From low-density neutron matter to the unitary limit

Various quantities of an attractively interacting fermion system at the unitary limit are determined by extrapolating Monte Carlo results of low-density neutron matter. Smooth extrapolation in terms of $1/(k_F{a}_0)$ ($k_F$ is the Fermi momentum, and $a_0$ is the ${}^1{S}_0$ scattering length) is found with the quantities examined: the ground-state energy, the pairing gap at $T\approx 0$, and the critical temperature of the normal-to-superfluid phase transition. We emphasize proximity of the physics of low-density neutron matter to that at the unitary limit. The extrapolated quantities are in a reasonable agreement with those in the literature.

Figure 1

Finite-size scaling of the energy parameter ξ. The thermodynamic limit is at $N_s\to \infty$. The Monte Carlo data of ξ with statistical uncertainties are shown at the Fermi momentum of $k_F=15$ (triangle), 30 (square), and 60 MeV (circle). The dotted lines are the $N_s^{-1/2}$ linear fits, Eq. (11), to the data of $N_s=4^3,6^3,8^3$, and ${10}^3$.

Figure 2

Energy parameter ξ as a function of the lattice filling $n$. Monte Carlo data (solid circles with statistical uncertainties) are for $n=1/16,1/8,3/16,1/4,3/8$, and $1/2$ on the $N_s=6^3$ lattice for $k_F=60$ MeV. The dashed line, Eq. (12), is the best fit to the data.

Figure 3

$\eta \equiv 1/(k_F{a}_0)$ dependence of the energy parameter ξ. The solid circles and the dashed curve are our Monte Carlo data and the best-fit, Eq. (14), respectively. Our ξ extrapolated to the unitary limit is shown by the open circle. For comparison, ξ obtained by other works are also shown: the dash-dotted curve is by the next-to-leading order ε expansion [23], the dotted curve is by the lattice Monte Carlo calculation with the symmetric heavy-light ansatz [22], the cross symbols are by the fixed-node diffusion Monte Carlo calculation [25], the solid up- and down-triangles are by the fixed-node Green's function Monte Carlo calculations at $\eta <0$ [26] and at $\eta =0$ [27], respectively, and the solid squares are by the determinantal quantum Monte Carlo calculation [28, 29].

Figure 4

ξ at the unitary limit. The values of ξ reported in the literature are shown in three groups from left to right: those determined by atomic Fermi-gas experiments (solid circles), by various Monte Carlo calculations (solid squares); and by ε expansions (triangles). The scale of the horizontal axis has no significance but for separating data. ξ's by the ε-expansion method on the right-hand side are divided into three subgroups as explained in the text. Our ξ from Eq. (14) is shown by the diamond at the right-most location in the Monte Carlo group. See the text for the reference to each value of ξ. The figure is an expanded version of a similar figure (Fig. 14) in Ref. [24].

Figure 5

Pairing gap Δ in the unit of ${\epsilon }_F$ as a function of $\eta \equiv 1/(k_F{a}_0)$. The solid circles are our Monte Carlo data, shown with statistical uncertainties [1]. The dashed curve is the best fit to the data by use of a quadratic function of $\eta$. Δ at the unitary limit $\eta \to 0$ determined by extrapolation is shown by an open circle. The solid squares, the solid up-triangles, the solid down-triangle, and the solid diamonds are the quantum Monte Carlo data by Bulgac et al. [28, 29], by Gezerlis et al. [26], by Carlson et al. [27], and by Chang et al. [31], respectively. Note that the data by Chang et al. at $\eta <0$ are those quoted in Refs. [28, 29].

Figure 6

$\eta$ dependence of the critical temperature $T_c$, shown in the unit of ${\epsilon }_F$. Our Monte Carlo data at the thermodynamic and continuum limits are shown by solid circles, and the extrapolated unitary-limit point is shown by an open circle. The extrapolation is made using the fit function, Eq. (16). For comparison, the solid squares show a recent quantum Monte Carlo result [29].

Figure 7

Critical temperature $T_c$ at the unitary limit appearing in the literature and our extrapolated $T_c$ (all shown in the unit of ${\epsilon }_F$). The solid squares (with statistical uncertainties) on the left-hand side of the figure are by Monte Carlo calculations [6, 11, 29, 37, 42, 43], and the solid triangles on the right-hand side are by ε expansion calculations [41]. Our $T_c$ is shown by the solid diamond with the statistical uncertainty. The horizontal scale is used merely for the separation of each $T_c$ value. The $T_c$'s by the ε expansion calculations are grouped into two in the same way as in Fig. 4. See the text for the reference of each $T_c$.