Local chiral effective field theory interactions and quantum Monte Carlo applications

We present details of the derivation of local chiral effective field theory interactions to next-to-next-to-leading order and show results for nucleon-nucleon phase shifts and deuteron properties for these potentials. We then perform systematic auxiliary-field diffusion Monte Carlo calculations for neutron matter based on the developed local chiral potentials at different orders. This includes studies of the effects of the spectral-function regularization and of the local regulators. For all orders, we compare the quantum Monte Carlo results with perturbative many-body calculations and find excellent agreement for low cutoffs.

Figure 1

Local chiral NN potentials $V\left(r\right)$ at $\mathrm{N}{}^2\mathrm{LO}$ for an SFR cutoff $\tilde{\Lambda }=1000\mathrm{MeV}$, decomposed into the central, central-isospin, spin, spin-isospin, spin-orbit, tensor, and tensor-isospin components, for cutoffs $R_0=0.9-1.2\mathrm{fm}$. For all components, we observe a softening of the potential going from a cutoff $R_0=0.9\mathrm{fm}$ to $R_0=1.2\mathrm{fm}$. We include the $R_0=0.9\mathrm{fm}$ potential for illustration, but as discussed in the text, do not use it in many-body calculations.

Figure 2

Local chiral NN potentials $V\left(r\right)$ at $\mathrm{N}{}^2\mathrm{LO}$ for an SFR cutoff $\tilde{\Lambda }=1000\mathrm{MeV}$ in the ${}^1{S}_0$ partial wave in the neutron-neutron system.

Figure 3

Phase shifts for the ${}^1{S}_0$ and ${}^3{S}_1{\text{−}}^3{D}_1$ partial waves at LO, NLO, and $\mathrm{N}{}^2\mathrm{LO}$ in comparison with the Nijmegen PWA [46]. The bands at each order correspond to the cutoff variation of $R_0=1.0-1.2\mathrm{fm}$. At NLO and $\mathrm{N}{}^2\mathrm{LO}$, we also vary the SFR cutoff from $\tilde{\Lambda }=1.0-1.4\mathrm{GeV}$.

Figure 4

Phase shifts for the ${}^1{P}_1,{}^3{P}_0,{}^3{P}_1$, and ${}^3{P}_2\text{−}{}^3{F}_2$ partial waves at LO, NLO, and $\mathrm{N}{}^2\mathrm{LO}$ in comparison with the Nijmegen PWA [46]. The bands are obtained as in Fig. 3.

Figure 5

Phase shifts for the ${}^1{D}_2,{}^3{D}_2,{}^1{F}_3,{}^3{F}_3,{}^1{G}_4$, and ${}^3{G}_4$ partial waves at LO, NLO, and $\mathrm{N}{}^2\mathrm{LO}$ in comparison with the Nijmegen PWA [46]. The bands are obtained as in Fig. 3.

Figure 6

Phase shifts for the ${}^3{D}_3\text{−}{}^3{G}_3$ and ${}^3{F}_4\text{−}{}^3{H}_4$ partial waves at LO, NLO, and $\mathrm{N}{}^2\mathrm{LO}$ in comparison with the Nijmegen PWA [46]. The bands are obtained in the same way as in Fig. 3.

Figure 7

Neutron-matter energy per particle $E/N$ as a function of density $n$ using AFDMC with the local chiral NN potentials at LO, NLO, and $\mathrm{N}{}^2\mathrm{LO}$. The bands are obtained by varying the cutoff $R_0=1.0-1.2\mathrm{fm}$ and the SFR cutoff $\tilde{\Lambda }=1000-1400\mathrm{MeV}$.

Figure 8

Neutron-matter energy per particle $E/N$ as a function of density $n$. We compare our AFDMC $\mathrm{N}{}^2\mathrm{LO}$ results of this work with the MBPT $\mathrm{N}{}^2\mathrm{LO}$ results of Ref. [67] using the momentum-space potentials of Ref. [3], the coupled-cluster results of Ref. [68] using the optimized $\mathrm{N}{}^2\mathrm{LO}$ potential of Ref. [47], the MBPT results of Ref. [69], and the CIMC results of Ref. [23], both using the same optimized $\mathrm{N}{}^2\mathrm{LO}$ potential.

Figure 9

Finite-size effects for the ground-state energy of neutron matter for a SFR cutoff $\tilde{\Lambda }=1000\mathrm{MeV}$ at $\mathrm{N}{}^2\mathrm{LO}$. Results are shown for different particle numbers for the $R_0=1.0\mathrm{fm}$ and the $R_0=1.2\mathrm{fm}$ potentials. We also show the kinetic energy, shifted down by $19\mathrm{MeV}$. The finite-size effects for the local chiral potentials follow the shell effects of the kinetic energy operator.

Figure 10

Ground-state energy of 66 neutrons at $\mathrm{N}{}^2\mathrm{LO}$. Shown are results for two SFR cutoffs, $\tilde{\Lambda }=1000\mathrm{MeV}$ and $\tilde{\Lambda }=1400\mathrm{MeV}$, and two different cutoffs, $R_0=1.0\mathrm{fm}$ (upper lines) and $R_0=1.2\mathrm{fm}$ (lower lines). The results exhibit a very weak $\tilde{\Lambda }$ dependence.

Figure 11

Results for MBPT and AFDMC calculations at LO, NLO, and $\mathrm{N}{}^2\mathrm{LO}$ for $R_0=1.0-1.2\mathrm{fm}$. For the MBPT results, we show the Hartree-Fock energies as well as the energy at second order and including third-order particle-particle and hole-hole corrections. The width of the bands includes a variation of the single-particle spectrum from a free to a Hartree-Fock spectrum. In addition, for both the MBPT and the AFDMC results we also vary the SFR cutoff $\tilde{\Lambda }=1000-1400\mathrm{MeV}$. For the LO $1.1-\mathrm{fm}$ results, the lower band corresponds to the second-order results.