Critical Phenomena

2025

1. Preprint

   

   Phase transition revealed by eigen microstate entropy

   Teng Liu, Xuezhi Niu, Mingli Zhang, Gaoke Hu, and 8 more authors

   arXiv preprint arXiv:2512.23086, 2025

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   We introduce the eigen microstate entropy (SEM), a novel metric of complexity derived from the probabilities of statistically independent eigen microstates. After establishing its scaling behavior in equilibrium systems and demonstrating its utility in critical phenomena (mean spherical, Ising, and Potts models), we apply SEM to non-equilibrium complex systems. Our analysis reveals a consistent precursor signal: a significant increase in SEM precedes major phase transitions. Specifically, we observe this entropy rise before biomolecular condensate formation in liquid-liquid phase separation in living cells and months ahead of El Niño events. These findings position SEM as a general framework for detecting and interpreting phase transitions in non-equilibrium systems.

   @article{liu2025phase,
     title = {Phase transition revealed by eigen microstate entropy},
     author = {Liu, Teng and Niu, Xuezhi and Zhang, Mingli and Hu, Gaoke and Chen, Yuhan and Zhang, Yongwen and Shi, Rui and Li, Jingyuan and Tan, Peng and Liu, Maoxin and Li, Hui and Chen, Xiaosong},
     journal = {arXiv preprint arXiv:2512.23086},
     doi = {10.48550/arXiv.2512.23086},
     year = {2025}
   }

2025

1. Chin. Phys. Lett.

   

   Finite-size scaling behaviors of eigen microstates near the critical point

   Gaoke Hu, Jia-Qi Dong, Yongwen Zhang, Teng Liu, and 3 more authors

   Chinese Physics Letters, Dec 2025

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   In the eigen microstate approach (EMA), phase emergence is signaled by the condensation of an eigen microstate with a finite eigenvalue. The associated eigenvalue acts as an order parameter, exhibiting finite-size scaling (FSS) behaviors within the critical regime. The eigen microstate derived from EMA precisely characterizes the spatial structure of the emergent phase, which remains less explored in systems lacking translational invariance. Here, we propose an FSS form for the eigen microstate itself and numerically validate it using the two-dimensional Ising model under various boundary conditions. This result enables the derivation of FSS relations for correlation functions, which we verify through numerical simulations. Our results complete the FSS formalism for the EMA.

   @article{hu2025finite,
     title = {Finite-size scaling behaviors of eigen microstates near the critical point},
     author = {Hu, Gaoke and Dong, Jia-Qi and Zhang, Yongwen and Liu, Teng and You, Wen-Long and Liu, Maoxin and Chen, Xiaosong},
     journal = {Chinese Physics Letters},
     doi = {10.1088/0256-307X/43/2/020001},
     dimension = {true},
     volume = {43},
     pages = {020001},
     publisher = {IOP Publishing},
     month = dec,
     year = {2025},
   }

2024

1. Phys. Rev. E.

   

   Eigen microstates in self-organized criticality

   Yongwen Zhang, Maoxin Liu, Gaoke Hu, Teng Liu, and 1 more author

   Physical Review E, Apr 2024

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   We employ the eigen microstates approach to explore the self-organized criticality (SOC) in two celebrated sandpile models, namely the BTW model and the Manna model. In both models, phase transitions from the absorbing state to the critical state can be understood by the emergence of dominant eigen microstates with significantly increased weights. Spatial eigen microstates of avalanches can be uniformly characterized by a linear system size rescaling. The first temporal eigen microstates reveal scaling relations in both models. Furthermore, by finite-size scaling analysis of the first eigen microstates, we numerically estimate critical exponents. Our findings could provide profound insights into eigen microstates of the universality and phase transition in nonequilibrium complex systems governed by self-organized criticality.

