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First-Principles AI finds crystallization of fractional quantum Hall liquids

Ahmed Abouelkomsan, Liang Fu

arXiv:2602.03927v2cond-mat.mes-hallcond-mat.str-elcs.AI

TL;DR

Determining when a fractional quantum Hall liquid crystallizes requires an unbiased way to treat fractionalization and charge ordering, particularly with strong Landau-level mixing. The paper introduces MagNet, a torus-based neural variational wavefunction, and finds that energy minimization discovers both FQH liquids and electron crystals across Landau-level mixing.

  • Problem

    Determining when FQH liquids crystallize remains challenging because fractionalization competes with charge ordering and strong Landau-level mixing must be treated without bias toward competing phases.

    Source: pp. 1–2

  • Method

    MagNet is a self-attention neural variational wavefunction on the torus that learns from microscopic-Hamiltonian energy minimization while respecting magnetic translation boundary conditions.

    Source: p. 1

  • Results

    MagNet discovers both FQH liquids and electron crystals across weak-to-strong Landau-level mixing, with correlations evolving toward crystalline modulations as κ increases.

    Source: pp. 1, 4

  • Takeaways & Limitations

    The results highlight first-principles AI as a tool for exploring quantum-matter phase diagrams and discovering correlated phases and microscopic organizing principles.

    Source: p. 2

Abstract

from arXiv · show

When does a fractional quantum Hall (FQH) liquid crystallize? Addressing this question requires a framework that treats fractionalization and crystallization on equal footing, especially in strong Landau-level mixing regime. Here, we introduce MagNet, a self-attention neural-network variational wavefunction designed for quantum systems in magnetic fields on the torus geometry. We show that MagNet provides a unifying and expressive ansatz capable of describing both FQH states and electron crystals within the same architecture. Trained solely by energy minimization of the microscopic Hamiltonian, MagNet discovers topological liquid and electron crystal ground states across a broad range of Landau-level mixing. Our results highlight the power of first-principles AI for solving strongly interacting many-body problems and finding competing phases without external training data or physics pre-knowledge.

Supplementary material for: “First-Principles AI finds crystallization of fractional

The supplementary material lists its authors and affiliation, and identifies additional technical details and variational energies provided there.

Supplementary material

Appendix A: Magnetic Boundary Conditions

The appendix describes how magnetic flux and gauge phases shape boundary conditions on the torus, including their consequences for winding and many-body translations.

Appendix A: Magnetic Boundary Conditions

  • The magnetic field is decomposed into a uniform component and a spatial fluctuation that averages to zero, with the associated vector potential’s uniform-field part linear in position.Source: Appendix A · p. 8
  • On the torus, net magnetic flux prevents the vector potential from being strictly periodic, so translations acquire position-dependent wavefunction phases to preserve physical observables and gauge covariance.Source: Appendix A · p. 8
  • Consistency of the boundary phases requires translations along the two torus directions to produce the same physical state regardless of path.Source: Appendix A · p. 8
  • The total flux through the torus is quantized, and transporting a particle around it gives a phase; a twist phase can also encode flux threaded through the torus handles.Source: Appendix A · p. 9
  • The net winding is fixed by the number of flux quanta and requires zeros, which may be vortices or antivortices; in the lowest Landau level, the wavefunction has Nϕ zeros.Source: Appendix A · p. 9
  • For N particles, translating any one particle around a torus lattice vector transforms the wavefunction by the twist and magnetic boundary phases.Source: Appendix A · p. 9

Appendix B: Neural Network Architecture

The architecture represents torus particle coordinates with periodic features, maps them into shared higher-dimensional embeddings, and processes those embeddings with attention and MLP layers.

Appendix B: Neural Network Architecture

  • For each particle, the architecture forms a periodic feature vector from its coordinates using sine and cosine components along the torus reciprocal vectors.Source: Appendix B · p. 9
  • The reciprocal vectors satisfy G_a · L_b = 2πδ_ab, and the feature vector is mapped into a higher-dimensional particle embedding.Source: Appendix B · p. 9
  • A shared transformation maps every particle coordinate to its initial vector, after which the vectors undergo multi-head self-attention followed by an MLP.Source: Appendix B · p. 9

1. Multilayer Perceptron (MLP)

The appendix specifies the MLP transformation and identifies its linear parameters and activation.

