Ab Initio Quadrupole-Invariant Analysis of Neutron-Rich Deformed Nuclei Around N = 28

Abstract

We present an ab initio quadrupole-invariant study of neutron-rich nuclei around $N=28$. Effective Hamiltonians and consistently evolved $E2$ operators are obtained from chiral NN+3N interactions using the ab initio valence-space in-medium similarity renormalization group, and the resulting $E2$ matrix elements are used to extract the deformation parameters. Along the even–even $N=28$ isotonic chain, the $\beta$ values increase systematically from ⁴⁸Ca to ⁴⁰Mg, providing a direct deformation signature of the weakening of the $N=28$ shell closure. The extracted $\beta$ and $\gamma$ values show a prolate–oblate–triaxial evolution of the $0_1^{+}$ states from ⁴⁰Mg to ⁴²Si and ⁴⁴S, while the $0_2^{+}$ states remain mainly in the prolate-to-triaxial region. Extending the analysis to the neighboring $N=27$ and 29 isotones, we find a systematic evolution from strongly prolate Mg isotopes, to oblate-side Si isotopes, and to prolate-to-triaxial S isotopes. These results provide a unified picture of quadrupole deformation, triaxiality, and configuration coexistence around $N=28$.

1. Introduction

Understanding how shell structure evolves far from stability is one of the central goals of modern nuclear structure physics. The traditional magic numbers, which are well established near the valley of stability [1,2], can be strongly modified in neutron-rich systems because of changes in the single-particle spectrum and the enhanced role of correlations [3,4,5,6,7,8]. Quadrupole collectivity is a particularly sensitive probe of such shell evolution: reduced low-lying excitation energies, enhanced electric-quadrupole transition strengths, and the appearance of deformed or coexisting structures all signal the weakening of spherical shell closures.

The neutron number $N=28$ offers a particularly important example. It is robust in the doubly magic nucleus ⁴⁸Ca, while below calcium the rapid lowering of the $2_1^{+}$ excitation energies, the enhancement of $E2$ transition strengths, and the appearance of low-lying coexisting structures indicate a strong reduction in the spherical shell gap [9,10,11,12,13,14,15,16,17,18,19,20,21]. The erosion of the gap between the neutron $\nu 0f_{7/2}$ and $\nu 1p_{3/2}$ orbitals favors quadrupole correlations, and the active proton degrees of freedom in the $sd$ shell further amplify collectivity through quasi-SU(3), pseudo-SU(3), and the Jahn–Teller effect [22,23,24,25,26,27,28]. As a consequence, the neutron-rich nuclei around $N=28$ display a rich interplay between shell erosion, deformation, and shape coexistence.

A number of theoretical studies have investigated the neutron-rich $N=28$ region, including shell-model (SM) calculations [22,23,24,25,26,27,29,30], beyond-mean-field (BMF) approaches [31,32,33,34], antisymmetrized molecular dynamics (AMD) calculations [28,35,36], and ab initio approaches [20,21,37]. These studies have suggested a rapid shape evolution in the even–even $N=28$ isotones, including well-developed deformation in ⁴⁰Mg and ⁴²Si, shape coexistence in ⁴⁴S, and a prolate–oblate–prolate evolution of the ground-state-associated structures from ⁴⁰Mg through ⁴²Si to ⁴⁴S [11,14,15,28,38,39,40]. Furthermore, previous SM analyses based on quadrupole rotational invariants have shown that the triaxial degree of freedom plays an important role in ^43,44S [30]. The low-lying structures in the sulfur isotopes have been interpreted in terms of shape and configuration coexistence, including the coexistence of the $3/2_1^{-}$ ground state with the $7/2_1^{-}$ isomer in ^43,45S and the appearance of low-lying deformed and triaxial structures in ⁴⁴S [21,30]. These interpretations are also connected to the rich experimental spectroscopy of sulfur isotopes in this mass region [41,42,43,44,45,46,47,48]. AMD and BMF calculations have also emphasized triaxiality, large shape fluctuations, and  competing configurations, especially in ⁴⁴S [28,33,34].

