An extension of FermiNet to discover quantum phase transitions
https://phys.org/news/2023-02-extension ... tions.html
by Ingrid Fadelli , Phys.org
Architectures based on artificial neural networks (ANNs) have proved to be very helpful in research settings, as they can quickly analyze vast amounts of data and make accurate predictions. In 2020, Google's British AI subsidiary DeepMind used a new ANN architecture dubbed the Fermionic neural network (FermiNet) to solve the Schrodinger equation for electrons in molecules, a central problem in the field of chemistry.
The Schroedinger equation is a partial differential equation based on well-established theory of energy conservation, which can be used to derive information about the behavior of electrons and solve problems related to the properties of matter. Using FermiNet, which is a conceptually simple method, DeepMind could solve this equation in the context of chemistry, attaining very accurate results that were comparable to those obtained using highly sophisticated quantum chemistry techniques.
Researchers at Imperial College London, DeepMind, Lancaster University, and University of Oxford recently adapted the FermiNet architecture to tackle a quantum physics problem. In their paper, published in Physical Review Letters, they specifically used FermiNet to calculate the ground states of periodic Hamiltonians and study the homogenous electron gas (HEG), a simplified quantum mechanical model of electrons interacting in solids.
"Molecules are nice, but physicists are more concerned with solving the Schrodinger equation for solid matter," Gino Cassella, one of the researchers who carried out the study, told Phys.org. "The field of 'condensed matter physics' centers around calculating the behavior of electrons in solid materials, from the wood of your desk to the silicon inside the transistors which power your phone. Naturally, then, we were curious to know if the FermiNet could yield equally accurate solutions to the Schrodinger equation for solids."
Initially, Cassella and his colleagues set out to study the HEG model. In contrast with real solids, this simplified model of solids does not contain atoms, but merely electrons that are whizzing around on a smeared-out positively charged background, which is sometimes referred to as 'jellium' (i.e., evoking the image of electrons embedded in a positively charged jelly).
"Despite its simplicity, the HEG exhibits one of the most important phenomena in the study of condensed matter physics: a quantum phase transition, known as the Wigner transition," Cassella explained. "As the density of the HEG decreases, it undergoes a transition from a 'gassy' state to a 'crystalline' state. We wanted to solve the Schrodinger equation with the FermiNet on either side of the Wigner transition and see how accurate the solutions we obtained are compared to current state-of-the-art methods."
Most deep learning methods used in physics research rely on the analysis of large amounts of data, yet FermiNet does not. In contrast, it leverages the 'variational principle' of quantum mechanics, which states that the energy of a guess for the wavefunction in a given system is always equal to or greater than the energy of the so-called 'ground-state wavefunction', and only equal when a guess is exactly the same as the ground-state wavefunction.
"This ground-state wavefunction and its corresponding energy is exactly the solution we are looking for," Cassella said. "What this means is that we can use the energy as an objective function that we want to make as low as possible, this is what machine learning practitioners would call a 'loss function'. In essence, we train our neural networks guided solely by the fundamental principles of quantum mechanics."