Emergent Wigner phases in moiré superlattice from deep learning

Abstract

Moiré superlattice designed in stacked van der Waals material provides a dynamic platform for hosting exotic and emergent condensed matter phenomena. However, the relevance of strong correlation effects and the large size of moiré unit cells pose significant challenges for traditional computational techniques. To overcome these challenges, we develop an unsupervised deep learning approach to uncover electronic phases emerging from moiré systems based on variational optimization of neural network many-body wavefunction. Our approach identifies diverse quantum states, including emergent phases such as generalized Wigner crystals, Wigner molecular crystals, and Wigner covalent crystals. These discoveries provide insights into recent experimental studies and suggest more phases for future exploration. They also highlight the crucial role of spin polarization in determining Wigner phases. More importantly, our proposed deep learning approach is proven general and efficient, offering a powerful framework for studying moiré physics.

Introduction

Moiré superlattices, spanning from twisted graphene^1,2,3, transition metal dichalcogenides (TMDs)^4,5,6,7,8,9, and other systems, have attracted significant research interest in condensed matter physics over the last few years¹⁰. They provide a flexible platform to tune the electronic, magnetic, and optical properties of materials, and explore strongly correlated and topological phenomena such as unconventional superconductivity^2,11 and correlated insulating behaviors^{2,3,4,5,6,7,8,9}. For example, recent discoveries of quantum phases, including the generalized Wigner crystal^6,7,8,9, known for their unique symmetry and particle arrangements, have provided deep insights into the behaviors of strongly correlated systems. These insights have been further enriched recently by Wigner molecular crystals emerging from multi-hole artificial atoms in twisted WS₂ homobilayer^12,13, underscoring the tunable feature of moiré superlattices.

Theoretically, a grand challenge remains that a general theoretical approach to handling strong electron correlation effects of moiré system is lacking, and the current investigations mostly rely on traditional computational methods, such as classic Monte Carlo⁶ and Hartree-Fock approximation^12,13,14. While these calculations may provide qualitative explanations for experimental results and make other predictions, it is questionable whether all essential correlated physics has been captured. The importance of electron correlation effects on the phase transitions in a moiré system is further highlighted with a combination of auxiliary field quantum Monte Carlo and diffusion Monte Carlo¹⁵. In recent years, powerful approaches based on deep learning architecture have been developed to treat quantum many-body problems. Based on the universality and expressiveness of neural networks, quantum many-body states can be well-represented without assuming a limited traditional ansatz, and more insights into the correlation effects are revealed. However, so far, these deep learning approaches can only be applied to simple lattice models^16,17,18, chemical molecules^19,20,21,22, uniform electron gases^{23,24,25,26,27,28,29}, small solids²³, and isolated Wigner molecules³⁰.

In this work, we develop a neural network based wavefunction methodology for moiré systems. The neural network wavefunction is trained via the variational quantum Monte Carlo method in an unsupervised manner, which has a quartic scaling with the number of electrons. Focusing on the family of TMD materials over a wide range of particle fillings, we unveil a sequence of intriguing phases. At fractional fillings, symmetry preserved and broken Wigner crystal emerges with varying fillings. At integer fillings, electron correlations lead to a rich diagram of Wigner phases, including the recently observed Wigner molecular crystals, and Wigner covalent crystals discovered in this work. This study underscores the pivotal role of electron correlations captured by deep learning in unlocking and leveraging the quantum phenomena inherent in moiré superlattices.