Recent ab initio valence-space in-medium similarity renormalization group (VS-IMSRG) calculations have provided a microscopic description of this region using consistently evolved Hamiltonians and electromagnetic operators [20,21,37]. Those studies used spectra, $B\left(E2\right)$ values, and spectroscopic quadrupole moments $Q_s$ to identify rotational sequences and configuration-coexisting structures in neutron-rich nuclei around $N=28$. These results showed that ab initio calculations can describe the emergence of quadrupole collectivity and provide microscopic insight into shape coexistence and isomeric structures in this mass region. However, $Q_s$ values and transition maps by themselves do not provide a direct state-by-state representation of deformation in the $\beta$–$\gamma$ plane. In particular, a small or vanishing $Q_s$ does not necessarily imply a spherical shape; it may also reflect triaxiality, K mixing, or fluctuations between prolate and oblate components. A complementary analysis is therefore useful to convert the ab initio $E2$-transition matrix elements into quadrupole invariants. Quadrupole invariants provide such a bridge between $E2$ matrix elements and intrinsic-shape language. Following the quadrupole-sum-rule approach of Kumar and Cline [49,50], the second- and third-order quadrupole invariants allow one to extract the magnitude of quadrupole collectivity $\beta$ and the triaxiality indicator $\gamma$ from $E2$ matrix elements, while higher-order invariants further provide estimates of the fluctuations in these deformation parameters [30,51,52].

In this work, we perform an ab initio quadrupole-invariant analysis of neutron-rich nuclei around $N=28$. Starting from chiral NN+3N interactions, we use the VS-IMSRG to consistently derive effective Hamiltonians and $E2$ operators, and then construct quadrupole invariants and extract deformation parameters $\beta$ and $\gamma$ for the low-lying states. We first examine the even–even $N=28$ isotones from ⁴⁸Ca to ⁴⁰Mg, focusing on the evolution of the $0_{1,2}^{+}$ and $2_{1,2}^{+}$ states. We then analyze the detailed low-lying structures of ⁴⁰Mg, ⁴²Si, and ⁴⁴S by combining $\beta$, $\gamma$, $Q_s$, and $E2$-transition maps. Finally, we extend the analysis to the neighboring $N=27$ and $N=29$ Mg, Si, and S isotopes to obtain a systematic $\beta$-$\gamma$ picture of shape evolution around $N=28$.

2. Materials and Methods

2.1. Valence-Space In-Medium Similarity Renormalization Group

We start from the intrinsic A-body Hamiltonian

$$H=\sum _{i<j}\left[{\displaystyle \frac{{({\overrightarrow{p}}_i-{\overrightarrow{p}}_j)}^2}{2mA}}+v_{ij}^{\mathrm{NN}}\right]+\sum _{i<j<k}{v}_{ijk}^{3\mathrm{N}},$$

(1)

where $\overrightarrow{p}$ is the nucleon momentum, m is the nucleon mass, and  $v^{\mathrm{NN}}$ and $v^{3\mathrm{N}}$ denote the NN and 3N interactions, respectively. In this work we employ the chiral NN+3N interaction EM1.8/2.0 [53,54,55,56], which is composed of a next-to-next-to-next-to-leading order NN interaction softened by the similarity renormalization group (SRG) evolution with momentum resolution scale $\lambda =$ 1.8 fm⁻¹ and a next-to-next-to-leading order 3N force with momentum cutoff $\Lambda =$ 2.0 fm⁻¹. The calculation is performed in a harmonic-oscillator basis at $\hslash \omega =16$ MeV with $e_{\max }=2n+𝓁=14$ and $E_{3\max }=14$. This model space was adopted in the previous VS-IMSRG calculations for neutron-rich nuclei around $N=28$ [20,21,57], which is sufficient for convergence.

In practical calculations, Hamiltonian (1) is transformed to a Hartree–Fock basis and rewritten in normal-ordered form,

$$H=E_0+\sum _{ij}{f}_{ij}\left\{a_i^{†}{a}_j\right\}+{\displaystyle \frac{1}{4}}\sum _{ijkl}{\Gamma }_{ijkl}\left\{a_i^{†}{a}_j^{†}{a}_l{a}_k\right\},$$

(2)

where the residual normal-ordered three-body term is neglected, while the dominant effects of the 3N interaction are retained through ensemble normal ordering. The VS-IMSRG aims to decouple the Hamiltonian from the large Hilbert space into a small valence space. This decoupling is achieved by solving the flow equation

$${\displaystyle \frac{dH\left(s\right)}{ds}}=[\eta \left(s\right),H\left(s\right)],$$

