Effective-interaction approach to the Fermi hard-sphere system

The formalism based on correlated basis functions and the cluster-expansion technique has been recently employed to derive an effective interaction from a realistic nuclear Hamiltonian. To gauge the reliability of this scheme, we perform a systematic comparison between the results of its application to the Fermi hard-sphere system and the predictions obtained from low-density expansions, as well as from other many-body techniques. The analysis of a variety of properties, including the ground-state energy, the effective mass, and the momentum distribution, shows that the effective-interaction approach is quite accurate, thus suggesting that it may be employed to achieve a consistent description of the structure and dynamics of nuclear matter in the density region relevant to astrophysical applications.

Figure 1

Radial dependence of the correlation functions obtained from the solution of the Euler–Lagrange equation (15). The dot-dashed, solid, and dashed lines correspond to $c=k_{F}a=0.3$, 0.5, and 0.7, respectively.

Figure 2

The full line shows the $c$ dependence of the dimensionless quantity $\zeta$, defined by Eq. (18), obtained within the FHNC approach for the system of hard spheres of radius $a=1\mathrm{fm}$, mass $m=1{\mathrm{fm}}^{-1}$, and degeneracy $\nu =4$. The results obtained from the low-density expansion of Eq. (19) are represented by the dashed line, while the diamonds correspond to the perturbative estimates of $\zeta$ computed, neglecting terms of order higher than $c^3$.

Figure 3

The full line shows the $c$ dependence of the correlation range, $d$, resulting from minimization of the ground-state energy of the hard-sphere system computed within the FHNC approach. The dashed line corresponds to the correlation range employed to obtain the CBF effective interaction of Eq. (14).

Figure 4

Radial dependence of the effective interaction defined by Eq. (14). The dot-dash, solid, and dashed lines correspond to $c=k_{F}a=0.3$, 0.5, and 0.7, respectively. The region $(r/a)<1$, where $v_{\mathrm{eff}}\left(r\right)=0$, is not shown.

Figure 5

Diagrammatic representation of the direct part of the first- and second-order contributions to the irreducible self-energy. Panels (a)–(c) correspond to the Hartree–Fock, polarization, and correlation terms, respectively. Dashed lines represent the CBF effective interaction, while upward- and downward-oriented solid lines depict the free propagation of particle and hole states, respectively.

Figure 6

Imaginary part of the quantities ${\Sigma }_{2h1p}(k<k_F,k^2/\left(2m\right))$ and ${\Sigma }_{2p1h}(k>k_F,k^2/\left(2m\right))$, computed at $c=0.3$ and displayed as a function of the dimensionless variable $k/k_F$. The solid and dashed lines correspond to the results obtained from the CBF effective interaction and from the low-density expansion of Ref. [16], respectively.

Figure 7

Energy dependence of the imaginary part of the polarization ($2mE/k_F^2>1$) and correlation ($2mE/k_F^2<1$) contributions to the self-energy of the Fermi hard-sphere system at $c=0.5$. The dashed, solid, and dot-dash lines correspond to $k/k_F=0.5$, 1, and 1.5, respectively.

Figure 8

Quasiparticle energy, computed from Eq. (28) at $c=0.2$ [panel (a)] and 0.5 [panel (b)]. The dashed and solid lines correspond to the first-order (i.e., Hartree–Fock) and second-order approximations to the self-energy, respectively. For comparison, the dot-dashed lines show the kinetic-energy spectrum.

Figure 9

$c$ dependence of the ratio $m^{☆}\left(k_F\right)/m$, obtained from Eqs. (3) and (28). The dot-dashed and solid lines represent the results of calculations carried out using the first- and second-order approximations to the self-energy. For comparison, the dashed line shows the results computed using the low-density expansion of Eq. (4).

Figure 10

Momentum distributions computed at second order in the CBF effective interactions, for three different values of $c=k_{F}a$. The values of the discontinuity corresponding $c=0.3$, 0.5, and 0.7 are 0.92, 0.72, and 0.28, respectively.

Figure 11

Discontinuity of the momentum distribution of the Fermi hard-sphere system, as a function of $c=k_{F}a$.

Figure 12

Momentum distribution of the Fermi hard-sphere system at $c=0.4$. Solid line shows results obtained at second order in the CBF effective interaction, dashed line shows variational results of Ref. [24], crosses show results of the low-density expansion at order $c^2$. All values of $n(k>k_F)$ are multiplied by a factor of ten.

Figure 13

Comparison between the momentum distribution obtained from the CBF effective interactions (crosses) and the low-density expansion discussed in Refs. [14, 16] (diamonds) for different values of the dimensionless parameter $c=k_{F}a$. Panels (a) and (b) correspond to the regions $k<k_F$ and $k>k_F$, respectively.

Figure 14

Comparison between the momentum distribution of the Fermi hard-sphere system obtained from the effective-interaction approach discussed in this article (squares) and that of isospin symmetric nuclear matter at equilibrium density reported in Ref. [29] (solid line), computed using correlated wave functions and second-order CBF perturbation theory.