Coupled-cluster calculations of nucleonic matter

Background: The equation of state (EoS) of nucleonic matter is central for the understanding of bulk nuclear properties, the physics of neutron star crusts, and the energy release in supernova explosions. Because nuclear matter exhibits a finely tuned saturation point, its EoS also constrains nuclear interactions.

Purpose: This work presents coupled-cluster calculations of infinite nucleonic matter using modern interactions from chiral effective field theory (EFT). It assesses the role of correlations beyond particle-particle and hole-hole ladders, and the role of three-nucleon forces (3NFs) in nuclear matter calculations with chiral interactions.

Methods: This work employs the optimized nucleon-nucleon ($NN$) potential NNLO${}_{\mathrm{opt}}$ at next-to-next-to leading order, and presents coupled-cluster computations of the EoS for symmetric nuclear matter and neutron matter. The coupled-cluster method employs up to selected triples clusters and the single-particle space consists of a momentum-space lattice. We compare our results with benchmark calculations and control finite-size effects and shell oscillations via twist-averaged boundary conditions.

Results: We provide several benchmarks to validate the formalism and show that our results exhibit a good convergence toward the thermodynamic limit. Our calculations agree well with recent coupled-cluster results based on a partial wave expansion and particle-particle and hole-hole ladders. For neutron matter at low densities, and for simple potential models, our calculations agree with results from quantum Monte Carlo computations. While neutron matter with interactions from chiral EFT is perturbative, symmetric nuclear matter requires nonperturbative approaches. Correlations beyond the standard particle-particle ladder approximation yield non-negligible contributions. The saturation point of symmetric nuclear matter is sensitive to the employed 3NFs and the employed regularization scheme. 3NFs with nonlocal cutoffs exhibit a considerably improved convergence than their local cousins. We are unable to find values for the parameters of the short-range part of the local 3NF that simultaneously yield acceptable values for the saturation point in symmetric nuclear matter and the binding energies of light nuclei.

Conclusions: Coupled-cluster calculations with nuclear interactions from chiral EFT yield nonperturbative results for the EoS of nucleonic matter. Finite-size effects and effects of truncations can be controlled. For the optimization of chiral forces, it might be useful to include the saturation point of symmetric nuclear matter.

Figure 1

Relative finite-size corrections for the kinetic energy in pure neutron matter at the Fermi momentum $k_F=1.6795{\mathrm{fm}}^{-1}$ vs the neutron number $A$. TABC10 are twist-averaged boundary conditions with 10 Gauss-Legendre points in each spatial direction.

Figure 2

Relative finite-size corrections for the Hartree-Fock energy of the NNLO${}_{\mathrm{opt}}$ (full line) and Minnesota (dashed line) potentials in pure neutron matter at the Fermi momentum $k_F=1.6795{\mathrm{fm}}^{-1}$ vs the neutron number $A$. TABC10 are twist-averaged boundary conditions with 10 Gauss-Legendre points in each spatial direction. The dashed-dotter line shows a power law fit to the NNLO${}_{\mathrm{opt}}$ results.

Figure 3

Relative finite-size corrections for the Hartree-Fock energy of the $NN$ potential NNLO${}_{\mathrm{opt}}$ and the 3NF potential in symmetric nuclear matter at the Fermi momentum $k_F=1.6{\mathrm{fm}}^{-1}$ vs the nucleon number $A$. PBC: periodic boundary conditions. TABC10 and TABC3 are twist-averaged boundary conditions with 10 and 3 Gauss-Legendre points in each spatial direction, respectively.

Figure 4

Energy per particle of neutron matter for NNLO${}_{\mathrm{opt}}$ computed in the CCD${}_{\mathrm{lad}}$ approximation with periodic boundary conditions (circles), twist-averaged boundary conditions (squares), and for infinite matter (diamonds). The latter results are from Ref. [36]. The calculations used $A=66$ neutrons and $n_{\max }=4$.

Figure 5

Energy per particle of symmetric nuclear matter for NNLO${}_{\mathrm{opt}}$ computed in the CCD approximation with periodic boundary conditions (diamonds), twist-averaged boundary conditions (squares), and with a special point and twisted boundary conditions (circles). The calculations used $A=132$ nucleons and $n_{\max }=4$.

Figure 6

Energy per particle of symmetric nuclear matter for NNLO${}_{\mathrm{opt}}$ computed in the CCD${}_{\mathrm{ladd}}$ approximation in the thermodynamic limit using partial-wave expansion (triangles) (partly adapted from Ref. [35]), with twist averaged boundary conditions (squares), and with a special point and twisted boundary conditions (circles). The calculations used $A=132$ nucleons and $n_{\max }=4$.

