Comparing Symmetrized Determinant Neural Quantum States for the Hubbard Model

TL;DR

This work tackles the challenge of simulating strongly correlated fermions in the square-lattice Hubbard model using neural quantum states. It systematically compares two determinant-based ansätze, the Hidden Fermion Determinant State (HFDS) and the Jastrow-Backflow (JBf), each parameterized by a Vision Transformer (ViT), and studies symmetry-projection strategies. In simulations on $8\times4$ cylinders and $8\times8$ toruses at $U/t=8$ and $n_h=1/8$, the two architectures achieve comparable variational energies and observables, while explicit symmetry averaging yields the lowest energies and reveals stripe order on cylinders and doped-Mott-insulator features on the torus. The results illustrate both the promise and current challenges of neural quantum states for correlated fermions, notably scalability with system size and the need for scalable, accurate symmetry incorporation.

Abstract

Accurate simulations of the Hubbard model are crucial to understanding strongly correlated phenomena, where small energy differences between competing orders demand high numerical precision. In this work, Neural Quantum States are used to probe the strongly coupled and underdoped regime of the square-lattice Hubbard model. We systematically compare the Hidden Fermion Determinant State and the Jastrow-Backflow ansatz, parametrized by a Vision Transformer, finding that in practice, their accuracy is similar. We also test different symmetrization strategies, finding that output averaging yields the lowest energies, though it becomes costly for larger system sizes. On cylindrical systems, we consistently observe filled stripes. On the torus, our calculations display features consistent with a doped Mott insulator, including antiferromagnetic correlations and suppressed density fluctuations. Our results demonstrate both the promise and current challenges of neural quantum states for correlated fermions.

Figures (7)

• Figure 1: Schematic of the ViT architecture adapted to the HFDS and JBf fermionic wavefunctions.
• Figure 2: Comparison of variational energies on a the $8 \times 4$ cylinder. The plot shows converged energies for both architectures as a function of the number of variational parameters and symmetrization methods. For HFDS-ViT (left), the number of hidden fermions $\tilde{N}$ is varied. For JBf-ViT, the number of determinants is varied. Symbols indicate different numbers of stacked encoder blocks, $n_l$, in the ViT.
• Figure 3: Comparison of variational energies on the $8 \times 4$ cylinder. The plot shows converged energies for the presented architectures as a function of the number of Monte Carlo samples and symmetrization methods.
• Figure 4: Comparison of physical observables on the $8 \times 4$ and $8 \times 8$ cylinders. The first row shows a horizontal cut of the spin-spin correlation. The $x$ axis corresponds to the site index and the size of the balls is proportional to the hole density at that site. The central (bottom) row plots show the observables in real space for the lowest energy solution on the $8 \times 4$ ($8 \times 8$) system.
• Figure 5: Comparison of physical observables on the $8 \times 8$ square lattice (PBC-PBC) with $U/t=8$ and $n_h= 1/8$ hole doping. The results show real space plots of the expectation values of the of spin-spin correlation function, the density-density correlation and the associated structure factors for both ansätze. The expectation values are computed from the converged Symm-ViT variational states, the symmetrization being with respect to the translation group in the $\vb*k = (\pi/2, \pi/2)$ which was identified as the lowest energy sector.