(3)

with the anti-Hermitian generator $\eta \left(s\right)$. In the present work, the Magnus formalism is employed with all operators truncated at the two-body level [58,59,60,61,62,63,64],

$$H_{\mathrm{eff}}=e^{\mathsf{\Omega }}H{e}^{-\mathsf{\Omega }},{\widehat{O}}_{\mathrm{eff}}=e^{\mathsf{\Omega }}\widehat{O}{e}^{-\mathsf{\Omega }},$$

(4)

where $\widehat{O}$ denotes an observable operator. In practical calculations, we take protons in the $sd$ shell and neutrons in the $pf$ shell above the ²⁸O core as the valence space. The effective Hamiltonians and $E2$ operators for this valence space are derived consistently by VS-IMSRG. The valence-space Hamiltonian is subsequently diagonalized with the large-scale shell-model code KSHELL [65]. $E2$ transition matrix elements calculated with the effective $E2$ operator are then used for the quadrupole-invariant analysis.

2.2. Quadrupole Invariants and Fluctuation Ranges

In the classical geometrical description, the nuclear shape is parametrized by the quadrupole deformation variables $\beta$ and $\gamma$, where $\beta$ specifies the overall deformation magnitude and $\gamma$ describes the axial asymmetry [66]. In the usual convention, $\gamma =0^{\circ }$ and ${60}^{\circ }$ correspond to axially symmetric prolate and oblate shapes, respectively, while intermediate values describe triaxial shapes. In the principal-axis frame, the  nonzero components of the electric quadrupole tensor are $Q_{20}={\displaystyle \frac{3Ze{R}_0^2}{4\pi }}\beta \cos \gamma$, and $Q_{2\pm 2}={\displaystyle \frac{3Ze{R}_0^2}{4\pi \sqrt{2}}}\beta \sin \gamma$. Scalar combinations constructed from this tensor are rotationally invariant, with the second- and third-order invariants being proportional to ${\beta }^2$ and ${\beta }^3\cos \left(3\gamma \right)$, respectively. In the rigid-rotor limit, these invariants recover the conventional geometrical deformation parameters. For  soft or configuration-mixed nuclei, however, they characterize moments of the deformation distribution rather than a unique static shape, while the higher-order quadrupole invariants determine the corresponding fluctuations in $\beta$ and $\gamma$.

In this work, the deformation parameters are extracted from the quadrupole-invariant method of Kumar and Cline [49,50], using the large-scale shell-model implementation of ref. [52]. Higher-order rotational invariants beyond the third order can be constructed through different J couplings, where J denotes the coupling of a pair of $E2$ operators [52]. In the present work we use the $J=0$ couplings [52]. For a state $|s⟩$, the required scalar invariants are written as

$$⟨Q^2⟩=\sqrt{5}{⟨s\left|\right|[E2\times E2]}^0\left|\right|s⟩,$$

(5)

$$⟨Q^3\cos \left(3\gamma \right)⟩=-\sqrt{{\displaystyle \frac{35}{2}}}⟨s\left|\right|{[{[E2\times E2]}^2\times E2]}^0\left|\right|s⟩,$$

(6)

$$⟨Q^4⟩=5⟨s\left|\right|{[{[E2\times E2]}^0\times {[E2\times E2]}^0]}^0\left|\right|s⟩,$$

(7)

$$⟨Q^6{\cos }^2\left(3\gamma \right)⟩={\displaystyle \frac{35}{2}}⟨s\left|\right|{[{[{[E2\times E2]}^2\times E2]}^0\times {[{[E2\times E2]}^2\times E2]}^0]}^0\left|\right|s⟩,$$

(8)

$$⟨Q^6⟩=5\sqrt{5}⟨s\left|\right|{[{[{[E2\times E2]}^0\times {[E2\times E2]}^0]}^0\times {[E2\times E2]}^0]}^0\left|\right|s⟩.$$

(9)

The quantity $⟨Q^3⟩$ entering the definition of $\gamma$ is estimated by the even-order invariants as

$$⟨Q^3⟩={\left[{\displaystyle \frac{1}{2}}({⟨Q^2⟩}^{1/2}+{⟨Q^4⟩}^{1/4})\right]}^3.$$

(10)

