Divergence of the isospin-asymmetry expansion of the nuclear equation of state in many-body perturbation theory

The isospin-asymmetry dependence of the nuclear-matter equation of state obtained from microscopic chiral two- and three-body interactions in second-order many-body perturbation theory is examined in detail. The quadratic, quartic, and sextic coefficients in the Maclaurin expansion of the free energy per particle of infinite homogeneous nuclear matter with respect to the isospin asymmetry are extracted numerically using finite differences, and the resulting polynomial isospin-asymmetry parametrizations are compared to the full isospin-asymmetry dependence of the free energy. It is found that in the low-temperature and high-density regime where the radius of convergence of the expansion is generically zero, the inclusion of higher-order terms beyond the leading quadratic approximation leads overall to a significantly poorer description of the isospin-asymmetry dependence. In contrast, at high temperatures and densities well below nuclear saturation density, the interaction contributions to the higher-order coefficients are negligible and the deviations from the quadratic approximation are predominantly from the noninteracting term in the many-body perturbation series. Furthermore, we extract the leading logarithmic term in the isospin-asymmetry expansion of the equation of state at zero temperature from the analysis of linear combinations of finite differences. It is shown that the logarithmic term leads to a considerably improved description of the isospin-asymmetry dependence at zero temperature.

Figure 1

Density and temperature dependence of the quadratic, quartic, and sextic Maclaurin coefficients for the first-order contributions from two- and three-body chiral forces (solid lines for n3lo414; dash-dotted lines for n3lo450). Also shown is the difference between the quadratic Maclaurin coefficient and the symmetry free energy.

Figure 2

Quadratic, quartic, and sextic Maclaurin coefficients for the $np$-channel second-order normal contribution (with $NN$ forces only). Also shown are the results for $F_{\text{sym}}(T,\rho )-A_2(T,\rho )$.

Figure 3

Same as Fig. 4 but for the quartic finite differences.

Figure 4

(Main plot) Sextic finite-difference results for the $np$-channel second-order $NN$ contribution at zero temperature (n3lo414, $\rho =0.20{\text{fm}}^{-3}$) and for the (suitably scaled) expression given by Eq. (26). (Top inset) Analogous results for the $np$-channel contribution with the logarithmic term subtracted. (Bottom inset) (Combined) results for the $nn$- and $pp$-channel contributions.

Figure 5

Results for the quadratic, quartic, and sextic Maclaurin coefficients for different densities and temperatures. Also shown is the difference between quadratic coefficient and the symmetry free energy (the dotted lines correspond to the results for a noninteracting nucleon gas).

Figure 6

Threshold line for the convergence of the isospin-asymmetry expansion of the free energy per particle; see text for details.

Figure 7

(Left column) Finite-difference results for the first- and second-order (normal) contributions to $A_6$ at $T=5,15\text{MeV}$ and $\rho =0.15{\text{fm}}^{-3}$ (calculated using n3lo414). The lines correspond to $N=2+n$, the points to $N=3+n$; in each the step-size variation extends from $\mathrm{\Delta }\delta =0.01$ to $\mathrm{\Delta }\delta =1/N$. At $T=15\text{MeV}$ the first-order $NN$ and $3N$ contributions are given in $10\text{-keV}$ units. The approximate equality of the first-order $\mathrm{DD}NN$ and the second-order $NN+\mathrm{DD}NN$ results at $T=15\text{MeV}$ is coincidental. (Right column) $A_6^{N,\mathrm{\Delta }\delta }$ and $A_8^{N,\mathrm{\Delta }\delta }$ for the second-order $NN$ contribution ($np$ channel only) at $T=4\text{MeV}$ and $\rho =0.30{\text{fm}}^{-3}$. Also shown are the results obtained from the iterative method; see Sec. 2c for details. Note that $A_8^{N,\mathrm{\Delta }\delta }$ is given in GeV.

Figure 8

Full isospin-asymmetry-dependent free energy per particle $F(T,\rho ,\delta )$ versus the quadratic, quartic, and sextic polynomial isospin-asymmetry approximations at $T=4\text{MeV}$ and $\rho =0.20{\text{fm}}^{-3}$. The inset shows the deviation $\mathrm{\Delta }F:=|F-F_{\left[2N\right]}|$ of the various approximations from the exact results for small neutron excesses $\delta \in [0,0.2]$ (results for n3lo414).

Figure 9

Full isospin-asymmetry-dependent free energy per particle $F(T,\rho ,\delta )$ at $T=0$ and $\rho =0.25{\text{fm}}^{-3}$ versus the quadratic and quartic (with and without the logarithmic term) isospin-asymmetry approximations based on Eq. (27). The inset shows the deviation of the various approximations from the exact results (results for n3lo450).