Figure 7

Energy per particle of neutron matter, computed with the Minnesota potential. Diamonds: AFDMC, circles: CCD, squares: CCD limited to $pp$ and $hh$ ladders.

Figure 8

Energy per particle of symmetric nuclear matter computed in the CCD approximation with NNLO${}_{\mathrm{opt}}+3$NF ($c_E=-0.389$, $c_D=-0.39$). The 3NF is included in the zero-body (black dashed-dotted line), one-body (red dashed line), and in the two-body (blue solid line) normal-ordered approximations. The calculations used $A=132$ nucleons and $n_{\max }=4$.

Figure 9

Energy per particle in neutron matter with NNLO${}_{\mathrm{opt}}$ (NN only). The black dashed line is Hartree-Fock (HF), the green dashed-dotted line is second-order many-body perturbation theory (MBPT2), the blue dashed line is coupled-cluster doubles ladder approximation (CCD${}_{\mathrm{ladd}}$), and the red solid line is coupled-cluster doubles (CCD). The calculations used $A=66$ neutrons, $n_{\max }=4$, and TABC3.

Figure 10

Energy per particle in neutron matter with NNLO${}_{\mathrm{opt}}$ ($NN$ only) and with inclusion of 3NF computed in the CCD and CCD(T) approximations. The 3NF LECs are given by $c_E=-0.389$ and $c_D=-0.39$. The calculations used $A=66$ neutrons, $n_{\max }=4$, and PBC and TABC. The QMC results are from Refs. [28, 30].

Figure 11

Energy per particle in symmetric nuclear matter with NNLO${}_{\mathrm{opt}}$ ($NN$ only) computed in the MBPT2 (triangles with green dashed-dotted line), CCD${}_{\mathrm{ladd}}$ (circles with blue dashed line), and CCD (squares with solid red line) approximations. The calculations used 132 nucleons, $n_{\max }=4$, and TABC3. Diamonds are results from self-consistent Green's function (SCGF) at the finite temperature $T=5$ MeV, taken from Ref. [34].

Figure 12

Energy per particle in symmetric nuclear matter with NNLO${}_{\mathrm{opt}}$ and 3NF computed in the MBPT2 (squares with green dashed-dotted line), CCD (circles with blue dashed line), and CCD(T) with 3NF in the normal-ordered two-body approximation (diamonds with solid red line), and including the residual 3NF in leading order (triangles with dotted black line). The 3NF LECs are given by $c_E=-0.389$ and $c_D=-0.39$. The calculations used 132 nucleons, $n_{\max }=4$, and PBC.

Figure 13

Energy per particle in pure neutron matter with NNLO${}_{\mathrm{opt}}$ and 3NF computed in the CCD(T) approximation including 3NFs in the normal ordered two-body approximation and including the residual 3NF in the CCD(T$:w{T}_2=0$) approximation. For the 3NF we used a local regulator with cutoffs $\Lambda =400$ and $\Lambda =500$ MeV. The 3NF LECs are given by $c_E=-0.389$ and $c_D=-0.39$ for the $\Lambda =500$ MeV local regulator, while for the $\Lambda =400$ MeV local regulator we used $c_E=-0.27$ and $c_D=-0.39$ with $c_E$ adjusted to the ${}^4$He binding energy. For the nonlocal regulator with $\Lambda =500$ MeV cutoff we used $c_E=-0.791$ and $c_D=-2$ adjusted to the triton and ${}^3$He binding energies. The calculations used 66 neutrons, $n_{\max }=4$, and PBC.

Figure 14

Same caption as in Fig. 13, except that the energy per particle is for symmetric nuclear matter. The calculations used 132 nucleons, $n_{\max }=4$, and PBC.

Figure 15

The blue band shows the region where the triton binding energy is reproduced within 5% of the experimental value. The red band shows the region where the saturation fermi momentum in symmetric nuclear matter is reproduced within 5% of its empirical value, and the green band shows the region where the energy per nucleon is within 5% of the empirical value.

Figure 16

Cutoff dependent fraction of the residual 3NF contribution to the MBPT2 energy per particle in symmetric nuclear matter for different regulators. Results are shown for the local regulator with $\Lambda =500$ MeV (diamonds), the local regulator with $\Lambda =400$ MeV (squares), and for the non-local regulator with $\Lambda =500$ MeV (circles). Calculations used 132 nucleons, $n_{\max }=4$, and PBC.