The invariant deformation parameters are defined as

$$\beta ={\displaystyle \frac{4\pi {⟨Q^2⟩}^{1/2}}{3Ze{R}_0^2}},\cos \left(3\gamma \right)={\displaystyle \frac{⟨Q^3\cos \left(3\gamma \right)⟩}{⟨Q^3⟩}}.$$

(11)

with $R_0=1.2A^{1/3}\mathrm{fm}$. The fluctuations of $\beta$ and $\cos \left(3\gamma \right)$ are defined by the higher-order invariants as

$$\Delta \beta ={\displaystyle \frac{\beta \sqrt{⟨Q^4⟩-{⟨Q^2⟩}^2}}{2⟨Q^2⟩}},\Delta \cos \left(3\gamma \right)=\sqrt{{\displaystyle \frac{⟨Q^6{\cos }^2\left(3\gamma \right)⟩}{⟨Q^6⟩}}-{\left({\displaystyle \frac{⟨Q^3\cos \left(3\gamma \right)⟩}{⟨Q^3⟩}}\right)}^2}.$$

(12)

Figure 1 shows the convergence of the quadrupole-invariant deformation parameters with respect to the number of intermediate states. For  the $0_1^{+}$ and $0_2^{+}$ states in ⁴⁴S, the calculated $\beta$, $\Delta \beta$, $\gamma$, and $\Delta \gamma$ are well converged with 100 states. The deformation parameters for the other nuclei studied in this work are also calculated using 100 states, with convergence checked in the same manner.

Figure 1. Convergence of the quadrupole-invariant deformation parameters for the $0_1^{+}$ and $0_2^{+}$ states in ⁴⁴S as a function of the number of intermediate states. The vertical ranges denote the corresponding fluctuation estimates.

3. Results

Extensive experimental and theoretical studies have shown that the robust $N=28$ shell closure in ⁴⁸Ca is strongly eroded toward the neutron-rich Mg, Si, and S isotopes, where quadrupole collectivity and deformation emerge [4,6,20,21]. Figure 2 provides an invariant-based visualization of this evolution. From ⁴⁸Ca to ⁴⁰Mg, the $\beta$ values of the $0_{1,2}^{+}$ states increase systematically. The very small $\beta$ value of the ⁴⁸Ca ground state reflects its nearly spherical doubly magic character, whereas the low-lying states in ⁴⁰Mg reach $\beta \approx 0.30$, indicating well-developed quadrupole collectivity. This trend gives a direct deformation signature of the weakening of the $N=28$ shell closure toward the neutron-rich region.

Figure 2. Calculated $\beta$ and $\gamma$ of the low-lying $0_{1,2}^{+}$ and $2_{1,2}^{+}$ states along the $N=28$ isotonic chain.

The low-lying spectra and $B\left(E2\right)$ strengths of ⁴⁰Mg, ⁴²Si, and ⁴⁴S, calculated using the same VS-IMSRG framework and EM1.8/2.0 interaction as in the present study, were compared in detail with the available experimental data in our previous work [20]. The low-lying spectra were reasonably reproduced, whereas the absolute $B\left(E2\right)$ strengths exhibited the systematic underestimation commonly found in VS-IMSRG calculations [67]. For example, the calculated excitation energy of the $2_1^{+}$ state in ⁴⁴S, 1.39 MeV, is close to the experimental value of 1.329 MeV, whereas the calculated $B(E2;2_1^{+}\to 0_1^{+})=30e^2{\mathrm{fm}}^4$ is below the experimental value of $44\left(6\right)e^2{\mathrm{fm}}^4$ [68]. Consequently, the absolute $\beta$ values extracted here may also be underestimated. In contrast, $\gamma$ is determined from ratios of quadrupole invariants and is therefore expected to be less sensitive to an approximately common rescaling of the $E2$ matrix elements. Nevertheless, nonuniform truncation effects among different matrix elements cannot be excluded, and their quantitative impact on the extracted $\beta$ and $\gamma$ values requires further investigation.

