Regular Article

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

¹ Computation-based Science and Technology Research Center, The Cyprus Institute, 20 Kavafi Str., 2121 Nicosia, Cyprus
² University of Cyprus, 1 Panepistimiou Avenue, 2109 Nicosia, Cyprus
³ Dipartimento di Fisica, Università di Pisa and INFN, Sezione di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy
⁴ Faculty of Mathematics and Natural Sciences, University of Wuppertal, Gaussstr. 20, 42119 Wuppertal, Germany

^a e-mail: andreas.athinodorou@pi.infn.it

Received: 21 October 2019
Received in final form: 20 September 2020
Accepted: 9 October 2020
Published online: 7 December 2020

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 T_c. 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 z^̃, 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 T_c(L) at different lattice sizes and that these values suffer from only small finite scaling effects. We demonstrate that T_c(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.