture the Kosterlitz-Thouless transition in the rotor plane. Our results further suggest that the small-worldness containing both locality and non-locality of the network topology can be a practical approach to extract characteristic quantities of topological states of matter.