Results

Deep learning architecture

Figure 1 summarizes the overall workflow. First of all, to explore the quantum phases of moiré materials, a neural network is employed to represent the many-body wavefunction. The neural network wavefunction Ψ_net has the following form

$$\begin{array}{c}{\Psi }_{{{{\rm{net}}}}}=\det [{e}^{i{{{{\bf{k}}}}}_{i}\cdot {{{{\bf{r}}}}}_{j}}{u}_{{{{{\bf{k}}}}}_{i}}({{{{\bf{r}}}}}_{1},...,{{{{\bf{r}}}}}_{N})]\,.\end{array}$$

(1)

Fig. 1: Workflow of deep learning approach.

a plotted structure of moiré materials, particles are constrained by moiré potential and interact with each other via Coulomb interaction. b a sketch of deep learning approach. The upper part illustrates a neural network wavefunction for a moiré superlattice. Particle features r are converted to be periodic and then fed into neural networks, forming the many-body wavefunction ansatzΨ. The neural network is then trained using unsupervised learning. Specifically, variational Monte Carlo is applied to sample patterns and minimize the energy expectation. During the training iterations, the neural network wavefunction gradually approaches the ground state. c emergent Wigner crystal phases in our simulation, including Mott insulator, Wigner molecular crystals, and other exotic phases.

Compared with traditional wavefunction, Ψ_net explicitly includes the features of all particles (r₁, . . . , r_N) in the formulation, permitting it to capture all the correlations. Specifically, particle features r_i are converted to be periodic and fed into permutation-equivalent neural networks^20,23 to construct the generalized Bloch functions ${u}_{{{{{\bf{k}}}}}_{i}}({{{{\bf{r}}}}}_{1},...,{{{{\bf{r}}}}}_{N})$ with plane-wave envelope e^ik⋅r in Eq. (1), which together with the Slater determinant form a periodic, complex-valued, and antisymmetric many-body wavefunction for superlattice. Millions of parameters are embedded in the neural network, making it a powerful and convenient tool for expressing quantum states of distinct symmetries.

Given the neural network wavefunction and the Hamiltonian of a real material, one can employ the variational Monte Carlo (VMC) method to efficiently optimize the neural network parameters. VMC is based on the variational principle of quantum mechanics, which states the ground state has the lowest energy among all the solutions to the stationary Schrödinger equation. Therefore, the VMC training of the neural network is an unsupervised process, which further guarantees the reliability of our deep learning approach.

Moiré superlattice

Moiré superlattice appears when two layers of van der Waals material overlap each other, with mismatched lattice constants and/or twisted angles. The superlattice formed features a significantly large lattice constant and contains thousands of atoms in the unit cell, which makes it unrealistic to employ the full ab initio Hamiltonian of a moiré material in the current workflow. Therefore, in this work, we focus on the low-energy effective Hamiltonian of TMDs⁴, which reads

$$\begin{array}{c}\hat{H}=\sum\limits_{i}\left[-\frac{{\Delta }_{i}}{2{m}^{* }}+{V}_{M}({{{{\bf{r}}}}}_{i})\right]+\frac{1}{2}\sum\limits_{i\ne j}\frac{1}{\epsilon | {{{{\bf{r}}}}}_{i}-{{{{\bf{r}}}}}_{j}| }\,,\\ {V}_{M}({{{\bf{r}}}})=-2V{\sum }_{i=1}^{3}\cos ({{{{\bf{b}}}}}_{i}\cdot {{{\bf{r}}}}+\phi )\,.\end{array}$$

(2)

This model describes doping holes of TMD materials near Fermi surface and m* refers to the effective mass of valence band edge. V_M is moiré potential experienced by holes across the whole moiré materials. b_i denotes the reciprocal vector of moiré cell, and V, ϕ are parameters depending on the material. When moiré materials get twisted to critical angles, flat band appears and Coulomb interaction becomes dominant, which accounts for the last term in Eq. (2) with ϵ referring to effective dielectric constant. A uniform charge background is also necessary to make the whole system neutral and remove divergence in Coulomb interactions.

