Hamiltonian learning for spin-spiral moiré magnets from electronic magnetotransport

Abstract: Two-dimensional noncollinear magnetic states, such as spin-spiral magnets, offer an excellent platform for investigating fundamental phenomena, with potential for advancing stray-field-free spintronics. However, detection and characterization of noncollinear magnetic states in two-dimensional systems remain challenging, motivating the development of alternative probing methods. Here, we present a methodology for extracting the spin-spiral $\mathbf{q}$ vector from lateral electronic transport measurements. Our approach leverages the magnetic field and bias dependence of the conductance to train a supervised machine learning algorithm, which enables us to extract the $\mathbf{q}$ vectors of arbitrary spin-spiral magnets. We demonstrate that this methodology is robust to the presence of impurities in the system and noise in the conductance data. Our findings show that the conductance pattern reveals a complex dependence on the $\mathbf{q}$ vector of the spin spiral, providing a new strategy to learn magnetic structures directly from transport experiments.

• The paper introduces an ML-based Hamiltonian learning framework that accurately reconstructs spin-spiral q-vectors from transport measurements in moiré magnets.
• A combination of PCA and a neural network enables near-perfect fidelity even under experimental noise, validating the approach.
• This methodology offers a scalable, non-invasive route for probing noncollinear magnetic orders in 2D van der Waals heterostructures.

Hamiltonian Learning of Spin-Spiral Moiré Magnets via Electronic Magnetotransport

Introduction

The paper "Hamiltonian learning for spin-spiral moiré magnets from electronic magnetotransport" (2604.02959) addresses the precision identification of noncollinear magnetic order in two-dimensional (2D) moiré magnets, specifically spin-spiral magnets (SSMs), through a Hamiltonian learning framework based on electronic magnetotransport measurements. Characterizing 2D noncollinear ordering poses significant challenges, especially in van der Waals (vdW) heterostructures, where stray-field-free control is crucial for spintronic applications. This manuscript develops and validates a ML methodology that leverages transport signatures in the Hofstadter regime for reconstructing the fundamental spin spiral $\mathbf{q}$-vector, which encodes both the wavevector magnitude and orientation of the spiral.

Figure 1: (a) Device schematic showing local moments on a twisted moiré lattice, with conductance probed via source/drain electrodes and perpendicular magnetic field. (b) Hamiltonian learning workflow extracting $\mathbf{q}$ from transport data.

Model and Physical Framework

The system consists of a twisted TMD bilayer structure comprising two functional layers: a Mott-insulating spin-spiral layer and a gate-tunable metallic probe layer. The hybridization between these layers results in a significant moiré superlattice, which magnifies magnetic-field-induced effects in electronic transport. The Hamiltonian captures nearest-neighbor hopping, spin-spiral exchange (proximity), chemical potential tunability, and disorder:

$$H = t\sum_{\langle\alpha\beta\rangle,s} (e^{i\phi_{\alpha\beta}} c^\dagger_{\alpha s} c_{\beta s} + h.c.) + \mu \sum_{\alpha, s} c^\dagger_{\alpha s} c_{\alpha s} + \sum_{\alpha, ss'} \left(\mathbf{J}_\alpha(\mathbf{q}) \cdot \vec{\sigma}\right)_{ss'} c^\dagger_{\alpha s} c_{\alpha s'} + \sum_{\alpha, s} W_\alpha c^\dagger_{\alpha s} c_{\alpha s}$$

The proximity-induced exchange field $\mathbf{J}_\alpha(\mathbf{q})$ is parameterized by a wavevector $\mathbf{q}$, giving direct access to the spin-spiral order. The model incorporates random onsite disorder, emulating real device imperfections.

Hofstadter Signatures and Transport Analysis

In the regime of phase-coherent, ballistic transport, the conductance spectra reflect the intricate interplay of moiré periodicity and external magnetic flux, realized in the much-studied Hofstadter butterfly. Modification of these patterns occurs due to exchange coupling with the SSM. The evaluation uses the Landauer formalism, with nonequilibrium Green's functions to compute $G(\mu, \phi)$ as a function of chemical potential and normalized flux.

Figure 2: (a) Hofstadter butterfly in density of states as a function of $\mu$ and normalized flux. (b) Conductance without exchange proximity. (c)-(f) Conductance changes $\Delta G$ for various $\mathbf{q}$, revealing strong dependence on spiral ordering.

Simulated datasets, capturing a range of $q_1, q_2$ spiral components, manifest distinct transport fingerprints for different spin-spiral states, allowing inference of the $\mathbf{q}$0-vector from experimental data.