Beyond the overall deformation magnitude, the evolution of $\gamma$ reveals a pronounced state dependence of the low-lying quadrupole structures. In the usual axial-rotor convention, the sign change of $Q_s\left(2_1^{+}\right)$ from negative in ⁴⁰Mg, to positive in ⁴²Si, and back to negative in ⁴⁴S suggests a prolate–oblate–prolate evolution of the ground-state-associated structures, as shown in Table 1. The invariant-derived $\beta$ and $\gamma$ values refine this picture. For ⁴⁰Mg, the $0_1^{+}$ state is characterized by a relatively small $\gamma$, indicating a prolate deformation. In contrast, the $0_1^{+}$ state of ⁴²Si moves toward the oblate deformation, while the $0_1^{+}$ state of ⁴⁴S lies close to $\gamma \approx {30}^{\circ }$, suggesting a more non-axial or triaxial character. The $0_2^{+}$ states exhibit a similar $\beta$ trend to the $0_1^{+}$ states, with increasing quadrupole collectivity from ⁴⁸Ca toward ⁴⁰Mg. Their $\gamma$ evolution, however, is not identical to that of the ground states. In ⁴⁰Mg, ⁴²Si, and ⁴⁴S, the $\gamma$ values of the $0_2^{+}$ states mostly lie in the $0^{\circ }$–${30}^{\circ }$ region, corresponding to prolate or triaxial deformation. In contrast, the AMD results suggest that the $0_1^{+}$ state of ⁴⁰Mg is prolate, while the $0_2^{+}$ state is oblate [28].

Table 1. Calculated excitation energies $E_x$, spectroscopic quadrupole moments $Q_s$, and deformation parameters $\beta$ and $\gamma$ for ⁴⁰Mg, ⁴²Si, and ⁴⁴S.

40Mg	$0_1^{+}$	$0_2^{+}$	$2_1^{+}$	$2_2^{+}$	$4_1^{+}$	$4_2^{+}$
$E_x$ (MeV)	0.00	1.48	0.68	1.99	1.92	3.16
$Q_s$ ($e{\mathrm{fm}}^2$)	-	-	−12.0	9.6	−15.5	19.4
$\beta$	0.31(2)	0.30(1)	0.30(2)	0.30(1)	0.29(2)	0.28(1)
$\gamma {(}^{\circ })$	${18}_{-1}^{+1}$	${20}_{-5}^{+4}$	${16}_{-1}^{+1}$	${21}_{-5}^{+5}$	${14}_{-2}^{+2}$	${14}_{-6}^{+4}$
42Si	$0_1^{+}$	$0_2^{+}$	$2_1^{+}$	$2_2^{+}$	$4_1^{+}$	$4_2^{+}$
$E_x$ (MeV)	0.00	2.82	0.81	3.45	2.35	4.25
$Q_s$ ($e{\mathrm{fm}}^2$)	-	-	13.2	−10.2	16.6	11.1
$\beta$	0.26(4)	0.24(6)	0.26(3)	0.23(5)	0.25(2)	0.20(3)
$\gamma {(}^{\circ })$	${54}_{-9}^{+6}$	${22}_{-22}^{+16}$	${55}_{-4}^{+5}$	${18}_{-18}^{+14}$	${56}_{-4}^{+4}$	${45}_{-7}^{+15}$
44S	$0_1^{+}$	$0_2^{+}$	$2_1^{+}$	$2_2^{+}$	$4_1^{+}$	$4_2^{+}$
$E_x$ (MeV)	0.00	1.31	1.39	2.02	2.76	3.09
$Q_s$ ($e{\mathrm{fm}}^2$)	-	-	−6.0	4.7	15.5	−14.3
$\beta$	0.21(2)	0.21(1)	0.21(1)	0.21(2)	0.19(1)	0.20(1)
$\gamma {(}^{\circ })$	${31}_{-7}^{+7}$	${25}_{-7}^{+6}$	${26}_{-7}^{+7}$	${30}_{-8}^{+8}$	${24}_{-6}^{+5}$	${14}_{-8}^{+5}$

In ⁴²Si, the $0_1^{+}$ and $0_2^{+}$ states have comparable $\beta$ values but clearly different $\gamma$ values. This indicates that the two $0^{+}$ states represent competing quadrupole structures with comparable collectivity and distinct triaxiality. In particular, the large $\gamma$-fluctuation range of the $0_2^{+}$ state in ⁴²Si indicates a broad distribution in the triaxial degree of freedom, suggesting a $\gamma$-soft character. However, the AMD results suggest that the $0_2^{+}$ state of ⁴²Si is mainly distributed in the small-$\beta$ spherical region [28]. Furthermore, our calculations show that the $\beta$ and $\gamma$ values of the $2_1^{+}$ and $2_2^{+}$ states in these nuclei closely follow those of the $0_1^{+}$ and $0_2^{+}$ states, respectively, supporting the interpretation of the $0_1^{+}$–$2_1^{+}$ and $0_2^{+}$–$2_2^{+}$ pairs as two correlated low-lying quadrupole structures.

