# The critical temperature of the 2D-Ising model through Deep Learning Autoencoders

## TL;DR

This paper demonstrates that deep learning autoencoders can effectively identify phase transitions and accurately estimate the critical temperature in 2D Ising models, including both ferromagnetic and anti-ferromagnetic cases.

## Contribution

The study introduces a novel autoencoder-based method to detect phase transitions and estimate critical temperatures in 2D Ising models, validated across different magnetic interactions.

## Key findings

- Autoencoders reveal ordered and disordered phases in the Ising model.
- The autoencoder's latent variable correlates with the critical temperature.
- Finite size scaling shows the method's predictions converge to theoretical values.

## 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 $\mathbb{Z}_2$ 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 ${\tilde z}$, 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 $L \to \infty$ 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.