1. Multilayer Perceptron (MLP)

  • The MLP transforms a generic input vector g1 into an output vector g2.Source: §1 · p. 9
  • The transformation uses a linear map W and bias vector b, both in the internal feature dimension.Source: §1 · p. 9
  • Its non-linear activation function F is chosen to be GELU.Source: §1 · p. 9

2. Self-attention

Particle-wise MLPs cannot capture correlations between particles, so MagNet uses self-attention to model how each particle is influenced by the others. Multiple attention heads are combined and passed through an MLP.

2. Self-attention

  • Self-attention captures interparticle correlations that particle-wise MLPs cannot describe.Source: §2 · p. 10
  • Each particle stream is assigned keys, queries, and values, with multiple self-attention operations applied in parallel heads.Source: §2 · p. 10
  • Key-query dot products compare particle streams, while an exponential factor weights correlations to identify relevant particles.Source: §2 · p. 10
  • Attention outputs from the heads are concatenated, transformed, and passed to an MLP.Source: §2 · p. 10

3. Mapping onto N × Ndet amplitudes

The network maps its final output to N × Ndet sets of amplitudes. These amplitudes are permutation invariant in the other particle coordinates, a condition used for a fermionic antisymmetric wavefunction.

3. Mapping onto N × Ndet amplitudes

  • The final network output is mapped to N × Ndet distinct sets of amplitudes.Source: §3 · p. 10
  • Matrices construct the real and imaginary parts of each amplitude.Source: §3 · p. 10
  • For the full ansatz to describe a fermionic antisymmetric wavefunction, each amplitude is chosen to be permutation invariant in the coordinates of the other particles.Source: §3 · p. 10

4. Mapping onto Nϕ × N × Ndet winding maps

The network output is mapped to winding maps indexed by Nϕ, N, and Ndet. This mapping constructs complex zero coordinates and fixes the twist phase to φ = 0.

4. Mapping onto Nϕ × N × Ndet winding maps

  • The final network output is also mapped to construct symmetric many-body functions for the winding maps.Source: §4 · p. 11
  • The map uses Nϕ × N × Ndet indexing, and matrices construct the real and imaginary parts of the complex coordinate of the zero.Source: §4 · p. 11
  • The twist phase is set to φ = 0.Source: §4 · p. 11

5. Jastrow factor

A Jastrow factor is used to enforce the Coulomb cusp condition, with a single trainable parameter.

5. Jastrow factor

  • A Jastrow factor enforces the Coulomb cusp condition.Source: §5 · p. 11
  • The factor uses a single trainable parameter, α.Source: §5 · p. 11

Appendix C: Optimization

Appendix C describes the training setup and shows a representative randomly initialized training curve for κ = 3.0.

Appendix C: Optimization

  • For κ = 3 and 9, training starts from random initialization; for κ = 12, 15, and 20, it starts from an optimized phase at κ = 3 or 9.The authors report that random and optimized-FQH initializations gave the same result in various checked cases.Source: Appendix C · p. 12
  • The network uses a small norm constraint to stabilize training.Table I lists the network hyperparameters.Source: Appendix C · p. 12
  • The representative curve uses N = 12, Nϕ = 36, κ = 3.0, and random initialization, with energy averaged over the previous 4000 steps.Energy is in units of ℏωc and excludes the finite-torus Madelung constant; the green line is lowest-Landau-level projected ED.Source: Figure S1 · p. 12, Appendix C · p. 12

Appendix D: Table of Energies

Appendix D lists energies for N = 9 and N = 12 at various Landau-level mixing values.

Appendix D: Table of Energies

  • The listed energies for N = 9 and N = 12 are measured in units of ℏωc and exclude the Madelung constant.The table is intended to facilitate comparison with exact diagonalization.Source: Appendix D · p. 12

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