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}

The functional is fitted to quantum Monte Carlo data, and for spin symmetric and uniform system it provides:

E/E_{\textrm{ffg}}=\xi=0.40(1),\qquad\Delta/\varepsilon_F=0.504(24).

For more info about fitting procedure and explicit form of functional terms see: arXiv:1008.3933.

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.
 * Parameters 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.

SLDA - superfluid local density approximation

The term in ASLDA functional that depends on the currents \bm{j}_{\sigma} introduces a significant cost to the computation. This term is responsible for maintaining Gallilean invariance of the ASLDA theory. The effective mass was found to be consistent with the bare mass to within 10% for a large range of polarizations, Therefore, in many applications, one can set \alpha_{\sigma}=1 without losing qualitative features of ASLDA theory. Under this ausumptin we obtain SLDA functional:

\mathcal{E}_{\textrm{edf}}(n,\nu,\ldots)  =  \frac{\tau_{\uparrow}}{2} + 
    \frac{\tau_{\downarrow}}{2} 
+
    D(n_{\uparrow},n_{\downarrow})
    +
    g(n_{\uparrow},n_{\downarrow})\nu^{\dagger}\nu 

Note 1: SLDA functional exhibits much better convergence properties than ASLDA, i.e st-wslda codes typically converge in a significantly smaller number of iterations.

Note 2: SLDA functional provides by factor about 2x better performance in case of td-wslda codes than ASLDA variant.

Below we provide plot showing relative energy change (in %) between simulations of SLDA and ASLDA that compute the energy of N fermions (both even and odd particle numbers) in the spin-symmetric unitary Fermi gas trapped in an isotropic harmonic oscillator. For more info see supplemental material of this paper.