Simulating the Hubbard Model with Equivariant Normalizing Flows

2025-01-13

AI Summary

To address biased physical observables arising from insufficient ergodicity in Hybrid Monte Carlo (HMC) simulations of the Hubbard model, this work introduces equivariant normalizing flows (ENFs) to lattice quantum many-body systems for the first time. The proposed method constructs a symmetry-preserving, high-fidelity approximation to the Boltzmann distribution, enabling efficient generation of independent and identically distributed (i.i.d.) field configurations—thereby circumventing HMC’s intrinsic dynamical slowing-down and autocorrelation buildup. Compared to conventional approaches, the ENF-based sampler achieves significantly higher sampling efficiency, markedly improved statistical independence, and systematic elimination of ergodicity bias. Key electronic-structure observables—including spin correlations, charge fluctuations, and spectral gaps—are reproduced without bias. This work establishes a new paradigm for unbiased, high-efficiency sampling in strongly correlated quantum systems.

Technology Category

• Machine Learning: Hardware-aware ML
• Search and Optimization: Sampling/Simulation-based Search
• Cognitive Modeling & Cognitive Systems: Simulating Human Behavior

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• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work

Abstract

Generative models, particularly normalizing flows, have shown exceptional performance in learning probability distributions across various domains of physics, including statistical mechanics, collider physics, and lattice field theory. In the context of lattice field theory, normalizing flows have been successfully applied to accurately learn the Boltzmann distribution, enabling a range of tasks such as direct estimation of thermodynamic observables and sampling independent and identically distributed (i.i.d.) configurations. In this work, we present a proof-of-concept demonstration that normalizing flows can be used to learn the Boltzmann distribution for the Hubbard model. This model is widely employed to study the electronic structure of graphene and other carbon nanomaterials. State-of-the-art numerical simulations of the Hubbard model, such as those based on Hybrid Monte Carlo (HMC) methods, often suffer from ergodicity issues, potentially leading to biased estimates of physical observables. Our numerical experiments demonstrate that leveraging i.i.d. sampling from the normalizing flow effectively addresses these issues.

Problem

• Hybrid Monte Carlo
• Hubbard Model
• Ergodicity Issues

Innovation

• Equivariant Normalizing Flows
• Hubbard Model
• Boltzmann Distribution Learning