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
Highlights
Trained neural networks can help distinguish phases in simple statistical systems – the structure of which is known – but, more importantly in more complex systems where the underlying phase structure is unknown
In order to identify signals of the phase structure of the 2D-Ising ferromagnetic model, as a first step, we investigate how the latent variable ziconf behaves as a function of the temperature T for each configuration
At low temperatures, by making use of the latent variable per configuration, the autoencoder predicts two states reflecting to the broken Z2 symmetry
Summary
Trained neural networks can help distinguish phases in simple statistical systems – the structure of which is known – but, more importantly in more complex systems where the underlying phase structure is unknown. This was carried out at one lattice volume, it is clear that since the latent variable is identified as the magnetization, the extracted critical temperature for finite volume will converge to the known, theoretically extracted, critical temperature at the infinite volume limit At this point, it should be made clear that goal of this work is not to use the autoencoder to reproduce the order parameter, but as a technical tool enabling to study particular features of the model and identify new quantities which might be proven useful for analyzing the phase behaviour of statistical and possibly gauge theoretical models.
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