Representability-Aware Neural Networks for Reduced Density Matrices: Application to Fractional Chern Insulators
Authors: Justin B. Hart, Awwab A. Azam, Thomas Li, Yunxuan Li, Ye Bi, Haining Pan, Jiabin Yu
Organizations: Department of Physics, University of Florida, Gainesville, FL, USA · Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL, USA · Palo Alto High School, Palo Alto, CA, USA · Google, Mountain View, CA, USA · Department of Animal Science, Iowa State University, Ames, IA, USA · Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, NJ, USA · Quantum Theory Project, University of Florida, Gainesville, FL, USA
Abstract
We develop a representability-aware and interpolable neural network (NN) framework for predicting two-particle reduced density matrices (2-RDMs). The NN incorporates a subset of representability conditions through its architecture and loss function, and can operate on different momentum meshes, enabling evaluating the representability conditions across multiple meshes, which we call interpolated representability condition. The framework can be used either to predict 2-RDMs on large momentum meshes by interpolating exact results from small meshes, or as a variational 2-RDM ansatz optimized by energy minimization on arbitrary meshes. We apply this approach to the fractional Chern insulator in the one-band projected model of twisted bilayer MoTe2 at twist angle 3.89∘ and hole filling 2/3. Trained on exact-diagonalization (ED) 2-RDMs from meshes with 12 or 18 momentum points using six different NN architectures, the best NN is the residual multilayer perceptron, which predicts the 6×6 2-RDM with 97.07%−98.18% accuracy relative to the ED 2-RDM but predicts an energy 77.353 meV above ED ground-state energy. We then variationally optimize the NN on several meshes including 6×6, predicting a 6×6 energy of just 0.104 meV below ED while maintaining 98.94%−98.96% accuracy. Compared with the conventional boundary-point semidefinite programming, which gives an energy 5.560 meV below ED with 96.40%−98.94% accuracy, the NN achieves a more accurate energy and similar accuracy while using only less than 1/20 as many parameters. Eventually, we add a symmetric mesh of 48 momentum points to the variational optimization of the NN, and provide a prediction of the many-body ground-state energy and the many-body quantum metric on that mesh.
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 WSe2/WS2 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 Sz=0 sector, removing the need to compute and compare multiple Sz sectors. This significantly reduces computational cost and paves the way for faster and more reliable magnetic material design.
We introduce QERNEL, a foundational neural wavefunction that variationally solves families of parameterized many-electron Hamiltonians and captures their ground states throughout parameter space within a single model. QERNEL combines FiLM-based parameter conditioning with scale-efficient architectural elements -- mixture of experts and grouped-query attention, substantially improving expressivity at low computational cost. We apply QERNEL to interacting electrons in semiconductor moiré heterobilayers, training a single weight-shared model for systems of up to 150 electrons. By solving the many-electron Schrödinger equation conditioned on moiré potential depth, QERNEL captures both quantum liquid and crystal states and discovers the sharp phase transition between them, marked by abrupt changes in interaction energy and charge density. Our work establishes a foundation model for moiré quantum materials and a scalable architecture toward a Large Electron Model for solids.
High-entropy alloys (HEAs) have attracted growing attention for their exceptional mechanical and thermal properties arising from complex atomic configurations. In this paper, we propose crystal fractional graph neural network for predicting the energy of high-entropy alloys by explicitly integrating both local atomic environments and global compositional information. The model consists of three components: a crystal graph neural network, which employs graph attention network layers to learn local interactions among 16 on-site atoms within the crystal lattice; fractional neural network, a fully connected network that embeds the global fraction of constituent elements; and feature fusion neural network, which fuses the outputs of the two submodels to predict the total crystal energy. We train the model on a dataset of 1,049 crystal structures and validate it on 198 quaternary structures, optimizing all hyperparameters via Optuna. Our results show that our model achieves an RMSE comparable to first-principles calculations and maintains high accuracy even for low-energy configurations. However, the model exhibits limitations in handling large crystal cells, which we aim to address in future work to extend its applicability to more complex systems.