The effective Hamiltonian has included the key feature of Coulomb interactions and greatly simplifies the problem, so it has been successfully used in studies of TMD heterobilayers such as WSe₂/WS₂ and Γ-valley homobilayers such as twisted WS₂^4,31. It is worth noting that electron correlations in such an effective Hamiltonian are as non-trivial as the ab initio Hamiltonian, and an exact solution remains extremely difficult. The most prevalent Hartree-Fock method treats particles independent from each other, which would lose the correlation effects. More accurate methods such as the full configuration interaction theory exist^14,32,33, but they are only applicable to isolated small molecules. Our deep learning wavefunction approach achieves an optimal balance of accuracy and efficiency, making it a very promising tool for studying moiré systems.

Generalized Wigner crystal

In 1934, Eugene Wigner imagined an electron-only crystal, where electrons spontaneously organize into special patterns as their density decreases. The pursuit of Wigner crystals has continued ever since, and driven significant condensed matter physics discoveries for decades. Recently, the investigation is expanded to generalized Wigner crystals in moiré materials^6,7,8,34. We first employ our deep learning methods to reproduce the experimentally observed Wigner phases in a WSe₂/WS₂ heterobilayer at fractional particle fillings ν ≤ 1, which serves as a validation of our methodology. The results are presented in Supplementary Fig. 5. In line with recent experimental results^6,7,8,9, we observe a Mott insulator phase instead of metal at ν = 1 manifesting the strong correlation characters of such systems. We also observe various Wigner crystal phases at ν = 1/3, 2/3 with C₃ symmetry, and symmetry-breaking stripe phases at ν = 1/2, 2/5, 1/6, which are also consistent with recent optical anisotropy and electronic compressibility measurements³⁴. It’s also worthy noting that Wigner crystal can be disordered if long range charge order is considered^14,35, and more theoretical and experiment efforts are needed to clarify this problem.

Wigner molecular crystal

Beyond the aforementioned fractional filling, recent research has been extended to multiple integer fillings of moiré superlattice, wherein multiple particles are assigned to a single moiré cell^12,13. These particles aggregate and form a “molecule", naturally residing at the minimum of the moiré potential. Upon the activation of Coulomb repulsion, these molecules slightly disperse and form the so-called Wigner molecular crystal¹². These molecular crystals are recently observed in 2H WS₂ homobilayer via scanning tunneling microscopy (STM)¹³, prompting us to employ our neural network to explain this observation.

Quantum states of 2H WS₂ with different fillings ν and spin quantum numbers S_z are investigated and the results are plotted in Fig. 2. At ν = 2 fillings, we find S_z = 0 state has the lowest energy in which two spin-opposite holes are all placed in AB region. If we align their spins parallel, one hole will depart due to Pauli exclusion and transfer to the B^S/S region, forming a charge-transfer insulator as an excited state³⁶. A more intricate situation happens at ν = 3. The previous understanding of the experimental pattern at ν = 3 presumes the ground state to be fully spin-polarized¹³, while we find that the S_z = 1/2 state presents a rather similar trimer pattern with S_z = 3/2 and a slightly lower energy 0.4 meV / hole (Supplementary Fig. 3). In either trimer pattern, particles mainly accumulate at the vertexes of the triangle, while the center remains empty to minimize Coulomb interactions. More detailed experimental characterization is desired to resolve such a delicate competition and confirm the true ground state. Concerning ν = 4, S_z = 1 state is identified as the ground state, exhibiting a more hollow triangle pattern. It is also intriguing to see a flipping of the triangle when the system enters a fully polarized S_z = 2 state, which can be further verified in experiment by applying a magnetic field. Overall, the predicted ground-state patterns from deep learning are perfectly consistent with recent STM observations at different fillings¹³, showing the reliability of our deep learning approach.

Fig. 2: Calculated density pattern of 2H WS₂ homobilayer.

a moiré potential of 2H WS₂ homobilayer. a_M denotes length of moiré cell and is set to 9.8nm. Colored circles denote regions with different stack patterns. b–c calculated ground state and excited state of 2H WS₂ at different particle fillings ν and magnetic numbers S_z. These are reported as (ν, S_z) on top of the sub-panels.

