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One body density matrix in a homogeneous system

The one body density matrix (OBDM) $g_1$ of a homogeneous system described by the many body wave function $\psi({\vec r_1}, ..., {\vec r}_N)$ according to (1.18) is equal to

$\displaystyle g_1(\vert\r~'-\r~''\vert)=N \frac{\int...\int\psi^*({\vec r}\,', ...
...\int\vert\psi({\vec r_1}, ..., {\vec r}_N)\vert^2\,d{\vec r_1} ... d{\vec r}_N}$     (2.139)

Since in DMC calculation does not sample directly the ground-state probability distribution $\phi_0^2$, but instead the mixed probability $\psi_T\phi_0$ (2.12) one obtains the mixed one-body density matrix as the output

$\displaystyle g_1^{mixed}(\r) =
N \frac{\int...\int \psi^*_T({\vec r}\,''+{\vec...
.....,{\vec r}_N)\phi_0({\vec r_1}, ...,{\vec r}_N)\,d{\vec r_1} ... d{\vec r}_N},$     (2.140)

This formula can be rewritten in a way convenient for the Monte Carlo sampling:

$\displaystyle g_1^{mixed}(r) =
n\frac{\int...\int[\psi^*_T({\vec r_1}+{\vec r},...
...c r}_N}{\int...\int f({\vec r_1}, ..., {\vec r}_N)d{\vec r_1} ... d{\vec r}_N},$     (2.141)

where we have used the asymptotic formula (2.19) and have taken into account that in a homogeneous system $g_2$ depends only on the module of the relative distance between two particles. If the trial wave function is chosen as a product of pair functions (2.37) then using the notation (2.38) $u(\vert{\vec r}_i
-{\vec r}_j\vert) = \ln f_2(\vert{\vec r}_i -{\vec r}_j\vert)$) and $f_1 \equiv 0$ one has $\psi_T({\vec r_1}, ...,{\vec r}_N)
= \prod\limits_{i<j}\exp\{u(\vert{\vec r}_i-{\vec r}_j\vert)\}$. Then the ratio of the trial wave function appearing in (2.141) becomes
$\displaystyle \frac{\psi_T({\vec r_1}+{\vec r}, ...,{\vec r}_N)}{\psi_T({\vec r...
...vert{\vec r_1}+\r-{\vec r}_j\vert)-\mu(\vert{\vec r_1}-{\vec r}_j\vert)\right\}$     (2.142)

In order to gain better statistics it is convenient to average over all possible pairs of particles

$\displaystyle g_1^{mixed}(r) =
\frac{1}{N} \sum\limits_{i=1}^N\frac{\psi_T({\ve...
...(\vert{\vec r}_i+\r-{\vec r}_j\vert)-u(\vert{\vec r}_i-{\vec r}_j\vert)\right\}$     (2.143)

The asymptotic limit of the OBDM gives the condensate density

$\displaystyle \lim\limits_{r\to\infty} g_2(r) = \frac{N_0}{V}$     (2.144)

and the condensate fraction is obtained by the calculating the asymptotic ratio
$\displaystyle \lim\limits_{r\to\infty} \frac{g_(r)}{n} = \frac{N_0}{N}$     (2.145)


next up previous contents
Next: One body density matrix Up: Measured quantities Previous: Static structure factor   Contents
G.E. Astrakharchik 15-th of December 2004