Simulating the Hubbard Model with Equivariant Normalizing Flows (2501.07371v1)

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.

• The paper introduces equivariant normalizing flows as an improved method for simulating the Hubbard model, addressing ergodicity issues faced by traditional Hybrid Monte Carlo (HMC) methods.
• This new approach incorporates symmetries inherent in the Hubbard model, like Z2 and translational symmetry, into the normalizing flow architecture to enhance efficiency and accuracy in learning the Boltzmann distribution.
• Numerical experiments demonstrate that equivariant normalizing flows significantly improve acceptance rates and reduce autocorrelation time compared to non-equivariant methods, showing scalability to larger lattices where other methods fail.

Simulating the Hubbard Model with Equivariant Normalizing Flows

This paper presents a methodological advancement in addressing the ergodicity issues faced by the Hybrid Monte Carlo (HMC) simulations applied to the Hubbard model. By employing generative models, specifically normalizing flows, the authors offer an innovative approach to efficiently learn the Boltzmann distribution for the Hubbard model. The Hubbard model is integral for exploring the electronic structure of materials such as graphene, and traditional methods like HMC often encounter significant challenges such as ergodicity problems due to the complex energy landscape and multi-modal nature of the model.

Methodological Advancements

The paper introduces normalizing flows as the primary tool to overcome these sampling difficulties. Normalizing flows are parametric bijective mappings that transform simple base distributions into complex target distributions. This adaptability makes them well-suited for approximating the Boltzmann distribution in the Hubbard model without the pitfalls of HMC. The key innovation here is the integration of equivariant properties into the normalizing flows, leveraging the inherent symmetries of the Hubbard action, including the $\mathbf{Z}_2$-symmetry, space-translation symmetry, and periodicity symmetry.

Equivariant Normalizing Flows

By embedding these symmetries into the architecture of the flow, the proposed approach reduces the complexity of the learning task, allowing the generative model to become more efficient and accurate. This transformation is crucial as it enables the normalizing flow to canonicalize the distribution, thus enhancing training efficiency and resulting in higher-quality samples, as demonstrated by the numerical experiments presented in the paper.

Numerical Results and Implications

The numerical experiments provide compelling evidence for the efficacy of equivariant normalizing flows. For a $2 \times 1$ lattice, the method drastically reduces training time, achieving a significantly higher acceptance rate compared to a non-equivariant approach. Moreover, it effectively mitigates the integrated autocorrelation time, indicating improved sampler efficiency. Notably, the normalizing flow method scales to larger time extents, such as the $2 \times 2$ lattice, where non-equivariant methods fail to learn the distribution.

These results underscore the capability of equivariant normalizing flows not only to handle the current limitations of HMC approaches effectively but also to provide a scalable framework that may impact the simulation of other lattice models or systems with complex symmetry properties.

Future Directions

This proof-of-concept work opens several avenues for future exploration. First, addressing the potential bijectivity challenges posed by canonical cells as the lattice size increases will be essential to extend these methods to larger and more complex systems. Additionally, exploration into various physical regimes of the Hubbard model, considering different coupling parameters, will further elucidate the versatility and efficacy of these flows.

Overall, this paper presents a strategically advantageous methodology for simulating models with complex energy landscapes, setting the stage for generative models' broader application in computational physics and potentially influencing the trajectory of AI methodologies applied to lattice field theories in the future.