Results

The FN-DMC energies for $N=66$ atoms and the potential $V(r)$ with $nR_0^3=10^{-6}$ are shown in Fig. 9.1 and in Table 31 as a function of the interaction parameter $-1/k_Fa$. The numerical simulations are carried out both with the BCS wave function, Eq. (9.3), and with the JS wave function, Eq. (9.4). For $-1/k_Fa>0.4$ we find that $\psi _{JS}$ gives lower energies, whereas for smaller values of $-1/k_Fa$, including the unitary limit and the BEC region, the function $\psi _{BCS}$ is preferable. This behavior reflects the level of accuracy of the variational ansatz for the nodal structure of the trial wave function. We believe that in the intersection region, $-1/k_Fa\sim 0.4$, both wave functions $\psi _{BCS}$ and $\psi _{JS}$ give a poorer description of the exact nodal structure of the state, resulting in a less accurate estimate of the energy. In the BCS region, $-1/k_Fa>1$, our results for $E/N$ are in agreement with the perturbation expansion of a weakly attractive Fermi gas^9.1 [HY57,LY57]

$$\frac{E}{N\epsilon_{FG}}= 1+\frac{10}{9\pi}k_Fa+\frac{4(11-2\log2)}{21\pi^2}(k_Fa)^2+... \;.$$ (9.5)

Table 9.1: Energy per particle and binding energy in the BEC-BCS crossover (energies are in units of $\epsilon _{FG}$).

$-1/k_Fa$	$E/N$	$\epsilon _b/2$	$E/N-\epsilon_b/2$
-6	-73.170(2)	-73.1804	0.010(2)
-4	-30.336(2)	-30.3486	0.013(2)
-2	-7.071(2)	-7.1018	0.031(2)
-1	-1.649(3)	-1.7196	0.071(3)
-0.4	-0.087(6)	-0.2700	0.183(6)
-0.2	0.223(1)	-0.0671	0.29(1)
0	0.42(1)	0	0.42(1)
0.2	0.62(3)	0	0.62(3)
0.4	0.72(3)	0	0.72(3)
1	0.79(2)	0	0.79(2)
2	0.87(1)	0	0.87(1)
4	0.92(1)	0	0.92(1)
6	0.94(1)	0	0.94(1)

In the unitary limit we find $E/N=\xi\epsilon_{FG}$, with $\xi=0.42(1)$. This result is compatible with the findings of Refs. [CCPS03,CPCS04] obtained using a different trial wave function which includes both Jastrow and BCS correlations. The value of the parameter $\beta=\xi-1$ has been measured in experiments with trapped Fermi gases [OHG+02,BAR+04b,BKC+04], but the precision is too low to make stringent comparisons with theoretical predictions. In the region of positive scattering length $E/N$ decreases by decreasing $k_Fa$. At approximately $-1/k_Fa\simeq-0.3$, the energy becomes negative, and by further decreasing $k_Fa$ it rapidly approaches the binding energy per particle $\epsilon _b/2$ indicating the formation of bound molecules [CPCS04]. The results with the binding energy subtracted from $E/N$ are shown in Fig. 9.2. In the BEC region, $-1/k_Fa<-1$, we find that the FN-DMC energies agree with the equation of state of a repulsive gas of molecules

$$\frac{E/N-\epsilon_b/2}{\epsilon_{FG}}=\frac{5}{18\pi}k_Fa_m\left[1+\frac{128}{15\sqrt{6\pi^3}}(k_Fa_m)^{3/2}+...\right] \;,$$ (9.6)

where the first term corresponds to the mean-field energy of a gas of molecules of mass $2m$ and density $n/2$ interacting with the positive molecule-molecule scattering length $a_m$, and the second term corresponds to the first beyond mean-field correction [LHY57]. If for $a_m$ we use the value calculated by Petrov et al. [PSS04] $a_m=0.6a$, we obtain the curves shown in Fig. 9.2. If, instead, we use $a_m$ as a fitting parameter to our FN-DMC results in the region $-1/k_Fa\le -1$, we obtain the value $a_m/a=0.62(1)$. From a best fit to the equation of state we calculate the chemical potential $\mu=dE/dN$ and the inverse compressibility $mc^2=n\partial\mu/\partial n$, where $c$ is the speed of sound. The results in units of the Fermi energy $\mu_F=\hbar^2k_F^2/2m$ and of the Fermi velocity $v_F=\hbar k_F/m$ are shown in Fig. 9.3. A detailed knowledge of the equation of state of the homogeneous system is important for the determination of the frequencies of collective modes in trapped systems [Str04], which have been recently measured in the BEC-BCS crossover regime [KTT04,KHG+04,BAR+04a].

