# Real-Space Visualization of Quantum Phase Transition by Network Topology

## TL;DR

This paper introduces a network-based approach to visualize and analyze quantum phase transitions in real space using quantum correlation functions, revealing changes in network topology that correspond to different quantum phases.

## Contribution

It presents a novel method of constructing real-space networks from quantum systems and demonstrates how network topology metrics can identify quantum phase transitions.

## Key findings

- Bose-Einstein condensate transition corresponds to a small-world network transition.
- Disorder-induced localization reduces network clustering and increases path length.
- Network measures effectively identify delocalized to localized regime crossover.

## Abstract

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 the three-dimensional bosonic system, the single-particle density matrix is adopted to construct the adjacency matrix. We show that the Bose-Einstein condensate transition can be interpreted as the transition into a small-world network, which is accurately captured by the small-world coefficient. For the one-dimensional disordered system, using the electron diffusion operator to build the adjacency matrix, we find that the 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 the loss of global connection, which is revealed by the small-world coefficient as well as other independent measures like the robustness, the efficiency, and the algebraic connectivity. Our results suggest that the quantum phase transitions can be visualized in real space and characterized by the network analysis with suitable choices of quantum correlation functions.