Decoding conformal field theories: from supervised to unsupervised learning

Abstract

We use machine learning to classify rational two-dimensional conformal field theories. We first use the energy spectra of these minimal models to train a supervised learning algorithm. We find that the machine is able to correctly predict the nature and the value of critical points of several strongly correlated spin models using only their energy spectra. This is in contrast to previous works that use machine learning to classify different phases of matter, but do not reveal the nature of the critical point between phases. Given that the ground-state entanglement Hamiltonian of certain topological phases of matter are also described by conformal field theories, we use supervised learning on Réyni entropies, and find that the machine is able to identify which conformal field theory describes the entanglement Hamiltonian with only the lowest few Réyni entropies to a high degree of accuracy. Finally, using autoencoders, an unsupervised learning algorithm, we find a hidden variable that has a direct correlation with the central charge and discuss prospects for using machine learning to investigate other conformal field theories, including higher-dimensional ones. Our results highlight that machine learning can be used to find and characterize critical points and also hint at the intriguing possibility to use machine learning to learn about more complex conformal field theories.

Machine Learning Symmetry

“…We hope that these promising results pave the way for a more exhaustive exploration of these landscapes using methods from data science and machine learning. We note especially the recent works on two dimensional CFTs using classifiers as well as autoencoders [43], along with the application of reinforcement learning to solve the equations of the conformal bootstrap itself [44]. It would also be interesting to examine if the methods of [21] could now be extended to these cases, and to glean interesting organizing principles by doing so.…”

Unsupervised Learning of Symmetry Protected Topological Phase Transitions

“…On the other hand over the past few years machine learning (ML) has emerged as a powerful tool to assist physicists to study a plethora of different problems in condensed matter and quantum sciences. A non-exhaustive list of notable examples include classifying phases of matter [21][22][23][24], studying non-equilibrium dynamics of physical systems [25][26][27], sim-ulating dynamics of quantum systems [28][29][30], and augmenting capabilities of quantum devices [31,32]. In particular in classifying applications, most of such techniques rely on supervised ML techniques where the ML algorithms after being trained with labeled systems learns to classify systems with new parameters [21,27,[33][34][35][36][37].…”