   @article{zhang2024eigen,
     title = {Eigen microstates in self-organized criticality},
     author = {Zhang, Yongwen and Liu, Maoxin and Hu, Gaoke and Liu, Teng and Chen, Xiaosong},
     journal = {Physical Review E},
     volume = {109},
     number = {4},
     pages = {044130},
     year = {2024},
     publisher = {APS},
     month = apr,
     dimensions = {true},
     doi = {10.1103/PhysRevE.109.044130},
   }

2022

1. Chin. Phys. Lett.

   

   Renormalization group theory of eigen microstates

   Teng Liu, Gaoke Hu, Jia-Qi Dong, Jingfang Fan, and 2 more authors

   Chinese Physics Letters, Jul 2022

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   We propose a renormalization group (RG) theory of eigen microstates, which are introduced in the statistical ensemble composed of microstates obtained from experiments or computer simulations. A microstate in the ensemble can be considered as a linear superposition of eigen microstates with probability amplitudes equal to their eigenvalues. Under the renormalization of a factor b, the largest eigenvalue σ1 has two trivial fixed points at low and high temperature limits and a critical fixed point with the RG relation, where β and ν are the critical exponents of order parameter and correlation length, respectively. With the Ising model in different dimensions, it has been demonstrated that the RG theory of eigen microstates is able to identify the critical point and to predict critical exponents and the universality class. Our theory can be used in research of critical phenomena both in equilibrium and non-equilibrium systems without considering the Hamiltonian, which is the foundation of Wilson’s RG theory and is absent for most complex systems.

   @article{liu2022renormalization,
     title = {Renormalization group theory of eigen microstates},
     author = {Liu, Teng and Hu, Gaoke and Dong, Jia-Qi and Fan, Jingfang and Liu, Maoxin and Chen, Xiaosong},
     journal = {Chinese Physics Letters},
     doi = {10.1088/0256-307X/39/8/080503},
     dimensions = {true},
     volume = {39},
     number = {8},
     pages = {080503},
     month = jul,
     year = {2022},
     publisher = {IOP Publishing},
   }

2023

1. SCPMA

   

   Universality class of machine learning for critical phenomena

   Gaoke Hu, Yu Sun, Teng Liu, Yongwen Zhang, and 4 more authors

   Science China Physics, Mechanics & Astronomy, Nov 2023

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   Herein, percolation phase transitions on a two-dimensional lattice were studied using machine learning techniques. Results reveal that different phase transitions belonging to the same universality class can be identified using the same neural networks (NNs), whereas phase transitions of different universality classes require different NNs. Based on this finding, we proposed the universality class of machine learning for critical phenomena. Furthermore, we investigated and discussed the NNs of different universality classes. Our research contributes to machine learning by relating the NNs with the universality class.

   @article{hu2023universality,
     title = {Universality class of machine learning for critical phenomena},
     author = {Hu, Gaoke and Sun, Yu and Liu, Teng and Zhang, Yongwen and Liu, Maoxin and Fan, Jingfang and Chen, Wei and Chen, Xiaosong},
     journal = {Science China Physics, Mechanics \& Astronomy},
     volume = {66},
     number = {12},
     pages = {120511},
     doi = {10.1007/s11433-023-2221-8},
     month = nov,
     dimensions = {true},
     year = {2023},
     publisher = {Springer},
   }

2019

1. SCPMA

   

   Condensation of eigen microstate in statistical ensemble and phase transition

   Gaoke Hu, Teng Liu, Maoxin Liu, Wei Chen, and 1 more author

   Science China Physics, Mechanics & Astronomy, Apr 2019

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   In a statistical ensemble with M microstates, we introduce an M×M correlation matrix with correlations among microstates as its elements. Eigen microstates of ensemble can be defined using eigenvectors of the correlation matrix. The eigenvalue normalized by M represents weight factor in the ensemble of the corresponding eigen microstate. In the limit M → ∞, weight factors drop to zero in the ensemble without localization of the microstate. The finite limit of the weight factor when M → ∞ indicates a condensation of the corresponding eigen microstate. This finding indicates a transition into a new phase characterized by the condensed eigen microstate. We propose a finite-size scaling relation of weight factors near critical point, which can be used to identify the phase transition and its universality class of general complex systems. The condensation of eigen microstate and the finite-size scaling relation of weight factors are confirmed using Monte Carlo data of one-dimensional and two-dimensional Ising models.

   @article{hu2019condensation,
     title = {Condensation of eigen microstate in statistical ensemble and phase transition},
     author = {Hu, Gaoke and Liu, Teng and Liu, Maoxin and Chen, Wei and Chen, Xiaosong},
     journal = {Science China Physics, Mechanics \& Astronomy},
     volume = {62},
     pages = {1--8},
     year = {2019},
     dimensions = {true},
     publisher = {Springer},
     month = apr,
     doi = {10.1007/s11433-018-9353-x},
   }