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arXiv:2610.08955

From continuum neural quantum states to effective lattice models: Fluctuating Wigner molecules in moiré superlattices

Ahmed Khalifa, Shubhayu Chatterjee

Published Oct 6, 2026• 0 citations
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Abstract

Effective descriptions of quantum materials rely on a separation of energy scales that identifies their low-energy degrees of freedom. When kinetic energy, lattice potential, and interactions compete, this hierarchy breaks down, and both the microscopic state and its effective description are unknown. Continuum neural quantum states (NQS) can find the ground state directly without projecting onto prescribed bands, orbitals, or spins, but their microscopic wavefunctions are difficult to interpret in terms of effective variables. Here, we introduce a wavefunction readout that maps the continuum NQS onto site-resolved charge and spin statistics, allowing effective degrees of freedom to emerge from the microscopic solution. Focusing on moiré transition metal dichalcogenides at filling $ν= 3$, the NQS reveals a low-spin ground state below the ferromagnetic crystal of spin-polarized Wigner molecules. Although the charge density retains triangular molecular motifs, our readout reveals significant charge fluctuations through bound nearest-neighbor doublon--hole pairs. Each molecular charge state has an inert spin-singlet core and rim electrons that carry the spin. Charge and spin fluctuations are therefore intertwined, with antiferromagnetic spin-correlations between sites, motivating an effective description by a spinful two-orbital Hubbard model. More generally, our results demonstrate a route from continuum wavefunctions to effective lattice models when competing scales preclude a controlled reduction.

DOI
10.48550/arxiv.2610.08955

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Effective descriptions of quantum materials rely on a separation of energy scales that identifies their low-energy degrees of freedom. When kinetic energy, lattice potential, and interactions compete, this hierarchy breaks down, and both the microscopic state and its effective description are unknown. Continuum neural quantum states (NQS) can find the ground state directly without projecting onto prescribed bands, orbitals, or spins, but their microscopic wavefunctions are difficult to interpret in terms of effective variables. Here, we introduce a wavefunction readout that maps the continuum NQS onto site-resolved charge and spin statistics, allowing effective degrees of freedom to emerge from the microscopic solution. Focusing on moiré transition metal dichalcogenides at filling ν=3ν= 3ν=3, the NQS reveals a low-spin ground state below the ferromagnetic crystal of spin-polarized Wigner molecules. Although the charge density retains triangular molecular motifs, our readout reveals significant charge fluctuations through bound nearest-neighbor doublon--hole pairs. Each molecular charge state has an inert spin-singlet core and rim electrons that carry the spin. Charge and spin fluctuations are therefore intertwined, with antiferromagnetic spin-correlations between sites, motivating an effective description by a spinful two-orbital Hubbard model. More generally, our results demonstrate a route from continuum wavefunctions to effective lattice models when competing scales preclude a controlled reduction.
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=2ν=2ν=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 sss-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.
Solving fractional electron states in twisted MoTe2_22​ with deep neural network
The emergence of moiré materials, such as twisted transition-metal dichalcogenides (TMDs), has created a fertile ground for discovering novel quantum phases of matter. However, solving many-electron problems in moiré systems presents significant challenges due to strong electron correlation and strong moiré band mixing. Recent advancements in neural quantum states hold the promise for accurate and unbiased variational solutions. Here, we introduce a powerful neural wavefunction to solve ground states of twisted MoTe2 across various fractional fillings, reaching unprecedented accuracy and system size. From the full structure factor and quantum weight, we conclude that our neural wavefunction accurately captures both the electron crystal at ν=1/3ν= 1/3ν=1/3 and various fractional quantum liquids in a unified manner.
Lattice models are powerful tools for studying strongly correlated quantum many-body systems, but their general lack of exact solutions motivates efforts to simulate them in tunable platforms. Recently, a promising new candidate has emerged for such platforms from two-dimensional materials. A subset of moiré systems can be effectively described as a two-dimensional electron gas (2D EG) subject to a moiré potential, with electron-electron interactions screened by nearby metallic gates. In this paper, we investigate the realization of lattice models in such systems. We show that, by controlling the gate separation, a 2D EG in a square moiré potential can be systematically tuned into a system whose ground state exhibits orders analogous to those of the square lattice Hubbard model, including the stripe phase. Furthermore, we study how variations in gate separation and moiré potential depth affect the ground-state orders. A number of antiferromagnetic phases, as well as a ferromagnetic phase and a paramagnetic phase, are identified. We then apply our quantitative downfolding approach to triangular moiré systems closer to current experimental conditions, compare them with the square lattice parameters studied, and outline routes for experimental realization of the phases.
Recent experiments in moiré materials have enabled the realization of a variety of exotic quantum phases. In this context, the Hofstadter-Hubbard model has been proposed as a possible setting for hosting chiral spin liquid. Concurrently, significant progress has been recently made in the computational methods for two-dimensional many-body fermion systems, which makes numerically studying this challenging model a real possibility in genuine 2D geometry. Motivated by these advances, we investigate the putative chiral spin liquid phase in the triangular-lattice Hofstadter-Hubbard model using variational Monte Carlo with neural quantum states (NQS) and projected entangled pair states (PEPS). We observe spin-charge separation directly in real space through numerical spin-pumping simulation and real-time spin and charge motion. In addition, in the context of anyonic superconductivity conjectured in this model, we find a positive two-electron binding energy on small systems, but it decreases below our numerical resolution as the system size increases. Our work demonstrates NQS and PEPS as powerful tools, capable of cross-checking each other, for diagnosing topological order and fractionalized excitations in strongly correlated electronic systems.
Motivated by twisted transition metal dichalcogenides (TMDs), we study an extended Hubbard model with both on-site and off-site repulsive interactions, in which Mott insulating states with concomitant charge order occur at fractional fillings. To resolve the charge ordering as well as the fate of the local moments formed thereby, we perform large-scale density matrix renormalization group calculations on cylindrical geometries for several filling fractions and ranges of interaction strength. Depending on the precise parameter regime, both antiferromagnetically ordered as well as quantum-disordered states are found, with a particularly prominent example being a quantum spin liquid-type ground state on top of charge-ordering with effective Kagomé geometry. We discuss the different mechanisms at play in stabilizing various electronic and magnetic states. The results suggest that moiré TMDs are a promising venue for emergent quantum magnetism of strongly interacting electrons.
Moiré systems provide a rich platform for studies of strong correlation physics. Recent experiments on heterobilayer transition metal dichalcogenide Moiré systems are exciting in that they manifest a relatively simple model system of an extended Hubbard model on a triangular lattice. Inspired by the prospect of the hetero-transition metal dichalcogenide Moiré system's potential as a solid-state-based quantum simulator, we explore the extended Hubbard model on the triangular lattice using the density matrix renormalization group. Specifically, we explore the two-dimensional phase space spanned by the key tuning parameters in the extended Hubbard model, namely, the kinetic energy strength and the further-range Coulomb interaction strengths. We find competition between Fermi fluid, chiral spin liquid, spin density wave, and charge order. In particular, our finding of the optimal further-range interaction for the chiral correlation presents a tantalizing possibility.