Predicting more phases

Having demonstrated the capability of our deep learning approach to reproduce the experimentally observed phases, we now proceed to further explore more phases. As an example, we examine the WS₂ homobilayer with 3R configuration, which differs from 2H configuration due to the opposite orientations of layers³¹. Notably, the energies of different Bernal regions (B^W/S and B^S/W) coincide with each other in 3R WS₂, resulting in an underlying C₆ symmetry (see Fig. 3a).

Fig. 3: Calculated density pattern of 3R WS₂ homobilayer.

a Moiré potential of 3R WS₂ homobilayer. a_M denotes length of moiré cell and is set to 16.51 nm. Colored circles denote different stacking patterns displayed in the lower panel. b Calculated ground states at different particle fillings ν and mangetic numbers S_z. These are reported as (ν, S_z) on top of the sub-panels. c Phase transition of (6, 3) state at different interaction ratio λ. λ measures the strength ratio between Coulomb repulsion and moiré potential attraction. Moiré potential amplitude V is tuned to control λ. The orientation of trimers get flipped with increasing λ to avoid strong Coulomb repulsion.

The calculated ground states at different particle fillings are plotted in Fig. 3b. The first phenomenon we observe is the spontaneous symmetry breaking in the moiré pattern. At ν = 1, holes are located at corners of the honeycomb and leave the adjacent sites empty, which breaks C₆ symmetry to C₃ and minimizes Coulomb interactions. This broken C₆ symmetry will then get restored if doping one more hole. All moiré potential minima will be occupied by holes and make a perfect honeycomb lattice. As the doping increases to ν = 3, two spin-opposite holes are forced to accompany each other in one corner of the honeycomb despite the Coulomb repulsion, forming a more diffuse pattern than the adjacent singly occupied site. This leads to two distinct density patterns appearing in the sites of the honeycomb, breaking the C₆ symmetry to C₃, and the symmetry will again get restored at ν = 4 since all the sites are doubly occupied then. At ν = 5, a special density pattern appears, which exhibits a mirror symmetry. It is also worth noting that (5, 3/2) state are nearly degenerate with the calculated ground state (5, 1/2) while showing a distinct C₃ symmetry in Supplementary Figs. 4 and 5.

When the filling ν reaches 6, trimers form in the sites of the honeycomb lattice. The trimers have different orientations and an intriguing phase transition can be induced by tuning either the Coulomb repulsion or the moiré potential attraction. In Fig. 3c, we plot the zoom-in pattern of the (6, 3) state at varying interactions, defined by the interaction ratio λ ~ ϵ⁻¹V^−1/4 12. When the moiré potential attraction dominates over the Coulomb repulsion, these two triangles get placed in a tip-to-tip manner, which resembles the shape of moiré potential. As the Coulomb repulsion gets stronger, triangles become larger and gradually melt into circles, finally getting flipped and forming a bottom-to-bottom pattern. Holes are then placed in the saddle points of moiré potential, which avoids strong Coulomb interaction compared to the tip-to-tip pattern.

Wigner covalent crystal

Up to now, most discovered Wigner phases appear as atomic crystals and molecular crystals, where “atoms” or “molecules” are isolated units that resemble chemical atoms and molecules. The equivalent of bonding commonly seen in molecules is notably absent in studies of Wigner phases. Here, by tuning the filling number and spin polarization, we show that it is possible to form covalent bonding in moiré systems. Figure 4a shows a schematic plot of the formation mechanism of a covalent bond between two Wigner molecules. Assuming each molecule already has several holes forming a relatively stable state, then excess holes would feel repulsive interactions from the existing holes. They will be pushed towards the middle regime of two molecules and reach equilibrium. This mechanism is similar to the “sharing” of electrons between two covalent bonded molecules, while the latter covalent bond is formed by overlapping orbitals of adjacent molecules. Such a mechanism can be further facilitated by enhancing Pauli exclusion, which is achievable by applying a magnetic field to align the spin of all holes in the same orientation (Fig. 4a).

