Scientists have uncovered a novel ground state of matter within artificial graphene, revealing a paired Wigner crystal formed through an unexpected self-assembly process. Conor Smith from the Center for Computational Quantum Physics at the Flatiron Institute and the Department of Electrical and Computer Engineering at the University of New Mexico, alongside Yubo Yang from the Center for Computational Quantum Physics, Flatiron Institute and the Department of Physics and Astronomy at Hofstra University, Zhou-Quan Wan, Yixiao Chen from ByteDance, Miguel A. Morales from the Center for Computational Quantum Physics, Flatiron Institute and the Department of Physics at the University of Toronto, and Shiwei Zhang utilised a neural-network-based Monte Carlo approach to identify this state in a two-dimensional electron gas subjected to a honeycomb moiré potential. This research demonstrates the spontaneous formation of molecules comprising paired electrons, which then organise into a Wigner crystal without any external guiding potential or attractive forces, offering a compelling example of emergent collective behaviour and opening avenues for the design of materials with unique electronic characteristics.
For decades, physicists have sought to understand how electrons arrange themselves in complex materials. Now, an artificial graphene system reveals an unexpected, self-organised pattern where electrons pair up and form crystalline structures, offering a fresh perspective on collective electron behaviour and potential control over material properties.
Scientists are increasingly focused on moiré systems as tunable platforms for investigating quantum matter — these artificially created structures, arising from the interference of two overlaid lattices, have already exhibited a range of exotic states. Prompting considerable research across experimental and theoretical physics. This new state emerges at a specific filling factor, where one electron occupies every four minima within the moiré pattern. Presents a unique arrangement of electron pairings.
Unlike many previously observed correlated states, this arrangement does not require external constraints or attractive forces to initiate molecular formation. Instead, electrons with opposing spins combine to form singlet-like valence bonds, restoring hexagonal symmetry within molecules that span multiple moiré minima. Then, these paired molecules self-assemble into a molecular Wigner crystal, a configuration where electrons are spatially separated due to strong repulsive interactions. With a quarter of the moiré minima remaining largely unoccupied.
This spontaneous organization represents a compelling example of collective behaviour in strongly interacting quantum systems. By understanding the full range of possibilities within these moiré landscapes remains a challenge, as conventional computational techniques often struggle to capture emergent orders. By integrating neural networks into a physically informed quantum Monte Carlo approach. The team achieved a balance between predictive accuracy and the ability to simulate sufficiently large systems.
The resulting neural network quantum state (NQS) solution revealed a stark contrast to the state predicted by conventional Diffusion Monte Carlo (DMC) methods. Here, this typically yield an antiferromagnetic valence bond solid. At low moiré potential depths, the system exhibited metallic behaviour with uniform electron density and no molecular formation — however, as the moiré potential deepened, both metallicity and molecular localization increased, indicating the formation of the paired Wigner crystal. Beyond expanding the known catalogue of moiré phases, this discovery highlights the promise of neural network techniques to uncover unexpected quantum phenomena and suggests avenues for engineering materials with tailored properties. The method integrates neural networks into a physically motivated variational ansatz, balancing predictive capability with the ability to model sufficiently large system sizes.
Through employing this method, researchers aimed to overcome limitations in conventional computational techniques that struggle with strongly correlated quantum many-body systems. To establish the precise electronic configuration required careful consideration of the system’s potential, focusing on a filling factor of one electron every four moiré minima and investigating the behaviour of electrons within the periodic potential created by the moiré pattern.
Once the system parameters were defined, the neural network was trained to accurately represent the many-body wavefunction, a complex mathematical description of the electrons’ collective state. At the heart of this approach lies a ‘backflow’ neural network, which transforms electron positions into quasi-positions using a self-attention graph neural network.
Accurately capturing the electron correlations demanded a sophisticated wavefunction ansatz. Instead of relying on traditional approximations, The project team used a neural network to learn an optimal wavefunction directly from the underlying physics, iteratively refining the network’s parameters to converge towards the ground state, the lowest energy configuration of the system.
