An integrated neural wavefunction solver for spinful Fermi systems

TL;DR

The paper addresses computing ground states of spinful fermionic systems combining continuous coordinates with discrete DOFs. It introduces a variational Monte Carlo framework that uses a transformer-based wavefunction to represent generalized orbitals and a spin-aware MCMC sampler to explore the extended configuration space, with $E_\theta = \mathbb{E}_{\{\\boldsymbol{\\xi}\} \sim |\\Psi_\\theta|^2}[E_{\\mathrm{loc},\\theta}(\\{\\boldsymbol{\\xi}\\})]$ and $H = \sum_i \\frac{\\boldsymbol{\\nabla}_i^2}{2m} + V(\\boldsymbol{r}) + H_{\\rm spin}$. The approach achieves universal approximation of spin–position orbitals and enables efficient energy estimation via spin–coordinate updates; it demonstrates accurate ground-state energies and spin textures for Rashba SOC, non-collinear spin textures, and 2D moiré antiferromagnets, with sector-preserving updates reducing optimization steps. The framework extends to additional isospin DOFs (layer, valley, sublattice) in 2D materials, offering a path toward scalable simulations of complex spinful quantum matter.

Abstract

We present an approach to solving the ground state of Fermi systems that contain spin or other discrete degrees of freedom in addition to continuous coordinates. The approach combines a Markov chain Monte Carlo sampling for energy estimation that we adapted to cover the extended configuration space with a transformer-based wavefunction to represent fermionic states. This sampling is necessary when the Hamiltonian contains explicit spin dependence and, for spin-independent Hamiltonians, we find that the inclusion of spin updates leads to faster convergence to an antiferromagnetic ground state. A transformer with both continuous position and discrete spin as inputs achieves universal approximation to spinful generalized orbitals. We validate the method on a range of two-dimensional material problems: a two-dimensional electron gas with Rashba spin-orbit coupling, a noncollinear spin texture, and a quantum antiferromagnet in a honeycomb moiré potential.

Figures (3)

• Figure 1: (a) Schematic representation of the VMC training loop. The wavefunction parameters $\theta$ and the batch configurations $\{\boldsymbol{\xi}_i\}$ are updated by a series of consecutive Monte Carlo sampling steps that modify the batch and optimizer steps that modify the wavefunction parameters. (b) Architecture of the neural network representing the wavefunction $\Psi_{\theta}$. Particle position and spin (or discrete DOFs) are processed in streams that affect each other through the attention mechanism.
• Figure 2: Optimization curves for (A) the Zeeman spin-spiral Hamiltonian, Eq. \ref{['eq:spin_spiral']}, with 3 electrons, showing the moving average of the energy over 5 optimization steps; (B) the Rashba Hamiltonian, Eq. \ref{['eq:rashba']}, with 5 electrons, showing the moving average of the energy over 20 optimization steps. In both cases, we used two-dimensional period systems with period 6 in each spatial direction and spin update probability $p=0.1$.
• Figure 3: Benchmark of spin updates in the Markov chain Monte Carlo routine at a two-dimensional electron gas with honeycomb potential: (a) Energy as a function of step with (black) no spin updates and (green) spin $s_z$ conserving updates with spin swap probability $p = 0.03$. The figure shows four runs with different random initialization for both Monte-Carlo procedures. (b) Final spin density, where the color indicates spin polarization $\frac{\langle n_\uparrow \rangle -\langle n_\downarrow \rangle}{\langle n_\uparrow \rangle +\langle n_\downarrow \rangle}$ and the saturation total density $\langle n_\uparrow \rangle +\langle n_\downarrow \rangle$.