It is also instructive to compare our results with the simple SU(3) description of the $sd$-shell valence protons. Assuming an $N=28$ neutron shell closure, the  ground-state bands of ⁴⁰Mg, ⁴²Si, and ⁴⁴S are expected to be described by the highest-weight SU(3) irreducible representations $(4,2)$, $(6,0)$, and  $(2,4)$, respectively, while the first excited $K=0^{+}$ bands, starting from the $0_2^{+}$ states, are associated with the next-highest-weight irreducible representations $(0,4)$, $(0,6)$, and $(4,0)$, respectively [69,70,71,72]. This simple picture is qualitatively consistent with our results for the prolate-side ground state of ⁴⁰Mg and the prolate-side $0_2^{+}$ state of ⁴⁴S. For ⁴²Si, it also predicts comparable deformation magnitudes but distinct shapes for the two $0^{+}$ states, although  their prolate–oblate ordering differs from our calculations. These differences may arise from the erosion of the $N=28$ shell closure and correlations beyond this simplified proton-only SU(3) description.

In the $N=28$ Mg, Si, and S isotopes, the low-lying structures differ not only in the magnitude of quadrupole collectivity, but also in their triaxiality indicators. Table 1 lists the detailed values of $E_x$, $Q_s$, $\beta$, and $\gamma$ for ⁴⁰Mg, ⁴²Si, and ⁴⁴S. In ⁴⁰Mg, the $0_1^{+}$, $2_1^{+}$, and $4_1^{+}$ states have similar large $\beta$ values, small $\gamma$ values, and negative $Q_s$ values for the $2_1^{+}$ and $4_1^{+}$ states, supporting a coherent prolate band. In contrast, the $0_2^{+}$, $2_2^{+}$, and $4_2^{+}$ states in ⁴⁰Mg have comparable $\beta$ values and also prolate-side $\gamma$ values, while the $Q_s$ values of the $2_2^{+}$ and $4_2^{+}$ states are positive. The positive $Q_s$ values of the $2_2^{+}$ and $4_2^{+}$ states indicate that the $0_2^{+}$-associated structure is distinct from the ground-state band. However, the invariant-derived $\gamma$ values of the $0_2^{+}$, $2_2^{+}$, and $4_2^{+}$ states in ⁴⁰Mg lie in the prolate-side to triaxial region, showing that this structure should not be interpreted as a simple rigid oblate band. Similar $\beta$ and $\gamma$ values are observed for the $0_1^{+}$, $2_1^{+}$, and $4_1^{+}$ states in ⁴²Si, but their $\gamma$ values lie close to the oblate side and the corresponding $Q_s$ values are positive.

For ⁴⁴S, the $E2$-transition pattern discussed below associates the $0_1^{+}$, $2_1^{+}$, and $4_2^{+}$ states with a common quadrupole-connected sequence, in agreement with the SM calculations [30]. This assignment is consistent with the signs of the spectroscopic quadrupole moments, since $Q_s\left(2_1^{+}\right)$ and $Q_s\left(4_2^{+}\right)$ are both negative, whereas $Q_s\left(4_1^{+}\right)$ is positive. However, the $\gamma$ value of the $4_2^{+}$ state differs from those of the $0_1^{+}$ and $2_1^{+}$ states, indicating that this sequence is a band-like structure with non-rigid collectivity. Therefore, the $\beta$ and $\gamma$ values provide state-resolved deformation information, while the $E2$-transition maps are required to identify the quadrupole-connected structures directly.

