Real-space visualization of quantum phase transitions by network topology

We demonstrate that with appropriate quantum correlation function, a real-space network model can be constructed to study the phase transitions in quantum systems. For a three-dimensional bosonic system, a single-particle density matrix is adopted to construct an adjacency matrix. We show that a Bose-Einstein condensate transition can be interpreted as a transition into a small-world network, which is accurately captured by a small-world coefficient. For a one-dimensional disordered system, using the electron diffusion operator to build the adjacency matrix, we find that Anderson localized states create many weakly linked subgraphs, which significantly reduces the clustering coefficient and lengthens the shortest path. We show that the crossover from delocalized to localized regimes as a function of the disorder strength can be identified as a loss of global connection, which is revealed by the small-world coefficient as well as other independent measures such as robustness, efficiency, and algebraic connectivity. Our results suggest that quantum phase transitions can be visualized in real space and characterized by network analysis with suitable choices of quantum correlation functions.