Variational calculation of nuclear matter in a finite particle number approach using the unitary correlation operator and high-momentum pair methods

We propose a new variational method for describing nuclear matter from nucleon-nucleon interaction. We use the unitary correlation operator method (UCOM) for central correlation to treat the short-range repulsion and further include the two-particle two-hole (2p2h) excitations of nucleon pair involving a large relative momentum, which is called high-momentum pair (HM). We describe nuclear matter in finite size with finite particle number on periodic boundary condition and increase the 2p2h configurations until we get the convergence of the total energy per particle. We demonstrate the validity of this UCOM+HM framework by applying it to the symmetric nuclear and neutron matters with the Argonne $\mathrm{V}{4}^{\prime }$ potential having short-range repulsion. The nuclear equations of state obtained in UCOM+HM are fairly consistent to those of other calculations such as Brueckner-Hartree-Fock and auxiliary field diffusion Monte Carlo in the overall density region.

Figure 1

Lattice in momentum space with Cartesian representation. The configuration of the occupied states with the number of grid points $N_g=7$ is shown using solid circles. Open circles indicate the unoccupied states. The grid spacing is $\mathrm{\Delta }k$.

Figure 2

Numbers of the 2p2h configurations $N_{2\mathrm{p}2\mathrm{h}}$ in symmetric nuclear matter (top) and neutron matter (bottom) with six kinds of magic numbers as functions of maximum mode of transfer momentum $n_q^{\max }$.

Figure 3

Energy surface of ${}^4\mathrm{He}$ with $\mathrm{AV}{4}^{\prime }$ potential as functions of radius. The units of energy and radius are MeV and fm, respectively. The red-solid (blue-dashed) line is the results with (without) UCOM. The dashed line is the results of GFMC [3].

Figure 4

Shift functions $R_{+}\left(r\right)$ in UCOM for each channel in symmetric nuclear matter (top) and neutron matter (bottom), respectively. The differences between $R_{+}\left(r\right)$ and the original distance $r$ are shown as the shift of the wave function.

Figure 5

Hamiltonian components per particle for the 0p0h state of symmetric nuclear matter without UCOM as functions of density $\rho$ with dots. Mass number is $A=28$ (top) and $A=132$ (bottom). Dots with red-circle, blue-diamond and green-square indicate the total ($E$), kinetic ($T$), and potential ($V$) energies, respectively. Results are compared with the Hartree-Fock solutions of Fermi sphere for infinite size with lines.

Figure 6

Hamiltonian components per particle for the 0p0h state of symmetric nuclear matter with UCOM. Mass numbers of $A=28$ (top) and $A=132$ (bottom). Notations are the same as those in Fig. 5.

Figure 7

Relative errors of the Hamiltonian components per particle in 0p0h+UCOM of symmetric nuclear matter with respect to the Hartree-Fock solutions of infinite matter with UCOM, as functions of mass number $A$. Kinetic (top) and potential (bottom) energies are shown with two kinds of densities $\rho$.

Figure 8

Dependence of the solutions of symmetric nuclear matter on the maximum mode of transfer momentum $n_q^{\max }$ with several mass numbers in UCOM+HM. Top panel shows the case of $A=132$, where $E$, $T$, and $V$ are the total, kinetic and potential energies, respectively. The term of $T_{\mathrm{uncorr}.}$ is the uncorrelated one-body kinetic energy without UCOM part. Bottom panel shows the total energies of six kinds of mass numbers. Dashed line is the energy of the Hartree-Fock calculation with UCOM for infinite matter.

Figure 9

Energies per particle of symmetric nuclear matter in UCOM+HM with mass number $A=132$ as functions of density $\rho$. We also compare the results with other theories, the values of which are taken from Refs. [23, 35].

Figure 10

Hamiltonian components per particle of neutron matter in the 0p0h configuration as functions of neutron density $\rho$ with dots in comparison with the Hartree-Fock solutions of infinite neutron matter with lines. Neutron number is $N=14$ (top) and 66 (bottom). The values with red-circle, blue-diamond, and green-square indicate the total ($E$), kinetic ($T$) and potential ($V$) energies, respectively.

Figure 11

Hamiltonian components per particle of neutron matter in the 0p0h+UCOM wave function for neutron numbers of $N=14$ (top) and $N=66$ (bottom). Notations are the same as those in Fig. 10.

Figure 12

Relative errors of the Hamiltonian components per particle in 0p0h+UCOM for neutron matter with respect to the Hartree-Fock solutions of infinite neutron matter with UCOM, as functions of neutron number $N$. Kinetic (top) and potential (bottom energies are shown with two kinds of neutron densities $\rho$.

Figure 13

Energies of neutron matter per particle increasing the maximum mode of transfer momentum $n_q^{\max }$ with several neutron numbers in UCOM+HM. Top panel shows the case of $N=66$, where $E$, $T$, and $V$ are the total, kinetic, and potential energies, respectively. The term of $T_{\mathrm{uncorr}.}$ is the uncorrelated kinetic energy without UCOM. Bottom panel shows the total energies of six kinds of neutron magic numbers. Dashed line is the total energy of the Hartree-Fock calculation with UCOM for infinite neutron matter.

Figure 14

Energies of neutron matter per particle in the present UCOM+HM method with a neutron number $N=66$ as functions of neutron density $\rho$. We compare the results with other theories, the values of which are taken from Ref. [23].