One body density matrix and condensate fraction

The one body density matrix (OBDM) of a homogeneous system described by the many body wavefunction $\psi(r_1, ..., r_N)$ is defined as follows

$$\rho(\vert\vec r~'-\vec r~''\vert) = N \frac{\int...\int \phi_0^*... ...t...\int \vert\phi_0(\vec r_1, ..., \vec r_N)\vert^2\,d\vec r_1 ... d\vec r_N}.$$ (3.82)

Since in DMC calculation one can not sample the ground-state probability distribution $\phi_0^2$, instead one samples the mixed probability $\psi_T\phi_0$ and one can calculate the mixed one-body density matrix

$$\rho_{mixed} (r) = N \frac{\int...\int \psi^*_T(\vec r\,''+\vec r... ...vec r_1, ...,\vec r_N)\phi_0(\vec r_1, ...,\vec r_N)\,d\vec r_1 ... d\vec r_N},$$ (3.83)

This formula can be further developed

$$\begin{array}{l} \displaystyle \rho_{mixed}(r) = N \frac{\int...\... ...(\vec r_1, ..., \vec r_N)d\vec r_1 ... d\vec r_N}, \quad t\to\infty \end{array}$$ (3.84)

where we have used the asymptotic formula (3.46). If the trial wavefunction is chosen as a product of pair functions (see eq. (3.31) then using the notation $\mu (\vert\vec r_i -\vec r_j\vert) = \ln g(\vert\vec r_i -\vec r_j\vert)$) one has

$$\psi_T(\vec r_1, ...,\vec r_N) = \prod\limits_{i<j} e^{\mu(\vert\vec r_i-\vec r_j\vert)}$$ (3.85)

Then, the ratio of trial wavefunction appearing in (3.84) becomes

$$\frac{\psi_T(\vec r_1+\vec r, ...,\vec r_N)}{\psi_T(\vec r_1, ...... ...mu(\vert\vec r_1+\vec r-\vec r_j\vert)-\mu(\vert\vec r_1-\vec r_j\vert)\right).$$ (3.86)

In order to gain better statistics one can average over all particles

$$\frac{1}{N} \sum\limits_{i=1}^N \frac{\psi_T(\vec r_1, ..., \vec ... ...\mu(\vert\vec r_i+\vec r-\vec r_j\vert)-\mu(\vert\vec r_i-\vec r_j\vert)\right)$$

The asymptotic limit of the OBDM gives the condensate density

$$\lim\limits_{r\to\infty} \rho(r) = \frac{N_0}{V}$$ (3.87)

and the condensate fraction is obtained by the calculating the asymptotic ratio

$$\lim\limits_{r\to\infty} \frac{\rho(r)}{\rho} = \frac{N_0}{N}$$ (3.88)