Figure 9.2: Energy per particle in the BEC-BCS crossover with the binding energy subtracted from $E/N$. Solid symbols: results with $\psi _{BCS}$, open symbols: results with $\psi _{JS}$. The red dot-dashed line is as in Fig. 9.1 and the blue dashed line corresponds to the expansion (9.6) holding in the BEC regime. Inset: enlarged view of the BEC regime $-1/k_Fa\le -1$. The solid blue line corresponds to the mean-field energy [first term in the expansion (9.6)], the dashed blue line includes the beyond mean-field correction Eq. 9.6).

Figure 9.3: Chemical potential $\mu$ (red solid line) and square of the speed of sound $c^2$ (blue long dashed line) in the BEC-BCS crossover calculated from a best fit to the equation of state. The blue short-dashed line and the blue dotted line correspond to $c^2$ calculated respectively from the expansion (9.5) and (9.6).

In Fig. 9.4 we show the results for the pair correlation function of parallel, $g_2^{\uparrow\uparrow}(r)$, and antiparallel spins, $g_2^{\uparrow\downarrow}(r)$. For parallel spins, $g_2^{\uparrow\uparrow}(r)$ must vanish at short distances due to the Pauli principle. In the BCS regime the effect of pairing is negligible and $g_2^{\uparrow\uparrow}(r)$ coincides with the prediction of a noninteracting Fermi gas $g_2^{\uparrow\uparrow}(r)=1-9/(k_Fr)^4[\sin(k_Fr)/k_Fr-\cos(k_Fr)]^2$. This result continues to hold in the case $-1/k_Fa=0$, where it is consistent with the picture of a gas in the unitary regime as a noninteracting Fermi gas with effective mass $m^\star=m/\xi$. In the BEC regime the static structure factor $S(k)$ of composite bosons can be estimated using the Bogoliubov result: $S(k)=\hbar^2k^2/[2M\omega(k)]$, where $\omega(k)=(\hbar^4k^4/4M^2+gn_m\hbar^2k^2/M)^{1/2}$ is the Bogoliubov dispersion relation for particles with mass $M=2m$, density $n_m=n/2$ and coupling constant $g=4\pi\hbar^2a_m/M$. The pair distribution function $g_2(r)$ of composite bosons, obtained through $g_2(r)=1+2/N\sum_{\bf k}[S(k)-1]e^{-i{\bf k}\cdot{\bf r}}$ using the value $a_m=0.6a$, is shown in Fig. 9.4 for $-1/k_Fa=-4$ and compared with the FN-DMC result. For large distances $r\gg a_m$, where Bogoliubov approximation is expected to hold, we find a remarkable agreement. This result is consistent with the equation of state in the BEC regime and shows that structural properties of the ground state of composite bosons are described correctly in our approach. For antiparallel spins, $g_2^{\uparrow\downarrow}(r)$ exhibits a large peak at short distances due to the attractive interaction. In the BEC regime the short range behavior is well described by the exponential decay $g_2^{\uparrow\downarrow}(r)\propto\exp(-2r\sqrt{\vert\epsilon_b\vert m}/\hbar)/r^2$ fixed by the molecular wave function $\phi_{bs}(r)$. In the unitary regime correlations extend over a considerably larger range compared to the tightly bound BEC regime. In the BCS regime the range of $g_2^{\uparrow\downarrow}(r)$ is much larger than $k_F^{-1}$ and is determined by the coherence length $\xi_0=\hbar^2 k_F/(m\Delta)$, where $\Delta$ is the gap parameter. In this regime the wave function we use does not account for pairing and is inadequate to investigate the behavior of $g_2^{\uparrow\downarrow}(r)$.

Figure: Pair correlation function of parallel, $g_2^{\uparrow\uparrow}(r)$, and (inset) of antiparallel spins, $g_2^{\uparrow\downarrow}(r)$, for $-1/k_Fa=0$ (unitary limit), $-1/k_Fa=-4$ (BEC regime), $-1/k_Fa=4$ (BCS regime) and for a noninteracting Fermi gas (FG). The dot-dashed line corresponds to the pair correlation function of a Bose gas with $a_m=0.6a$ and $-1/k_Fa=-4$ calculated using the Bogoliubov approximation.