Simulating Correlated Electrons with Symmetry-Enforced Normalizing Flows

2025-06-20

AI Summary

In simulating the Boltzmann distribution of the fermionic Hubbard model, conventional methods—such as Hybrid Monte Carlo (HMC)—suffer from bias and inefficiency in the continuous-time limit due to poor ergodicity. To address this, we propose the first symmetry-aware normalizing flow model. Leveraging group representation theory, we design an invertible neural architecture explicitly constrained by physical symmetries; combined with variational inference and Monte Carlo acceleration, it enables unbiased and efficient modeling of the target distribution. Crucially, the model generates independent and identically distributed (i.i.d.) samples, eliminating autocorrelation errors inherent in Markov chain methods. In electronic structure calculations for strongly correlated systems—including graphene—our approach achieves over an order-of-magnitude improvement in sampling efficiency compared to HMC, significantly enhancing both the accuracy and robustness of physical observable estimation.

Technology Category

• Machine Learning: Hardware-aware ML
• Search and Optimization: Sampling/Simulation-based Search
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work

Abstract

We present the first proof of principle that normalizing flows can accurately learn the Boltzmann distribution of the fermionic Hubbard model - a key framework for describing the electronic structure of graphene and related materials. State-of-the-art methods like Hybrid Monte Carlo often suffer from ergodicity issues near the time-continuum limit, leading to biased estimates. Leveraging symmetry-aware architectures as well as independent and identically distributed sampling, our approach resolves these issues and achieves significant speed-ups over traditional methods.

Problem

• Accurately simulate fermionic Hubbard model distributions
• Address ergodicity issues in current simulation methods
• Achieve faster sampling compared to traditional techniques

Innovation

• Normalizing flows learn fermionic Hubbard model
• Symmetry-aware architectures resolve ergodicity issues
• IID sampling achieves significant speed-ups