First-Principles AI finds crystallization of fractional quantum Hall liquids

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.

💡 Research Summary

The authors address a long‑standing problem in two‑dimensional electron systems under strong magnetic fields: determining the conditions under which a fractional quantum Hall (FQH) liquid gives way to a crystalline (Wigner) phase, especially when Landau‑level (LL) mixing is strong. To treat fractionalization and crystallization on an equal footing they introduce MagNet, a self‑attention neural‑network variational wavefunction specifically designed for torus geometry, where periodic boundary conditions and magnetic translation symmetries are exact.

The many‑body Hamiltonian consists of kinetic energy (scale K = ℏωc) and Coulomb interaction (scale U), with the dimensionless mixing parameter κ = U/K controlling the degree of LL mixing. In the limits κ → 0 and κ → ∞ the ground state is respectively a lowest‑LL FQH liquid and a classical Wigner crystal; intermediate κ is a non‑perturbative regime where the competition is poorly understood.

MagNet builds on the “Fermi‑Set” universal fermionic network idea but adds crucial innovations. The wavefunction is written as a Jastrow factor times a determinant of generalized orbitals ϕₙⱼ(r_i; {r_{≠i}}). Each orbital is factorized into a periodic part Fₙⱼ and a quasi‑periodic part χₙⱼ that carries Nϕ vortices (zeros) required by the torus magnetic boundary condition. The positions of these zeros are not fixed; they are represented by winding maps η_{n,α}^j({r}), which are fully many‑body, periodic, symmetric functions learned by the neural network. By allowing η to depend on all particle coordinates, χ becomes non‑holomorphic, enabling the ansatz to describe states beyond the lowest LL.

The network input consists of sin(G_a·r_i) and cos(G_a·r_i) for the two reciprocal lattice vectors G₁, G₂, guaranteeing periodicity. These inputs pass through L layers of self‑attention followed by multilayer perceptrons. The final layer outputs the parameters that define η (the zero maps) and the amplitudes of Fₙⱼ. The full set of variational parameters {θ} is optimized solely by minimizing the expectation value of the microscopic Hamiltonian, using stochastic reconfiguration or similar gradient‑based methods. No external training data, Laughlin‑type trial functions, or explicit LL truncation are supplied.

The method is benchmarked at filling ν = 1/3 with N = 12 electrons on a hexagonal torus (Nϕ = 36). For κ ≈ 3 the optimized wavefunction reproduces hallmark liquid signatures: the pair‑correlation function g(r) is featureless, the static structure factor S(q) shows no Bragg peaks, and decomposition into center‑of‑mass momentum sectors yields weight only in three sectors, exactly matching the topological degeneracy of the ν = 1/3 Laughlin state. Energy comparison with lowest‑LL projected exact diagonalization (ED) shows MagNet achieving lower variational energies across κ, demonstrating its ability to incorporate LL mixing without truncation.

Increasing κ gradually transforms g(r) from a smooth liquid profile to one with pronounced oscillations, while S(q) develops a peak at the crystal ordering vector K. Finite‑size scaling of the peak height S(K) for κ = 12 and 15 shows only modest growth with system size, suggesting that in the thermodynamic limit the transition may be weakly first‑order or that larger systems are needed to resolve true long‑range order. Nonetheless, the continuous evolution captured by a single variational family provides a unified picture of the FQH‑to‑crystal transition.

In summary, MagNet constitutes a first‑principles AI solver for a paradigmatic strongly correlated problem. It respects exact magnetic translation symmetry on the torus, encodes the required winding number through learnable many‑body zero maps, and, via self‑attention, attains sufficient expressive power to represent both topologically ordered liquids and charge‑ordered crystals. The work showcases how neural‑network wavefunctions can explore phase diagrams without bias, discover competing phases, and offer microscopic insight into the interplay of topology and crystallization. Future extensions could address larger particle numbers, other filling fractions, and more exotic competing orders such as Hall crystals or composite fermion crystals.