Quantum Monte Carlo Calculations of Light Nuclei with Fully Propagated Theoretical Uncertainties

Abstract

We report on the first quantum Monte Carlo calculations of helium isotopes with fully propagated theoretical uncertainties from the interaction to the many-body observables. To achieve this, we build emulators for solutions to the Faddeev equations for the binding energy and Gamow-Teller matrix element of $^3\text{H}$, as well as for auxiliary-field diffusion Monte Carlo calculations of the $^4\text{He}$ charge radius, employing local two- and three-body interactions up to next-to-next-to-leading order in chiral effective field theory. We use these emulators to determine the posterior distributions for all low-energy couplings that appear in the interaction up to this order using Bayesian inference while accounting for theoretical uncertainties. We then build emulators for auxiliary-field diffusion Monte Carlo for helium isotopes and propagate the full posterior distributions to these systems. Our approach serves as a framework for $\textit{ab initio}$ studies of atomic nuclei with consistently treated and correlated theoretical uncertainties.

Figures (4)

• Figure 1: Corner plot for the posterior distributions of the ground-state energies of Helium isotopes with fully propagated uncertainties. Fit 1 adjusts 3N forces to the $^3$He energy and GT matrix element, while fit 2 also includes the $^4$He charge radius. The 2D panels show the posterior at 50% and 90% credible intervals. We show the mean values with 68% confidence interval for our posteriors (error bars) and compare with experiment Wang_Huang_Kondev_etal_2021 and AFDMC calculations of Ref. Lonardoni_Gandolfi_Lynn_etal_2018.
• Figure 2: Percent error of the emulators for the $^3\text{H}$ energy as a function of the number of training points. The percent error is computed by comparing the emulator predictions against a set of 75 exact Faddeev calculations, which remain fixed for all $N_{\text{train}}$ and PMM dimension. Both the PMM and EC emulators are tuned to reproduce the $^3\text{H}$ energy across the 11 dimensional LEC space for the full N$^2$LO. The training points are chosen uniformly from the LEC prior distributions. For the EC it was also necessary to choose training points from this set such that the matrices did not become ill-conditioned.
• Figure 3: Posteriors for the $^3$H energy, GT matrix element ratio to experiment, and $^4$He charge radius used to constrain the 3N couplings $c_D$ and $c_E$ in our Bayesian fit. For the $^3$H properties, we compare predictions for both the EC and PMM emulators, and find that despite the EC's significantly smaller emulator error (see Fig. \ref{['fig:ntrain']}) differences of the resulting distributions are negligible. The shaded regions in the correlation plots cover 50% and 90% of the total probability. Experimental values for fits from Wang_Huang_Kondev_etal_2021Gazit_Quaglioni_Navratil_2009Krauth_Schuhmann_Ahmed_etal_2021
• Figure 4: Normalized posterior distributions for all LECs at N$^2$LO with $R_0=0.6\ \text{fm}$ and the $\text{E}_{\tau}$ form of the 3N interaction Lynn_Tews_Carlson_etal_2016. The initial uniform prior for the NN LECs is shown in red. The posterior of the NN Bayesian fit from Ref. Somasundaram_Lynn_Huth_etal_2024 together with uniform priors for $c_D$ and $c_E$ is shown in yellow, and the new result for the full posterior for all 11 LECs is shown in blue (fit 1 to only $^3$H $E_0$ and GT matrix element) and green (fit 2 with the additional $^4$He charge radius constraint).