Predicting magnetism with first-principles AI

TL;DR

This work directly solves the many-electron Schrodinger equation with neural-network variational Monte Carlo, which provides a highly expressive variational wavefunction for strongly correlated systems and predicts itinerant ferromagnetism and an antiferromagnetic insulator in twisted $\Gamma$-valley homobilayer.

Abstract

Computational discovery of magnetic materials remains challenging because magnetism arises from the competition between kinetic energy and Coulomb interaction that is often beyond the reach of standard electronic-structure methods. Here we tackle this challenge by directly solving the many-electron Schrödinger equation with neural-network variational Monte Carlo, which provides a highly expressive variational wavefunction for strongly correlated systems. Applying this technique to transition metal dichalcogenide moiré semicondutors, we predict itinerant ferromagnetism in WSe$_2$/WS$_2$ and an antiferromagnetic insulator in twisted $Γ$-valley homobilayer, using the same neural network without any physics input beyond the microscopic Hamiltonian. Crucially, both types of magnetic states are obtained from a single calculation within the $S_z=0$ sector, removing the need to compute and compare multiple $S_z$ sectors. This significantly reduces computational cost and paves the way for faster and more reliable magnetic material design.

Figures (6)

• Figure 1: (a) Workflow. Defining the quantum system with its first-principles Hamiltonian, the NNVMC technique optimizes a neural network to represent its ground state by energy minimization. The total magnetization $\langle \hat{\bm S}^2\rangle$ and staggered magnetization $\langle (\hat{\bm S}^A - \hat{\bm S}^B) \rangle$ identifies the magnetic order of the ground state. (b) Band structure of the moiré semiconductor model for WSe$_2$/WS$_2$. (c) Spin density in the itinerant ferromagnetic state obtained from the variational solution with $28$ electrons in 21 unit cells, for visualization shown for the $S_z = 7\, \hbar$ result where magnetization is aligned with the computational $S_z$ basis. Color indicates spin polarization $\frac{\langle \hat{n}_\uparrow \rangle - \langle \hat{n}_\downarrow \rangle}{\langle \hat{n}_\uparrow \rangle + \langle \hat{n}_\downarrow \rangle}$ and opacity total density $\langle \hat{n}_\uparrow \rangle + \langle \hat{n}_\downarrow \rangle$.
• Figure 2: Itinerant magnetism. (a) Ground state energy in different $S_z$ sectors and (b) total spin $\langle S^2\rangle$ in different $S_z$ sectors in the model for WSe$_2$/WS$_2$ semiconductor with 28 particles in 21 unit cells. Panels (c) and (d) show the total occupation $n_\sigma(\bm k) = \sum_\alpha \langle c_{\bm k, \alpha, \sigma}^\dagger c_{\bm k, \alpha, \sigma}\rangle$ of modes resolved by spin $\sigma$ and momentum $\bm k$ quantum numbers in the first Brillouin zone, visualized for the $S_z = 7\, \hbar$ sector. For visualization, we included the values at all $K$ points on the first Brillouin zone corners.
• Figure 3: Antiferromagnetic insulator. (a) Band structure and (b) spin density for the homobilayer semiconductor model with honeycomb potential $\varphi = \pi$, Eq. \ref{['eq:system-hamiltonian']}, for $V = 15\, {\rm meV}$, $\epsilon = 6$, and $54$ electrons. (c) Staggered magnetization $M_z = \langle S_z^A - S_z ^B \rangle$ as a function of potential strength computed for 18 electrons in 9 unit cells. (d) Long-wavelength behavior of the static structure factor $4 \pi S(q) / q^2$ for different potential strength $V$ computed for variable system sizes of 9 ("$\triangle$"), 12 ("$\triangledown$"), 16 ("$\times$"), 21 ("$+$"), and 27 ("$\cdot$") unit cells and two electrons per unit cell, with $V_c = 7.7\, {\rm meV}$.
• Figure 4: Spin density along the $S_z$ axis in the itinerant ferromagnet as discussed in Fig. \ref{['fig:1']}(c) and Fig. \ref{['fig:2']} in the main text, computed in the $S_z = 0$ sector for the system with 28 electrons in 21 unit cells.
• Figure 5: Spin density-density correlation in the itinerant ferromagnet as discussed in Fig. \ref{['fig:2']} in the main text. The data shown here is computed from the system with 28 electrons in 21 unit cells at $S_z = 7\, \hbar$.