Neural Network Method for Calculation of the Curie Point of the Two-Dimensional Ising Model
https://doi.org/10.25205/2541-9447-2022-17-2-5-15
Abstract
The authors describe a method for determining the critical point of a second order phase transitions using a convolutional neural network based on the Ising model on a square lattice. Data for training and analysis were obtained using Monte Carlo simulations. The neural network was trained on the data corresponding to the low-temperature phase, that is a ferromagnetic one and high-temperature phase, that is a paramagnetic one, respectively. After training, the neural network analyzed input data from the entire temperature range: from 0.1 to 5.0 (in dimensionless units J) and determined the Curie point Tc.
Supporting agencies: This work was supported by a grant from the President of the Russian Federation for State Support of Leading Scientific Schools of the Russian Federation No. NSh-2559.2022.1.2.
About the Authors
A. O. Korol
Far Eastern Federal University
Russian Federation
Russian Federation
Alena O. Korol, master, research-engineer, Department of Theoretical Physics and Intelligent Technologies, Institute of High Technologies and Advanced Materials
Vladivostok
K. V. Nevedev
Far Eastern Federal University
Russian Federation
Russian Federation
Konstantin V. Nefedev, Doctor of Science (Physics and Mathematics), professor, director, Department of Theoretical Physics and Intelligent Technologies, Institute of High Technologies and Advanced Materials
Vladivostok
V. Yu. Kapitan
Far Eastern Federal University
Russian Federation
Russian Federation
Vitalii Yu. Kapitan, Candidate of Science (Physics and Mathematics), Associate professor, Information Security Department, Institute of Mathematics and Computer Technology
Vladivostok
References
1. Cipra B.A. An introduction to the Ising model // The American Mathematical Monthly, 1987, vol. 94, no. 10, pp. 937–959.
2. Fisher D. S., Huse D. A. Ordered phase of short-range Ising spin-glasses // Physical review letters, 1986, vol. 56, no. 15, p. 1601.
3. Kruis J., Maris G. Three representations of the Ising model // Scientific reports, 2016, vol. 6, no. 1, pp. 1–11.
4. Padalko M. A. et al. An accelerated exhaustive enumeration algorithm in the Ising model // Dal’nevostochnyi Matematicheskii Zhurnal [Far Eastern Mathematical Journal], 2019, vol. 19, no. 2, pp. 235–244.
5. Prudnikov V. V. et al. Marginal Effect of Structural Defects on the Nonequilibrium Critical Behavior of the Two-Dimensional Ising Model // JETP, 2020, vol. 130, no. 2, p. 258–273.
6. Anderson P. W. Ordering and antiferromagnetism in ferrites // Physical Review, 1956, vol. 102, no. 4, pp. 1008.
7. Narita N. et al. Design and numerical study of flux control effect dominant MAMR head: FC writer // IEEE Transactions on Magnetics, 2020, vol. 57, no. 3, pp. 1–5.
8. Vasil’ev E. V. et al. Numerical simulation of two-dimensional magnetic skyrmion structures // Computer Research and Modeling, 2020, vol. 12, no. 5, pp. 1051–1061.
9. Soldatov K. S. et al. Finite-size scaling in ferromagnetic spin systems on the pyrochlore lattice // Dal’nevostochnyi Matematicheskii Zhurnal [Far Eastern Mathematical Journal], 2020, vol. 20, no. 2, pp. 255–266.
10. Landau D., Binder K. A guide to Monte Carlo simulations in statistical physics. Cambridge university press, 2003.
11. Shapovalova K. V. et al. Why does Far East Federal University need supercomputer? // Sovremennye naukoemkie tekhnologii, 2017, no. 1, pp. 81–87. (in Russ.)
12. Shapovalova K. V. et al. Methods of canonical and multicanonical sampling of the space of vector models // Dal’nevostochnyi Matematicheskii Zhurnal, 2017, vol. 17, no. 1, p. 124–130. (in Russ.)
13. Makarov A. G. et al. On the Numerical Calculation of Frustrations in the Ising Model // JETP Letters, 2019, vol. 110, no. 10, pp. 702–706.
14. Carrasquilla J., Melko R.G. Machine learning phases of matter // Nature Physics, 2017, vol. 13, no. 5, pp. 431–434.
15. Kenta S. et al. Machine-Learning Studies on Spin Models // Scientific reports, 2020, vol. 10, no. 1, pp. 1–6.
16. Onsager L. A Two-Dimensional Model with an Order-Disorder Transition // Phys. Rev., 1944, vol. 65, pp. 117–149.
17. Kapitan V. Yu., Shevchenko Yu. A., Andriushchenko P. D., Nefedev K. V. Supercomputer modeling and numerical solutions of statistical physics problems. Publishing house of FEFU, Vladivostok, 2017. 195 p.
18. Goodfellow I., Bengio Y., Courville A. Deep learning. MIT press, 2016.
19. LeCun Y. et al. Handwritten digit recognition with a back-propagation network // Advances in Neural Information Processing Systems (NIPS), 1989, pp. 396–404.
20. LeCun Y. et al. Gradient-based learning applied to document recognition // Proceedings of the IEEE, 1998, vol. 86, no. 11, pp. 2278–2324.
21. Abadi M. et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems // arXiv preprint arXiv: 1603.04467, 2016.
22. Glorot X., Bordes A., Bengio Y. Deep sparse rectifier neural networks // Proceedings of the fourteenth international conference on artificial intelligence and statistics. – JMLR Workshop and Conference Proceedings, 2011, pp. 315–323.
Review
For citations:
Korol A.O., Nevedev K.V., Kapitan V.Yu. Neural Network Method for Calculation of the Curie Point of the Two-Dimensional Ising Model. SIBERIAN JOURNAL OF PHYSICS. 2022;17(2):5-15. (In Russ.) https://doi.org/10.25205/2541-9447-2022-17-2-5-15
Views:
327
Email this article 