Most of the properties of materials, from their strength and hardness to their conductivity and colour, are determined by the behaviour of the electrons they contain. It is the electrons that form the chemical bonds holding molecules and solids together, that carry currents, and that gain energy when photons are absorbed. Electrons have so little mass that they behave almost entirely quantum mechanically, propagating through solids and molecules more like waves than particles. Understanding and predicting materials properties thus requires solving the quantum mechanical Schrödinger equation, a formidably complicated problem to which ARCHER2 is well suited. This Materials Chemistry Consortium project used ARCHER2 and the Tier-2 Baskerville GPU-based machine in Birmingham to solve the many-electron Schrödinger equation for an interacting electron liquid using the new, neural-network-based, “FermiNet” approach [1] we introduced in 2020. The most exciting result was our demonstration that FermiNet is capable of discovering quantum phase transitions unaided [2]. Although we “discovered” a quantum phase transition (from an electron liquid to a Wigner crystal) that was already well known, we have since gone on to discover a hitherto unknown phase of the spin-imbalanced two-dimensional Fermi gas [3]. In the new phase, the minority-spin electrons form a Wigner crystal bathed in a liquid-like sea of majority-spin electrons. We have also used FermiNet to investigate superfluidity [4] and positronic chemistry [5].

As the density of a classical liquid is lowered, it undergoes a phase transition to become a gas. A quantum mechanical liquid of electrons behaves quite differently: as the electron density is lowered, the electrons spontaneously organize themselves into a so-called Wigner crystal.

An illustration of how the FermiNet neural network learns the wave functions of many-electron systems. In this case, the quantum mechanical Hamiltonian operator Ĥ describes a box containing a fixed number of interacting electrons. The image on the right illustrates how the neural network learns a liquid-like wave function when the box is small (the electron density is high) and a crystalline wave function when the box is large (the electron density is low). The density is expressed in terms of r_s, the radius of a sphere containing one electron on average. The value of r_s  is chosen when we start the simulation.

A FermiNet is a neural network that takes electron coordinates as inputs and computes the corresponding value of the ground-state wave function ψ(r₁, σ₁, r₂, σ₂, …, r_N, σ_N), where  r₁, r₂, …, r_N and σ₁, σ₂, …, σ_N are the position vectors and spin projections of the N electrons in the molecule or solid. The training of neural networks usually requires vast archives of data – large language models such as ChatGPT are trained on most of the internet. This would be problematic in many-particle quantum theory because the accurate data required are unavailable. Fortunately, quantum mechanics comes equipped with a built-in loss function: the exact ground-state wave function is the one that minimises the value of the energy expectation value – a high-dimensional integral we can evaluate numerically using a Markov-chain Monte Carlo algorithm. Thus, even if our starting neural network provides a very bad approximation to ψ(r₁, σ₁, r₂, σ₂, …, r_N, σ_N), we can home in on the exact ground state by adjusting the weights and biases to lower the energy expectation value. No external source of data is required.

New electronic states (superconductors, heavy fermions, Kondo insulators, Mott insulators, fractional quantum Hall states, topological insulators, quantum magnets, altermagnets, …) are almost always discovered experimentally because we lack theoretical and computational tools able to identify phases we do not already understand. For the same reason, newly discovered phases often take decades to explain: superconductivity was discovered by Kammerlingh Onnes in 1911 but not fully understood until the advent of BCS theory in 1957. The neural wave function approach demonstrated in this work promises to make the computational discovery of new phases much easier in future.

References

1. David Pfau, James S. Spencer, Alexander G.D.G. Matthews, and W.M.C. Foulkes, Ab initio solution of the many-electron Schrödinger equation with deep neural networks, Rev. Research 2, 033429 (2020).
2. Gino Cassella, Halvard Sutterud, Sam Azadi, N.D. Drummond, David Pfau, James S. Spencer, and W.M.C. Foulkes, Discovering quantum phase transitions with fermionic neural networks, Rev. Lett. 130, 036401 (2023).
3. Wan Tong Lou, James S. Spencer, David Pfau, W.M.C. Foulkes, and Johannes Knolle, The phase diagram of the spin imbalanced Fermi gas, in preparation (2025).
4. Wan Tong Lou, Halvard Sutterud, Gino Cassella, W.M.C. Foulkes, Johannes Knolle, David Pfau, and James S. Spencer, Neural wave functions for superfluids, Rev. X 14, 021030 (2024).
5. Cassella, W.M.C. Foulkes, D. Pfau, and J.S. Spencer, Neural network variational Monte Carlo for positronic chemistry, Nat. Common. 15, 5214 (2024).