Fig. 4: Calculated covalent states of 3R WS₂ homobilayer.

a Schematic plot of covalent bond formation in Wigner phases. Upper panel: formation of covalent bond in a spin polarized system. Colored circles denote B^W/S and B^S/W regions of 3R WS₂, respectively. Gray circles denote regions around the centered sublattices. Lower panel: particle patterns of the monomer, dimer and trimer bond. Black dots denote electrons around sublattices. b particle density pattern of several calculated covalent states. a_M is set to 16.51 nm. Tuples above the sub-panels denote the particle fillings and the magnetic number of spin (ν, S_z). In each sub-panel, the right figure shows a zoom-in of the covalent bond.

Figure 4b demonstrates the emergence of covalent bonds in fully polarized Wigner phases. Specifically, the first panel shows the formation of covalent bonds in a fully polarized ν = 3 state when an additional hole is doped into the C₆ symmetric ν = 2 state. Consequently, the honeycomb lattice of the ν = 2 state deforms due to Coulomb interactions, breaking its C₆ symmetry down to C₂. Similar situations occur in polarized ν = 5 and 7 states, except that the honeycomb sites are now occupied by dimers and trimers, respectively. As we further increase the filling to ν = 9, honeycomb sites are connected in a network of covalent bonds, forming a novel crystal, which we dub “Wigner covalent crystal” following the existing names of Wigner crystal and Wigner molecular crystal.

Conclusions

Moiré materials host various novel phenomena, tightly related to strong correlations. Despite the rapid development of experimental observation, the simulation approach of these materials remains at an early stage, which hinders a thorough understanding of the strongly correlated phenomena. In this work, we described a deep learning approach to simulate moiré materials with a rigorous treatment of electron correlations. Our simulations reveal numerous exotic phases with varying symmetries, demonstrating the generality, accuracy, and efficiency of this approach for exploring moiré materials. Besides providing novel moiré patterns, our framework can also be extended to predict more material behaviors and properties, bypassing certain experimental constraints. This work will help accelerate the discovery of moiré materials, including multi-layer systems, and their applications in the future. Furthermore, our approach shows promise for broader research areas, such as the quantum transparency phenomenon and the anomalous Hall effect.

Methods

Effective model of TMD

Moiré materials are known to contain enormous atom numbers in the unit cell, which pose a great challenge for direct ab initio simulations. Nonetheless, an effective model can be derived for doping holes in TMD materials to simplify the problem^4,31.

According to the density functional theory (DFT) result^4,31, heterobilayers such as WSe₂/WS₂ exhibit valence band maximums located at K and $K^{\prime}$ points of the Brillouin zone, which have the same energy but opposite spin due to spin-orbit coupling. On the other hand, homobilayers such as WS₂ show valence band maximums located at Γ point with spin degeneracy, as a result of Kramers’ theorem. In either case, the most active valence states show a spin degeneracy, which can be treated as a quasi-particle with spin degrees as shown in Eq. (2). Furthermore, the effective mass m* is fitted to describe the dispersion of DFT valence band edge and moiré potential is derived to mimic the interaction between doping holes and inertia electrons far below Fermi surface, see refs. ^4,31,37 for more detailed derivations. When the moiré system gets twisted to critical angles, flat bands appear indicating that the effective mass of the particle approaches infinity. Kinetic energy vanishes and Coulomb interaction dominates in the Hamiltonian, which requires high-accuracy wavefunction methods to incorporate correlations among particles.

