ASLDA - asymmetric superfluid local density approximation

The functional has the generic form:

\begin{aligned}
\mathcal{E}_{\textrm{edf}}(n,\nu,\ldots) & =  \alpha_{\uparrow}(n_{\uparrow},n_{\downarrow})\frac{\tau_{\uparrow}}{2} + 
    \alpha_{\downarrow}(n_{\uparrow},n_{\downarrow})\frac{\tau_{\downarrow}}{2} \\
&+
    D(n_{\uparrow},n_{\downarrow})
    +
    g(n_{\uparrow},n_{\downarrow})\nu^{\dagger}\nu \\ 
    &+ [1-\alpha_{\uparrow}(n_{\uparrow},n_{\downarrow})]\dfrac{\bm{j}_{\uparrow}^2}{2n_{\uparrow}}
    + [1-\alpha_{\downarrow}(n_{\uparrow},n_{\downarrow})]\dfrac{\bm{j}_{\downarrow}^2}{2n_{\downarrow}}
\end{aligned}

TODO

Stabilization of ASLDA functional

In the case of calculations for trapped system term \frac{\bm{j}_{\sigma}^2}{2n_{\sigma}} is source of numerical instabilities. Precisely, for small density regions we have:

• n_{\sigma}\rightarrow 0,
• \bm{j}_{\sigma}\rightarrow 0,
• \frac{\bm{j}_{\sigma}^2}{2n_{\sigma}}\rightarrow 0.

However, division of very small numbers is numerically not stable operations. For this reason, we introduce stabilization procedure:

\dfrac{\bm{j}_{\sigma}^2}{2n_{\sigma}}\longrightarrow f_{\textrm{reg.}}(n_{\sigma})\dfrac{\bm{j}_{\sigma}^2}{2n_{\sigma}}

Role of f_{\textrm{reg.}} is to exclude from computation regions of small density. The function is controlled by two parameters in predefines.h file:

/**
 * Meaningful only in case of ASLDA.
 * Paremeters defining stabilization procedure of ASLDA functional. 
 * For regions with density smaller than ASLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY 
 * contribution from current term j^2/2n is assumed to be zero. 
 * For regions with density above ASLDA_STABILIZATION_RETAIN_ABOVE_DENSITY 
 * the contribution is assumed to be intact by stabilization procedure. 
 * */
#define ASLDA_STABILIZATION_RETAIN_ABOVE_DENSITY  1.0e-5
#define ASLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY 1.0e-7

There meaning is presented on the figure below.
Smooth transition between densities ASLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY and ASLDA_STABILIZATION_RETAIN_ABOVE_DENSITY is introduced in order to avoid discontinuities for quantities, that may lead to divergences when computing derivatives.