Neural Wave Functions for Superfluids

TL;DR

The paper tackles the challenge of describing the superfluid ground state of the unitary Fermi gas (UFG), a universal system with $a^{-1}=0$ and $r_e=0$, for which the ground-state energy obeys $E=\xi E_{FG}$ with $E_{FG}=\frac{3}{5}\frac{\hbar^2 k_F^2}{2m}$. It employs variational Monte Carlo with the Fermionic Neural Network (FermiNet) and reveals that the original Slater FermiNet struggles to describe large, paired systems, motivating a shift to antisymmetric geminal power (AGP) based wave functions implemented within the FermiNet framework. The AGPs FermiNet delivers substantially improved energies, often beating fixed-node diffusion Monte Carlo for the same model interaction, and yields finite pairing gaps $\Delta$ and condensate fractions $c$ that align with experimental and AFMC benchmarks. This work provides a general, basis-free, variational approach to studying superfluids and suggests broad applicability to other pairing symmetries and quantum many-body systems, with potential extensions to Pfaffian forms and non-$s$-wave superfluids.

Abstract

Understanding superfluidity remains a major goal of condensed matter physics. Here we tackle this challenge utilizing the recently developed Fermionic neural network (FermiNet) wave function Ansatz [D. Pfau et al., Phys. Rev. Res. 2, 033429 (2020).] for variational Monte Carlo calculations. We study the unitary Fermi gas, a system with strong, short-range, two-body interactions known to possess a superfluid ground state but difficult to describe quantitatively. We demonstrate key limitations of the FermiNet Ansatz in studying the unitary Fermi gas and propose a simple modification based on the idea of an antisymmetric geminal power singlet (AGPs) wave function. The new AGPs FermiNet outperforms the original FermiNet significantly in paired systems, giving results which are more accurate than fixed-node diffusion Monte Carlo and are consistent with experiment. We prove mathematically that the new Ansatz, which only differs from the original Ansatz by the method of antisymmetrization, is a strict generalization of the original FermiNet architecture, despite the use of fewer parameters. Our approach shares several advantages with the original FermiNet: the use of a neural network removes the need for an underlying basis set; and the flexibility of the network yields extremely accurate results within a variational quantum Monte Carlo framework that provides access to unbiased estimates of arbitrary ground-state expectation values. We discuss how the method can be extended to study other superfluids.

Figures (8)

• Figure 1: Comparison between results obtained using the AGPs FermiNet and the Slater FermiNet for different numbers of particles, $N$, with $r_s=1$ and $\mu = 12$. All simulations used 32 determinants, 300,000 optimization steps, and the same hyperparameters, which are detailed in the Appendix.
• Figure 2: Comparison of the system-size dependent values of the Bertsch parameter, $\xi$, as calculated using the AGPs FermiNet and FN-DMC, with $k_F=1$ and $\mu = 12$. According to the variational principle, lower values are better. The error bars on the AGPs FermiNet results are smaller than the sizes of the crosses. Inset: difference between the AGPs and FN-DMC values of the Bertsch parameter. The errors in the inset are obtained by adding the standard errors of the FN-DMC and AGPs FermiNet results in quadrature. The latter are obtained by computing the standard error of the MCMC-averaged Bertsch parameter accumulated over 50,000 inference steps.
• Figure 3: Pairing gaps calculated with the Slater FermiNet and the AGPs FermiNet for different numbers of particles $N$, with $r_s=1$ and $\mu = 12$.
• Figure 4: Comparison of the TBDM estimators calculated using the AGPs FermiNet and the Slater FermiNet with $N=38$, $r_s=1$ and $\mu = 12$. The error bars show the standard error of the TBDM estimator, accumulated over 2,000 inference steps. Most of the error bars are so small that they are obscured the symbols.
• Figure 5: Number of block-diagonal determinants required for the multi-determinant Slater FermiNet energy to fall within 5% and 10% of the energy obtained using a single-determinant/single-geminal AGPs FermiNet.