It’s also worth noting that the moiré potential of 3R materials shows a slightly different form from Eq. (2)³¹, which reads

$${V}_{M}^{{{{\rm{3R}}}}}({{{\bf{r}}}}) = -2 {\sum }_{i=1}^{3} {\sum }_{s=1}^{3}{V}_{s}\cos ({{{{\bf{b}}}}}_{{{{\bf{i}}}}}^{{{{\bf{s}}}}}\cdot {{{\bf{r}}}}+\phi )\,,\\ {{{{\bf{b}}}}}_{i}^{1} = \frac{4\pi }{\sqrt{3}{a}_{M}}(\sin \frac{2\pi i}{3},\cos \frac{2\pi i}{3})\,,\\ {{{{\bf{b}}}}}_{i}^{2} = \frac{4\pi }{{a}_{M}}\left[\sin (\frac{2\pi i}{3}+\frac{\pi }{6}),\cos (\frac{2\pi i}{3}+\frac{\pi }{6})\right]\,,\\ {{{{\bf{b}}}}}_{i}^{3} = \frac{8\pi }{\sqrt{3}{a}_{M}}(\sin \frac{2\pi i}{3},\cos \frac{2\pi i}{3})\,,$$

(3)

where ${{{{\bf{b}}}}}_{i}^{s}$ denote reciprocal vectors of different shells. Specific parameters of various materials are summarized in Supplementary Note 3.

Neural network architecture

Our neural network resembles similar architecture to previously proposed networks for solid materials and uniform electron gas^23,25.

Particle features r_i are combined with each other and form permutation equivalent features f_i²⁰

$${{{{\bf{f}}}}}_{i} = \;{{{\rm{concat}}}}\left[{{{{\bf{g}}}}}_{i}^{p},{{{{\bf{g}}}}}_{i}^{S}\right],\\ {{{{\bf{g}}}}}_{i}^{p} = \;{{{\rm{concat}}}}\left[\sin ({{{{\bf{r}}}}}_{i}\cdot {{{{\bf{b}}}}}^{p})\,{{{{\bf{a}}}}}^{p},\sum\limits_{j}\sin ({{{{\bf{r}}}}}_{j}\cdot {{{{\bf{b}}}}}^{p})\,{{{{\bf{a}}}}}^{p}\right],\\ {{{{\bf{g}}}}}_{i}^{S} = \;{{{\rm{concat}}}}\left[\sum\limits_{j}\sin [({{{{\bf{r}}}}}_{i}-{{{{\bf{r}}}}}_{j})\cdot {{{{\bf{b}}}}}^{S}]\,{{{{\bf{a}}}}}^{S},\sum\limits_{j}d({{{{\bf{r}}}}}_{i}-{{{{\bf{r}}}}}_{j})\right],$$

(4)

where a, b denote lattice vectors and reciprocal lattice vectors, and subscripts denote primitive cell and supercell. To ensure the periodic boundary condition, we employ the triangle distance input feature d(r), which reads²⁵

$$4{\pi }^{2}{d}^{2}({{{\bf{r}}}}) = \sum\limits_{ij}\sin ({{{\bf{r}}}}\cdot {{{{\bf{b}}}}}_{i}^{S})\sin ({{{\bf{r}}}}\cdot {{{{\bf{b}}}}}_{j}^{S})\,{{{{\bf{a}}}}}_{i}^{S}\cdot {{{{\bf{a}}}}}_{j}^{S}\\ +[1-\cos ({{{\bf{r}}}}\cdot {{{{\bf{b}}}}}_{i}^{S})][1-\cos ({{{\bf{r}}}}\cdot {{{{\bf{b}}}}}_{j}^{S})]\,{{{{\bf{a}}}}}_{i}^{S}\cdot {{{{\bf{a}}}}}_{j}^{S}\,.\\ $$

(5)

These features are subsequently fed into a neural network to construct effective orbital functions ${u}_{{{{{\bf{k}}}}}_{i}}$ which capture correlations between particles

$${u}_{{{{{\bf{k}}}}}_{i}}({{{{\bf{r}}}}}_{1},...,{{{{\bf{r}}}}}_{N})={{{\rm{Network}}}}({{{{\bf{f}}}}}_{j})\,.$$

(6)

These orbitals are then combined with a momentum dependent phase factor ${e}^{i{{{{\bf{k}}}}}_{i}\cdot {{{{\bf{r}}}}}_{j}}$ to become elements of determinants, which forms a legal antisymmetric many-body wavefunction for particles in moiré superlattice.

