Quantum Monte Carlo calculations of two neutrons in finite volume

Ab initio calculations provide direct access to the properties of pure neutron systems that are challenging to study experimentally. In addition to their importance for fundamental physics, their properties are required as input for effective field theories of the strong interaction. In this work, we perform auxiliary-field diffusion Monte Carlo calculations of the ground state and first excited state of two neutrons in a finite box, considering a simple contact potential as well as chiral effective field theory interactions. We compare the results against exact diagonalizations and present a detailed analysis of the finite-volume effects, whose understanding is crucial for determining observables from the calculated energies. Using the Lüscher formula, we extract the low-energy $S$-wave scattering parameters from ground- and excited-state energies for different box sizes.

Figure 1

AFDMC results for the energy of two neutrons in the ground state in finite volume with the contact potential (15) for different box sizes $L$ compared with the Lüscher formula. $C_0$ is adjusted to give the physical $nn$ scattering length $a=-18.9$ fm (closed circles and solid line) and to give a very large scattering length $a=-101.7$ fm (open circles and dashed line). The gray band shows a fit (as described in the text) to the AFDMC results for $a=-101.7$ fm. The energies are given in terms of the dimensionless quantity $q^2=EM{L}^2/\left(4{\pi }^2\right)$.

Figure 2

AFDMC results for the energy of two neutrons in the first excited state in finite volume with the potential (15) for different box sizes $L$ (red circles) compared with the Lüscher formula (solid line). The error bars on the AFDMC results with a spherical nodal surface include both statistical uncertainties and a systematic uncertainty of 1% discussed in the text in Sec. 4b. $C_0$ is adjusted to give the physical $nn$ scattering length $a=-18.9$ fm. The energies are given in terms of the dimensionless quantity $q^2=EM{L}^2/\left(4{\pi }^2\right)$. Also shown are the energies calculated by exact diagonalization (blue diamonds) as discussed in Sec. 4b.

Figure 3

The nodal surfaces $r_{\text{node}}(\theta ,\varphi )$ of the first excited states of the contact potential (15) with the physical $nn$ scattering length $a=-18.9$ fm for different box sizes $L$. The wave functions from which the nodal surfaces are extracted are obtained via diagonalization. See text for details.

Figure 4

The coefficients $c_l$ of the decomposition of the nodal surfaces in terms of cubic harmonics $Y_l^c$ as in Eq. (21) as a function of $l$. The coefficients for each box size shown are normalized such that $c_0=1$. The $y$ axis is log scaled showing the power-law suppression of the two coefficients $c_4$ and $c_6$ with comparison to $c_0$.

Figure 5

AFDMC results for the energy of two neutrons in the ground state in finite volume with the LO chiral EFT interaction compared with the Lüscher formula for different box sizes $L$. The cutoffs $R_0=1.0$ fm (red circles and solid line) and $R_0=1.2$ fm (blue diamonds and dashed line) are used. The energies are given in terms of the dimensionless quantity $q^2=EM{L}^2/\left(4{\pi }^2\right)$. The region where $\left|p\right|>m_{\pi }/2$ is indicated by the gray band; see Sec. 2.

Figure 6

AFDMC results for the energy of two neutrons in the ground state in finite volume with the NLO and ${\mathrm{N}}^2\mathrm{LO}$ chiral EFT interactions with cutoff $R_0=1.0$ fm compared with the Lüscher formula for different box sizes $L$. The results at NLO (${\mathrm{N}}^2\mathrm{LO}$) are given as the red circles and solid line (blue diamonds and dashed line). The dark gray band shows a fit (as described in the text) to the AFDMC results for the ${\mathrm{N}}^2\mathrm{LO}$ chiral potential. The energies are given in terms of the dimensionless quantity $q^2=EM{L}^2/\left(4{\pi }^2\right)$. Points in the region $\left|p\right|>m_{\pi }/2$ indicated by the gray band are not included in the fit; see Sec. 2.

Figure 7

AFDMC results for the energy of two neutrons in the first excited state in finite volume with the LO chiral EFT interaction with cutoff $R_0=1.0$ fm (red circles) compared with the Lüscher formula (solid line) for different box sizes $L$. The error bars on the AFDMC results with a spherical nodal surface include both statistical uncertainties and a systematic uncertainty of 1% discussed in the text in Sec. 4b. The dark gray band shows a combined fit (as described in the text) to the ground and first excited-state AFDMC results for the LO chiral potential. The energies are given in terms of the dimensionless quantity $q^2=EM{L}^2/\left(4{\pi }^2\right)$. Points in the region $\left|p\right|>m_{\pi }/2$ indicated by the gray band are not included in the fit; see Sec. 2.

Figure 8

${}^{1}S_{0}nn$ phase shift ${\delta }_0\left(p\right)$ from AFDMC results for the first excited state in finite volume with the LO chiral EFT interaction with cutoff $R_0=1.0$ fm with spherical nodal surface (red circles) and nonspherical nodal surface (orange squares) compared with the phase shift obtained from $nn$ scattering in infinite volume (solid line). The error bars on the AFDMC results include both statistical uncertainties and a systematic uncertainty of 1% discussed in the text in Sec. 4b. The dashed line shows the phase shift obtained from the effective range expansion (ERE) with $a=-18.9$ fm and $r_{\text{e}}=2.01$ fm. The region where $\left|p\right|>m_{\pi }/2$ is indicated by the gray band; see Sec. 2.