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Next: Results Up: BEC-BCS crossover Previous: Introduction   Contents

Model

The homogeneous two-component Fermi gas is described by the Hamiltonian

\begin{displaymath}
\hat H=-\frac{\hbar^2}{2m}\left( \sum_{i=1}^{N_\uparrow}\nab...
...bla^2_{i^\prime}\right)
+\sum_{i,i^\prime}V(r_{ii^\prime}) \;,
\end{displaymath} (9.2)

where $m$ denotes the mass of the particles, $i,j,...$ and $i^\prime,j^\prime,...$ label, respectively, spin-up and spin-down particles and $N_\uparrow=N_\downarrow=N/2$, $N$ being the total number of atoms. We model the interspecies interatomic interactions using an attractive square-well potential: $V(r)=-V_0$ for $r<R_0$, and $V(r)=0$ otherwise. In order to ensure that the mean interparticle distance is much larger than the range of the potential we use $nR_0^3=10^{-6}$, where $n=k_F^3/(3\pi^2)$ is the gas number density. By varying the depth $V_0$ of the potential one can change the value of the $s$-wave scattering length, which for this potential is given by $a=R_0[1-\tan(K_0R_0)/(K_0R_0)]$, where $K_0^2=mV_0/\hbar^2$. We vary $K_0$ is the range: $0<K_0<\pi/R_0$. For $K_0R_0<\pi/2$ the potential does not support a two-body bound state and $a<0$. For $K_0R_0>\pi/2$, instead, the scattering length is positive, $a>0$, and a molecular state appears whose binding energy $\epsilon_b$ is determined by the trascendental equation $\sqrt{\vert\epsilon_b\vert m/\hbar^2}R_0\tan(\bar{K}R_0)/(\bar{K}R_0)=-1$, where $\bar{K}^2=K_0^2-\vert\epsilon_b\vert m/\hbar^2$. The value $K_0=\pi/(2R_0)$ corresponds to the unitary limit where $\vert a\vert=\infty$ and $\epsilon_b=0$.

In the present study we resort to the Fixed Node Monte Carlo technique described in Sec. 2.4. We make use of the following trial wave functions. A BCS wave function

\begin{displaymath}
\psi_{BCS}({\bf R})={\cal A} \left( \phi(r_{11^\prime})\phi(r_{22^\prime})...\phi(r_{N_\uparrow N_\downarrow})\right) \;,
\end{displaymath} (9.3)

and a Jastrow-Slater (JS) wave function
\begin{displaymath}
\psi_{JS}({\bf R})=\prod_{i,i^\prime}\varphi(r_{ii^\prime}) ...
...\alpha}
e^{i{\bf k}_\alpha\cdot{\bf r}_{i^\prime}} \right] \;,
\end{displaymath} (9.4)

where ${\cal A}$ is the antisymmetrizer operator ensuring the correct antisymmetric properties under particle exchange. In the JS wave function, Eq. (9.4), the plane wave orbitals have wave vectors ${\bf k}_\alpha=2\pi/L(\ell_{\alpha x}
\hat{x}+\ell_{\alpha y}\hat{y}+\ell_{\alpha z}\hat{z})$, where $L$ is the size of the periodic cubic box fixed by $nL^3=N$, and $\ell$ are integer numbers. The correlation functions $\phi(r)$ and $\varphi(r)$ in Eqs. (9.3)-(9.4) are constructed from solutions of the two-body Schrödinger equation with the square-well potential $V(r)$. In particular, in the region $a>0$ we take for the function $\phi(r)$ the bound-state solution $\phi_{bs}(r)$ with energy $\epsilon_b$ and in the region $a<0$ the unbound-state solution corresponding to zero scattering energy: $\phi_{us}(r)=(R_0-a)\sin(K_0r)/[r\sin(K_0R_0)]$ for $r<R_0$ and $\phi_{us}(r)=1-a/r$ for $r>R_0$. In the unitary limit, $\vert a\vert\to\infty$, $\phi_{bs}(r)=\phi_{us}(r)$.

The JS wave function $\psi _{JS}$, Eq. (9.4), is used only in the region of negative scattering length, $a<0$, with a Jastrow factor $\varphi(r)=\phi_{us}(r)$ for $r<\bar{R}$. In order to reduce possible size effects due to the long range tail of $\phi_{us}(r)$, we have used $\varphi(r)=C_1+C_2\exp(-\alpha r)$ for $r>\bar{R}$, with $\bar{R}<L/2$ a matching point. The coefficients $C_1$ and $C_2$ are fixed by the continuity condition for $\varphi(r)$ and its first derivative at $r=\bar{R}$, whereas the parameter $\alpha>0$ is chosen in such a way that $\varphi(r)$ goes rapidly to a constant. Residual size effects have been finally determined carrying out calculations with an increasing number of particles $N=14$, 38, and 66. In the inset of Fig. 9.1 we show the dependence of the energy per particle $E/N$ on $N$ in the unitary limit. Similar studies carried out in the BEC and BCS regime show that the value $N=66$ is optimal since finite-size corrections in the energy are below the reported statistical error in the whole BEC-BCS crossover. We have also checked that effects due to the finite range $R_0$ of the potential are negligible.

Figure: Energy per particle in the BEC-BCS crossover. Solid symbols refer to results obtained with the trial wave function $\psi _{BCS}$, open symbols refer to the ones obtained with $\psi _{JS}$. The red dot-dashed line is the expansion (9.5) holding in the BCS region and the blue dotted line corresponds to the binding energy $\epsilon _b/2$. Inset: finite size effects in the unitary limit $-1/k_Fa=0$.
\includegraphics[width=0.6\textwidth]{figBECBCS1.eps}


next up previous contents
Next: Results Up: BEC-BCS crossover Previous: Introduction   Contents
G.E. Astrakharchik 15-th of December 2004