Machine Learning-Based Hamiltonian Inference

An ML pipeline is developed to reconstruct the spin-spiral $\mathbf{q}$1-vector from transport data. The approach involves the following stages:

• Dataset Construction: 10,000 conductance maps parametrized by random $\mathbf{q}$2 vectors with realistic disorder and exchange parameters.
• Dimensionality Reduction: Principal Component Analysis (PCA) reduces high-dimensional conductance data to the principal components capturing $\mathbf{q}$3 of variance.
• Supervised Learning: A feedforward neural network with two hidden layers (100 neurons each, ReLU activation) is trained to map the reduced conductance features to $\mathbf{q}$4. Training uses Adam optimization and MSE loss minimization on both training and validation sets.

Figure 3: (a) Predicted vs. true $\mathbf{q}$5 values. (b) Predicted vs. true $\mathbf{q}$6 for previously unseen test data.

Quantitatively, the trained model achieves extremely high fidelity: $\mathbf{q}$7 and $\mathbf{q}$8 for test data unseen during training, highlighting the strong predictive mapping from transport observables to underlying spin-spiral order.

Robustness and Noise Analysis

Assessment of the approach under experimental imperfections is conducted by adding controlled, bounded multiplicative noise to the conductance data, both during training and inference. The algorithm displays a nuanced tradeoff: higher training noise impairs performance on noiseless data but improves robustness against matching or higher noise levels at inference.

Figure 4: (a)-(d) Fidelity deterioration as function of test noise $\mathbf{q}$9 for various training noise strengths. (e)-(f) Fidelity dependence on exchange coupling $$H = t\sum_{\langle\alpha\beta\rangle,s} (e^{i\phi_{\alpha\beta}} c^\dagger_{\alpha s} c_{\beta s} + h.c.) + \mu \sum_{\alpha, s} c^\dagger_{\alpha s} c_{\alpha s} + \sum_{\alpha, ss'} \left(\mathbf{J}_\alpha(\mathbf{q}) \cdot \vec{\sigma}\right)_{ss'} c^\dagger_{\alpha s} c_{\alpha s'} + \sum_{\alpha, s} W_\alpha c^\dagger_{\alpha s} c_{\alpha s}$$0 in the training and testing data.

Exchange coupling $$H = t\sum_{\langle\alpha\beta\rangle,s} (e^{i\phi_{\alpha\beta}} c^\dagger_{\alpha s} c_{\beta s} + h.c.) + \mu \sum_{\alpha, s} c^\dagger_{\alpha s} c_{\alpha s} + \sum_{\alpha, ss'} \left(\mathbf{J}_\alpha(\mathbf{q}) \cdot \vec{\sigma}\right)_{ss'} c^\dagger_{\alpha s} c_{\alpha s'} + \sum_{\alpha, s} W_\alpha c^\dagger_{\alpha s} c_{\alpha s}$$1 is another critical parameter. High $$H = t\sum_{\langle\alpha\beta\rangle,s} (e^{i\phi_{\alpha\beta}} c^\dagger_{\alpha s} c_{\beta s} + h.c.) + \mu \sum_{\alpha, s} c^\dagger_{\alpha s} c_{\alpha s} + \sum_{\alpha, ss'} \left(\mathbf{J}_\alpha(\mathbf{q}) \cdot \vec{\sigma}\right)_{ss'} c^\dagger_{\alpha s} c_{\alpha s'} + \sum_{\alpha, s} W_\alpha c^\dagger_{\alpha s} c_{\alpha s}$$2 values ensure prominent proximity effects, leading to reliable reconstruction of $$H = t\sum_{\langle\alpha\beta\rangle,s} (e^{i\phi_{\alpha\beta}} c^\dagger_{\alpha s} c_{\beta s} + h.c.) + \mu \sum_{\alpha, s} c^\dagger_{\alpha s} c_{\alpha s} + \sum_{\alpha, ss'} \left(\mathbf{J}_\alpha(\mathbf{q}) \cdot \vec{\sigma}\right)_{ss'} c^\dagger_{\alpha s} c_{\alpha s'} + \sum_{\alpha, s} W_\alpha c^\dagger_{\alpha s} c_{\alpha s}$$3 even at moderate mismatch between $$H = t\sum_{\langle\alpha\beta\rangle,s} (e^{i\phi_{\alpha\beta}} c^\dagger_{\alpha s} c_{\beta s} + h.c.) + \mu \sum_{\alpha, s} c^\dagger_{\alpha s} c_{\alpha s} + \sum_{\alpha, ss'} \left(\mathbf{J}_\alpha(\mathbf{q}) \cdot \vec{\sigma}\right)_{ss'} c^\dagger_{\alpha s} c_{\alpha s'} + \sum_{\alpha, s} W_\alpha c^\dagger_{\alpha s} c_{\alpha s}$$4 and $$H = t\sum_{\langle\alpha\beta\rangle,s} (e^{i\phi_{\alpha\beta}} c^\dagger_{\alpha s} c_{\beta s} + h.c.) + \mu \sum_{\alpha, s} c^\dagger_{\alpha s} c_{\alpha s} + \sum_{\alpha, ss'} \left(\mathbf{J}_\alpha(\mathbf{q}) \cdot \vec{\sigma}\right)_{ss'} c^\dagger_{\alpha s} c_{\alpha s'} + \sum_{\alpha, s} W_\alpha c^\dagger_{\alpha s} c_{\alpha s}$$5. For weak $$H = t\sum_{\langle\alpha\beta\rangle,s} (e^{i\phi_{\alpha\beta}} c^\dagger_{\alpha s} c_{\beta s} + h.c.) + \mu \sum_{\alpha, s} c^\dagger_{\alpha s} c_{\alpha s} + \sum_{\alpha, ss'} \left(\mathbf{J}_\alpha(\mathbf{q}) \cdot \vec{\sigma}\right)_{ss'} c^\dagger_{\alpha s} c_{\alpha s'} + \sum_{\alpha, s} W_\alpha c^\dagger_{\alpha s} c_{\alpha s}$$6, fidelity drops significantly, emphasizing the necessity of strong spiral-probe exchange in experimental realization.