Figure 3 displays the $E2$-transition maps for the $N=27$, 28, and 29 Mg, Si, and S isotopes. The $E2$ connectivity clearly identifies the main band-like sequences. In ⁴⁰Mg and ⁴²Si, the strong $E2$ links among the $0_1^{+}$, $2_1^{+}$, and $4_1^{+}$ states support their assignment to the ground-state band. In ⁴⁴S, however, the corresponding sequence is formed by the $0_1^{+}$, $2_1^{+}$, and $4_2^{+}$ states, while the $4_1^{+}$ state has weak downward $E2$ transitions and is interpreted as an isomeric bandhead, as discussed in Ref. [20]. Similarly, in ⁴³S and ⁴⁵S, the $7/2_1^{-}$ states can be regarded as candidate isomeric states, coexisting with the ground-state $3/2_1^{-}$ [21].

Figure 3. Calculated $E2$-transition maps for Mg, Si, and S isotopes around $N=28$. The size of each dot indicates the magnitude of the $Q_s$ and the color denotes its sign, with blue (red) corresponding to negative (positive) values. Line widths indicate downward $B\left(E2\right)$ strengths.

To show the shape evolution more clearly, Figure 4 presents the $\beta$-$\gamma$ plane for the $N=27$, 28, and 29 Mg, Si, and S isotopes. The low-lying states in the Mg isotopes are located around $\beta \approx 0.28$–$0.31$ and $\gamma \approx {10}^{\circ }$–${20}^{\circ }$, indicating a common strong prolate deformation across the $N=27$, 28, and 29 Mg systems. The Si isotopes show a markedly different pattern. The low-lying negative-parity states in ^41,43Si appear on the oblate side, and the $0_1^{+}$, $2_1^{+}$, and $4_1^{+}$ states in ⁴²Si are also located near large $\gamma$ values, around $\gamma \approx {55}^{\circ }$. In contrast, the $0_2^{+}$ and $2_2^{+}$ states of ⁴²Si move to the smaller $\gamma$ region, with a large fluctuation range, indicating a $\gamma$-soft structure. Furthermore, the low-lying states in ^43,44,45S mainly lie at $\beta \approx 0.18$–$0.21$ and $\gamma \approx {15}^{\circ }$–${30}^{\circ }$, indicating prolate-side to triaxial deformation.

Figure 4. State-resolved $\beta$-$\gamma$ fan plot for Mg, Si, and S isotopes. The radial coordinate denotes $\beta$ and the angular coordinate denotes $\gamma$.

4. Conclusions

We have applied the quadrupole-invariant method to ab initio VS-IMSRG calculations of neutron-rich nuclei around $N=28$. Starting from chiral NN+3N interactions, effective valence-space Hamiltonians and consistently evolved $E2$ operators are obtained and used to extract deformation parameters $\beta$ and $\gamma$. Along the even–even $N=28$ isotonic chain, the systematic increase of $\beta$ from ⁴⁸Ca to ⁴⁰Mg provides a direct deformation signature of the weakening of the $N=28$ shell closure and the growth of quadrupole collectivity. The extracted $\beta$ and $\gamma$ values further indicate a prolate–oblate–triaxial evolution of the $0_1^{+}$ states from ⁴⁰Mg to ⁴²Si and ⁴⁴S. In contrast, the $0_2^{+}$ states show comparable quadrupole collectivity but follow a different $\gamma$ evolution, mainly lying in the prolate-to-triaxial region. In particular, the large $\gamma$-fluctuation range of the $0_2^{+}$ state in ⁴²Si suggests a $\gamma$-soft structure. By combining the extracted $\beta$ and $\gamma$ values with $Q_s$ systematics and the calculated $E2$-transition maps, we analyzed the state-by-state organization of the low-lying quadrupole structures. For ⁴⁴S, although  the $0_1^{+}$, $2_1^{+}$, and $4_2^{+}$ states form a quadrupole-connected band-like sequence, the $\gamma$ value of the $4_2^{+}$ state differs from those of the $0_1^{+}$ and $2_1^{+}$ states, indicating non-rigid collectivity. Thus, the $\beta$ and $\gamma$ values provide state-resolved deformation information, while the $E2$-transition maps are needed to identify the quadrupole-connected structures directly. Extending the analysis to the neighboring $N=27$ and $N=29$ isotopes, we find a systematic shape evolution across the Mg, Si, and S chains: strong prolate deformation in Mg, oblate structures in Si, and prolate-to-triaxial deformation in S. These results demonstrate that quadrupole invariants built from ab initio $E2$ matrix elements offer a useful probe of shape evolution and shape coexistence in the neutron-rich $N\approx 28$ region.