Twisted Sector Ground States in Moiré TMDs

• Twisted Sector Ground States are unique quantum states induced by twist, showing electron crystalline and fractional liquid behaviors in moiré TMD systems.
• Variational neural quantum state methods, using message-passing networks and neural backflow, enable precise determination of ground state properties in large moiré clusters.
• The approach extracts key observables like static structure factors and quantum weight tensors, facilitating direct comparisons with experimental data on twisted materials.

Twisted sector ground states arise in systems where either geometric twist, layer misalignment, nontrivial boundary conditions, or symmetry-enforced band mixing lead to ground states distinct from those found in untwisted, translation-invariant settings. In the context of quantum materials—specifically, twisted moiré transition-metal dichalcogenides (TMDs) such as MoTe₂—these ground states reflect the interplay between strong correlations, band topology, and spatially modulated potentials. Recent theoretical advances employ highly expressive neural quantum states to elucidate the complex ordering and fractionalization in these systems, allowing for accurate variational determination of both crystalline and fractional liquid ground states across large-scale clusters (Luo et al., 17 Mar 2025).

1. Variational Neural Wavefunction Approach

Recent progress in solving many-electron problems in twisted moiré materials leverages neural quantum state (NQS) methodologies, which combine physical symmetry constraints with highly non-linear, machine-learned functionals. The method begins with a single-particle Bloch basis: $$\varphi_{k}(r_i, l_i) = \sum_{g} u^{l_i}_{k g}\, e^{i (k+g) \cdot r_i}$$ where $r_i$ denotes the spatial coordinate of electron $i$, $l_i$ encodes a discrete layer/spin label, $k$ indexes the Brillouin zone mesh, and $u^{l_i}_{k g}$ are mean-field Bloch coefficients.

The many-body correlated ansatz is built by passing the coordinates and quantum numbers through a message-passing graph neural network, which computes non-linear, high-dimensional node features $V^{t}_i$ after several update layers. The single-particle positions are then "dressed" by a neural backflow transformation: $$\mathbf{r}_i \to \overline{\mathbf{r}}_i = \mathbf{r}_i + W V^{t}_i$$ where $W$ is a learnable complex-valued matrix. An additional neural orbital transformation and momentum-space mixing concludes the ansatz construction.

This approach is permutation-equivariant, preserves translation symmetry (by construction), and admits efficient scalable sampling via Markov Chain Monte Carlo and stochastic reconfiguration.

2. Fractional Electron States and Moiré Lattice Fillings

The neural variational method uncovers a hierarchy of ground states as the filling $\nu$ of the moiré superlattice is varied. Key findings include:

• At fractional filling $\nu = 1/3$, the ground state exhibits strong spatial ordering indicative of an electron (Wigner) crystal, as seen from prominent Bragg peaks in the calculated structure factor.
• At other non-integral fillings, the neural ansatz variationally converges to ground states with the haLLMarks of fractional quantum liquids—i.e., states without long-range crystalline order but exhibiting properties congruent with Laughlin-like or fractional quantum Hall states adapted to a lattice.

The ansatz resolves both types of correlated phases within the same computational framework, capturing essential interaction-driven physics of twisted TMDs.

3. Extraction of Structure Factor and Quantum Weight

The full static structure factor $S(\mathbf{q})$, computed from the variational ground state, directly characterizes the internal correlations: $$S(\mathbf{q}) \approx K_{\alpha\beta} q_\alpha q_\beta + O(q^3)$$ with quantum weight tensor $K_{\alpha\beta}$. For crystalline (symmetry-broken) states, $S(\mathbf{q})$ displays sharp Bragg peaks. In fractional liquid phases, the low-$q$ behavior is quadratic, with the coefficient $K$ tightly constrained by topological invariants (such as the many-body Chern number), reflecting incompressibility and fractionalization.

These observables are accessed by sampling the neural wavefunction with Monte Carlo, allowing for large system sizes (up to 36 moiré unit cells in the cited work), thus approaching the thermodynamic limit.

4. Unified Description of Crystalline and Fractional Quantum Liquids

The neural architecture's capacity includes both spatial symmetry–broken (electron crystal) and topologically ordered (fractional quantum liquid) ground states without bias. Notably:

• At $\nu = 1/3$, the "twisted sector ground state" is a generalized electron crystal, stabilized by strong moiré band mixing and Coulomb interactions.
• Away from commensurate fillings, energy minimization leads to liquid-like, gapped quantum Hall–type states. The ansatz captures quantum fluctuations and topological signatures (such as quantum weight saturation), distinguishing them from classical or mean-field ordered phases.

This demonstrates that a single variational representation can encompass a rich variety of twisted sector ground states, each dictated by the filling and interplay of intra- and inter-band quantum correlations.

5. Implications and Broader Impact for Twisted Moiré Materials

This methodology for solving twisted sector ground states in MoTe₂ has broader consequences:

• The approach is "materials-agnostic" within the class of moiré systems, implying direct applicability to other twisted TMDs, heterobilayers, and multilayers with nontrivial twist angles or mismatched period lattices.
• By achieving unbiased accuracy and scalability, the method provides a route toward quantitative theory-experiment comparison (e.g., structure factor, ground-state phase diagrams, transport gaps) in the presence of strong band mixing and electron correlation.
• The framework is well-suited for exploration of more exotic phases, e.g., non-Abelian fractional quantum Hall or topological superconductors, by systematic extension to include spin, valley, or time-reversal-breaking perturbations.
• The quantitative diagnostics—particularly the quantum weight and full structure factor—facilitate detection and characterization of novel topological and symmetry-broken phases in experiment and inform design principles for devices based on tunable correlated ground states in twisted quantum materials.

6. Technical Advances and Outlook

The deep neural network ansatz employed here incorporates physically motivated feature engineering (sinusoidal functions respecting the lattice, symmetry constraints embedded in the message-passing layers) with variational flexibility, enabling efficient representation of high-entanglement, strongly correlated ground states. The success in achieving accurate energies and diagnosing crystalline and fractional phases at scale suggests that neural quantum states are a promising tool for the paper of twisted sector ground states in complex quantum materials (Luo et al., 17 Mar 2025).

This comprehensive neural variational framework thereby marks a critical advance in the understanding and control of strongly correlated and topologically nontrivial phases in twisted two-dimensional quantum materials.