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arXiv:2610.03145

Correlated Metals, Metamagnetism, and Orbital Nematic Order in moiré Materials with Neural Quantum States

Wei Zhang, Ataç İmamoğlu, Tao Shi, Eugene Demler, Ivan Morera

Published Oct 2, 2026• 0 citations
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Abstract

In this work, we investigate a continuum model of two-dimensional electrons interacting via long-range Coulomb forces and subject to a triangular periodic potential at filling factor $ν=2$. Using neural quantum states based on message-passing neural networks optimized with variational Monte Carlo, we determine the ground-state phase diagram as a function of periodic potential strength and Zeeman field. Although the noninteracting system is expected to form a conventional band insulator, we show that strong correlations can qualitatively reshape the insulating state and, in part of the phase diagram, suppress the band insulator in favor of a correlated metallic phase, with a first-order transition separating the two regimes. Moreover, the metal--insulator transition exhibits a strong dependence on spin polarization: the unpolarized system becomes insulating at substantially weaker periodic potentials through the formation of an $s$-wave molecular insulator, where opposite-spin electrons bind near each potential minimum, whereas the fully polarized system requires a significantly deeper potential and instead forms a symmetry-broken orbital nematic molecular crystal. This spin-dependent molecular transition gives rise to discontinuous magnetization curves, metamagnetic transitions, and several multicritical points in the phase diagram, together with clear signatures in the density profiles, real-space correlation functions, and structure factors. To further elucidate the role of the long-range character of the Coulomb interaction, we also study short-range repulsive interactions, which can be realized with ultracold-atom platforms. We find that the qualitative structure of the many-body phase diagram remains unchanged, demonstrating the robustness of our results across different interaction models.

DOI
10.48550/arxiv.2610.03145

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We use a convolutional restricted Boltzmann machine (CRBM) neural network to construct a variational wave function (WF) for the Hubbard model on a square lattice and study it using the variational Monte Carlo (VMC) method. In the wave function, the CRBM acts as a correlation factor to a mean-field BCS state. The number of variational parameters in the WF does not grow automatically with the lattice size and it is computationally much more efficient compared with other neural-network-based WFs. We find that, in the intermediate- to strong-coupling regime of the model at half filling, the wave function outperforms even the highly accurate long-range backflow-Jastrow correlated wave function. Using the WF, we study the ground state of the half filled model as a function of onsite Coulomb repulsion UUU. We consider two cases for the next-nearest-neighbor hopping parameter, e.g., ${t}^{\ensuremath{'}}=0$ as well as a frustrated model case with ${t}^{\ensuremath{'}}\ensuremath{\ne}0$. By examining several quantities, e.g., double occupancy, charge gap, momentum distribution, and spin-spin correlations, we find that the weekly correlated phase in both cases is paramagnetic metallic (PM). As UUU is increased, the system undergoes a first-order Mott transition to an insulating state at a critical Uc{U}_{c}Uc​, the value of which depends on ${t}^{\ensuremath{'}}$. The Mott state in both cases is spin gapped with long-range antiferromagnetic (AFM) order. Remarkably, the AFM order emerges spontaneously from the wave function which does not have any explicitly broken symmetry in it. Apart from some quantitative differences in the results for the two values of ${t}^{\ensuremath{'}}$, we find some interesting qualitative differences in the way the Mott transition takes place in the two cases.
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