Asymmetric nuclear matter based on chiral two- and three-nucleon interactions

We calculate the properties of isospin-asymmetric nuclear matter based on chiral nucleon-nucleon (NN) and three-nucleon (3N) interactions. To this end, we develop an improved normal-ordering framework that allows us to include general 3N interactions starting from a plane-wave partial-wave-decomposed form. We present results for the energy per particle for general isospin asymmetries based on a set of different Hamiltonians, study their saturation properties, the incompressibility, symmetry energy, and also provide an analytic parametrization for the energy per particle as a function of density and isospin asymmetry.

Figure 1

The upper panel shows the matrix elements of the effective interaction ${\overline{V}}_{\text{3N}}={\overline{V}}_{3\mathrm{N}}/9$ in the ${}^1{S}_0$ channel with $m_T=-1$ as a function of the center-of-mass momentum $P$ for fixed relative momenta, $p=p^{\prime }=1{\mathrm{fm}}^{-1}$ (solid) and $p=2p^{\prime }=1{\mathrm{fm}}^{-1}$ (dashed line), and proton fractions $x$ at a neutron Fermi momentum $k_{\mathrm{F}}^n=1.4{\mathrm{fm}}^{-1}$. For the color code, see the legend in the lower panel. The lower panel shows the diagonal matrix elements times $p^2$ as a function of the relative momentum $p$. In the first- and second-order many-body contributions, the value of $P$ is kinematically limited to $P\le k_F^{{\tau }_1}+k_F^{{\tau }_2}$, so for $m_T=-1$ to $P\le 2k_F^n$.

Figure 2

Comparison of 3N Hartree–Fock energies based on Hamiltonian 2 in Table 1 for the $P=0$ (red dashed) and $P$-average approximation (blue solid line) for the effective interaction ${\overline{V}}_{3\mathrm{N}}^{\mathrm{as}}$. Results are shown as differences to the exact Hartree–Fock energy for three proton fractions, $x=0$ (left), $x=0.3$ (center), and $x=0.5$ (right panel). The $P=0$ values give larger deviations above saturation density, whereas the $P$-average approximation behaves more systematic over the entire density range.

Figure 3

Partial-wave convergence of the ${\mathrm{N}}^2\mathrm{LO}$ 3N contributions at the Hartree–Fock level in neutron matter (top) and symmetric matter (bottom) for Hamiltonian 2 of Table 1. For neutron matter the contributions for $\mathcal{J}>5/2$ are very small, so the individual lines are nearly indistinguishable.

Figure 4

Energy per particle of nuclear matter as a function of the total density $n=n_n+n_p$ for various proton fractions $x$. The two approximations of the effective NN potential, $P=0$ and $P$ average, and two approximations for the single-particle energies, free and Hartree–Fock, are shown. The energy range is based on the set of Hamiltonians listed in Table 1. The excluded Hamiltonian 6* has no influence on the uncertainty bands. For a better view, the area between the dashed lines are not filled in the case of a free spectrum. For $x=0.5$ we also show the empirical saturation point (see text for details).

Figure 5

Correlation between the saturation density and energy for the seven Hamiltonians of Table 1, indicated by the figure. The green area highlights the obtained Coester band based on independently fitting the saturation points for a free and a Hartree–Fock spectrum. The gray lines connect the two spectra. As discussed in the text, the empirical saturation point (gray box) is given by the range of 14 selected energy-density functionals. The region is in good agreement with our calculated Coester band. See the text for details of Hamiltonian 6*.

Figure 6

Contributions at third order in the perturbative expansion to the energy of neutron matter (top panel) and symmetric nuclear matter (lower panel). The color coding and line styles are the same as in Fig. 4. Accordingly, the uncertainty estimates are based the Hamiltonians of Table 1.

Figure 7

Interaction energy at first, second, and third order in symmetric nuclear matter ($x=0.5$) for the employed Hamiltonians of Table 1. As discussed in the text Hamiltonian 6* is excluded. The suppression of higher orders in the perturbative expansion demonstrates the convergence of many-body perturbation theory and suggests that the sensitivity of the second-order results on the single-particle energies provide a conservative estimate for the many-body uncertainties.