Phase determination with and without deep learning

Detection of phase transitions is a critical task in statistical physics, traditionally pursued through analytic methods and direct numerical simulations. Recently, machine-learning techniques have emerged as promising tools in this context, with a particular focus on supervised and unsupervised learning methods, along with nonlearning approaches. In this paper, we study the performance of unsupervised learning in detecting phase transitions in the J_{1}-J_{2} Ising model on the square lattice. The model is chosen due to its simplicity and complexity, thus providing an understanding of the application of machine-learning techniques in both straightforward and challenging scenarios. We propose a simple method based on a direct comparison of configurations. The reconstruction error, defined as the mean-squared distance between two configurations, is used to determine the critical temperatures. The results from the comparison of configurations are contrasted with those of the configurations generated by variational autoencoders. Our findings highlight that for certain systems a simpler method can yield results comparable to more complex neural networks. This paper contributes to the broader understanding of machine-learning applications in statistical physics and introduces an efficient approach to the detection of phase transitions using machine determination techniques.