Quantum Monte Carlo Calculations with Chiral Effective Field Theory Interactions

We present the first quantum Monte Carlo (QMC) calculations with chiral effective field theory (EFT) interactions. To achieve this, we remove all sources of nonlocality, which hamper the inclusion in QMC calculations, in nuclear forces to next-to-next-to-leading order. We perform auxiliary-field diffusion Monte Carlo (AFDMC) calculations for the neutron matter energy up to saturation density based on local leading-order, next-to-leading order, and next-to-next-to-leading order nucleon-nucleon interactions. Our results exhibit a systematic order-by-order convergence in chiral EFT and provide nonperturbative benchmarks with theoretical uncertainties. For the softer interactions, perturbative calculations are in excellent agreement with the AFDMC results. This work paves the way for QMC calculations with systematic chiral EFT interactions for nuclei and nuclear matter, for testing the perturbativeness of different orders, and allows for matching to lattice QCD results by varying the pion mass.

Figure 1

Neutron-proton phase shifts as a function of laboratory energy $E_{\mathrm{lab}}=2p^2/m$ in the ${}^{1}S_0$, ${}^{3}P_0$, ${}^{3}P_1$, and ${}^{3}P_2$ partial waves (from left to right) in comparison to the Nijmegen partial-wave analysis (PWA) [43]. The LO, NLO, and ${\mathrm{N}}^2\mathrm{LO}$ bands are obtained by varying $R_0$ between 0.8–1.2 fm (with a spectral-function cutoff $\tilde{\Lambda }=800\mathrm{MeV}$).

Figure 2

Neutron matter energy per particle $E/N$ as a function of density $n$ calculated using AFDMC with chiral EFT $NN$ interactions at LO, NLO, and ${\mathrm{N}}^2\mathrm{LO}$. The statistical errors are smaller than the points shown. The lines give the range of the energy band obtained by varying $R_0$ between 0.8–1.2 fm (as for the phase shifts in Fig. 1), which provides an estimate of the theoretical uncertainty at each order. The ${\mathrm{N}}^2\mathrm{LO}$ band is comparable to the one at NLO due to the large $c_i$ couplings in the ${\mathrm{N}}^2\mathrm{LO}$ two-pion exchange.

Figure 3

The AFDMC ${\mathrm{N}}^2\mathrm{LO}$ band of Fig. 2 in comparison to perturbative calculations of the neutron matter energy using the same local ${\mathrm{N}}^2\mathrm{LO}$ $NN$ interactions. The lower (upper) limit of the AFDMC ${\mathrm{N}}^2\mathrm{LO}$ band is for $R_0=1.2\mathrm{fm}$ ($R_0=0.8\mathrm{fm}$), corresponding to a momentum cutoff $\Lambda \sim 400\mathrm{MeV}$ ($\Lambda \sim 600\mathrm{MeV}$). Perturbative results are shown for Hartree-Fock plus second-order contributions (2nd order) and including third-order particle-particle and hole-hole corrections (3rd order). The bands at 2nd and 3rd order are obtained by using a Hartree-Fock or free single-particle spectrum. For the softer $R_0=1.2\mathrm{fm}$ interaction (narrow purple bands), the third-order corrections are small and the perturbative third-order energy is in excellent agreement with the AFDMC results, while for the harder $R_0=0.8\mathrm{fm}$ interaction (light red bands), the convergence is clearly slow. At low densities, we also show the QMC (2010) results of Refs. 28, 51.