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VNU Journal of Science Mathematics - Physics

Machine Learning of the Two-Dimensional Diluted Bond Ising Model

Nguyen Trong Luu, Pham Long Nhat, Nguyen Vo Nguyen Huy, Hoang Mai Anh, Tran Truong Giang, Duong Xuan Nui, Hoang Van Nam, Vu Hai, Dao Xuan Viet, Nguyen Duc Dung

Published Jul 30, 2025• 0 citations
Publisher

Abstract

The bond-diluted 2D Ising model is essential for understanding the effects of disorder on magnetic systems. Although the T-p phase diagram of this model has been developed through analytical methods and Monte Carlo simulations, the region near the percolation threshold pc remains insufficiently explored. In this study, we reinvestigate the impact of bond dilution on phase transitions in the bond-diluted 2D Ising model using a machine learning approach. A convolutional neural network, initially trained on data from the pure (undiluted) 2D Ising model, is employed through transfer learning to analyze bond-diluted systems. The numerical results show that as bond concentration decreases, the critical temperature also decreases, in agreement with previous Monte Carlo simulation results. Moreover, these findings are instrumental in estimating the critical bond dilution near the percolation threshold pc. As a result, a comprehensive phase diagram for the bond-diluted 2D Ising model has been constructed.

DOI
10.25073/2588-1124/vnumap.5018

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The bond-diluted 2D Ising model is essential for understanding the effects of disorder on magnetic systems. Although the T-p phase diagram of this model has been developed through analytical methods and Monte Carlo simulations, the region near the percolation threshold pc remains insufficiently explored. In this study, we reinvestigate the impact of bond dilution on phase transitions in the bond-diluted 2D Ising model using a machine learning approach. A convolutional neural network, initially trained on data from the pure (undiluted) 2D Ising model, is employed through transfer learning to analyze bond-diluted systems. The numerical results show that as bond concentration decreases, the critical temperature also decreases, in agreement with previous Monte Carlo simulation results. Moreover, these findings are instrumental in estimating the critical bond dilution near the percolation threshold pc. As a result, a comprehensive phase diagram for the bond-diluted 2D Ising model has been constructed.
We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent α\alphaα . According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, starting from the pure model limit down to the neighbourhood of the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions to check the stability of the disorder fixed point. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted q = 4 Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is indeed governed by the same universality class as the site-diluted model.
We have performed Monte Carlo calculations of the thermodynamic properties of the bond-dilute Ising model on the square lattice. The results are obtained for the magnetization, internal energy, specific heat, and susceptibility for several concentrations of missing bonds. The dependence of the Curie temperature on dilution is in good agreement with the results obtained by other methods. The phase transition is found to remain sharp with the same value of the susceptibility exponent as for the pure Ising model.
In order to study the influence of quenched disorder on second-order phase transitions, high-temperature series expansions of the susceptibility and the free energy are obtained for the quenched bond-diluted Ising model in d=3−−5d=3--5d=3−−5 dimensions. They are analyzed using different extrapolation methods tailored to the expected singularity behaviors. In d=4d=4d=4 and 5 dimensions we confirm that the critical behavior is governed by the pure fixed point up to dilutions near the geometric bond percolation threshold. The existence and form of logarithmic corrections for the pure Ising model in d=4d=4d=4 are confirmed, and our results for the critical behavior of the diluted system are in agreement with the type of singularity predicted by renormalization group considerations. In three dimensions we find large crossover effects between the pure Ising, percolation, and random fixed points. We estimate the critical exponent of the susceptibility to be $\ensuremath{\gamma}=1.305(5)$ at the random fixed point.
Randomness is an important subject in the study of phase transition as defect and impurity may present in any real material. The pre-existing ordered phase of a pure system can be affected or even ruined by the presence of randomness. Here we study ferromagnetic Ising model on a square lattice with a presence of randomness in the form of bond dilution. The pure system of this model is known to experience second order phase transition, separating between the high temperature paramagnetic and low-temperature ferromagnetic phase. We used Wang-Landau algorithm of Monte Carlo method to obtain the density of states from which we extract the ensemble average of energy and the specific heat. We observed the signature of phase transition indicated by the diverging peak of the specific heat as system sizes increase. These peaks shift to the lower temperature side as the dilution increases. The lower temperature ordered phase preserves up to certain concentration of dilution and is totally ruined when the bonds no longer percolates.
The critical relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.