For comparison, a standard Diffusion Monte Carlo (DMC) projection using a Slater, Jastrow, backflow wavefunction was also performed — assessing the resulting electronic state involved analysing both metallicity and molecular localization. Metallicity was quantified via the complex polarization, a measure of how easily electrons can move through the material, and meanwhile, molecular localization was determined by calculating the degree to which electrons remain confined within the ring-like motifs formed by the paired electrons. Through tracking these quantities as the moiré potential depth was varied, while researchers could observe the emergence of the paired Wigner crystal and confirm its insulating behaviour.
Singlet Pairing and Molecular Wigner Crystal Formation in a Honeycomb Moiré Potential
At a filling factor of one electron every four moiré minima. Researchers discovered a new ground state within the two-dimensional electron gas residing in a honeycomb moiré potential. Here, this state features oppositely-spinning electrons pairing to form singlet-like valence bonds, restoring hexagonal symmetry within molecules that span moiré minima. These paired molecules then assemble into a molecular Wigner crystal, depleting approximately one quarter of the moiré minima.
In turn, the formation of this paired Wigner crystal, occurring without any pre-existing confining potential or attractive interactions, represents a compelling example of collective behaviour in strongly interacting systems. Detailed analysis of intra-molecular correlations reveals that opposite-spin electrons within the same molecule tend to occupy opposite ends, maximising their separation at an angular separation of π.
Configurations with angular separations of 2Ï€/3 and 4Ï€/3 occur with reduced probability, while arrangements where both electrons reside on the same side of the ring are strongly suppressed. These observations support a model of each singlet molecule behaving as an effective electric dipole, possessing a centre-of-mass position and a dipole moment that can rotate freely to preserve C6 symmetry.
As the moiré potential depth increases, the dipole moment grows and the dipole structure becomes more defined. Also, the centres-of-mass of these molecules arrange themselves in a triangular lattice, as evidenced by their pair correlation function. When examining the alignment angle between neighboring singlet dipoles, perfect alignment and anti-alignment correspond to specific angles. Distributions of alignment angles at several moiré depths show clear peaks indicative of ordered dipole arrangements.
Beyond this specific state, The project identifies at least two distinct phases depending on the potential depth — transitioning from a metallic liquid at low potential to a paired Wigner crystal with global charge localization at larger moiré promise strengths. Through replacing a Slater determinant with a projected-BCS structure lowered the energy at an intermediate prospect depth, and the possibility of superconductivity or supersolid-like behaviour, prompting further investigation with more refined detection measures and larger system sizes. At the same time, the effort highlights the prevalence of long-range charge order alongside short-range spin correlations in the form of singlet pairing across a wide range of promise depths. Inspiring both theoretical exploration and facilitating experimental detection.
Electron pairs self-assemble into crystalline structures within twisted materials
Scientists are beginning to uncover unexpected order within what appears to be electronic chaos — recent work detailing the behaviour of electrons trapped within moiré superlattices, artificial structures created when two materials are twisted relative to one another. Reveals a novel state of matter: a ‘molecular Wigner crystal’. This isn’t simply electrons arranging themselves neatly. Instead, pairs of electrons bind together, forming molecules that then organise into a crystalline pattern.
For years, achieving such a configuration has proven difficult, as strong interactions between electrons typically lead to disorder rather than predictable arrangements. With advanced computational techniques, researchers have bypassed the need for external forces or attractive interactions to ‘pre-assemble’ these molecular units. Instead, the system self-organises, driven by the unique geometry of the moiré potential.
Beyond the immediate discovery, this effort offers a new avenue for controlling electron behaviour, potentially leading to devices with tailored electrical properties. The reliance on computationally intensive methods presents a limitation, as simulating these systems remains a significant challenge. The biggest question remains how these findings translate to real materials and devices.
While moiré superlattices offer a promising platform, fabricating them with sufficient precision and scale is not trivial — unlike previous observations of Wigner crystals requiring extreme conditions, this new state emerges at relatively accessible temperatures. Once further refined, this could open doors to exploring correlated electron phenomena at higher temperatures, simplifying experimental setups, and this represents a shift from simply observing exotic states to actively designing materials that host them. The next step will likely involve exploring similar arrangements with different filling factors and lattice geometries.