Ground state energy at unitarity

We consider two-component fermions on the lattice in the unitarity limit. This is an idealized limit of attractive fermions where the range of the interaction is zero and the scattering length is infinite. Using Euclidean time projection, we compute the ground state energy using four computationally different but physically identical auxiliary-field methods. The best performance is obtained using a bounded continuous auxiliary field and a nonlocal updating algorithm called the hybrid Monte Carlo. With this method, we calculate results for 10 and 14 fermions at lattice volumes $4^3,5^3,6^3,7^3,8^3$ and extrapolate to the continuum limit. For 10 fermions in a periodic cube, the ground state energy is $0.292(12)$ times the ground state energy for noninteracting fermions. For 14 fermions, the ratio is $0.329(5)$.

Figure 1

Schematic diagram showing four equivalent Euclidean time lattice formulations. The numbered arrows indicate the order of discussion in the text.

Figure 2

Sum of bubble diagrams contributing to two-particle scattering.

Figure 3

Monte Carlo results for ${\xi }_{5,5}(L_t{\alpha }_t)$ for $L=5$ and $L_t=24$. $P_r$ is the rejection probability, and $P_s$ is the singular matrix probability.

Figure 4

Monte Carlo results for ${\xi }_{5,5}(L_t{\alpha }_t)$ for $L=5$ and $L_t=48$.

Figure 5

Monte Carlo results for ${\xi }_{5,5}(L_t{\alpha }_t)$ for $L=5$ and $L_t=72$.

Figure 6

Lattice data for ${\xi }_{5,5}(t)$ vs $E_{F}t$ for $L=4,5,6,7,8$. Also shown are the results of the two-parameter fits.

Figure 7

Lattice data for ${\xi }_{7,7}(t)$ vs $E_{F}t$ for $L=4,5,6,7,8$. Also shown are the results of the two-parameter fits.

Figure 8

Results from the two-parameter fits for ${\xi }_{5,5}$ and ${\xi }_{7,7}$ at finite $L$ and the corresponding continuum limit extrapolations.

Figure 9

Comparison of results from Fig. 8 with Hamiltonian lattice results using the symmetric heavy-light ansatz in the lowest filling approximation.

Figure 10

Results for ${\xi }_{N,N}$ at unitarity using the symmetric heavy-light ansatz in the lowest filling approximation [41].