Toward Scalable Normalizing Flows for the Hubbard Model

TL;DR

The paper targets scalable sampling of the Hubbard model using generative methods and NE-MCMC. It introduces a sausage-based formulation and QR stabilization to improve numerical stability and reduce cost, and analyzes how (stochastic) normalizing flows and non-equilibrium MCMC scale with lattice size and temperature. Results show linear-scaling advantages for NE-MCMC with ESS behaving as a function of $n_{step}/N_x$, while NF/SNF training costs grow exponentially with spatial extent at fixed MC depth; SNFs may reach linear scaling if MC depth increases with system size. Overall, NE-MCMC currently offers the most robust scalability, with SNFs holding promise under adjusted training dynamics for fully scalable Hubbard-model sampling.

Abstract

Normalizing flows have recently demonstrated the ability to learn the Boltzmann distribution of the Hubbard model, opening new avenues for generative modeling in condensed matter physics. In this work, we investigate the steps required to extend such simulations to larger lattice sizes and lower temperatures, with a focus on enhancing stability and efficiency. Additionally, we present the scaling behavior of stochastic normalizing flows and non-equilibrium Markov chain Monte Carlo methods for this fermionic system.

Figures (3)

• Figure 1: Scaling of computational cost (a) and memory consumption (b) for the full fermion matrix (blue) and the sausage formalism (orange) for different $N_t$ at fixed $N_x = 4$. Dashed lines correspond to fits based on the scaling behavior from \ref{['tab:action_scaling']}. Error bars are smaller than the marker size.
• Figure 2: Precision of the Hubbard action in the sausage formalism with (orange) and without (blue) QR decomposition for different $\beta$. The black line marks graphene at room temperature, and the red shaded area indicates impracticable precision. The system is evaluated at $U=2$, $N_x = 20$, and $\delta = 0.5$. For most data points, the error bars are smaller than the markers.
• Figure 3: Scaling of the training cost of (S)NFs for a fixed architecture (left) and scaling of the sampling cost of NE-MCMC (right) at $U=2$.