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Many-body systems when continuous phase transition occurs are mainly built inthe interrelationship between particles, implemented through many-bodycorrelations. Some of them may exhibit so-called topological order hardlymeasured by experiments. Therefore we need, beyond mean-field theory, thecomplex-systems approach that stresses the systemic complexity of many-bodynetwork at criticality. According to our previous study, network spaceexperiences the homogeneous-heterogeneous transition invisible in traditionalphase transitions. The network robustness can be a useful indicator to capturethe critical phenomena of phase transitions with/without symmetry breaking. Inthis work, we demonstrate the idea of the change of robust networks issuccessfully applied to the well-known 1D quantum and 2D classical XY models.

Network topology transition at criticality