Neural Network Discovery of Paired Wigner Crystals in Artificial Graphene

TL;DR

This work identifies a novel Paired Wigner Crystal phase in a honeycomb moiré lattice at filling $ν_m=1/4$ using neural-network quantum Monte Carlo. Opposite-spin electrons form singlet pairs that delocalize over hexagonal rings, and these molecular units crystallize into a triangular lattice, all without confinement or attraction. The result, validated against conventional DMC and supported by BCS-type variational improvements, demonstrates the power of expressive neural-network wavefunctions for uncovering unexpected quantum orders in strongly correlated moiré systems. The findings broaden the phase diagram of moiré materials and suggest experimental avenues to observe paired molecular solids and related exotic states.

Abstract

Moiré systems have emerged as an exciting tunable platform for engineering and probing quantum matter. A large number of exotic states have been observed, stimulating intense efforts in experiment, theory, and simulation. Utilizing a neural-network-based quantum Monte Carlo approach, we discover a new ground state of the two-dimensional electron gas in a honeycomb moire potential at a filling factor of $ν_m =1/4$ (one electron every four moiré minima). In this state, two opposite-spin electrons pair to form a singlet-like valence bond state which restores local $C_6$ symmetry in hexagonal molecules each spanning $6$ moiré minima. These molecules of pairs then form a molecular Wigner crystal, leaving one quarter of the moiré minima mostly depleted. The formation of such a paired Wigner crystal, absent any confining potential or attractive interaction to facilitate "pre-assembling" the molecule, provides a fascinating case of collective phenomena in strongly interacting quantum many-body systems, and opportunities to engineer exotic properties.

Figures (9)

• Figure 1: Discovery of the Paired Wigner crystal state.a, Schematics for realization of a honeycomb lattice in a twisted TMD bilayer device. b, Illustration of the ground state obtained using the conventional DMC approach, which has each electron distributing over two moiré minima and forming an AFM valence bond solid. The top panel shows the landscape of the honeycomb moié potential and electron configurations at quarter filling ($\nu_m=1/4$). c, Illustration of our NQS with a backflow neural network, which transforms electron positions into quasi-positions with a self-attention graph neural network. d, Illustration of the NQS solution, which predicts a triangular Wigner crystal of ring motifs, each made up of a pair of up and down electrons.
• Figure 2: Metallicity and molecular localization.a,b,c, The electron densities for three values of $V_m/W$, as indicated by the dashed lines. Metallicity is measured via the complex polarization $\vert Z\vert$, which is zero for a metal and asymptotically unity for an insulator. The molecular localization $f_m$ measures the degree of localization on the ring motifs, characterized by the ratio of the integrated charge density inside and outside the ring. This ratio will reach unity if no electron density "leaks out of a molecule". At low moiré potential depth, both quantities are zero, giving a metallic state with uniform density and no molecular formation. As the moiré potential deepens, both metallicity and localization become finite as the paired Wigner crystal forms.
• Figure 3: Charge and spin densities and correlation functions.a, The charge (solid) and spin (dashed) densities are plotted along the path depicted in the top-right inset for various potential depths. The color indicates the value of $V_m/W$ as shown in the legend box. The average charge and spin densities are indicated by the arrows. The bottom right inset shows A schematic illustration of the singlet spin state of each molecule at large $V_m/W$. b, Charge-charge correlation function with $V_m/W=0.5$ on the left half and $V_m/W=8.0$ on the right half; c, the corresponding spin-spin correlations.
• Figure 4: Structure and correlations of the pairs.a, Intra-molecule correlation between the pair of up and down electrons on the same ring is characterized by the distribution of their relative angle $\theta$, as illustrated by the inset. Curves of different colors show different moiré well depths. b, Orientational correlation between the nearest-neighbor pairs. The colors of the curves indicate potential well depths, as given in the legend in panel a. The configuration corresponding to each peak is depicted at the top of the plot. c, Correlation between the center-of-mass positions.
• Figure S1: Eigenvalues of orbitals in the projected-BCS wavefunction