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arXiv:2202.05183 · Physical Review Letters

Discovering Quantum Phase Transitions with Fermionic Neural Networks

Gino Cassella, Halvard Sutterud, Sam Azadi, N. D. Drummond, David Pfau, James S. Spencer, W. M. C. Foulkes

Published Jan 20, 2023• 79 citations
arXivHTMLPDF

Abstract

Deep neural networks have been very successful as highly accurate wave function Ansätze for variational Monte Carlo calculations of molecular ground states. We present an extension of one such Ansatz, FermiNet, to calculations of the ground states of periodic Hamiltonians, and study the homogeneous electron gas. FermiNet calculations of the ground-state energies of small electron gas systems are in excellent agreement with previous initiator full configuration interaction quantum Monte Carlo and diffusion Monte Carlo calculations. We investigate the spin-polarized homogeneous electron gas and demonstrate that the same neural network architecture is capable of accurately representing both the delocalized Fermi liquid state and the localized Wigner crystal state. The network converges on the translationally invariant ground state at high density and spontaneously breaks the symmetry to produce the crystalline ground state at low density, despite being given no a priori knowledge that a phase transition exists.

DOI
10.1103/physrevlett.130.036401

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We design a neural network Ansatz for variationally finding the ground-state wave function of the Homogeneous Electron Gas, a fundamental model in the physics of extended systems of interacting fermions. We study the spin-polarised and paramagnetic phases with 7, 14 and 19 electrons over a broad range of densities from rs=1r_s=1rs​=1 to rs=100r_s=100rs​=100, obtaining similar or higher accuracy compared to a state-of-the-art iterative backflow baseline even in the challenging regime of very strong correlation. Our work extends previous applications of neural network Ansätze to molecular systems with methods for handling periodic boundary conditions, and makes two notable changes to improve performance: splitting the pairwise streams by spin alignment and generating backflow coordinates for the orbitals from the network. We illustrate the advantage of our high quality wave functions in computing the reduced single particle density matrix. This contribution establishes neural network models as flexible and high precision Ansätze for periodic electronic systems, an important step towards applications to crystalline solids.
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The authors introduce here WAPNET, a periodic neural network (NN) variational ansatz for solving the ground state of a homogeneous electron gas with high accuracy over a broad range of the density coupling constant rs{r}_{s}rs​. Going beyond recent work for molecules, this contribution establishes NN models as flexible and powerful ansatz for electronic structure calculations in extended systems. In all density regimes, WAPNET-based variational Monte Carlo results are comparable to or better than state-of-the-art benchmarks obtained by diffusion Monte Carlo with iterative backflow.
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The search for exotic phases of matter in quantum systems remains a central goal in condensed matter physics. Recent progress in machine learning has driven the development of powerful new computational methods in quantum chemistry and physics. In particular, the application of neural networks to solve the Schrödinger equation within the variational Monte Carlo (VMC) framework has led to the development of the successful neural network VMC (NNVMC) method. This thesis extends and explores the application of NNVMC to the study of strongly interacting Fermi gases. We first investigated the unitary Fermi gas (UFG)---a system with strong, short-range attractions known to possess a superfluid ground state that is difficult to describe quantitatively. By applying the NNVMC method with the FermiNet Ansatz, we first identified its key limitations before proposing a simple modification. This change led to a substantial improvement in accuracy over the original FermiNet in the UFG, outperforming results from the state-of-the-art simulations while being consistent with experiments. Next, we explored the phases of the 2D spin-imbalanced Fermi gas. The combined power of NNVMC and our modified FermiNet enabled us not only to observe the long-predicted Fulde-Ferrell-Larkin-Ovchinnikov phase but also to discover a previously unknown phase, characterised by the crystallisation of tightly bound Cooper pairs within a uniform sea of unpaired fermions. Lastly, we studied an anisotropic Fermi gas featuring spin-dependent splitting in momentum space --- a prime signature of the recently discovered form of collinear magnetism, termed altermagnetism. We mapped its 2D phase diagram, providing unbiased estimations for the locations of various phases beyond the mean-field approximation. In conclusion, our work has demonstrated that the NNVMC method is a powerful and versatile tool for investigating strongly correlated systems, confirming its potential to uncover novel phenomena in the vast landscape of quantum systems.