At integer fillings, the flatness of the energy bands in moiré systems suppresses kinetic energy, so we assume the bands are fully occupied, leading to an insulating state. To model these cases, we select the k-points in phase factors forming closed shells within the Brillouin zone. For fractional fillings, where classical Coulomb interactions dominate the system’s behavior, a single-reference method remains suitable for identifying the correct charge order³⁸. Accordingly, we also employ closed-shell configurations in these simulations. Additionally, we use an enlarged primitive cell, commensurate with possible charge orders, to construct the supercell. Specific hyperparameters of network are given in Supplementary Note 1.

Neural network optimization

Considering the wide range of length scale in moiré superlattice, we rescale the effective model ${{{\bf{r}}}}\to {a}_{M}{{{\bf{r}}}}^{\prime}$ for the convenience of Monte Carlo sampling, which reads

$${\widehat{H}}^{{\prime} }\equiv {\sum}_{i}\left[-\frac{{\Delta }_{i}^{{\prime} }}{2}+{m}^{* }{a}_{M}^{2}{V}_{M}({{{{\bf{r}}}}}_{i}^{{\prime} })\right]+\frac{1}{2}{\sum}_{i\ne j}\frac{{m}^{* }{a}_{M}}{\epsilon | {{{{\bf{r}}}}}_{i}^{{\prime} }-{{{{\bf{r}}}}}_{j}^{{\prime} }| }.$$

(7)

Our networks is optimized to minimize the scaled Hamiltonian ${\hat{H}}^{{\prime} }$, whose gradient reads

$$\nabla \langle {E}_{l}\rangle = \,2\,{{{\rm{Re}}}}\left[\langle {E}_{l}\nabla {{{\rm{ln}}}}{\Psi }^{* }\rangle -\langle {E}_{l}\rangle \langle \nabla {{{\rm{ln}}}}{\Psi }^{* }\rangle \right],\\ \,{E}_{l}={\Psi }^{-1}{\hat{H}}^{{\prime} }\Psi ,$$

(8)

and 〈. . . 〉 denotes the expectations of operators with ∣Ψ∣² distribution. Moreover, Kronecker factored curvature estimator (KFAC) optimizer³⁹ is employed to train the neural network towards the ground states at different fillings and spin polarization.

Moiré pattern analysis

To visualize the Wigner phases, we plot the particle density ρ(r) derived from many-body wavefunction Ψ

$$\rho ({{{\bf{r}}}})=N\int{d}^{3}{{{{\bf{r}}}}}_{2}\cdots {d}^{3}{{{{\bf{r}}}}}_{N}\,| \Psi ({{{\bf{r}}}},{{{{\bf{r}}}}}_{2},\cdots \,,{{{{\bf{r}}}}}_{N}){| }^{2}\,.$$

(9)

In practice, ρ is evaluated by accumulating Monte Carlo samples of particles on a 100 × 100 uniform grid over the moiré cell.

Workflow and computational details

Supercell approximation is employed in our simulations. Particles are initialized uniformly in the moiré supercell, and gradually form moiré patterns during energy minimization, see Supplementary Note 2 for more details. The expectations of operators are evaluated via the Monte Carlo approach. Forward Laplacian technique is employed to speed up the simulation⁴⁰. Most simulations in this work are performed on eight A800 graphics processing units within several hours. Training curves of each system are plotted in Supplementary Figs. 2–4. Other excited states are plotted in Supplementary Fig. 6. Calculated energy and geometry are listed in Supplementary Note 4.