Complex-energy eigenvector continuation for nuclear many-body broad resonances

TL;DR

Broad resonances in nuclear systems couple to the continuum, making high-dimensional, potentially unstable calculations challenging. The authors extend the complex-energy eigenvector continuation approach to the Berggren basis within the Gamow shell model, by scaling the Hamiltonian as $H(\alpha)=H_0+\alpha H_1$ and projecting the target problem onto a subspace spanned by training solutions, yielding energies of the form $\tilde{E}=\tilde{E}_R-i\tilde{\Gamma}/2$. They benchmark the method on ${}^{4}\mathrm{H}$, ${}^{4}\mathrm{n}$, ${}^{6}\mathrm{He}$, and ${}^{7}\mathrm{He}$, showing that including bound and narrow-resonance training states is essential to accurately reproduce both the resonance energy $E_R$ and width $\Gamma$ for broad states. The results demonstrate that complex-energy EC provides a stable, efficient extrapolation framework for open quantum systems with broad resonances, with remaining uncertainties largely tied to the choice of NN interactions and SRG evolution, and with potential guidance for future experiments.

Abstract

Broad resonances are a unique phenomenon in nuclear many-body systems. Theoretical studies usually involve the continuum degree of freedom, which drastically increases the model space of calculations, and may lead to non-convergence or instability of computations. In this paper, we present the extension of the eigenvector continuation (EC) method to the complex-energy space to treat the broad resonances of open quantum systems of nuclei. EC provides an efficient method to predict the solution of a large-space many-body problem within a small subspace. Using only a few bound and narrow resonance solutions as input in EC, we can obtain the solution of a broad resonance. We have applied the complex-energy EC to the broad resonances of $^4$H, four-neutron $^4n$, $^6$He and $^7$He systems.

Figures (4)

• Figure 1: $^4$H EC extrapolation to the $2^{-}$ g.s. broad resonance based on NCGSM with $\mathrm{{N^3LO}_{EM}}$. Left panels show the real part $E_R$ and resonance width $\Gamma$ of the eigen energy obtained using the EC emulator based on $\alpha=1.7, 1.8, 1.9, 2.0$ bound snapshots [denoted by EC($\mathcal{B}$)], and using the emulator with $\alpha=1.4,1.5$ narrow resonances incorporated [denoted by EC($\mathcal{B+R}$)], compared with exact NCGSM calculations. The right panel shows norm matrix elements $\tilde{N}_{ij}$ in the grid of the control parameter $\alpha$ with the chosen snapshots indicated by white dots. The bound and resonance regimes are indicated by $\mathcal{B}$ and $\mathcal{R}$, respectively. Note that norm matrix elements are complex numbers, therefore we plot their absolute values.
• Figure 2: EC emulated energy $E_R$ (relative to $^4$He) and resonance width $\Gamma$ of the $^6$He $2^+_1$ excited state using three different choices of four snapshots, compared with exact GSM calculations and experimental data 6He_exp. The deviation from the exact width at the target point $\alpha_*=1.0$ is defined by $\sigma=|\Gamma^{\mathrm{EC}}-\Gamma^{\mathrm{Exact}}|/\Gamma^{\mathrm{Exact}}$. EC calculation with the choice of only the four bound snapshots leads to a bound extrapolation result, as shown in the leftmost panel.
• Figure 3: EC calculations of ${}^{4}n$ energy $E_\textsc{R}$ and resonance width $\Gamma$, compared with the data of the observed resonance-like peak Duer2022. For $\mathrm{N^3LO_{EM}}$, we present detailed EC result as a function of $\alpha$, with theoretical uncertainties at the target $\alpha_*=1.0$ indicated by error bars. EC extrapolations at the target with $\mathrm{N^2LO_{opt}}$, $\mathrm{N^4LO_{EMN}}$ and $\mathrm{N^3LO_{local}}$ are shown by symbols $\bm{\vartriangle}$, $\bm{\triangledown}$ and $\bm{\Box}$, respectively, in the right. Other calculations based on NCGSM ($\pmb{\pentagon}$) PhysRevC.100.054313, NCSM ($\raisebox{-0.35ex}{$\bm{\circ}$}$) PhysRevLett.117.182502 and QMC ($\raisebox{-0.3ex}{${\triangleright}$}$) PhysRevLett.118.232501 are also included in the rightmost for comparisons.
• Figure 4: EC emulations for ${}^{7}$He $3/2^-$ g.s., $1/2^-$ and $5/2^-$ excited states, with theoretical uncertainties at the target $\alpha_*=1.0$ indicated by error bars. Red diamonds indicate chosen EC snapshots which belong to exact GSM solutions with $^4$He core. Experimental data are taken from Ref. TILLEY20023, but for the $3/2^-$ resonance different experiments TILLEY20023CAO201246PhysRevC.109.L061602 gave different widths indicated by a blue shadowing bar. Note that different axis scales are used for different panels.