First-Principles AI finds crystallization of fractional quantum Hall liquids

TL;DR

It is shown that MagNet provides a unifying and expressive ansatz capable of describing both FQH states and electron crystals within the same architecture, highlighting 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.

Abstract

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.

Figures (5)

• Figure 1: Schematic illustration of the MagNet architecture: Electron coordinates (in the presence of magnetic field) are mapped to higher dimensional feature space which is passed to $L$ layers of successive self-attention and multi-layer perceptron blocks. The final output undergo two different projections to form the generalized orbitals \ref{['eq:phiproduct']} which build the full variational ansatz. For simplicity, we show a plane pierced by a magnetic field but our ansatz is designed for the torus geometry.
• Figure 2: (a) Pair correlation function $g({\boldsymbol{r}})$ at $\kappa = 3.0$. (b) The structure factor $S({\boldsymbol{q}})$ along a line cut at $\kappa = 3.0$. (c) The overlap $|\langle \Psi|\Phi_{K_1} \rangle|^2$ of the optimized wavefunction $\Psi$ at $\kappa = 3.0$ with its projection $\Phi_{K_1}$ defined as the eigenstates of CM magnetic translation operator along $L_1$. (d) Comparison of the variational energy of the NN ansatz $E_{\rm NN}$ (in units of $\hbar \omega_c$) against the energies obtained from exact diagonalization $E_{\rm LLL-ED}$ projected onto the lowest Landau level for various values of $\kappa$. All calculations are performed at filling $\nu=1/3$ ($N = 12$ and $N_{\phi} = 36$) on a hexagonal torus with equal aspect ratio.
• Figure 3: Pair correlation function $g({\boldsymbol{r}})$ at $\nu = 1/3$ for various values of the Landau level mixing parameter $\kappa$. All calculations are performed with $N = 12$ and $N_\phi = 36$ on a hexagonal torus with equal aspect ratio.
• Figure 4: Structure factor $S({\boldsymbol{q}})$ evaluated at the ordering wavevector ${\boldsymbol{q}}={\boldsymbol{K}}$ corresponding to a crystal with one electron per unit cell, plotted as a function of particle number $N$ for various values of $\kappa$. The reported $S( {\boldsymbol{q}} = {\boldsymbol{K}})$ is rotationally averaged over the six $C_6$-related ordering wavevectors.
• Figure S1: Training curve for $N = 12$ and $N = 36$ at $\kappa = 3.0$ with random initialization. Energy at a given step is averaged over the previous 4000 steps and it is in units of $\hbar \omega_c$ and does not include the Madelung constant of this finite torus. The green line denotes the energy of the lowest landau level projected ED.