Tackling the Sign Problem in the Doped Hubbard Model with Normalizing Flows

Abstract

The Hubbard model at finite chemical potential is a cornerstone for understanding doped correlated systems, but simulations are severely limited by the sign problem. In the auxiliary-field formulation, the spin basis mitigates the sign problem, yet severe ergodicity issues have limited its use. We extend recent advances with normalizing flows at half-filling to finite chemical potential by introducing an annealing scheme enabling ergodic sampling. Compared to state-of-the-art hybrid Monte Carlo in the charge basis, our approach accurately reproduces exact diagonalization results while reducing statistical uncertainties by an order of magnitude, opening a new path for simulations of doped correlated systems.

Figures (8)

• Figure 1: Schematic illustration of the annealing procedure controlled by the parameter $\lambda$. At $\lambda=0$, the fermion determinant is absent and the target distribution is Gaussian. As $\lambda$ is increased, fermionic effects are gradually introduced, deforming the distribution until the full interacting Hubbard action with its multimodal structure is recovered at $\lambda=1$.
• Figure 2: Average sign $\Sigma$ as a function of the chemical potential $\mu$ for an eight-site hexagonal lattice with $U = 2$, $\beta = 8$, and $N_t = 16$. Shown are results from the annealing-based NF approach in the spin basis (NF SP, yellow) and optimized HMC simulations in the charge basis Gantgen:2023byf (O+HMC CH, grey).
• Figure 3: One-body correlation functions for an eight-site hexagonal lattice with $U=2$, $\beta=8$, $\mu=1.75$, and $N_t=40$. Shown are continous exact diagonalization results (ED, grey dashed), state-of-the-art optimized HMC in the charge basis (O+HMC CH, grey), and our annealing-based NF ansatz in the spin basis (NF SP, yellow). Both HMC and NF results are generated with 1M samples each.
• Figure 4: One-body correlation functions for an 18-site hexagonal lattice with $U=2$, $\beta=8$, $\mu=2.5$, and $N_t=40$. Shown are state-of-the-art optimized HMC in the charge basis (O+HMC CH, grey) and our annealing-based NF ansatz in the spin basis (NF SP, yellow). Both HMC and NF results are generated with 1M samples each.
• Figure 5: Distribution of the two-site Hubbard model in the spin basis ($U=3$, $\beta=5$, and $N_t$=20) for $\mu=0$ (left) and $\mu=2$ (right). Shown are the fields summed over the temporal direction, i.e., $\phi_x = \sum_{t=0}^{N_t-1} \phi_{xt}$.