Complex-scaled no-core shell model calculations of bound and unbound nuclear states in light nuclei

TL;DR

This work develops a complex-scaled no-core shell model (CS-NCSM) by applying complex scaling to the intrinsic Hamiltonian, enabling ab initio calculation of bound and resonant states in light nuclei. By combining an HO basis with a Lawson correction to remove center-of-mass spuriosity and using TBMEs that are either rotated in coordinate space or handled via a separable expansion, the intrinsic Hamiltonian yields complex-symmetric matrices whose eigenvalues reveal resonance energies and widths through a theta-trajectory stabilization approach. Numerical results for $^2$H–$^5$He/$^5$Li with the Daejeon16 interaction show good agreement with prior methods and demonstrate convergence with model-space size, while illustrating practical limits in width handling and the need for larger spaces or three-body forces in future work. The CS-NCSM offers a computationally competitive path to ab initio resonance properties in light nuclei, with potential impact on low-energy nuclear structure and reactions.

Abstract

The complex scaling method is commonly used to describe decaying states, but its applications are limited because the Hamiltonian operator must contain only relative coordinates. This has hindered the use of complex scaling in models defined with laboratory single-particle coordinates, and in particular one of the most important model in low-energy nuclear physics, the no-core shell model. We will then present a straightforward procedure for introducing complex scaling in the no-core shell model in order to calculate nuclear resonance states. For that matter, the complex-scaled two-body matrix elements must firstly be determined, and the resulting many-body Hamiltonian complex symmetric matrix must be diagonalized afterwards. Applications pertain to the bound ground states of the lightest nuclei $^2{\rm H}$, $^3{\rm H}$, $^3{\rm He}$, and $^4{\rm He}$, as well as the resonance ground states of $^5$He and $^5$Li, whereby the realistic interaction Daejeon16 is utilized.

Figures (4)

• Figure 1: The schematic spectrum of the CS intrinsic Hamiltonian. It is assumed that the intrinsic Hamiltonian has only one bound state, one resonance state and two thresholds. The bound and resonance states are displayed with black and red circles. The thresholds are indicated by green circles, and thick, solid lines sign the rotated-down continuums. The dotted circular arc designates an angular value of $2\theta$.
• Figure 2: Schematic spectrum of the CS Lawson-corrected Hamiltonian. The intrinsic Hamiltonian only has two thresholds. Subsystems do not have resonances and the system have one bound and one resonance state. The bound and resonance states are represented by black and red circles. The thresholds are denoted by green circles, and the rotated down continuums are represented by thick solid lines. Only the part of the spectrum is shown where the maximum excitation of the c.m. motion is $2\hbar \omega$. The dotted circular arc represents an angle of $2\theta$.
• Figure 3: The $1^+$ spectrum of the deuteron nucleus in the CS-NCSM model space of $12 \hbar \omega$ and a CS angle of $\theta=5$ degrees is depicted. Black dots represent the eigenvalues obtained from the numerical diagonalization of the CS Hamiltonian matrix. The colored lines correspond to the theoretical continuum spectrum after rotation. The solid line arc represents the angle of $2\theta$ between the rotated contour and the real axis. Theoretical shifts of $\hbar\omega$ (contours) and $2\hbar\omega$ (ground states) are depicted (see text for details).
• Figure 4: Complex eigenenergies of $^5$He (full squares) and $^5$Li (full circles) in the CS-NCSM model using $6 \hbar \omega, 10 \hbar \omega, 12 \hbar \omega$ model spaces (top, middle, bottom, repsectively) as a function of CS angles of $\theta=7.5-16$ degrees. Energies are provided with respect to $^4$He. A polynomial fit linking the different complex eigenenergies is illustrated by dashed and solid lines for $^5$He and $^5$Li, respectively. Results are not provided for the $8 \hbar \omega$ model space because complex energies do not present optima in $\theta$ contours (see text for details).