# Discovering Quantum Phase Transitions with Fermionic Neural Networks

arXiv: https://arxiv.org/abs/2202.05183 · HTML: https://mlsanity.com/paper/2202.05183/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 31 references that are in the map.
- First posted on arXiv 10 Feb 2022.

## Builds on (most cited first, 1 of 1)

- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)

## Built on by (most cited first, 26 of 26)

- [A Self-Attention Ansatz for Ab-initio Quantum Chemistry](https://mlsanity.com/paper/2211.13672/) (2022)
- [Ab-initio quantum chemistry with neural-network wavefunctions](https://mlsanity.com/paper/2208.12590/) (2022)
- [Generalizing Neural Wave Functions](https://mlsanity.com/paper/2302.04168/) (2023)
- [Gold-standard solutions to the Schr"odinger equation using deep learning: How much physics do we need?](https://mlsanity.com/paper/2205.09438/) (2022)
- [A Kaczmarz-inspired approach to accelerate the optimization of neural network wavefunctions](https://mlsanity.com/paper/2401.10190/) (2024)
- [Sampling-free Inference for Ab-Initio Potential Energy Surface Networks](https://mlsanity.com/paper/2205.14962/) (2022)
- [An ab initio foundation model of wavefunctions that accurately describes chemical bond breaking](https://mlsanity.com/paper/2506.19960/) (2025)
- [Pairing-based graph neural network for simulating quantum materials](https://mlsanity.com/paper/2311.02143/) (2023)
- [Variational Monte Carlo on a Budget -- Fine-tuning pre-trained Neural Wavefunctions](https://mlsanity.com/paper/2307.09337/) (2023)
- [Deep Variational Free Energy Approach to Dense Hydrogen](https://mlsanity.com/paper/2209.06095/) (2022)
- [Towards a Foundation Model for Neural Network Wavefunctions](https://mlsanity.com/paper/2303.09949/) (2023)
- [Transferable Neural Wavefunctions for Solids](https://mlsanity.com/paper/2405.07599/) (2024)
- [Forward Laplacian: A New Computational Framework for Neural Network-based Variational Monte Carlo](https://mlsanity.com/paper/2307.08214/) (2023)
- [First-Principles AI finds crystallization of fractional quantum Hall liquids](https://mlsanity.com/paper/2602.03927/) (2026)
- [QERNEL: a Scalable Large Electron Model](https://mlsanity.com/paper/2604.26018/) (2026)
- [Convergence of variational Monte Carlo simulation and scale-invariant pre-training](https://mlsanity.com/paper/2303.11602/) (2023)
- [Is attention all you need to solve the correlated electron problem?](https://mlsanity.com/paper/2502.05383/) (2025)
- [The neural networks with tensor weights and emergent fermionic Wick rules in the large-width limit](https://mlsanity.com/paper/2507.05303/) (2025)
- [Artificial intelligence for artificial materials: moir'e atom](https://mlsanity.com/paper/2303.08162/) (2023)
- [Diagonal Symmetrization of Neural Network Solvers for the Many-Electron Schr"odinger Equation](https://mlsanity.com/paper/2502.05318/) (2025)
- [Revisiting the Broken Symmetry Phase of Solid Hydrogen: A Neural Network Variational Monte Carlo Study](https://mlsanity.com/paper/2512.17703/) (2025)
- [Ground state phases of the two-dimension electron gas with a unified variational approach](https://mlsanity.com/paper/2405.19397/) (2024)
- [A Theoretical Framework for an Efficient Normalizing Flow-Based Solution to the Electronic Schrodinger Equation](https://mlsanity.com/paper/2406.00047/) (2024)
- [Accurate and thermodynamically consistent hydrogen equation of state for planetary modeling with flow matching](https://mlsanity.com/paper/2501.10594/) (2025)
- [Predicting magnetism with first-principles AI](https://mlsanity.com/paper/2602.09093/) (2026)
- [Honeycomb Crystal in a Mass-Asymmetric Electron-Hole Bilayer](https://mlsanity.com/paper/2610.17030/) (2026)

## Abstract (arXiv)

Deep neural networks have been extremely successful as highly accurate wave function ans"atze 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 is given no a priori knowledge that a phase transition exists, but converges on the translationally invariant ground state at high density and spontaneously breaks the symmetry to produce the crystalline ground state at low density.