The critical temperature of the 2D-Ising model through deep learning autoencoders

Abstract

We investigate deep learning autoencoders for the unsupervised recognition of phase transitions in physical systems formulated on a lattice. We focus our investigation on the 2-dimensional ferromagnetic Ising model and then test the application of the autoencoder on the anti-ferromagnetic Ising model. We use spin configurations produced for the 2-dimensional ferromagnetic and anti-ferromagnetic Ising model in zero external magnetic field. For the ferromagnetic Ising model, we study numerically the relation between one latent variable extracted from the autoencoder to the critical temperature Tc. The proposed autoencoder reveals the two phases, one for which the spins are ordered and the other for which spins are disordered, reflecting the restoration of the ℤ2 symmetry as the temperature increases. We provide a finite volume analysis for a sequence of increasing lattice sizes. For the largest volume studied, the transition between the two phases occurs very close to the theoretically extracted critical temperature. We define as a quasi-order parameter the absolute average latent variable z̃, which enables us to predict the critical temperature. One can define a latent susceptibility and use it to quantify the value of the critical temperature Tc(L) at different lattice sizes and that these values suffer from only small finite scaling effects. We demonstrate that Tc(L) extrapolates to the known theoretical value as L →∞ suggesting that the autoencoder can also be used to extract the critical temperature of the phase transition to an adequate precision. Subsequently, we test the application of the autoencoder on the anti-ferromagnetic Ising model, demonstrating that the proposed network can detect the phase transition successfully in a similar way. Graphical abstract

Application of the Variational Autoencoder to Detect the Critical Points of the Anisotropic Ising Model

“…3, we see a sharp change in the τ 0 around the value of K c . This result is consistent with the accomplishments of prior published works for the isotropic Ising model on a square lattice [23][24][25]50].…”

“…VAEs are being successfully applied recently to detect phase transitions in classical spin models [23][24][25]. The input data sets are given by Monte Carlo method and then unsupervised machine learning, such as the VAE is used for deciphering and distinguishing different physics in the input data sets.…”

Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology

“…There is an emerging body of work exploring the use of machine learning and other data analysis methods to detect and classify phase transitions in statistical physics systems. An incomplete list of references includes [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. One of the motivations of this approach is to develop methodologies which require minimal a priori knowledge about the systems in question.…”

Machine learning approaches to the QCD transition

“…Ref. [2,3] for a small subset of available studies, which contains a more complete set of references. Analysis of the thermal transition in Yang-Mills have appeared [4,5], but the investigations of gauge models, and of the even more complex fermion-gauge models such as QCD are scarce.…”