/*! \page TBDMDoc two-body density matrix Keyword: TBDM \section description Description Spherical average of the so-called projected two-body density matrix \f$ \rho^P_{2,\uparrow\downarrow}(|{\bf r}|) \f$, \f[ \rho_{2,\uparrow\downarrow}({\bf r}_1,{\bf r}_2;{\bf r}_1',{\bf r}_2') =\int d^3r_3\ldots d^3r_N\, \Psi^*({\bf r}_1,{\bf r}_2,{\bf r}_3,\ldots,{\bf r}_N) \Psi({\bf r}_1',{\bf r}_2',{\bf r}_3,\ldots,{\bf r}_N)\,, \f] \f[ \rho^P_{2,\uparrow\downarrow}(|{\bf r}|) =\frac1{4\pi}\int d\Omega_{\bf r}\int d^3r_1d^3r_2\, \rho_{2,\uparrow\downarrow}({\bf r}_1+{\bf r},{\bf r}_2+{\bf r}; {\bf r}_1,{\bf r}_2)\,, \f] calculated on a one-dimensional grid. The many-body wave function \f$ \Psi \f$ in the above definition is assumed normalized to unity. Normalization is chosen so that \f$ \rho^P_{2,\uparrow\downarrow}(0)=1 \f$. In homogeneous and isotropic systems, such as homogeneous electron gas, the quantity is spherically symmetrical even before spherical averaging. The averaging step only reduces error bars in such cases. The projected two-body density matrix \f$ \rho^P_{2,\uparrow\downarrow}(|{\bf r}|) \f$ is a means to calculate BCS-like condensate fraction \f$ \alpha \f$. [see for instance Astrakharchik et al., Phys. Rev. Lett. 95, 230405 (2005) and references therein]. In unpolarized systems, \f$ N_{\uparrow}=N_{\downarrow}=N/2 \f$, we have \f[ \alpha=\lim_{r\to\infty} \frac{N}{2}\, \rho^P_{2,\uparrow\downarrow}(r) \f] or \f[ \alpha=\lim_{r\to\infty} \frac{N}{2} \Bigl[ \rho^P_{2,\uparrow\downarrow}(r) -\rho_{1\uparrow}^{sph}(r)\,\rho_{1\downarrow}^{sph}(r) \Bigr]\, . \f] The latter expression converges faster to the thermodynamic limit. \section options Options \subsection reqopt Required None. \subsection optopt Optional
Option | Type | Default | Description |
---|---|---|---|
CUTOFF | Float | half of the smallest distance in the simulation cell | Largest distance at which the density matrix is calculated. Values much larger than the default value have little physical meaning. |
NGRID | Integer | 5 | Number of points in the interval [0;CUTOFF] where the density matrix is calculated. The first point is CUTOFF/NGRID, the last point is CUTOFF. |
AIP | Integer | 1 | Number of directions for spherical averaging using a Gaussian quadrature rule. Available are Gaussian rules with 4, 6, 12, 18, 26 and 32 points. Value AIP=1 disables spherical averaging and the vector \f$ \bf r \f$ then points in the x-direction. |