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Physica A Statistical Mechanics and its Applications

Similarity learning for detecting criticality and classifying phase transition orders in q-state Potts models

Jianmin Shen, Xiangna Chen, Dian Xu, Shanshan Wang, Feng Gao, Qiuping Alexandre Wang, Wei Li

Published Oct 23, 2026• 0 citations
Publisher
DOI
10.1016/j.physa.2026.132166

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Layered Potts models on a square lattice where ferromagnetic and antiferromagnetic bonds are regularly disposed are studied by means of an interface method, analytically and employing Monte Carlo simulations. They are classified into three types depending on the arrangement of bonds and the number of Potts spin states, q : frustrated (FR) models that have frustration, degenerate (DG) models that have a high degree of degeneracy, and fitting models that have a unique ground state apart from the trivial degeneracy. Both FR and DG models are shown to exhibit phase transitions due to entropy gains, as previously predicted. The q dependence of the interface tension and the critical temperature is clarified. We find that there are very similar properties between FR models and the corresponding Ising models. The models with q >4 are suggested to undergo first order phase transitions.
We numerically investigate the two-dimensional q-state quantum Potts model on the infinite square lattice by using the infinite projected entangled-pair state (iPEPS) algorithm. We show that the quantum fidelity, defined as an overlap measurement between an arbitrary reference state and the iPEPS ground state of the system, can detect q-fold degenerate ground states for the Z_{q} broken-symmetry phase. Accordingly, a multiple bifurcation of the quantum ground-state fidelity is shown to occur as the transverse magnetic field varies from the symmetry phase to the broken-symmetry phase, which means that a multiple-bifurcation point corresponds to a critical point. A (dis)continuous behavior of quantum fidelity at phase transition points characterizes a (dis)continuous phase transition. Similar to the characteristic behavior of the quantum fidelity, the magnetizations, as order parameters, obtained from the degenerate ground states exhibit multiple bifurcation at critical points. Each order parameter is also explicitly demonstrated to transform under the Z_{q} subgroup of the symmetry group of the Hamiltonian. We find that the q-state quantum Potts model on the square lattice undergoes a discontinuous (first-order) phase transition for q=3 and q=4 and a continuous phase transition for q=2 (the two-dimensional quantum transverse Ising model).
Conventionally, the training of a neural network for learning phases of matter uses real physical quantities as the training set. However, it has been demonstrated in several studies that this may not be required. Here we investigate the phase transitions of the two-dimensional (2D) q-state Potts models on the square lattice using a pre-trained neural network (NN). The employed NN was trained previously using two artifically made configurations as the training set. Hence no training is conducted for the present study. Remarkably, the used NN not only calculates the critical points of the considered phase transitions precisely, but also determines the nature of these phase transition definitely without ambiguity. Our results as well as the outcomes found previously suggest that training a NN using real physical quantities as the training set is not required to obtain a working NN. These unconventional studies also imply that if the configuration space of a system can be classified into two categories, then the NN considered here is likely applicable to study the phase transition of that system. Comparison between our approach and other known NN methods for studying the 2D q-state Potts models is briefly discussed as well.