Coupled-cluster studies of infinite nuclear matter

Background: Coupled-cluster (CC) theory is a widely used many-body method for studying strongly correlated many-fermion systems. It allows for systematic inclusions of complicated many-body correlations beyond a mean field. Recent applications to finite nuclei have shown that first-principles approaches like CC theory can be extended to studies of medium-heavy nuclei, with excellent agreement with experiment. However, CC calculations of properties of infinite nuclear matter are rather few and date back more than 30 yr.

Purpose: The aim of this work is thus to develop the relevant formalism for performing CC calculations in nuclear matter and neutron-star matter, including thereby important correlations to infinite order in the interaction and testing modern nuclear forces based on chiral effective field theory. Our formalism includes also the exact treatment of the so-called Pauli operator in a partial-wave expansion of the equation of state.

Methods: Nuclear- and neutron-matter calculations are done using a coupled particle-particle and hole-hole ladder approximation. The coupled ladder equations are derived as an approximation of CC theory, leaving out particle-hole and nonlinear diagrams from the CC doubles amplitude equation. This study is a first step toward CC calculations for nuclear and neutron matter.

Results: We present results for both symmetric nuclear matter and pure neutron matter employing state-of-the-art nucleon-nucleon interactions based on chiral effective field theory. We employ also the newly optimized chiral interaction [Ekström et al., Phys. Rev. Lett. 110, 192502 (2013)] to study infinite nuclear matter. The ladder approximation method and corresponding results are compared with conventional Brueckner-Hartree-Fock theory. The ladder approximation is derived and studied using both exact and angular-averaged Pauli exclusion operators, with angular-averaged input momenta for the single-particle potentials in all calculations. The inclusion of an exact treatment of the Pauli operators in a partial-wave expansion yields corrections of the order of $1.7\%–2\%$ of the total energy in symmetric nuclear matter. Similarly, the inclusion of both hole-hole and particle-particle ladders result in corrections of the order $0.7\%–2\%$ compared to the approximation with only particle-particle ladders. Including these effects, we get at most almost a $6\%$ difference between our CC calculation and the standard Brueckner-Hartree-Fock approach.

Conclusions: We have performed CC calculations of symmetric nuclear matter and pure neutron matter including particle-particle and hole-hole diagrams to infinite order using an exact Pauli operator and angular-averaged single-particle energies. The contributions from hole-hole diagrams and exact Pauli operators add important changes to the final energies per particle.

Figure 1

Diagrammatic representation of the ${\widehat{T}}_2$ amplitude equation in the CCD approximation. The coupled particle-particle and hole-hole ladder (PPHH-LAD) equations are obtained by leaving out the sixth diagram, which has summation over one particle and one hole state, and all the nonlinear diagrams, that is, the four last diagrams. The particle-particle ladder (PP-LAD) equations are otherwise equal to the PPHH-LAD equations; apart from that in the former case also the fifth diagram is neglected. The two-particle interaction is illustrated by a dashed line and the $t$ amplitude by a solid line. The dashed line with a cross at one vertex represents the Fock operator. Lines with arrows pointing upwards represent particles, whereas lines with arrows pointing downwards represent holes. The interaction and $t$ amplitude are assumed to be antisymmetric.

Figure 2

The CCD energy equation, given in terms of diagrams. The two-particle interaction is given by a dashed line and the $t$ amplitude by a solid line. The dashed line with a cross at one vertex represents the kinetic energy operator. Lines with arrows pointing upwards represent particles, lines with arrows pointing downwards represent holes, and circles are always hole lines.

Figure 3

The BHF energy equation in terms of diagrams. The waved line represents the $G$ matrix, and a circle means summation over hole states.

Figure 4

Definition of the $G$ matrix, expressed using diagrams. The wavy line represents the $G$-matrix interaction. All other parts are defined as in Fig. 1.

Figure 5

Total energy per nucleon of symmetric nuclear matter in the Hartree-Fock approximation, given as a function of Fermi momentum $k_F$. The calculation was done with a bare N${}^3$LO two-body interaction, and the total angular momentum truncation was set to $\mathcal{J}\le$ 24.

Figure 6

Convergence of the correlation energy of symmetric nuclear matter, given as a function of maximum total angular momentum ${\mathcal{J}}_{\text{max}}$. The correlation energy was calculated for the Fermi momentum $k_F=$ 1.8 fm${}^{-1}$, using the PPHH-LAD approximation and exact Pauli exclusion operators.

Figure 7

Total energy per particle as a function of the Fermi momentum, calculated for symmetric nuclear matter. The two approximations of the PPHH-LAD equations, average and exact, are compared with a BHF calculation with angular-averaged Pauli exclusion operator. These calculations were done with the N${}^3$LO [64] two-body interaction. The box denotes the uncertainty region for the experimental saturation point of symmetric nuclear matter, as obtained by extrapolating from observables of finite nuclei.

Figure 8

The exact PPHH-LAD and PP-LAD calculations of energy per nucleon as a function of Fermi momentum. The equation of state of a BHF calculation is also given. The box denotes the uncertainty region for the experimental saturation point of symmetric nuclear matter. All results were obtained with the N${}^3$LO interaction [64].

Figure 9

Single-particle potential as a function of laboratory frame momentum, calculated at Fermi momentum 1.8 fm${}^{-1}$ with a $G$ matrix and with a bare interaction. In both cases, we used the N${}^3$LO two-body interaction and the total angular momentum cutoff ${\mathcal{J}}_{\text{max}}=9$. The only difference between the BHF and PP-LAD equations is the single-particle energy: In BHF theory the single-particle potential is calculated using a self-consistent $G$ matrix, whereas in the PP-LAD approximation the single-particle energy is obtained by replacing the $G$ matrix with a bare interaction.

Figure 10

Energy per particle for symmetric nuclear matter, as calculated in the HF and PPHH-LAD approximations using the N${}^3$LO and NNLO${}_{\text{opt}}$ two-body interactions. In the PPHH-LAD approximation, the angular momentum cutoff was set to $\mathcal{J}\le 8$ and the calculations were done with exact Pauli exclusion operators.

Figure 11

Energy per particle for pure neutron matter given as a function of Fermi momentum $k_F$. The figure shows results of calculations in Hartree-Fock (HF) and coupled PPHH-LAD, using the N${}^3$LO and NNLO${}_{\text{opt}}$ two-body interactions. The neutron-matter results obtained with exact Pauli operators have been published in Ref. [63].

Figure 12

Potential energy per particle for the ${}^1{S}_0$ partial wave as function of the Fermi momentum $k_F$ for pure neutron matter. We plot the Hartree-Fock potential energy and the total potential energy obtained by adding the correlation energies obtained from PPHH-LAD approximation with angular-averaged Pauli exclusion operators. Both potential models have been employed.

Figure 13

Potential energy per particle for the ${}^3{P}_0$ partial wave as function of the Fermi momentum $k_F$ for pure neutron matter. We plot the Hartree-Fock potential energy and the total potential energy obtained by adding the correlation energies obtained from PPHH-LAD approximation with angular-averaged Pauli exclusion operators. Both potential models have been employed.

Figure 14

Symmetry energy as function of density, calculated in the PPHH-LAD approximation with exact Pauli exclusion operators. The empirical saturation density of symmetric nuclear matter is approximately 0.17 fm${}^{-3}$ [78].