Introduction

Recent experiments on two-component ultracold atomic Fermi gases near a Feshbach resonance have opened the possibility of investigating the crossover from a Bose-Einstein condensate (BEC) to a Bardeen-Cooper-Schrieffer (BCS) superfluid. In these systems the strength of the interaction can be varied over a very wide range by magnetically tuning the two-body scattering amplitude. For positive values of the $s$-wave scattering length $a$, atoms with different spins are observed to pair into bound molecules which, at low enough temperature, form a Bose condensate [JBA+03,GRJ03,ZSS+03]. The molecular BEC state is adiabatically converted into an ultracold Fermi gas with $a<0$ and $k_F\vert a\vert\ll 1$ [BAR+04a,BKC+04], where standard BCS theory is expected to apply. In the crossover region the value of $\vert a\vert$ can be orders of magnitude larger than the inverse Fermi wave vector $k_F^{-1}$ and one enters a new strongly-correlated regime known as unitary limit [OHG+02,BAR+04b,BKC+04]. In dilute systems, for which the effective range of the interaction $R_0$ is much smaller than the mean interparticle distance, $k_FR_0\ll 1$, the unitary regime is believed to be universal [Hei01,Bru04,PCHK04,DH04]. In this regime, the only relevant energy scale should be given by the energy of the noninteracting Fermi gas,

$$\epsilon_{FG}=\frac{3}{10}\frac{\hbar^2k_F^2}{m} \;.$$ (9.1)

The unitary regime presents a challenge for many-body theoretical approaches because there is not any obvious small parameter to construct a well-posed theory. The first theoretical studies of the BEC-BCS crossover at zero temperature are based on the mean-field BCS equations [Leg80,NSR85,ERdM97]. More sophisticated approaches take into account the effects of fluctuations [PS00,PPS04], or include explicitly the bosonic molecular field [HKCW01,OG03]. These theories provide a correct description in the deep BCS regime, but are only qualitatively correct in the unitary limit and in the BEC region. In particular, in the BEC regime the dimer-dimer scattering length has been calculated exactly from the solution of the four-body problem, yielding $a_m=0.6a$ [PSS04]. Available results for the equation of state in this regime do not describe correctly the repulsive molecule-molecule interactions [HMV04].

Quantum Monte Carlo techniques are the best suited tools for treating strongly-correlated systems. These methods have already been applied to ultracold degenerate Fermi gases in a recent work by Carlson et al. [CCPS03]. In this study the energy per particle of a dilute Fermi gas in the unitary limit is calculated with the fixed-node Green's function Monte Carlo method (FN-GFMC) giving the result $E/N=\xi\epsilon_{FG}$ with $\xi=0.44(1)$. In a subsequent work [CPCS04], the same authors have extended the FN-GFMC calculations to investigate the equation of state in the BCS and BEC regimes. Their results in the BEC limit are compatible with a repulsive molecular gas, but the equation of state has not been extracted with enough precision.

In the present Chapter, we report results for the equation of state of a Fermi gas in the BEC-BCS crossover region using the fixed-node diffusion Monte Carlo method (FN-DMC). The interaction strength is varied over a very broad range from $-6\le -1/k_Fa\le 6$, including the unitary limit and the deep BEC and BCS regimes. In the unitary and in the BCS limit we find agreement, respectively, with the results of Ref. [CCPS03] and with the known perturbation expansion of a weakly attractive Fermi gas [HY57,LY57]. In the BEC regime, we find a gas of molecules whose repulsive interactions are well described by the dimer-dimer scattering length $a_m=0.6a$. Results for the pair correlation functions of parallel and antiparallel spins are reported in the various regimes. In the BEC regime we find agreement with the pair correlation function of composite bosons calculated using the Bogoliubov approximation.