Quantum Monte Carlo calculations of neutron matter with chiral three-body forces

Chiral effective field theory (EFT) enables a systematic description of low-energy hadronic interactions with controlled theoretical uncertainties. For strongly interacting systems, quantum Monte Carlo (QMC) methods provide some of the most accurate solutions, but they require as input local potentials. We have recently constructed local chiral nucleon-nucleon (NN) interactions up to next-to-next-to-leading order (${\mathrm{N}}^2\mathrm{LO}$). Chiral EFT naturally predicts consistent many-body forces. In this paper, we consider the leading chiral three-nucleon (3N) interactions in local form. These are included in auxiliary field diffusion Monte Carlo (AFDMC) simulations. We present results for the equation of state of neutron matter and for the energies and radii of neutron drops. In particular, we study the regulator dependence at the Hartree-Fock level and in AFDMC and find that present local regulators lead to less repulsion from 3N forces compared to the usual nonlocal regulators.

Figure 1

Contributions to 3N forces at ${\mathrm{N}}^2\mathrm{LO}$. These include a two-pion-exchange part given by the couplings $c_1,c_3,$ and $c_4$, a one-pion-exchange–contact interaction given by $c_D$, and a 3N contact interaction given by $c_E$.

Figure 2

Contributions to the neutron-matter energy per particle $E/N$ as a function of density $n$ at the Hartree-Fock level. The black band shows the energy obtained using a nonlocal regulator, as in Ref. [34], with a 3N cutoff $400–500\mathrm{MeV}$. The blue band corresponds to the LR part of the two-pion-exchange interaction $V_C$ with the local regulator used here, the red band to the SR part of $V_C$, and the green band to the IR of $V_C$. For these bands, the cutoff in the local regulator is varied with $R_{\text{3N}}=1.0–1.2\mathrm{fm}$. The dashed-dotted line corresponds to the results for $V_C$ using the local momentum-space regulator of Ref. [31] with a cutoff ${\mathrm{\Lambda }}_{\text{3N}}=500\mathrm{MeV}$. This shows that these local 3N forces provide less repulsion at the Hartree-Fock level than with nonlocal regulators. The dashed lines show the results for $V_C$ with the local regulator and small $R_{\text{3N}}=0.5\mathrm{fm}$.

Figure 3

Contributions to the energy per particle $E/N$ at saturation density as a function of the cutoff $R_{\text{3N}}$. The lines show the LR, SR, and IR parts of the two-pion-exchange interaction $V_C$ with the local regulator used here, calculated at the Hartree-Fock level. The bands are the contributions of the corresponding 3N parts to the AFDMC energies for a variation of the NN cutoff $R_0=1.0–1.2\mathrm{fm}$; see also Fig. 4.

Figure 4

Variation of the AFDMC energy per particle at saturation density as a function of the 3N cutoff $R_{\text{3N}}$ for an NN cutoff $1.0\mathrm{fm}$ (black lines in the upper part) and $1.2\mathrm{fm}$ (red lines in the lower part). The horizontal lines correspond to the NN-only energy. The squares are for the $c_1$ and LR $c_3$ part of $V_C$, the crosses include also the SR $c_3$ part of $V_C$, and the circles include all parts of $V_C$.

Figure 5

Dependence of the AFDMC energy per particle at saturation density as a function of the 3N cutoff $R_{\text{3N}}$ on different long-range regulators. Results are shown for an NN cutoff $R_0=1.2\mathrm{fm}$. The long-range regulator is given by ${\left[1-e^{-{(r/R_{3\text{N}})}^{n_1}}\right]}^{n_2}$ with different parameters $n_1$ and $n_2$.

Figure 6

Energy per particle as a function of density for neutron matter at ${\mathrm{N}}^2\mathrm{LO}$, including NN forces and the 3N $V_C$ interaction in AFDMC. Results are shown for an NN cutoff $R_0=1.0–1.2\mathrm{fm}$ and $R_{3\text{N}}$ in the same range.

Figure 7

Comparison of the neutron-matter energy at ${\mathrm{N}}^2\mathrm{LO}$ based on the local chiral NN+3N potentials in AFDMC (this work) with the ${\mathrm{N}}^2\mathrm{LO}$ calculation of Ref. [10] based on the EGM ${\mathrm{N}}^2\mathrm{LO}$ potentials and using MBPT, with the particle-particle (pp) ladder results of Ref. [14] based on the EM ${\mathrm{N}}^2\mathrm{LO}$ potential, and with results based on the ${\mathrm{N}}^2{\mathrm{LO}}_{\text{opt}}$ potential using CC theory [11] and the SCGF methods [13].

Figure 8

Energies and radii of neutron drops with $N=8$ and 20 neutrons in a harmonic oscillator potential with an oscillator parameter $\hslash \omega =10\mathrm{MeV}$ using AFDMC. The same cutoff is used in the NN and 3N interactions. We give the results at different orders in the chiral expansion, and at ${\mathrm{N}}^2\mathrm{LO}$, for NN forces only, plus only the LR $c_1$ and $c_3$ parts of $V_C$, and including also the SR and IR parts of $V_C$. The bands are given by the cutoff variation $R_0=R_{\text{3N}}=1.0–1.2\mathrm{fm}$. At LO with the softer cutoff and $N=20$, the system collapses. We compare our results with the calculations of Ref. [40], using coupled-cluster theory at the $\mathrm{\Lambda }\mathrm{CCSD}$ level, where the band is given by two different similarity renormalization group (SRG) evolution scales (for one initial NN+3N Hamiltonian).