Precision benchmark calculations for four particles at unitarity

The unitarity limit describes interacting particles where the range of the interaction is zero and the scattering length is infinite. We present precision benchmark calculations for two-component fermions at unitarity using three different ab initio methods: Hamiltonian lattice formalism using iterated eigenvector methods, Euclidean lattice formalism with auxiliary-field projection Monte Carlo methods, and continuum diffusion Monte Carlo methods with fixed and released nodes. We have calculated the ground-state energy of the unpolarized four-particle system in a periodic cube as a dimensionless fraction of the ground-state energy for the noninteracting system. We obtain values of $0.211(2)$ and $0.210(2)$ using two different Hamiltonian lattice representations, $0.206(9)$ using Euclidean lattice formalism, and an upper bound of $0.212(2)$ from fixed-node diffusion Monte Carlo methods. Released-node calculations starting from the fixed-node result yield a decrease of less than $0.002$ over a propagation of $0.4E_F^{-1}$ in Euclidean time, where $E_F$ is the Fermi energy. We find good agreement among all three ab initio methods.

Figure 1

Plot of $p\cot {\delta }_0(p)$ versus $p^2$ for the lattice Hamiltonians $H_1$ and $H_2$.

Figure 2

Ground-state energy ratio ${\xi }_{2,2}$ for lattice Hamiltonians $H_1$ and $H_2$. We show results for values of $L=4,5,6,7,8$ and extrapolate to the infinite volume limit.

Figure 3

Plot of $p\cot {\delta }_0(p)$ versus $p^2$ for the Euclidean time lattice formalism.

Figure 4

The lattice data for ${\xi }_{2,2}(t)$ versus $E_{F}t$ for $L=4,5,6,7,8$. Also shown are the results of the asymptotic fits.

Figure 5

Results for ${\xi }_{2,2}$ for $L=4,5,6,7,8$ plotted versus $L^{-1}$. The lattice results are extrapolated to the continuum limit $L\to \infty .$

Figure 6

Fixed-node and released-node diffusion Monte Carlo results for $r_0/L=0.025$ and $\alpha =0.01$, $0.05$, $0.2.$

Figure 7

Fixed-node diffusion Monte Carlo results for ${\xi }_{2,2}$ versus $r_0/L$. The data are extrapolated to the limit of zero effective range.