Prediction results with both training and testing at finite noise remain robust:

Figure 5: Prediction results for $$H = t\sum_{\langle\alpha\beta\rangle,s} (e^{i\phi_{\alpha\beta}} c^\dagger_{\alpha s} c_{\beta s} + h.c.) + \mu \sum_{\alpha, s} c^\dagger_{\alpha s} c_{\alpha s} + \sum_{\alpha, ss'} \left(\mathbf{J}_\alpha(\mathbf{q}) \cdot \vec{\sigma}\right)_{ss'} c^\dagger_{\alpha s} c_{\alpha s'} + \sum_{\alpha, s} W_\alpha c^\dagger_{\alpha s} c_{\alpha s}$$7 and $$H = t\sum_{\langle\alpha\beta\rangle,s} (e^{i\phi_{\alpha\beta}} c^\dagger_{\alpha s} c_{\beta s} + h.c.) + \mu \sum_{\alpha, s} c^\dagger_{\alpha s} c_{\alpha s} + \sum_{\alpha, ss'} \left(\mathbf{J}_\alpha(\mathbf{q}) \cdot \vec{\sigma}\right)_{ss'} c^\dagger_{\alpha s} c_{\alpha s'} + \sum_{\alpha, s} W_\alpha c^\dagger_{\alpha s} c_{\alpha s}$$8 under $$H = t\sum_{\langle\alpha\beta\rangle,s} (e^{i\phi_{\alpha\beta}} c^\dagger_{\alpha s} c_{\beta s} + h.c.) + \mu \sum_{\alpha, s} c^\dagger_{\alpha s} c_{\alpha s} + \sum_{\alpha, ss'} \left(\mathbf{J}_\alpha(\mathbf{q}) \cdot \vec{\sigma}\right)_{ss'} c^\dagger_{\alpha s} c_{\alpha s'} + \sum_{\alpha, s} W_\alpha c^\dagger_{\alpha s} c_{\alpha s}$$9.

Optimization and Training Dynamics

Dimensionality reduction via PCA is shown to retain essential information for learning, with $\mathbf{J}_\alpha(\mathbf{q})$0 achieving near-total variance capture. Model convergence during training exhibits monotonic decrease in MSE for both train and validation sets, with no overfitting observed.

Figure 6: (a) Cumulative explained variance vs. number of PCA components. (b) Train/validation MSE loss across epochs.

Implications and Future Directions

This methodology demonstrates that electronic transport, traditionally a probe for charge and topological phenomena, is sensitive enough to reveal the complex, noncollinear magnetic ordering in vdW moiré magnets via data-driven Hamiltonian learning. The possibility of reconstructing vectorial order parameters of magnetic textures from mesoscopic transport marks an important advance, particularly given the resilience to moderate noise and disorder.

Practically, this framework can be integrated into experimental platforms, enabling high-throughput, non-invasive probing of 2D magnetic phases. Theoretically, it suggests extensions to more complex, possibly topological or skyrmionic, spin-textures, and invites the integration of more sophisticated inference architectures (e.g. graph neural networks or invertible neural operators) for richer Hamiltonian extraction in correlated or interacting regimes.

Conclusion

The work establishes an ML-based Hamiltonian learning pipeline for extracting spin-spiral $\mathbf{J}_\alpha(\mathbf{q})$1-vectors in moiré magnets through carefully designed electronic transport experiments. The results demonstrate high fidelity, substantial robustness to various noise sources, and resilience to device imperfections, offering a viable blueprint for the direct identification of noncollinear magnetic orderings in 2D systems by electronic means (2604.02959). This approach provides a scalable route for the systematic study of emergent magnetism in low-dimensional materials and has broad implications for both spintronics and fundamental condensed matter research.