Simulating the Hubbard Model with Equivariant Normalizing Flows

TL;DR

The paper tackles ergodicity challenges in simulating the Hubbard model by using equivariant normalizing flows to learn the Boltzmann distribution $p(\phi) \propto e^{-S[\phi]}$ and generate i.i.d. samples. It introduces a symmetry-aware flow architecture that enforces Hubbard model symmetries (Z2, space-translation, and approximate periodicity) to improve training efficiency and sampling quality. Empirical results on small 1+1D lattices show substantial gains over standard HMC, with higher acceptance rates and dramatically reduced autocorrelation times, especially as the temporal extent grows. This work demonstrates a promising, symmetry-informed generative approach for sampling from challenging distributions in strongly correlated lattice systems, motivating further scaling and regime exploration.

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.

Figures (5)

• Figure 1: HMC configurations for a $(N_x,N_t) = (2,1)$ lattice with parameters $\beta=\kappa=1$ and $U=18$, generated with a leapfrog integrator with an integration step of $\epsilon = 0.1$ (left) and $\epsilon = 1.0$ (right). The integrated autocorrelation time $\tau_\mathrm{int}$ and the acceptance rate $a$ are $\tau_{\mathrm{int}} = 83 \pm 14$ and $a = 99.9\%$ (left) and $\tau_\mathrm{int}= 346\pm100$ and $a = 86.9\%$ (right), respectively. Histograms of the magnetization, obtained by summing along each dimension, respectively, are shown at the top and on the right hand side of the figures. The analytical contours are exact for $N_t=1$ and in the strong-coupling limit, i.e. $U/\kappa\rightarrow \infty$, for $N_t>1$. The colormap ranging from purple to yellow represents the areas of low and high density, respectively.
• Figure 2: Schematic representation of a coupling-based normalizing flow. Starting with a Gaussian prior distribution $q_Z(z)$ (left), the normalizing flow $f_\theta(z)$ maps the prior samples $z$ to the target samples $\phi$ through a set of invertible functions $f^i$, each taking as input the output of the previous layer, $y^{i-1}$, ultimately approximating the target distribution $p(\phi)$ (right). In the context of lattice field theories, this target distribution corresponds to the path integral distribution $p(\phi) = Z^{-1} e^{-S[\phi]}$, where $S[\phi]$ is the action of the theory.
• Figure 3: Transformation of the prior distribution into the canonical cell of the theory. Starting from a Gaussian prior distribution (left), we apply three types of symmetry transformations (middle): $\mathbf{Z}_2$ symmetry (top), space-translation symmetry (center), and periodicity symmetry (bottom). This results in a canonical cell of triangular shape (right). The small black arrows indicate the mapping of three exemplary samples under these transformation, indicating their position before and after the transformation has been applied.
• Figure 4: Field configurations obtained with non-equivariant (left), equivariant (middle), and equivariant and metropolized (right) normalizing flows. The marginalized magnetization, shown at the top and on the right of each plot, is exact for $N_t=1$ and in the strong-coupling limit for $N_t>1$. The top and bottom rows show results for $(N_x,N_t) = (2,1)$ and $(N_x,N_t) = (2,2)$ lattices, respectively. For $N_t\geq 1$, $\phi_{\{1,2\}}$ is understood to be the sum in the temporal direction, i.e., $\phi_{\{1,2\}} = \sum_{i=1}^{N_t} \phi_{\{1,2\}t}$. While the equivariant approach is able to learn the structure for both lattice sizes to high precision, the non-equivariant flow is unable to learn an approximate to the target distribution for $N_t> 1$. Note that the slightly visible lines in the equivariant distributions originate from a penalty term necessary to keep the normalizing flow bijective while applying the symmetry transformation $T$. The right-most plots on both rows have been obtained by performing metropolization, i.e., filtering i.i.d. samples through a metropolis accept-reject step. This ensures that the sampling from the approximate model $q_\theta$ is asymptotically unbiased.
• Figure 5: Acceptance rate vs. training steps for a non-equivariant (red) and an equivariant (purple) normalizing flow for a $(N_x,N_t)=(2,1)$ lattice. The equivariant approach yields acceptance rates above $80\%$ after four thousand training steps, while the non-equivariant only reaches $40\%$. In order to to reach an acceptance rate of$~70\%$, the non-equivariant flow would need more than seven-hundred thousand training steps.