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

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

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

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

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

$$\frac{\psi_T({\vec r_1}+{\vec r}, ...,{\vec r}_N)}{\psi_T({\vec r... ...vert{\vec r_1}+\r-{\vec r}_j\vert)-\mu(\vert{\vec r_1}-{\vec r}_j\vert)\right\}$$ (2.142)

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

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

The asymptotic limit of the OBDM gives the condensate density

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

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

$$\lim\limits_{r\to\infty} \frac{g_(r)}{n} = \frac{N_0}{N}$$ (2.145)